Optimal. Leaf size=181 \[ \frac{\sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^2 \sqrt{n}}+\frac{\sqrt{\frac{\pi }{2}} g e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^2 \sqrt{n}} \]
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Rubi [A] time = 0.272842, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {2401, 2389, 2300, 2180, 2204, 2390, 2310} \[ \frac{\sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^2 \sqrt{n}}+\frac{\sqrt{\frac{\pi }{2}} g e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^2 \sqrt{n}} \]
Antiderivative was successfully verified.
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Rule 2401
Rule 2389
Rule 2300
Rule 2180
Rule 2204
Rule 2390
Rule 2310
Rubi steps
\begin{align*} \int \frac{f+g x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx &=\int \left (\frac{e f-d g}{e \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{g (d+e x)}{e \sqrt{a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx\\ &=\frac{g \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{e}+\frac{(e f-d g) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{e}\\ &=\frac{g \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^2}+\frac{(e f-d g) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^2}\\ &=\frac{\left (g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^2 n}+\frac{\left ((e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^2 n}\\ &=\frac{\left (2 g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b n}+\frac{2 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b e^2 n}+\frac{\left (2 (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b n}+\frac{x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b e^2 n}\\ &=\frac{e^{-\frac{a}{b n}} (e f-d g) \sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^2 \sqrt{n}}+\frac{e^{-\frac{2 a}{b n}} g \sqrt{\frac{\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^2 \sqrt{n}}\\ \end{align*}
Mathematica [A] time = 0.140126, size = 164, normalized size = 0.91 \[ \frac{\sqrt{\pi } e^{-\frac{2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (2 e^{\frac{a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac{1}{n}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+\sqrt{2} g (d+e x) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )\right )}{2 \sqrt{b} e^2 \sqrt{n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.472, size = 0, normalized size = 0. \begin{align*} \int{(gx+f){\frac{1}{\sqrt{a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) }}}}\, dx \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 \frac{g x + f}{\sqrt{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 \frac{f + g x}{\sqrt{a + b \log{\left (c \left (d + e x\right )^{n} \right )}}}\, 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 \frac{g x + f}{\sqrt{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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