Optimal. Leaf size=94 \[ \frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac{4 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{3 e}-\frac{4 b d n \sqrt{d+e x}}{3 e}-\frac{4 b n (d+e x)^{3/2}}{9 e} \]
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Rubi [A] time = 0.042146, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2319, 50, 63, 208} \[ \frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac{4 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{3 e}-\frac{4 b d n \sqrt{d+e x}}{3 e}-\frac{4 b n (d+e x)^{3/2}}{9 e} \]
Antiderivative was successfully verified.
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Rule 2319
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac{(2 b n) \int \frac{(d+e x)^{3/2}}{x} \, dx}{3 e}\\ &=-\frac{4 b n (d+e x)^{3/2}}{9 e}+\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac{(2 b d n) \int \frac{\sqrt{d+e x}}{x} \, dx}{3 e}\\ &=-\frac{4 b d n \sqrt{d+e x}}{3 e}-\frac{4 b n (d+e x)^{3/2}}{9 e}+\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac{\left (2 b d^2 n\right ) \int \frac{1}{x \sqrt{d+e x}} \, dx}{3 e}\\ &=-\frac{4 b d n \sqrt{d+e x}}{3 e}-\frac{4 b n (d+e x)^{3/2}}{9 e}+\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac{\left (4 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^2}\\ &=-\frac{4 b d n \sqrt{d+e x}}{3 e}-\frac{4 b n (d+e x)^{3/2}}{9 e}+\frac{4 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{3 e}+\frac{2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}\\ \end{align*}
Mathematica [A] time = 0.0679848, size = 77, normalized size = 0.82 \[ \frac{2 \left (\sqrt{d+e x} \left (3 a (d+e x)+3 b (d+e x) \log \left (c x^n\right )-2 b n (4 d+e x)\right )+6 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right )}{9 e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.588, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \sqrt{ex+d}\, 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 = 1.42478, size = 475, normalized size = 5.05 \begin{align*} \left [\frac{2 \,{\left (3 \, b d^{\frac{3}{2}} n \log \left (\frac{e x + 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) -{\left (8 \, b d n - 3 \, a d +{\left (2 \, b e n - 3 \, a e\right )} x - 3 \,{\left (b e x + b d\right )} \log \left (c\right ) - 3 \,{\left (b e n x + b d n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{9 \, e}, -\frac{2 \,{\left (6 \, b \sqrt{-d} d n \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) +{\left (8 \, b d n - 3 \, a d +{\left (2 \, b e n - 3 \, a e\right )} x - 3 \,{\left (b e x + b d\right )} \log \left (c\right ) - 3 \,{\left (b e n x + b d n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{9 \, e}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.88579, size = 102, normalized size = 1.09 \begin{align*} \frac{2 \left (\frac{a \left (d + e x\right )^{\frac{3}{2}}}{3} + 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 )\right )}{e} \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 \sqrt{e x + d}{\left (b \log \left (c x^{n}\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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