Optimal. Leaf size=53 \[ \frac {b F^a \log (F) \text {Ei}\left (b (c+d x)^2 \log (F)\right )}{2 d}-\frac {F^{a+b (c+d x)^2}}{2 d (c+d x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2214, 2210} \[ \frac {b F^a \log (F) \text {Ei}\left (b (c+d x)^2 \log (F)\right )}{2 d}-\frac {F^{a+b (c+d x)^2}}{2 d (c+d x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^3} \, dx &=-\frac {F^{a+b (c+d x)^2}}{2 d (c+d x)^2}+(b \log (F)) \int \frac {F^{a+b (c+d x)^2}}{c+d x} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{2 d (c+d x)^2}+\frac {b F^a \text {Ei}\left (b (c+d x)^2 \log (F)\right ) \log (F)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 47, normalized size = 0.89 \[ \frac {F^a \left (b \log (F) \text {Ei}\left (b (c+d x)^2 \log (F)\right )-\frac {F^{b (c+d x)^2}}{(c+d x)^2}\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.43, size = 100, normalized size = 1.89 \[ \frac {{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} F^{a} {\rm Ei}\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F)\right ) \log \relax (F) - F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 53, normalized size = 1.00 \[ -\frac {b \,F^{a} \Ei \left (1, -\left (d x +c \right )^{2} b \ln \relax (F )\right ) \ln \relax (F )}{2 d}-\frac {F^{a} F^{\left (d x +c \right )^{2} b}}{2 \left (d x +c \right )^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.68, size = 51, normalized size = 0.96 \[ -\frac {F^a\,\left (F^{b\,{\left (c+d\,x\right )}^2}+b\,\ln \relax (F)\,\mathrm {expint}\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2\right )\,{\left (c+d\,x\right )}^2\right )}{2\,d\,{\left (c+d\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + b \left (c + d x\right )^{2}}}{\left (c + d x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________