Optimal. Leaf size=66 \[ \frac {\log (f) f^{-\frac {b^2}{4 c}} \text {Ei}\left (\frac {(b+2 c x)^2 \log (f)}{4 c}\right )}{16 c^2}-\frac {f^{b x+c x^2}}{4 c (b+2 c x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2239, 2238} \[ \frac {\log (f) f^{-\frac {b^2}{4 c}} \text {Ei}\left (\frac {(b+2 c x)^2 \log (f)}{4 c}\right )}{16 c^2}-\frac {f^{b x+c x^2}}{4 c (b+2 c x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2238
Rule 2239
Rubi steps
\begin {align*} \int \frac {f^{b x+c x^2}}{(b+2 c x)^3} \, dx &=-\frac {f^{b x+c x^2}}{4 c (b+2 c x)^2}+\frac {\log (f) \int \frac {f^{b x+c x^2}}{b+2 c x} \, dx}{4 c}\\ &=-\frac {f^{b x+c x^2}}{4 c (b+2 c x)^2}+\frac {f^{-\frac {b^2}{4 c}} \text {Ei}\left (\frac {(b+2 c x)^2 \log (f)}{4 c}\right ) \log (f)}{16 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 77, normalized size = 1.17 \[ \frac {f^{-\frac {b^2}{4 c}} \left (\log (f) (b+2 c x)^2 \text {Ei}\left (\frac {(b+2 c x)^2 \log (f)}{4 c}\right )-4 c f^{\frac {(b+2 c x)^2}{4 c}}\right )}{16 c^2 (b+2 c x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 100, normalized size = 1.52 \[ -\frac {4 \, c f^{c x^{2} + b x} - \frac {{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} {\rm Ei}\left (\frac {{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \relax (f)}{4 \, c}\right ) \log \relax (f)}{f^{\frac {b^{2}}{4 \, c}}}}{16 \, {\left (4 \, c^{4} x^{2} + 4 \, b c^{3} x + b^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{c x^{2} + b x}}{{\left (2 \, c x + b\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 74, normalized size = 1.12 \[ -\frac {f^{-\frac {b^{2}}{4 c}} f^{\frac {\left (2 c x +b \right )^{2}}{4 c}}}{4 \left (2 c x +b \right )^{2} c}-\frac {f^{-\frac {b^{2}}{4 c}} \Ei \left (1, -\frac {\left (2 c x +b \right )^{2} \ln \relax (f )}{4 c}\right ) \ln \relax (f )}{16 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{c x^{2} + b x}}{{\left (2 \, c x + b\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {f^{c\,x^2+b\,x}}{{\left (b+2\,c\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{b x + c x^{2}}}{\left (b + 2 c x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________