Optimal. Leaf size=200 \[ \frac {2 b^2 d^2 \log ^2(F) (d e-c f)^2 \text {Int}\left (\frac {F^{a+b (c+d x)^2}}{e+f x},x\right )}{f^4}+\frac {b d^2 \log (F) \text {Int}\left (\frac {F^{a+b (c+d x)^2}}{e+f x},x\right )}{f^2}-\frac {\sqrt {\pi } b^{3/2} d^2 F^a \log ^{\frac {3}{2}}(F) (d e-c f) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{f^4}+\frac {b d \log (F) (d e-c f) F^{a+b (c+d x)^2}}{f^3 (e+f x)}-\frac {F^{a+b (c+d x)^2}}{2 f (e+f x)^2} \]
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Rubi [A] time = 0.31, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^3} \, dx &=-\frac {F^{a+b (c+d x)^2}}{2 f (e+f x)^2}+\frac {\left (b d^2 \log (F)\right ) \int \frac {F^{a+b (c+d x)^2}}{e+f x} \, dx}{f^2}-\frac {(b d (d e-c f) \log (F)) \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx}{f^2}\\ &=-\frac {F^{a+b (c+d x)^2}}{2 f (e+f x)^2}+\frac {b d (d e-c f) F^{a+b (c+d x)^2} \log (F)}{f^3 (e+f x)}+\frac {\left (b d^2 \log (F)\right ) \int \frac {F^{a+b (c+d x)^2}}{e+f x} \, dx}{f^2}-\frac {\left (2 b^2 d^3 (d e-c f) \log ^2(F)\right ) \int F^{a+b (c+d x)^2} \, dx}{f^4}+\frac {\left (2 b^2 d^2 (d e-c f)^2 \log ^2(F)\right ) \int \frac {F^{a+b (c+d x)^2}}{e+f x} \, dx}{f^4}\\ &=-\frac {F^{a+b (c+d x)^2}}{2 f (e+f x)^2}+\frac {b d (d e-c f) F^{a+b (c+d x)^2} \log (F)}{f^3 (e+f x)}-\frac {b^{3/2} d^2 (d e-c f) F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \log ^{\frac {3}{2}}(F)}{f^4}+\frac {\left (b d^2 \log (F)\right ) \int \frac {F^{a+b (c+d x)^2}}{e+f x} \, dx}{f^2}+\frac {\left (2 b^2 d^2 (d e-c f)^2 \log ^2(F)\right ) \int \frac {F^{a+b (c+d x)^2}}{e+f x} \, dx}{f^4}\\ \end {align*}
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Mathematica [A] time = 1.16, size = 0, normalized size = 0.00 \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (f x + e\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {F^{a +\left (d x +c \right )^{2} b}}{\left (f x +e \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (f x + e\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {F^{a+b\,{\left (c+d\,x\right )}^2}}{{\left (e+f\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + b \left (c + d x\right )^{2}}}{\left (e + f x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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