Integrand size = 13, antiderivative size = 119 \[ \int f^{a+\frac {b}{x^2}} x^6 \, dx=\frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {2}{35} b f^{a+\frac {b}{x^2}} x^5 \log (f)+\frac {4}{105} b^2 f^{a+\frac {b}{x^2}} x^3 \log ^2(f)+\frac {8}{105} b^3 f^{a+\frac {b}{x^2}} x \log ^3(f)-\frac {8}{105} b^{7/2} f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right ) \log ^{\frac {7}{2}}(f) \]
[Out]
Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2245, 2237, 2242, 2235} \[ \int f^{a+\frac {b}{x^2}} x^6 \, dx=-\frac {8}{105} \sqrt {\pi } b^{7/2} f^a \log ^{\frac {7}{2}}(f) \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )+\frac {8}{105} b^3 x \log ^3(f) f^{a+\frac {b}{x^2}}+\frac {4}{105} b^2 x^3 \log ^2(f) f^{a+\frac {b}{x^2}}+\frac {1}{7} x^7 f^{a+\frac {b}{x^2}}+\frac {2}{35} b x^5 \log (f) f^{a+\frac {b}{x^2}} \]
[In]
[Out]
Rule 2235
Rule 2237
Rule 2242
Rule 2245
Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {1}{7} (2 b \log (f)) \int f^{a+\frac {b}{x^2}} x^4 \, dx \\ & = \frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {2}{35} b f^{a+\frac {b}{x^2}} x^5 \log (f)+\frac {1}{35} \left (4 b^2 \log ^2(f)\right ) \int f^{a+\frac {b}{x^2}} x^2 \, dx \\ & = \frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {2}{35} b f^{a+\frac {b}{x^2}} x^5 \log (f)+\frac {4}{105} b^2 f^{a+\frac {b}{x^2}} x^3 \log ^2(f)+\frac {1}{105} \left (8 b^3 \log ^3(f)\right ) \int f^{a+\frac {b}{x^2}} \, dx \\ & = \frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {2}{35} b f^{a+\frac {b}{x^2}} x^5 \log (f)+\frac {4}{105} b^2 f^{a+\frac {b}{x^2}} x^3 \log ^2(f)+\frac {8}{105} b^3 f^{a+\frac {b}{x^2}} x \log ^3(f)+\frac {1}{105} \left (16 b^4 \log ^4(f)\right ) \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx \\ & = \frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {2}{35} b f^{a+\frac {b}{x^2}} x^5 \log (f)+\frac {4}{105} b^2 f^{a+\frac {b}{x^2}} x^3 \log ^2(f)+\frac {8}{105} b^3 f^{a+\frac {b}{x^2}} x \log ^3(f)-\frac {1}{105} \left (16 b^4 \log ^4(f)\right ) \text {Subst}\left (\int f^{a+b x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {2}{35} b f^{a+\frac {b}{x^2}} x^5 \log (f)+\frac {4}{105} b^2 f^{a+\frac {b}{x^2}} x^3 \log ^2(f)+\frac {8}{105} b^3 f^{a+\frac {b}{x^2}} x \log ^3(f)-\frac {8}{105} b^{7/2} f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right ) \log ^{\frac {7}{2}}(f) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.72 \[ \int f^{a+\frac {b}{x^2}} x^6 \, dx=\frac {1}{105} f^a \left (-8 b^{7/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right ) \log ^{\frac {7}{2}}(f)+f^{\frac {b}{x^2}} x \left (15 x^6+6 b x^4 \log (f)+4 b^2 x^2 \log ^2(f)+8 b^3 \log ^3(f)\right )\right ) \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.84
method | result | size |
meijerg | \(\frac {f^{a} \ln \left (f \right )^{\frac {7}{2}} b^{3} \sqrt {-b}\, \left (-\frac {2 x^{7} \left (\frac {8 b^{3} \ln \left (f \right )^{3}}{15 x^{6}}+\frac {4 b^{2} \ln \left (f \right )^{2}}{15 x^{4}}+\frac {2 b \ln \left (f \right )}{5 x^{2}}+1\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{7 \left (-b \right )^{\frac {7}{2}} \ln \left (f \right )^{\frac {7}{2}}}+\frac {16 b^{\frac {7}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{105 \left (-b \right )^{\frac {7}{2}}}\right )}{2}\) | \(100\) |
risch | \(\frac {f^{a} x^{7} f^{\frac {b}{x^{2}}}}{7}+\frac {2 f^{a} \ln \left (f \right ) b \,x^{5} f^{\frac {b}{x^{2}}}}{35}+\frac {4 f^{a} \ln \left (f \right )^{2} b^{2} x^{3} f^{\frac {b}{x^{2}}}}{105}+\frac {8 f^{a} \ln \left (f \right )^{3} b^{3} x \,f^{\frac {b}{x^{2}}}}{105}-\frac {8 f^{a} \ln \left (f \right )^{4} b^{4} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{105 \sqrt {-b \ln \left (f \right )}}\) | \(111\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.72 \[ \int f^{a+\frac {b}{x^2}} x^6 \, dx=\frac {8}{105} \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} b^{3} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{3} + \frac {1}{105} \, {\left (15 \, x^{7} + 6 \, b x^{5} \log \left (f\right ) + 4 \, b^{2} x^{3} \log \left (f\right )^{2} + 8 \, b^{3} x \log \left (f\right )^{3}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]
[In]
[Out]
\[ \int f^{a+\frac {b}{x^2}} x^6 \, dx=\int f^{a + \frac {b}{x^{2}}} x^{6}\, dx \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.24 \[ \int f^{a+\frac {b}{x^2}} x^6 \, dx=\frac {1}{2} \, f^{a} x^{7} \left (-\frac {b \log \left (f\right )}{x^{2}}\right )^{\frac {7}{2}} \Gamma \left (-\frac {7}{2}, -\frac {b \log \left (f\right )}{x^{2}}\right ) \]
[In]
[Out]
\[ \int f^{a+\frac {b}{x^2}} x^6 \, dx=\int { f^{a + \frac {b}{x^{2}}} x^{6} \,d x } \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08 \[ \int f^{a+\frac {b}{x^2}} x^6 \, dx=\frac {f^a\,f^{\frac {b}{x^2}}\,x^7}{7}-\frac {8\,f^a\,x^7\,\sqrt {\pi }\,{\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}^{7/2}}{105}+\frac {8\,f^a\,x^7\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \left (f\right )}{x^2}}\right )\,{\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}^{7/2}}{105}+\frac {8\,b^3\,f^a\,f^{\frac {b}{x^2}}\,x\,{\ln \left (f\right )}^3}{105}+\frac {4\,b^2\,f^a\,f^{\frac {b}{x^2}}\,x^3\,{\ln \left (f\right )}^2}{105}+\frac {2\,b\,f^a\,f^{\frac {b}{x^2}}\,x^5\,\ln \left (f\right )}{35} \]
[In]
[Out]