Optimal. Leaf size=84 \[ -\frac{x^6 f^{a+b x^3}}{b^2 \log ^2(f)}+\frac{2 x^3 f^{a+b x^3}}{b^3 \log ^3(f)}-\frac{2 f^{a+b x^3}}{b^4 \log ^4(f)}+\frac{x^9 f^{a+b x^3}}{3 b \log (f)} \]
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Rubi [A] time = 0.0950641, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ -\frac{x^6 f^{a+b x^3}}{b^2 \log ^2(f)}+\frac{2 x^3 f^{a+b x^3}}{b^3 \log ^3(f)}-\frac{2 f^{a+b x^3}}{b^4 \log ^4(f)}+\frac{x^9 f^{a+b x^3}}{3 b \log (f)} \]
Antiderivative was successfully verified.
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Rule 2212
Rule 2209
Rubi steps
\begin{align*} \int f^{a+b x^3} x^{11} \, dx &=\frac{f^{a+b x^3} x^9}{3 b \log (f)}-\frac{3 \int f^{a+b x^3} x^8 \, dx}{b \log (f)}\\ &=-\frac{f^{a+b x^3} x^6}{b^2 \log ^2(f)}+\frac{f^{a+b x^3} x^9}{3 b \log (f)}+\frac{6 \int f^{a+b x^3} x^5 \, dx}{b^2 \log ^2(f)}\\ &=\frac{2 f^{a+b x^3} x^3}{b^3 \log ^3(f)}-\frac{f^{a+b x^3} x^6}{b^2 \log ^2(f)}+\frac{f^{a+b x^3} x^9}{3 b \log (f)}-\frac{6 \int f^{a+b x^3} x^2 \, dx}{b^3 \log ^3(f)}\\ &=-\frac{2 f^{a+b x^3}}{b^4 \log ^4(f)}+\frac{2 f^{a+b x^3} x^3}{b^3 \log ^3(f)}-\frac{f^{a+b x^3} x^6}{b^2 \log ^2(f)}+\frac{f^{a+b x^3} x^9}{3 b \log (f)}\\ \end{align*}
Mathematica [A] time = 0.01129, size = 53, normalized size = 0.63 \[ \frac{f^{a+b x^3} \left (b^3 x^9 \log ^3(f)-3 b^2 x^6 \log ^2(f)+6 b x^3 \log (f)-6\right )}{3 b^4 \log ^4(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 52, normalized size = 0.6 \begin{align*}{\frac{ \left ({b}^{3}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{3}-3\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}+6\,b{x}^{3}\ln \left ( f \right ) -6 \right ){f}^{b{x}^{3}+a}}{3\,{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.31256, size = 84, normalized size = 1. \begin{align*} \frac{{\left (b^{3} f^{a} x^{9} \log \left (f\right )^{3} - 3 \, b^{2} f^{a} x^{6} \log \left (f\right )^{2} + 6 \, b f^{a} x^{3} \log \left (f\right ) - 6 \, f^{a}\right )} f^{b x^{3}}}{3 \, b^{4} \log \left (f\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79495, size = 128, normalized size = 1.52 \begin{align*} \frac{{\left (b^{3} x^{9} \log \left (f\right )^{3} - 3 \, b^{2} x^{6} \log \left (f\right )^{2} + 6 \, b x^{3} \log \left (f\right ) - 6\right )} f^{b x^{3} + a}}{3 \, b^{4} \log \left (f\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.135558, size = 68, normalized size = 0.81 \begin{align*} \begin{cases} \frac{f^{a + b x^{3}} \left (b^{3} x^{9} \log{\left (f \right )}^{3} - 3 b^{2} x^{6} \log{\left (f \right )}^{2} + 6 b x^{3} \log{\left (f \right )} - 6\right )}{3 b^{4} \log{\left (f \right )}^{4}} & \text{for}\: 3 b^{4} \log{\left (f \right )}^{4} \neq 0 \\\frac{x^{12}}{12} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22751, size = 112, normalized size = 1.33 \begin{align*} \frac{b^{3} f^{b x^{3}} f^{a} x^{9} \log \left (f\right )^{3} - 3 \, b^{2} f^{b x^{3}} f^{a} x^{6} \log \left (f\right )^{2} + 6 \, b f^{b x^{3}} f^{a} x^{3} \log \left (f\right ) - 6 \, f^{b x^{3}} f^{a}}{3 \, b^{4} \log \left (f\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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