Optimal. Leaf size=96 \[ \frac {2 F^{a+b (c+d x)^3}}{3 b^3 d \log ^3(F)}-\frac {2 (c+d x)^3 F^{a+b (c+d x)^3}}{3 b^2 d \log ^2(F)}+\frac {(c+d x)^6 F^{a+b (c+d x)^3}}{3 b d \log (F)} \]
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Rubi [A] time = 0.21, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ -\frac {2 (c+d x)^3 F^{a+b (c+d x)^3}}{3 b^2 d \log ^2(F)}+\frac {2 F^{a+b (c+d x)^3}}{3 b^3 d \log ^3(F)}+\frac {(c+d x)^6 F^{a+b (c+d x)^3}}{3 b d \log (F)} \]
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)^8 \, dx &=\frac {F^{a+b (c+d x)^3} (c+d x)^6}{3 b d \log (F)}-\frac {2 \int F^{a+b (c+d x)^3} (c+d x)^5 \, dx}{b \log (F)}\\ &=-\frac {2 F^{a+b (c+d x)^3} (c+d x)^3}{3 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^3} (c+d x)^6}{3 b d \log (F)}+\frac {2 \int F^{a+b (c+d x)^3} (c+d x)^2 \, dx}{b^2 \log ^2(F)}\\ &=\frac {2 F^{a+b (c+d x)^3}}{3 b^3 d \log ^3(F)}-\frac {2 F^{a+b (c+d x)^3} (c+d x)^3}{3 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^3} (c+d x)^6}{3 b d \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 0.58 \[ \frac {F^{a+b (c+d x)^3} \left (b^2 \log ^2(F) (c+d x)^6-2 b \log (F) (c+d x)^3+2\right )}{3 b^3 d \log ^3(F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 172, normalized size = 1.79 \[ \frac {{\left ({\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \relax (F)^{2} - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \relax (F) + 2\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \, b^{3} d \log \relax (F)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 705, normalized size = 7.34 \[ \frac {b^{2} d^{6} x^{6} 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) + b c^{3} \log \relax (F) + a \log \relax (F)\right )} \log \relax (F)^{2} + 6 \, b^{2} c d^{5} x^{5} 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) + b c^{3} \log \relax (F) + a \log \relax (F)\right )} \log \relax (F)^{2} + 15 \, b^{2} c^{2} d^{4} x^{4} 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) + b c^{3} \log \relax (F) + a \log \relax (F)\right )} \log \relax (F)^{2} + 20 \, b^{2} c^{3} d^{3} x^{3} 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) + b c^{3} \log \relax (F) + a \log \relax (F)\right )} \log \relax (F)^{2} + 15 \, b^{2} c^{4} d^{2} x^{2} 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) + b c^{3} \log \relax (F) + a \log \relax (F)\right )} \log \relax (F)^{2} + 6 \, b^{2} c^{5} d x 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) + b c^{3} \log \relax (F) + a \log \relax (F)\right )} \log \relax (F)^{2} + b^{2} c^{6} 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) + b c^{3} \log \relax (F) + a \log \relax (F)\right )} \log \relax (F)^{2} - 2 \, b d^{3} x^{3} 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) + b c^{3} \log \relax (F) + a \log \relax (F)\right )} \log \relax (F) - 6 \, b c d^{2} x^{2} 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) + b c^{3} \log \relax (F) + a \log \relax (F)\right )} \log \relax (F) - 6 \, b c^{2} d x 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) + b c^{3} \log \relax (F) + a \log \relax (F)\right )} \log \relax (F) - 2 \, b c^{3} 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) + b c^{3} \log \relax (F) + a \log \relax (F)\right )} \log \relax (F) + 2 \, 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) + b