Optimal. Leaf size=47 \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (F)} (a+b x)^{n/2}\right )}{b \sqrt{c} n \sqrt{\log (F)}} \]
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Rubi [A] time = 0.0651868, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2211, 2204} \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (F)} (a+b x)^{n/2}\right )}{b \sqrt{c} n \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
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Rule 2211
Rule 2204
Rubi steps
\begin{align*} \int F^{c (a+b x)^n} (a+b x)^{-1+\frac{n}{2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int F^{c x^2} \, dx,x,(a+b x)^{n/2}\right )}{b n}\\ &=\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{c} (a+b x)^{n/2} \sqrt{\log (F)}\right )}{b \sqrt{c} n \sqrt{\log (F)}}\\ \end{align*}
Mathematica [A] time = 0.0117368, size = 47, normalized size = 1. \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (F)} (a+b x)^{n/2}\right )}{b \sqrt{c} n \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.177, size = 36, normalized size = 0.8 \begin{align*}{\frac{\sqrt{\pi }}{bn}{\it Erf} \left ( \sqrt{-c\ln \left ( F \right ) } \left ( bx+a \right ) ^{{\frac{n}{2}}} \right ){\frac{1}{\sqrt{-c\ln \left ( F \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{\frac{1}{2} \, n - 1} F^{{\left (b x + a\right )}^{n} c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65197, size = 128, normalized size = 2.72 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-c \log \left (F\right )} \operatorname{erf}\left ({\left (b x + a\right )} \sqrt{-c \log \left (F\right )}{\left (b x + a\right )}^{\frac{1}{2} \, n - 1}\right )}{b c n \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{c \left (a + b x\right )^{n}} \left (a + b x\right )^{\frac{n}{2} - 1}\, dx \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{\left (b x + a\right )}^{\frac{1}{2} \, n - 1} F^{{\left (b x + a\right )}^{n} c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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