Optimal. Leaf size=74 \[ \frac{x^{n/2} f^{a+b x^n}}{b n \log (f)}-\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{2 b^{3/2} n \log ^{\frac{3}{2}}(f)} \]
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Rubi [A] time = 0.065467, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2213, 2211, 2204} \[ \frac{x^{n/2} f^{a+b x^n}}{b n \log (f)}-\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{2 b^{3/2} n \log ^{\frac{3}{2}}(f)} \]
Antiderivative was successfully verified.
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Rule 2213
Rule 2211
Rule 2204
Rubi steps
\begin{align*} \int f^{a+b x^n} x^{-1+\frac{3 n}{2}} \, dx &=\frac{f^{a+b x^n} x^{n/2}}{b n \log (f)}-\frac{\int f^{a+b x^n} x^{\frac{1}{2} (-2+n)} \, dx}{2 b \log (f)}\\ &=\frac{f^{a+b x^n} x^{n/2}}{b n \log (f)}-\frac{\operatorname{Subst}\left (\int f^{a+b x^2} \, dx,x,x^{1+\frac{1}{2} (-2+n)}\right )}{b n \log (f)}\\ &=-\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} x^{n/2} \sqrt{\log (f)}\right )}{2 b^{3/2} n \log ^{\frac{3}{2}}(f)}+\frac{f^{a+b x^n} x^{n/2}}{b n \log (f)}\\ \end{align*}
Mathematica [A] time = 0.0087414, size = 39, normalized size = 0.53 \[ -\frac{f^a x^{3 n/2} \text{Gamma}\left (\frac{3}{2},-b \log (f) x^n\right )}{n \left (-b \log (f) x^n\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 67, normalized size = 0.9 \begin{align*}{\frac{{f}^{a}{f}^{b{x}^{n}}}{\ln \left ( f \right ) bn}{x}^{{\frac{n}{2}}}}-{\frac{{f}^{a}\sqrt{\pi }}{2\,\ln \left ( f \right ) bn}{\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.
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Maxima [A] time = 1.28449, size = 45, normalized size = 0.61 \begin{align*} -\frac{f^{a} x^{\frac{3}{2} \, n} \Gamma \left (\frac{3}{2}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac{3}{2}} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60275, size = 182, normalized size = 2.46 \begin{align*} \frac{2 \, b x^{\frac{1}{2} \, n} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )} \log \left (f\right ) + \sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x^{\frac{1}{2} \, n}\right )}{2 \, b^{2} n \log \left (f\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{b x^{n} + a} x^{\frac{3}{2} \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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