Optimal. Leaf size=62 \[ \frac {(c+d x)^3 F^{a+b (c+d x)^3}}{3 b d \log (F)}-\frac {F^{a+b (c+d x)^3}}{3 b^2 d \log ^2(F)} \]
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Rubi [A] time = 0.14, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ \frac {(c+d x)^3 F^{a+b (c+d x)^3}}{3 b d \log (F)}-\frac {F^{a+b (c+d x)^3}}{3 b^2 d \log ^2(F)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^3} (c+d x)^5 \, dx &=\frac {F^{a+b (c+d x)^3} (c+d x)^3}{3 b d \log (F)}-\frac {\int F^{a+b (c+d x)^3} (c+d x)^2 \, dx}{b \log (F)}\\ &=-\frac {F^{a+b (c+d x)^3}}{3 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^3} (c+d x)^3}{3 b d \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 40, normalized size = 0.65 \[ \frac {F^{a+b (c+d x)^3} \left (b \log (F) (c+d x)^3-1\right )}{3 b^2 d \log ^2(F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 84, normalized size = 1.35 \[ \frac {{\left ({\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \relax (F) - 1\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \, b^{2} d \log \relax (F)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 891, normalized size = 14.37 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 89, normalized size = 1.44 \[ \frac {\left (b \,d^{3} x^{3} \ln \relax (F )+3 b c \,d^{2} x^{2} \ln \relax (F )+3 b \,c^{2} d x \ln \relax (F )+b \,c^{3} \ln \relax (F )-1\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 b^{2} d \ln \relax (F )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.09, size = 133, normalized size = 2.15 \[ \frac {{\left (F^{b c^{3} + a} b d^{3} x^{3} \log \relax (F) + 3 \, F^{b c^{3} + a} b c d^{2} x^{2} \log \relax (F) + 3 \, F^{b c^{3} + a} b c^{2} d x \log \relax (F) + F^{b c^{3} + a} b c^{3} \log \relax (F) - F^{b c^{3} + a}\right )} e^{\left (b d^{3} x^{3} \log \relax (F) + 3 \, b c d^{2} x^{2} \log \relax (F) + 3 \, b c^{2} d x \log \relax (F)\right )}}{3 \, b^{2} d \log \relax (F)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 95, normalized size = 1.53 \[ \frac {F^{b\,d^3\,x^3}\,F^{3\,b\,c^2\,d\,x}\,F^a\,F^{b\,c^3}\,F^{3\,b\,c\,d^2\,x^2}\,\left (b\,\ln \relax (F)\,c^3+3\,b\,\ln \relax (F)\,c^2\,d\,x+3\,b\,\ln \relax (F)\,c\,d^2\,x^2+b\,\ln \relax (F)\,d^3\,x^3-1\right )}{3\,b^2\,d\,{\ln \relax (F)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 144, normalized size = 2.32 \[ \begin {cases} \frac {F^{a + b \left (c + d x\right )^{3}} \left (b c^{3} \log {\relax (F )} + 3 b c^{2} d x \log {\relax (F )} + 3 b c d^{2} x^{2} \log {\relax (F )} + b d^{3} x^{3} \log {\relax (F )} - 1\right )}{3 b^{2} d \log {\relax (F )}^{2}} & \text {for}\: 3 b^{2} d \log {\relax (F )}^{2} \neq 0 \\c^{5} x + \frac {5 c^{4} d x^{2}}{2} + \frac {10 c^{3} d^{2} x^{3}}{3} + \frac {5 c^{2} d^{3} x^{4}}{2} + c d^{4} x^{5} + \frac {d^{5} x^{6}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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