Optimal. Leaf size=53 \[ \frac{b F^a \log (F) \text{Ei}\left (b (c+d x)^3 \log (F)\right )}{3 d}-\frac{F^{a+b (c+d x)^3}}{3 d (c+d x)^3} \]
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Rubi [A] time = 0.125406, antiderivative size = 53, normalized size of antiderivative = 1., 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 = {2214, 2210} \[ \frac{b F^a \log (F) \text{Ei}\left (b (c+d x)^3 \log (F)\right )}{3 d}-\frac{F^{a+b (c+d x)^3}}{3 d (c+d x)^3} \]
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)^4} \, dx &=-\frac{F^{a+b (c+d x)^3}}{3 d (c+d x)^3}+(b \log (F)) \int \frac{F^{a+b (c+d x)^3}}{c+d x} \, dx\\ &=-\frac{F^{a+b (c+d x)^3}}{3 d (c+d x)^3}+\frac{b F^a \text{Ei}\left (b (c+d x)^3 \log (F)\right ) \log (F)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0316843, size = 47, normalized size = 0.89 \[ \frac{F^a \left (b \log (F) \text{Ei}\left (b (c+d x)^3 \log (F)\right )-\frac{F^{b (c+d x)^3}}{(c+d x)^3}\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{3}}}{ \left ( dx+c \right ) ^{4}}}\, 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 )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.49807, size = 315, normalized size = 5.94 \begin{align*} \frac{{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\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 ) - F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \,{\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} 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 )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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