Optimal. Leaf size=43 \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{\sqrt{b} n \sqrt{\log (f)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0363298, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2211, 2204} \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{\sqrt{b} n \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2211
Rule 2204
Rubi steps
\begin{align*} \int f^{a+b x^n} x^{-1+\frac{n}{2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int f^{a+b x^2} \, dx,x,x^{n/2}\right )}{n}\\ &=\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} x^{n/2} \sqrt{\log (f)}\right )}{\sqrt{b} n \sqrt{\log (f)}}\\ \end{align*}
Mathematica [A] time = 0.008519, size = 43, normalized size = 1. \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{\sqrt{b} n \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.05, size = 32, normalized size = 0.7 \begin{align*}{\frac{{f}^{a}\sqrt{\pi }}{n}{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }{x}^{{\frac{n}{2}}} \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 = 1.14225, size = 51, normalized size = 1.19 \begin{align*} \frac{\sqrt{\pi } f^{a} x^{\frac{1}{2} \, n}{\left (\operatorname{erf}\left (\sqrt{-b x^{n} \log \left (f\right )}\right ) - 1\right )}}{\sqrt{-b x^{n} \log \left (f\right )} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.52508, size = 109, normalized size = 2.53 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x x^{\frac{1}{2} \, n - 1}\right )}{b n \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 f^{b x^{n} + a} x^{\frac{1}{2} \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]