Optimal. Leaf size=200 \[ -\frac {F^{a+b (c+d x)^2}}{2 f (e+f x)^2}+\frac {b d (d e-c f) F^{a+b (c+d x)^2} \log (F)}{f^3 (e+f x)}-\frac {b^{3/2} d^2 (d e-c f) F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \log ^{\frac {3}{2}}(F)}{f^4}+\frac {b d^2 \log (F) \text {Int}\left (\frac {F^{a+b (c+d x)^2}}{e+f x},x\right )}{f^2}+\frac {2 b^2 d^2 (d e-c f)^2 \log ^2(F) \text {Int}\left (\frac {F^{a+b (c+d x)^2}}{e+f x},x\right )}{f^4} \]
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Rubi [A]
time = 0.21, 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 = {}
\begin {gather*} \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^3} \, dx \end {gather*}
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 = 1.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {F^{a +b \left (d x +c \right )^{2}}}{\left (f x +e \right )^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
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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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {F^{a + b \left (c + d x\right )^{2}}}{\left (e + f x\right )^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {F^{a+b\,{\left (c+d\,x\right )}^2}}{{\left (e+f\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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