Optimal. Leaf size=44 \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d \sqrt{\log (F)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0122826, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2204} \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2204
Rubi steps
\begin{align*} \int F^{a+b (c+d x)^2} \, dx &=\frac{F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right )}{2 \sqrt{b} d \sqrt{\log (F)}}\\ \end{align*}
Mathematica [A] time = 0.0058501, size = 44, normalized size = 1. \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 58, normalized size = 1.3 \begin{align*} -{\frac{\sqrt{\pi }{F}^{{c}^{2}b+a}{F}^{-{c}^{2}b}}{2\,d}{\it Erf} \left ( -d\sqrt{-b\ln \left ( F \right ) }x+{bc\ln \left ( F \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01588, size = 78, normalized size = 1.77 \begin{align*} \frac{\sqrt{\pi } F^{b c^{2} + a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} d x - \frac{b c \log \left (F\right )}{\sqrt{-b \log \left (F\right )}}\right )}{2 \, \sqrt{-b \log \left (F\right )} F^{b c^{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.58251, size = 123, normalized size = 2.8 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b d^{2} \log \left (F\right )} F^{a} \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right )}{2 \, b d^{2} \log \left (F\right )} \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 F^{a + b \left (c + d x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.25157, size = 49, normalized size = 1.11 \begin{align*} -\frac{\sqrt{\pi } F^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} d{\left (x + \frac{c}{d}\right )}\right )}{2 \, \sqrt{-b \log \left (F\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]