Optimal. Leaf size=67 \[ \frac {2 f^{a+b x^3}}{3 b^3 \log ^3(f)}-\frac {2 x^3 f^{a+b x^3}}{3 b^2 \log ^2(f)}+\frac {x^6 f^{a+b x^3}}{3 b \log (f)} \]
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Rubi [A] time = 0.07, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 2209
Rule 2212
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.42, size = 39, normalized size = 0.58 \[ \frac {{\left (b^{2} x^{6} \log \relax (f)^{2} - 2 \, b x^{3} \log \relax (f) + 2\right )} f^{b x^{3} + a}}{3 \, b^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 61, normalized size = 0.91 \[ \frac {b^{2} f^{b x^{3}} f^{a} x^{6} \log \relax (f)^{2} - 2 \, b f^{b x^{3}} f^{a} x^{3} \log \relax (f) + 2 \, f^{b x^{3}} f^{a}}{3 \, b^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 0.60 \[ \frac {\left (b^{2} x^{6} \ln \relax (f )^{2}-2 b \,x^{3} \ln \relax (f )+2\right ) f^{b \,x^{3}+a}}{3 b^{3} \ln \relax (f )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 47, normalized size = 0.70 \[ \frac {{\left (b^{2} f^{a} x^{6} \log \relax (f)^{2} - 2 \, b f^{a} x^{3} \log \relax (f) + 2 \, f^{a}\right )} f^{b x^{3}}}{3 \, b^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.47, size = 39, normalized size = 0.58 \[ \frac {f^{b\,x^3+a}\,\left (\frac {b^2\,x^6\,{\ln \relax (f)}^2}{3}-\frac {2\,b\,x^3\,\ln \relax (f)}{3}+\frac {2}{3}\right )}{b^3\,{\ln \relax (f)}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 54, normalized size = 0.81 \[ \begin {cases} \frac {f^{a + b x^{3}} \left (b^{2} x^{6} \log {\relax (f )}^{2} - 2 b x^{3} \log {\relax (f )} + 2\right )}{3 b^{3} \log {\relax (f )}^{3}} & \text {for}\: 3 b^{3} \log {\relax (f )}^{3} \neq 0 \\\frac {x^{9}}{9} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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