Optimal. Leaf size=84 \[ -\frac {2 f^{a+b x^3}}{b^4 \log ^4(f)}+\frac {2 x^3 f^{a+b x^3}}{b^3 \log ^3(f)}-\frac {x^6 f^{a+b x^3}}{b^2 \log ^2(f)}+\frac {x^9 f^{a+b x^3}}{3 b \log (f)} \]
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Rubi [A] time = 0.10, antiderivative size = 84, normalized size of antiderivative = 1.00, 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 2209
Rule 2212
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.43, size = 51, normalized size = 0.61 \[ \frac {{\left (b^{3} x^{9} \log \relax (f)^{3} - 3 \, b^{2} x^{6} \log \relax (f)^{2} + 6 \, b x^{3} \log \relax (f) - 6\right )} f^{b x^{3} + a}}{3 \, b^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 83, normalized size = 0.99 \[ \frac {b^{3} f^{b x^{3}} f^{a} x^{9} \log \relax (f)^{3} - 3 \, b^{2} f^{b x^{3}} f^{a} x^{6} \log \relax (f)^{2} + 6 \, b f^{b x^{3}} f^{a} x^{3} \log \relax (f) - 6 \, f^{b x^{3}} f^{a}}{3 \, b^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.62 \[ \frac {\left (b^{3} x^{9} \ln \relax (f )^{3}-3 b^{2} x^{6} \ln \relax (f )^{2}+6 b \,x^{3} \ln \relax (f )-6\right ) f^{b \,x^{3}+a}}{3 b^{4} \ln \relax (f )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 62, normalized size = 0.74 \[ \frac {{\left (b^{3} f^{a} x^{9} \log \relax (f)^{3} - 3 \, b^{2} f^{a} x^{6} \log \relax (f)^{2} + 6 \, b f^{a} x^{3} \log \relax (f) - 6 \, f^{a}\right )} f^{b x^{3}}}{3 \, b^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.46, size = 51, normalized size = 0.61 \[ -\frac {f^{b\,x^3+a}\,\left (-\frac {b^3\,x^9\,{\ln \relax (f)}^3}{3}+b^2\,x^6\,{\ln \relax (f)}^2-2\,b\,x^3\,\ln \relax (f)+2\right )}{b^4\,{\ln \relax (f)}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 68, normalized size = 0.81 \[ \begin {cases} \frac {f^{a + b x^{3}} \left (b^{3} x^{9} \log {\relax (f )}^{3} - 3 b^{2} x^{6} \log {\relax (f )}^{2} + 6 b x^{3} \log {\relax (f )} - 6\right )}{3 b^{4} \log {\relax (f )}^{4}} & \text {for}\: 3 b^{4} \log {\relax (f )}^{4} \neq 0 \\\frac {x^{12}}{12} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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