Optimal. Leaf size=39 \[ \frac{f^{a-\frac{b^2}{4 c}} \text{Ei}\left (\frac{(b+2 c x)^2 \log (f)}{4 c}\right )}{4 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0371917, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2238} \[ \frac{f^{a-\frac{b^2}{4 c}} \text{Ei}\left (\frac{(b+2 c x)^2 \log (f)}{4 c}\right )}{4 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2238
Rubi steps
\begin{align*} \int \frac{f^{a+b x+c x^2}}{b+2 c x} \, dx &=\frac{f^{a-\frac{b^2}{4 c}} \text{Ei}\left (\frac{(b+2 c x)^2 \log (f)}{4 c}\right )}{4 c}\\ \end{align*}
Mathematica [A] time = 0.0545218, size = 39, normalized size = 1. \[ \frac{f^{a-\frac{b^2}{4 c}} \text{Ei}\left (\frac{(b+2 c x)^2 \log (f)}{4 c}\right )}{4 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 40, normalized size = 1. \begin{align*} -{\frac{1}{4\,c}{f}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}{\it Ei} \left ( 1,-{\frac{ \left ( 2\,cx+b \right ) ^{2}\ln \left ( f \right ) }{4\,c}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{c x^{2} + b x + a}}{2 \, c x + b}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.50717, size = 105, normalized size = 2.69 \begin{align*} \frac{{\rm Ei}\left (\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (f\right )}{4 \, c}\right )}{4 \, c f^{\frac{b^{2} - 4 \, a c}{4 \, c}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + b x + c x^{2}}}{b + 2 c x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{c x^{2} + b x + a}}{2 \, c x + b}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]