Optimal. Leaf size=109 \[ -\frac {2 b d \log (F) (d e-c f) \text {Int}\left (\frac {F^{a+b (c+d x)^2}}{e+f x},x\right )}{f^2}-\frac {F^{a+b (c+d x)^2}}{f (e+f x)}+\frac {\sqrt {\pi } \sqrt {b} d F^a \sqrt {\log (F)} \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{f^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, 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 = {} \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx &=-\frac {F^{a+b (c+d x)^2}}{f (e+f x)}+\frac {\left (2 b d^2 \log (F)\right ) \int F^{a+b (c+d x)^2} \, dx}{f^2}-\frac {(2 b d (d e-c f) \log (F)) \int \frac {F^{a+b (c+d x)^2}}{e+f x} \, dx}{f^2}\\ &=-\frac {F^{a+b (c+d x)^2}}{f (e+f x)}+\frac {\sqrt {b} d F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \sqrt {\log (F)}}{f^2}-\frac {(2 b d (d e-c f) \log (F)) \int \frac {F^{a+b (c+d x)^2}}{e+f x} \, dx}{f^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.78, size = 0, normalized size = 0.00 \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {F^{a +\left (d x +c \right )^{2} b}}{\left (f x +e \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {F^{a+b\,{\left (c+d\,x\right )}^2}}{{\left (e+f\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + b \left (c + d x\right )^{2}}}{\left (e + f x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________