Optimal. Leaf size=67 \[ -\frac{2 x^3 f^{a+b x^3}}{3 b^2 \log ^2(f)}+\frac{2 f^{a+b x^3}}{3 b^3 \log ^3(f)}+\frac{x^6 f^{a+b x^3}}{3 b \log (f)} \]
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Rubi [A] time = 0.0702023, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ -\frac{2 x^3 f^{a+b x^3}}{3 b^2 \log ^2(f)}+\frac{2 f^{a+b x^3}}{3 b^3 \log ^3(f)}+\frac{x^6 f^{a+b x^3}}{3 b \log (f)} \]
Antiderivative was successfully verified.
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Rule 2212
Rule 2209
Rubi steps
\begin{align*} \int f^{a+b x^3} x^8 \, dx &=\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*}
Mathematica [A] time = 0.008727, size = 41, normalized size = 0.61 \[ \frac{f^{a+b x^3} \left (b^2 x^6 \log ^2(f)-2 b x^3 \log (f)+2\right )}{3 b^3 \log ^3(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 40, normalized size = 0.6 \begin{align*}{\frac{ \left ({b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}-2\,b{x}^{3}\ln \left ( f \right ) +2 \right ){f}^{b{x}^{3}+a}}{3\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09035, size = 63, normalized size = 0.94 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82486, size = 100, normalized size = 1.49 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.128307, size = 54, normalized size = 0.81 \begin{align*} \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}\: 3 b^{3} \log{\left (f \right )}^{3} \neq 0 \\\frac{x^{9}}{9} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23852, size = 82, normalized size = 1.22 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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