Optimal. Leaf size=87 \[ -\frac {F^{a+b (c+d x)^3}}{6 d (c+d x)^6}-\frac {b F^{a+b (c+d x)^3} \log (F)}{6 d (c+d x)^3}+\frac {b^2 F^a \text {Ei}\left (b (c+d x)^3 \log (F)\right ) \log ^2(F)}{6 d} \]
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Rubi [A]
time = 0.13, antiderivative size = 87, 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 = {2245, 2241}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2241
Rule 2245
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*}
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Mathematica [A]
time = 0.22, size = 64, normalized size = 0.74 \begin {gather*} \frac {F^a \left (b^2 \text {Ei}\left (b (c+d x)^3 \log (F)\right ) \log ^2(F)-\frac {F^{b (c+d x)^3} \left (1+b (c+d x)^3 \log (F)\right )}{(c+d x)^6}\right )}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {F^{a +b \left (d x +c \right )^{3}}}{\left (d x +c \right )^{7}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 269 vs.
\(2 (81) = 162\).
time = 0.35, size = 269, normalized size = 3.09 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.88, size = 76, normalized size = 0.87 \begin {gather*} -\frac {F^a\,b^2\,{\ln \left (F\right )}^2\,\left (\frac {\mathrm {expint}\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}{2}+F^{b\,{\left (c+d\,x\right )}^3}\,\left (\frac {1}{2\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3}+\frac {1}{2\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^6}\right )\right )}{3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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