Optimal. Leaf size=46 \[ -\frac {\sqrt {\pi } F^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 46, 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 = {2211, 2204} \[ -\frac {\sqrt {\pi } F^a \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2204
Rule 2211
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^2} \, dx &=-\frac {\operatorname {Subst}\left (\int F^{a+b x^2} \, dx,x,\frac {1}{c+d x}\right )}{d}\\ &=-\frac {F^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{2 \sqrt {b} d \sqrt {\log (F)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 46, normalized size = 1.00 \[ -\frac {\sqrt {\pi } F^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 45, normalized size = 0.98 \[ \frac {\sqrt {\pi } F^{a} \sqrt {-\frac {b \log \relax (F)}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \relax (F)}{d^{2}}}}{d x + c}\right )}{2 \, b \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 35, normalized size = 0.76 \[ -\frac {\sqrt {\pi }\, F^{a} \erf \left (\frac {\sqrt {-b \ln \relax (F )}}{d x +c}\right )}{2 \sqrt {-b \ln \relax (F )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.50, size = 35, normalized size = 0.76 \[ -\frac {F^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \relax (F)}{\sqrt {b\,\ln \relax (F)}\,\left (c+d\,x\right )}\right )}{2\,d\,\sqrt {b\,\ln \relax (F)}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{\left (c + d x\right )^{2}}}}{\left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________