Integrand size = 13, antiderivative size = 86 \[ \int f^{a+b x^2} x^7 \, dx=-\frac {3 f^{a+b x^2}}{b^4 \log ^4(f)}+\frac {3 f^{a+b x^2} x^2}{b^3 \log ^3(f)}-\frac {3 f^{a+b x^2} x^4}{2 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^6}{2 b \log (f)} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2243, 2240} \[ \int f^{a+b x^2} x^7 \, dx=-\frac {3 f^{a+b x^2}}{b^4 \log ^4(f)}+\frac {3 x^2 f^{a+b x^2}}{b^3 \log ^3(f)}-\frac {3 x^4 f^{a+b x^2}}{2 b^2 \log ^2(f)}+\frac {x^6 f^{a+b x^2}}{2 b \log (f)} \]
[In]
[Out]
Rule 2240
Rule 2243
Rubi steps \begin{align*} \text {integral}& = \frac {f^{a+b x^2} x^6}{2 b \log (f)}-\frac {3 \int f^{a+b x^2} x^5 \, dx}{b \log (f)} \\ & = -\frac {3 f^{a+b x^2} x^4}{2 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^6}{2 b \log (f)}+\frac {6 \int f^{a+b x^2} x^3 \, dx}{b^2 \log ^2(f)} \\ & = \frac {3 f^{a+b x^2} x^2}{b^3 \log ^3(f)}-\frac {3 f^{a+b x^2} x^4}{2 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^6}{2 b \log (f)}-\frac {6 \int f^{a+b x^2} x \, dx}{b^3 \log ^3(f)} \\ & = -\frac {3 f^{a+b x^2}}{b^4 \log ^4(f)}+\frac {3 f^{a+b x^2} x^2}{b^3 \log ^3(f)}-\frac {3 f^{a+b x^2} x^4}{2 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^6}{2 b \log (f)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.62 \[ \int f^{a+b x^2} x^7 \, dx=\frac {f^{a+b x^2} \left (-6+6 b x^2 \log (f)-3 b^2 x^4 \log ^2(f)+b^3 x^6 \log ^3(f)\right )}{2 b^4 \log ^4(f)} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(\frac {\left (b^{3} x^{6} \ln \left (f \right )^{3}-3 b^{2} x^{4} \ln \left (f \right )^{2}+6 b \,x^{2} \ln \left (f \right )-6\right ) f^{b \,x^{2}+a}}{2 \ln \left (f \right )^{4} b^{4}}\) | \(52\) |
risch | \(\frac {\left (b^{3} x^{6} \ln \left (f \right )^{3}-3 b^{2} x^{4} \ln \left (f \right )^{2}+6 b \,x^{2} \ln \left (f \right )-6\right ) f^{b \,x^{2}+a}}{2 \ln \left (f \right )^{4} b^{4}}\) | \(52\) |
meijerg | \(\frac {f^{a} \left (6-\frac {\left (-4 b^{3} x^{6} \ln \left (f \right )^{3}+12 b^{2} x^{4} \ln \left (f \right )^{2}-24 b \,x^{2} \ln \left (f \right )+24\right ) {\mathrm e}^{b \,x^{2} \ln \left (f \right )}}{4}\right )}{2 b^{4} \ln \left (f \right )^{4}}\) | \(59\) |
parallelrisch | \(\frac {f^{b \,x^{2}+a} x^{6} \ln \left (f \right )^{3} b^{3}-3 f^{b \,x^{2}+a} x^{4} \ln \left (f \right )^{2} b^{2}+6 f^{b \,x^{2}+a} x^{2} \ln \left (f \right ) b -6 f^{b \,x^{2}+a}}{2 \ln \left (f \right )^{4} b^{4}}\) | \(80\) |
norman | \(-\frac {3 \,{\mathrm e}^{\left (b \,x^{2}+a \right ) \ln \left (f \right )}}{\ln \left (f \right )^{4} b^{4}}+\frac {x^{6} {\mathrm e}^{\left (b \,x^{2}+a \right ) \ln \left (f \right )}}{2 b \ln \left (f \right )}+\frac {3 x^{2} {\mathrm e}^{\left (b \,x^{2}+a \right ) \ln \left (f \right )}}{\ln \left (f \right )^{3} b^{3}}-\frac {3 x^{4} {\mathrm e}^{\left (b \,x^{2}+a \right ) \ln \left (f \right )}}{2 \ln \left (f \right )^{2} b^{2}}\) | \(91\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.59 \[ \int f^{a+b x^2} x^7 \, dx=\frac {{\left (b^{3} x^{6} \log \left (f\right )^{3} - 3 \, b^{2} x^{4} \log \left (f\right )^{2} + 6 \, b x^{2} \log \left (f\right ) - 6\right )} f^{b x^{2} + a}}{2 \, b^{4} \log \left (f\right )^{4}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77 \[ \int f^{a+b x^2} x^7 \, dx=\begin {cases} \frac {f^{a + b x^{2}} \left (b^{3} x^{6} \log {\left (f \right )}^{3} - 3 b^{2} x^{4} \log {\left (f \right )}^{2} + 6 b x^{2} \log {\left (f \right )} - 6\right )}{2 b^{4} \log {\left (f \right )}^{4}} & \text {for}\: b^{4} \log {\left (f \right )}^{4} \neq 0 \\\frac {x^{8}}{8} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.72 \[ \int f^{a+b x^2} x^7 \, dx=\frac {{\left (b^{3} f^{a} x^{6} \log \left (f\right )^{3} - 3 \, b^{2} f^{a} x^{4} \log \left (f\right )^{2} + 6 \, b f^{a} x^{2} \log \left (f\right ) - 6 \, f^{a}\right )} f^{b x^{2}}}{2 \, b^{4} \log \left (f\right )^{4}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.64 \[ \int f^{a+b x^2} x^7 \, dx=\frac {{\left (b^{3} x^{6} \log \left (f\right )^{3} - 3 \, b^{2} x^{4} \log \left (f\right )^{2} + 6 \, b x^{2} \log \left (f\right ) - 6\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{2 \, b^{4} \log \left (f\right )^{4}} \]
[In]
[Out]
Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60 \[ \int f^{a+b x^2} x^7 \, dx=-\frac {f^{b\,x^2+a}\,\left (-\frac {b^3\,x^6\,{\ln \left (f\right )}^3}{2}+\frac {3\,b^2\,x^4\,{\ln \left (f\right )}^2}{2}-3\,b\,x^2\,\ln \left (f\right )+3\right )}{b^4\,{\ln \left (f\right )}^4} \]
[In]
[Out]