Optimal. Leaf size=422 \[ \frac{6 \sqrt{3 \pi } g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^4 n^{3/2}}+\frac{6 \sqrt{2 \pi } g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^4 n^{3/2}}+\frac{2 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^4 n^{3/2}}+\frac{4 \sqrt{\pi } g^3 e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^4 n^{3/2}}-\frac{2 (d+e x) (f+g x)^3}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]
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Rubi [A] time = 1.31534, antiderivative size = 422, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2400, 2401, 2389, 2300, 2180, 2204, 2390, 2310} \[ \frac{6 \sqrt{3 \pi } g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^4 n^{3/2}}+\frac{6 \sqrt{2 \pi } g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^4 n^{3/2}}+\frac{2 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^4 n^{3/2}}+\frac{4 \sqrt{\pi } g^3 e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^4 n^{3/2}}-\frac{2 (d+e x) (f+g x)^3}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]
Antiderivative was successfully verified.
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Rule 2400
Rule 2401
Rule 2389
Rule 2300
Rule 2180
Rule 2204
Rule 2390
Rule 2310
Rubi steps
\begin{align*} \int \frac{(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx &=-\frac{2 (d+e x) (f+g x)^3}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{8 \int \frac{(f+g x)^3}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b n}-\frac{(6 (e f-d g)) \int \frac{(f+g x)^2}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e n}\\ &=-\frac{2 (d+e x) (f+g x)^3}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{8 \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}{b n}-\frac{(6 (e f-d g)) \int \left (\frac{(e f-d g)^2}{e^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{2 g (e f-d g) (d+e x)}{e^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{g^2 (d+e x)^2}{e^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{b e n}\\ &=-\frac{2 (d+e x) (f+g x)^3}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (8 g^3\right ) \int \frac{(d+e x)^3}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^3 n}-\frac{\left (6 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}{b e^3 n}+\frac{\left (24 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}{b e^3 n}-\frac{\left (12 g (e f-d g)^2\right ) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^3 n}+\frac{\left (24 g (e f-d g)^2\right ) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^3 n}-\frac{\left (6 (e f-d g)^3\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^3 n}+\frac{\left (8 (e f-d g)^3\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^3 n}\\ &=-\frac{2 (d+e x) (f+g x)^3}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (8 g^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^4 n}-\frac{\left (6 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 )}{b e^4 n}+\frac{\left (24 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 )}{b e^4 n}-\frac{\left (12 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 )}{b e^4 n}+\frac{\left (24 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 )}{b e^4 n}-\frac{\left (6 (e f-d g)^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^4 n}+\frac{\left (8 (e f-d g)^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^4 n}\\ &=-\frac{2 (d+e x) (f+g x)^3}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (8 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 )}{b e^4 n^2}-\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 \frac{e^{\frac{3 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^4 n^2}+\frac{\left (24 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 )}{b e^4 n^2}-\frac{\left (12 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 )}{b e^4 n^2}+\frac{\left (24 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 )}{b e^4 n^2}-\frac{\left (6 (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 )}{b e^4 n^2}+\frac{\left (8 (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 )}{b e^4 n^2}\\ &=-\frac{2 (d+e x) (f+g x)^3}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (16 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^2 e^4 n^2}-\frac{\left (12 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^2 e^4 n^2}+\frac{\left (48 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^2 e^4 n^2}-\frac{\left (24 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^2 e^4 n^2}+\frac{\left (48 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^2 e^4 n^2}-\frac{\left (12 (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^2 e^4 n^2}+\frac{\left (16 (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^2 e^4 n^2}\\ &=\frac{2 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 )}{b^{3/2} e^4 n^{3/2}}+\frac{4 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 )}{b^{3/2} e^4 n^{3/2}}+\frac{6 e^{-\frac{2 a}{b n}} g (e f-d g)^2 \sqrt{2 \pi } (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 )}{b^{3/2} e^4 n^{3/2}}+\frac{6 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 )}{b^{3/2} e^4 n^{3/2}}-\frac{2 (d+e x) (f+g x)^3}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}\\ \end{align*}
Mathematica [B] time = 2.7476, size = 1281, normalized size = 3.04 \[ \frac{2 \left (2 e^{-\frac{4 a}{b n}} g^3 \sqrt{\pi } (d+e x)^4 \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \sqrt{a+b \log \left (c (d+e x)^n\right )} \left (c (d+e x)^n\right )^{-4/n}-3 d e^{-\frac{3 a}{b n}} g^3 \sqrt{3 \pi } (d+e x)^3 \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \sqrt{a+b \log \left (c (d+e x)^n\right )} \left (c (d+e x)^n\right )^{-3/n}+3 e e^{-\frac{3 a}{b n}} f g^2 \sqrt{3 \pi } (d+e x)^3 \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \sqrt{a+b \log \left (c (d+e x)^n\right )} \left (c (d+e x)^n\right )^{-3/n}+3 d^2 e^{-\frac{2 a}{b n}} g^3 \sqrt{2 \pi } (d+e x)^2 \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \sqrt{a+b \log \left (c (d+e x)^n\right )} \left (c (d+e x)^n\right )^{-2/n}-6 d e e^{-\frac{2 a}{b n}} f g^2 \sqrt{2 \pi } (d+e x)^2 \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \sqrt{a+b \log \left (c (d+e x)^n\right )} \left (c (d+e x)^n\right )^{-2/n}+3 e^2 e^{-\frac{2 a}{b n}} f^2 g \sqrt{2 \pi } (d+e x)^2 \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \sqrt{a+b \log \left (c (d+e x)^n\right )} \left (c (d+e x)^n\right )^{-2/n}-d^3 e^{-\frac{a}{b n}} g^3 \sqrt{\pi } (d+e x) \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \sqrt{a+b \log \left (c (d+e x)^n\right )} \left (c (d+e x)^n\right )^{-1/n}+3 d^2 e e^{-\frac{a}{b n}} f g^2 \sqrt{\pi } (d+e x) \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \sqrt{a+b \log \left (c (d+e x)^n\right )} \left (c (d+e x)^n\right )^{-1/n}-6 d e^2 e^{-\frac{a}{b n}} f^2 g \sqrt{\pi } (d+e x) \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \sqrt{a+b \log \left (c (d+e x)^n\right )} \left (c (d+e x)^n\right )^{-1/n}+\sqrt{b} e^3 e^{-\frac{a}{b n}} f^3 \sqrt{n} (d+e x) \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \sqrt{-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}} \left (c (d+e x)^n\right )^{-1/n}+3 \sqrt{b} d e^2 e^{-\frac{a}{b n}} f^2 g \sqrt{n} (d+e x) \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \sqrt{-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}} \left (c (d+e x)^n\right )^{-1/n}-\sqrt{b} e^4 g^3 \sqrt{n} x^4-\sqrt{b} d e^3 g^3 \sqrt{n} x^3-3 \sqrt{b} e^4 f g^2 \sqrt{n} x^3-3 \sqrt{b} d e^3 f g^2 \sqrt{n} x^2-3 \sqrt{b} e^4 f^2 g \sqrt{n} x^2-\sqrt{b} e^4 f^3 \sqrt{n} x-3 \sqrt{b} d e^3 f^2 g \sqrt{n} x-\sqrt{b} d e^3 f^3 \sqrt{n}\right )}{b^{3/2} e^4 n^{3/2} \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.727, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{3} \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{-{\frac{3}{2}}}}\, 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}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{3}{2}}}\,{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}}{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac{3}{2}}}\, 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}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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