Optimal. Leaf size=87 \[ \frac {b^2 F^a \log ^2(F) \text {Ei}\left (b (c+d x)^2 \log (F)\right )}{4 d}-\frac {F^{a+b (c+d x)^2}}{4 d (c+d x)^4}-\frac {b \log (F) F^{a+b (c+d x)^2}}{4 d (c+d x)^2} \]
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Rubi [A] time = 0.20, 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 = {2214, 2210} \[ \frac {b^2 F^a \log ^2(F) \text {Ei}\left (b (c+d x)^2 \log (F)\right )}{4 d}-\frac {F^{a+b (c+d x)^2}}{4 d (c+d x)^4}-\frac {b \log (F) F^{a+b (c+d x)^2}}{4 d (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^5} \, dx &=-\frac {F^{a+b (c+d x)^2}}{4 d (c+d x)^4}+\frac {1}{2} (b \log (F)) \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^3} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{4 d (c+d x)^4}-\frac {b F^{a+b (c+d x)^2} \log (F)}{4 d (c+d x)^2}+\frac {1}{2} \left (b^2 \log ^2(F)\right ) \int \frac {F^{a+b (c+d x)^2}}{c+d x} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{4 d (c+d x)^4}-\frac {b F^{a+b (c+d x)^2} \log (F)}{4 d (c+d x)^2}+\frac {b^2 F^a \text {Ei}\left (b (c+d x)^2 \log (F)\right ) \log ^2(F)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 64, normalized size = 0.74 \[ \frac {F^a \left (b^2 \log ^2(F) \text {Ei}\left (b (c+d x)^2 \log (F)\right )-\frac {F^{b (c+d x)^2} \left (b \log (F) (c+d x)^2+1\right )}{(c+d x)^4}\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 183, normalized size = 2.10 \[ \frac {{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} F^{a} {\rm Ei}\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F)\right ) \log \relax (F)^{2} - {\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F) + 1\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{4 \, {\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 6 \, c^{2} d^{3} x^{2} + 4 \, c^{3} d^{2} x + c^{4} d\right )}} \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 86, normalized size = 0.99 \[ -\frac {b^{2} F^{a} \Ei \left (1, -\left (d x +c \right )^{2} b \ln \relax (F )\right ) \ln \relax (F )^{2}}{4 d}-\frac {b \,F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )}{4 \left (d x +c \right )^{2} d}-\frac {F^{a} F^{\left (d x +c \right )^{2} b}}{4 \left (d x +c \right )^{4} d} \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.76, size = 76, normalized size = 0.87 \[ -\frac {F^a\,b^2\,{\ln \relax (F)}^2\,\left (\frac {\mathrm {expint}\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2\right )}{2}+F^{b\,{\left (c+d\,x\right )}^2}\,\left (\frac {1}{2\,b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2}+\frac {1}{2\,b^2\,{\ln \relax (F)}^2\,{\left (c+d\,x\right )}^4}\right )\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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