Integrand size = 13, antiderivative size = 67 \[ \int f^{a+b x^3} x^8 \, dx=\frac {2 f^{a+b x^3}}{3 b^3 \log ^3(f)}-\frac {2 f^{a+b x^3} x^3}{3 b^2 \log ^2(f)}+\frac {f^{a+b x^3} x^6}{3 b \log (f)} \]
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Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2243, 2240} \[ \int f^{a+b x^3} x^8 \, dx=\frac {2 f^{a+b x^3}}{3 b^3 \log ^3(f)}-\frac {2 x^3 f^{a+b x^3}}{3 b^2 \log ^2(f)}+\frac {x^6 f^{a+b x^3}}{3 b \log (f)} \]
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Rule 2240
Rule 2243
Rubi steps \begin{align*} \text {integral}& = \frac {f^{a+b x^3} x^6}{3 b \log (f)}-\frac {2 \int f^{a+b x^3} x^5 \, dx}{b \log (f)} \\ & = -\frac {2 f^{a+b x^3} x^3}{3 b^2 \log ^2(f)}+\frac {f^{a+b x^3} x^6}{3 b \log (f)}+\frac {2 \int f^{a+b x^3} x^2 \, dx}{b^2 \log ^2(f)} \\ & = \frac {2 f^{a+b x^3}}{3 b^3 \log ^3(f)}-\frac {2 f^{a+b x^3} x^3}{3 b^2 \log ^2(f)}+\frac {f^{a+b x^3} x^6}{3 b \log (f)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.61 \[ \int f^{a+b x^3} x^8 \, dx=\frac {f^{a+b x^3} \left (2-2 b x^3 \log (f)+b^2 x^6 \log ^2(f)\right )}{3 b^3 \log ^3(f)} \]
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Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(\frac {\left (b^{2} x^{6} \ln \left (f \right )^{2}-2 b \,x^{3} \ln \left (f \right )+2\right ) f^{b \,x^{3}+a}}{3 \ln \left (f \right )^{3} b^{3}}\) | \(40\) |
risch | \(\frac {\left (b^{2} x^{6} \ln \left (f \right )^{2}-2 b \,x^{3} \ln \left (f \right )+2\right ) f^{b \,x^{3}+a}}{3 \ln \left (f \right )^{3} b^{3}}\) | \(40\) |
meijerg | \(-\frac {f^{a} \left (2-\frac {\left (3 b^{2} x^{6} \ln \left (f \right )^{2}-6 b \,x^{3} \ln \left (f \right )+6\right ) {\mathrm e}^{b \,x^{3} \ln \left (f \right )}}{3}\right )}{3 b^{3} \ln \left (f \right )^{3}}\) | \(47\) |
parallelrisch | \(\frac {f^{b \,x^{3}+a} x^{6} b^{2} \ln \left (f \right )^{2}-2 f^{b \,x^{3}+a} x^{3} b \ln \left (f \right )+2 f^{b \,x^{3}+a}}{3 \ln \left (f \right )^{3} b^{3}}\) | \(59\) |
norman | \(\frac {2 \,{\mathrm e}^{\left (b \,x^{3}+a \right ) \ln \left (f \right )}}{3 \ln \left (f \right )^{3} b^{3}}+\frac {x^{6} {\mathrm e}^{\left (b \,x^{3}+a \right ) \ln \left (f \right )}}{3 b \ln \left (f \right )}-\frac {2 x^{3} {\mathrm e}^{\left (b \,x^{3}+a \right ) \ln \left (f \right )}}{3 \ln \left (f \right )^{2} b^{2}}\) | \(68\) |
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Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.58 \[ \int f^{a+b x^3} x^8 \, dx=\frac {{\left (b^{2} x^{6} \log \left (f\right )^{2} - 2 \, b x^{3} \log \left (f\right ) + 2\right )} f^{b x^{3} + a}}{3 \, b^{3} \log \left (f\right )^{3}} \]
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Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.79 \[ \int f^{a+b x^3} x^8 \, dx=\begin {cases} \frac {f^{a + b x^{3}} \left (b^{2} x^{6} \log {\left (f \right )}^{2} - 2 b x^{3} \log {\left (f \right )} + 2\right )}{3 b^{3} \log {\left (f \right )}^{3}} & \text {for}\: b^{3} \log {\left (f \right )}^{3} \neq 0 \\\frac {x^{9}}{9} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int f^{a+b x^3} x^8 \, dx=\frac {{\left (b^{2} f^{a} x^{6} \log \left (f\right )^{2} - 2 \, b f^{a} x^{3} \log \left (f\right ) + 2 \, f^{a}\right )} f^{b x^{3}}}{3 \, b^{3} \log \left (f\right )^{3}} \]
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Time = 0.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int f^{a+b x^3} x^8 \, dx=\frac {b^{2} f^{b x^{3}} f^{a} x^{6} \log \left (f\right )^{2} - 2 \, b f^{b x^{3}} f^{a} x^{3} \log \left (f\right ) + 2 \, f^{b x^{3}} f^{a}}{3 \, b^{3} \log \left (f\right )^{3}} \]
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Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.58 \[ \int f^{a+b x^3} x^8 \, dx=\frac {f^{b\,x^3+a}\,\left (\frac {b^2\,x^6\,{\ln \left (f\right )}^2}{3}-\frac {2\,b\,x^3\,\ln \left (f\right )}{3}+\frac {2}{3}\right )}{b^3\,{\ln \left (f\right )}^3} \]
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