c^{3} \log \relax (F) + a \log \relax (F)\right )}}{3 \, b^{3} d \log \relax (F)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 200, normalized size = 2.08 \[ \frac {\left (b^{2} d^{6} x^{6} \ln \relax (F )^{2}+6 b^{2} c \,d^{5} x^{5} \ln \relax (F )^{2}+15 b^{2} c^{2} d^{4} x^{4} \ln \relax (F )^{2}+20 b^{2} c^{3} d^{3} x^{3} \ln \relax (F )^{2}+15 b^{2} c^{4} d^{2} x^{2} \ln \relax (F )^{2}+6 b^{2} c^{5} d x \ln \relax (F )^{2}+b^{2} c^{6} \ln \relax (F )^{2}-2 b \,d^{3} x^{3} \ln \relax (F )-6 b c \,d^{2} x^{2} \ln \relax (F )-6 b \,c^{2} d x \ln \relax (F )-2 b \,c^{3} \ln \relax (F )+2\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 b^{3} d \ln \relax (F )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.13, size = 308, normalized size = 3.21 \[ \frac {{\left (F^{b c^{3} + a} b^{2} d^{6} x^{6} \log \relax (F)^{2} + 6 \, F^{b c^{3} + a} b^{2} c d^{5} x^{5} \log \relax (F)^{2} + 15 \, F^{b c^{3} + a} b^{2} c^{2} d^{4} x^{4} \log \relax (F)^{2} + F^{b c^{3} + a} b^{2} c^{6} \log \relax (F)^{2} - 2 \, F^{b c^{3} + a} b c^{3} \log \relax (F) + 2 \, {\left (10 \, F^{b c^{3} + a} b^{2} c^{3} d^{3} \log \relax (F)^{2} - F^{b c^{3} + a} b d^{3} \log \relax (F)\right )} x^{3} + 3 \, {\left (5 \, F^{b c^{3} + a} b^{2} c^{4} d^{2} \log \relax (F)^{2} - 2 \, F^{b c^{3} + a} b c d^{2} \log \relax (F)\right )} x^{2} + 6 \, {\left (F^{b c^{3} + a} b^{2} c^{5} d \log \relax (F)^{2} - F^{b c^{3} + a} b c^{2} d \log \relax (F)\right )} x + 2 \, F^{b c^{3} + a}\right )} e^{\left (b d^{3} x^{3} \log \relax (F) + 3 \, b c d^{2} x^{2} \log \relax (F) + 3 \, b c^{2} d x \log \relax (F)\right )}}{3 \, b^{3} d \log \relax (F)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 196, normalized size = 2.04 \[ F^{b\,d^3\,x^3}\,F^{3\,b\,c^2\,d\,x}\,F^a\,F^{b\,c^3}\,F^{3\,b\,c\,d^2\,x^2}\,\left (\frac {b^2\,c^6\,{\ln \relax (F)}^2-2\,b\,c^3\,\ln \relax (F)+2}{3\,b^3\,d\,{\ln \relax (F)}^3}+\frac {d^5\,x^6}{3\,b\,\ln \relax (F)}+\frac {2\,c^2\,x\,\left (b\,c^3\,\ln \relax (F)-1\right )}{b^2\,{\ln \relax (F)}^2}+\frac {2\,c\,d^4\,x^5}{b\,\ln \relax (F)}+\frac {2\,d^2\,x^3\,\left (10\,b\,c^3\,\ln \relax (F)-1\right )}{3\,b^2\,{\ln \relax (F)}^2}+\frac {5\,c^2\,d^3\,x^4}{b\,\ln \relax (F)}+\frac {c\,d\,x^2\,\left (5\,b\,c^3\,\ln \relax (F)-2\right )}{b^2\,{\ln \relax (F)}^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 306, normalized size = 3.19 \[ \begin {cases} \frac {F^{a + b \left (c + d x\right )^{3}} \left (b^{2} c^{6} \log {\relax (F )}^{2} + 6 b^{2} c^{5} d x \log {\relax (F )}^{2} + 15 b^{2} c^{4} d^{2} x^{2} \log {\relax (F )}^{2} + 20 b^{2} c^{3} d^{3} x^{3} \log {\relax (F )}^{2} + 15 b^{2} c^{2} d^{4} x^{4} \log {\relax (F )}^{2} + 6 b^{2} c d^{5} x^{5} \log {\relax (F )}^{2} + b^{2} d^{6} x^{6} \log {\relax (F )}^{2} - 2 b c^{3} \log {\relax (F )} - 6 b c^{2} d x \log {\relax (F )} - 6 b c d^{2} x^{2} \log {\relax (F )} - 2 b d^{3} x^{3} \log {\relax (F )} + 2\right )}{3 b^{3} d \log {\relax (F )}^{3}} & \text {for}\: 3 b^{3} d \log {\relax (F )}^{3} \neq 0 \\c^{8} x + 4 c^{7} d x^{2} + \frac {28 c^{6} d^{2} x^{3}}{3} + 14 c^{5} d^{3} x^{4} + 14 c^{4} d^{4} x^{5} + \frac {28 c^{3} d^{5} x^{6}}{3} + 4 c^{2} d^{6} x^{7} + c d^{7} x^{8} + \frac {d^{8} x^{9}}{9} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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