Optimal. Leaf size=383 \[ \frac{\sqrt{3 \pi } g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}+\frac{3 \sqrt{\frac{\pi }{2}} g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}+\frac{\sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}+\frac{\sqrt{\pi } g^3 e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 \sqrt{b} e^4 \sqrt{n}} \]
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Rubi [A] time = 0.727602, antiderivative size = 383, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {2401, 2389, 2300, 2180, 2204, 2390, 2310} \[ \frac{\sqrt{3 \pi } g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}+\frac{3 \sqrt{\frac{\pi }{2}} g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}+\frac{\sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}+\frac{\sqrt{\pi } g^3 e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 \sqrt{b} e^4 \sqrt{n}} \]
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)^3}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx &=\int \left (\frac{(e f-d g)^3}{e^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{3 g (e f-d g)^2 (d+e x)}{e^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{3 g^2 (e f-d g) (d+e x)^2}{e^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{g^3 (d+e x)^3}{e^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx\\ &=\frac{g^3 \int \frac{(d+e x)^3}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}+\frac{\left (3 g^2 (e f-d g)\right ) \int \frac{(d+e x)^2}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}+\frac{\left (3 g (e f-d g)^2\right ) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}+\frac{(e f-d g)^3 \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}\\ &=\frac{g^3 \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}+\frac{\left (3 g^2 (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}+\frac{\left (3 g (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}+\frac{(e f-d g)^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}\\ &=\frac{\left (g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{4 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac{\left (3 g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{3 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac{\left (3 g (e f-d g)^2 (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^4 n}+\frac{\left ((e f-d g)^3 (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^4 n}\\ &=\frac{\left (2 g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{4 a}{b n}+\frac{4 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b e^4 n}+\frac{\left (6 g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b n}+\frac{3 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b e^4 n}+\frac{\left (6 g (e f-d g)^2 (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^4 n}+\frac{\left (2 (e f-d g)^3 (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^4 n}\\ &=\frac{e^{-\frac{a}{b n}} (e f-d g)^3 \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^4 \sqrt{n}}+\frac{e^{-\frac{4 a}{b n}} g^3 \sqrt{\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 \sqrt{b} e^4 \sqrt{n}}+\frac{3 e^{-\frac{2 a}{b n}} g (e f-d g)^2 \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^4 \sqrt{n}}+\frac{e^{-\frac{3 a}{b n}} g^2 (e f-d g) \sqrt{3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}\\ \end{align*}
Mathematica [A] time = 0.417212, size = 331, normalized size = 0.86 \[ \frac{\sqrt{\pi } e^{-\frac{4 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-4/n} \left (2 \sqrt{3} g^2 e^{\frac{a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{\frac{1}{n}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+3 \sqrt{2} g e^{\frac{2 a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+2 e^{\frac{3 a}{b n}} (e f-d g)^3 \left (c (d+e x)^n\right )^{3/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+g^3 (d+e x)^3 \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )\right )}{2 \sqrt{b} e^4 \sqrt{n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.741, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{3}{\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{{\left (g x + f\right )}^{3}}{\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{\left (f + g x\right )^{3}}{\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{{\left (g x + f\right )}^{3}}{\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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