Optimal. Leaf size=146 \[ -\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{32 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{3 e^3}+\frac{20 b d n \sqrt{d+e x}}{3 e^3}-\frac{4 b n (d+e x)^{3/2}}{9 e^3} \]
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Rubi [A] time = 0.161974, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {43, 2350, 12, 897, 1153, 208} \[ -\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{32 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{3 e^3}+\frac{20 b d n \sqrt{d+e x}}{3 e^3}-\frac{4 b n (d+e x)^{3/2}}{9 e^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2350
Rule 12
Rule 897
Rule 1153
Rule 208
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx &=-\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-(b n) \int \frac{2 \left (-8 d^2-4 d e x+e^2 x^2\right )}{3 e^3 x \sqrt{d+e x}} \, dx\\ &=-\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{(2 b n) \int \frac{-8 d^2-4 d e x+e^2 x^2}{x \sqrt{d+e x}} \, dx}{3 e^3}\\ &=-\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{(4 b n) \operatorname{Subst}\left (\int \frac{-3 d^2-6 d x^2+x^4}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{3 e^4}\\ &=-\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{(4 b n) \operatorname{Subst}\left (\int \left (-5 d e+e x^2-\frac{8 d^2}{-\frac{d}{e}+\frac{x^2}{e}}\right ) \, dx,x,\sqrt{d+e x}\right )}{3 e^4}\\ &=\frac{20 b d n \sqrt{d+e x}}{3 e^3}-\frac{4 b n (d+e x)^{3/2}}{9 e^3}-\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{\left (32 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{3 e^4}\\ &=\frac{20 b d n \sqrt{d+e x}}{3 e^3}-\frac{4 b n (d+e x)^{3/2}}{9 e^3}-\frac{32 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{3 e^3}-\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}\\ \end{align*}
Mathematica [A] time = 0.0897264, size = 124, normalized size = 0.85 \[ \frac{-48 a d^2-24 a d e x+6 a e^2 x^2-6 b \left (8 d^2+4 d e x-e^2 x^2\right ) \log \left (c x^n\right )-96 b d^{3/2} n \sqrt{d+e x} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+56 b d^2 n+52 b d e n x-4 b e^2 n x^2}{9 e^3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.557, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45716, size = 784, normalized size = 5.37 \begin{align*} \left [\frac{2 \,{\left (24 \,{\left (b d e n x + b d^{2} n\right )} \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) +{\left (28 \, b d^{2} n - 24 \, a d^{2} -{\left (2 \, b e^{2} n - 3 \, a e^{2}\right )} x^{2} + 2 \,{\left (13 \, b d e n - 6 \, a d e\right )} x + 3 \,{\left (b e^{2} x^{2} - 4 \, b d e x - 8 \, b d^{2}\right )} \log \left (c\right ) + 3 \,{\left (b e^{2} n x^{2} - 4 \, b d e n x - 8 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{9 \,{\left (e^{4} x + d e^{3}\right )}}, \frac{2 \,{\left (48 \,{\left (b d e n x + b d^{2} n\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) +{\left (28 \, b d^{2} n - 24 \, a d^{2} -{\left (2 \, b e^{2} n - 3 \, a e^{2}\right )} x^{2} + 2 \,{\left (13 \, b d e n - 6 \, a d e\right )} x + 3 \,{\left (b e^{2} x^{2} - 4 \, b d e x - 8 \, b d^{2}\right )} \log \left (c\right ) + 3 \,{\left (b e^{2} n x^{2} - 4 \, b d e n x - 8 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{9 \,{\left (e^{4} x + d e^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 60.8541, size = 262, normalized size = 1.79 \begin{align*} \frac{- \frac{2 a d^{2}}{\sqrt{d + e x}} - 4 a d \sqrt{d + e x} + \frac{2 a \left (d + e x\right )^{\frac{3}{2}}}{3} + 2 b d^{2} \left (\frac{2 n \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{\sqrt{- d}} - \frac{\log{\left (c \left (- \frac{d}{e} + \frac{d + e x}{e}\right )^{n} \right )}}{\sqrt{d + e x}}\right ) - 4 b d \left (\sqrt{d + e x} \log{\left (c \left (- \frac{d}{e} + \frac{d + e x}{e}\right )^{n} \right )} - \frac{2 n \left (\frac{d e \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{\sqrt{- d}} + e \sqrt{d + e x}\right )}{e}\right ) + 2 b \left (\frac{\left (d + e x\right )^{\frac{3}{2}} \log{\left (c \left (- \frac{d}{e} + \frac{d + e x}{e}\right )^{n} \right )}}{3} - \frac{2 n \left (\frac{d^{2} e \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{\sqrt{- d}} + d e \sqrt{d + e x} + \frac{e \left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{3 e}\right )}{e^{3}} \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 (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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