Optimal. Leaf size=87 \[ \frac{b^2 F^a \log ^2(F) \text{Ei}\left (b (c+d x)^3 \log (F)\right )}{6 d}-\frac{F^{a+b (c+d x)^3}}{6 d (c+d x)^6}-\frac{b \log (F) F^{a+b (c+d x)^3}}{6 d (c+d x)^3} \]
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Rubi [A] time = 0.192794, antiderivative size = 87, normalized size of antiderivative = 1., 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 = {2214, 2210} \[ \frac{b^2 F^a \log ^2(F) \text{Ei}\left (b (c+d x)^3 \log (F)\right )}{6 d}-\frac{F^{a+b (c+d x)^3}}{6 d (c+d x)^6}-\frac{b \log (F) F^{a+b (c+d x)^3}}{6 d (c+d x)^3} \]
Antiderivative was successfully verified.
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Rule 2214
Rule 2210
Rubi steps
\begin{align*} \int \frac{F^{a+b (c+d x)^3}}{(c+d x)^7} \, dx &=-\frac{F^{a+b (c+d x)^3}}{6 d (c+d x)^6}+\frac{1}{2} (b \log (F)) \int \frac{F^{a+b (c+d x)^3}}{(c+d x)^4} \, dx\\ &=-\frac{F^{a+b (c+d x)^3}}{6 d (c+d x)^6}-\frac{b F^{a+b (c+d x)^3} \log (F)}{6 d (c+d x)^3}+\frac{1}{2} \left (b^2 \log ^2(F)\right ) \int \frac{F^{a+b (c+d x)^3}}{c+d x} \, dx\\ &=-\frac{F^{a+b (c+d x)^3}}{6 d (c+d x)^6}-\frac{b F^{a+b (c+d x)^3} \log (F)}{6 d (c+d x)^3}+\frac{b^2 F^a \text{Ei}\left (b (c+d x)^3 \log (F)\right ) \log ^2(F)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.0756531, size = 64, normalized size = 0.74 \[ \frac{F^a \left (b^2 \log ^2(F) \text{Ei}\left (b (c+d x)^3 \log (F)\right )-\frac{F^{b (c+d x)^3} \left (b \log (F) (c+d x)^3+1\right )}{(c+d x)^6}\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.084, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{3}}}{ \left ( dx+c \right ) ^{7}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.58922, size = 567, normalized size = 6.52 \begin{align*} \frac{{\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 )} F^{a}{\rm Ei}\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 \left (F\right )\right ) \log \left (F\right )^{2} -{\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 \left (F\right ) + 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}}{6 \,{\left (d^{7} x^{6} + 6 \, c d^{6} x^{5} + 15 \, c^{2} d^{5} x^{4} + 20 \, c^{3} d^{4} x^{3} + 15 \, c^{4} d^{3} x^{2} + 6 \, c^{5} d^{2} x + c^{6} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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