We report the experimental realization of efficient tunable nanosources of second harmonic light with individual multiresonant log-periodic optical antennas. By designing the nanoantenna with a bandwidth of several octaves, simultaneous enhancement of fundamental and harmonic fields is observed over a broad range of frequencies, leading to a high second harmonic conversion efficiency, together with an effective second order susceptibility within the range of values provided by widespread inorganic crystals. Moreover, the geometrical configuration of the nanoantenna makes the generated second harmonic signal independent from the polarization of the fundamental excitation. These results open new possibilities for the development of efficient integrated nonlinear nanodevices with high frequency tunability.