Optimal. Leaf size=115 \[ \frac{2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac{4 b d^2 n \sqrt{d+e x}}{5 e}+\frac{4 b d^{5/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{5 e}-\frac{4 b d n (d+e x)^{3/2}}{15 e}-\frac{4 b n (d+e x)^{5/2}}{25 e} \]
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Rubi [A] time = 0.050536, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2319, 50, 63, 208} \[ \frac{2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac{4 b d^2 n \sqrt{d+e x}}{5 e}+\frac{4 b d^{5/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{5 e}-\frac{4 b d n (d+e x)^{3/2}}{15 e}-\frac{4 b n (d+e x)^{5/2}}{25 e} \]
Antiderivative was successfully verified.
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Rule 2319
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac{(2 b n) \int \frac{(d+e x)^{5/2}}{x} \, dx}{5 e}\\ &=-\frac{4 b n (d+e x)^{5/2}}{25 e}+\frac{2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac{(2 b d n) \int \frac{(d+e x)^{3/2}}{x} \, dx}{5 e}\\ &=-\frac{4 b d n (d+e x)^{3/2}}{15 e}-\frac{4 b n (d+e x)^{5/2}}{25 e}+\frac{2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac{\left (2 b d^2 n\right ) \int \frac{\sqrt{d+e x}}{x} \, dx}{5 e}\\ &=-\frac{4 b d^2 n \sqrt{d+e x}}{5 e}-\frac{4 b d n (d+e x)^{3/2}}{15 e}-\frac{4 b n (d+e x)^{5/2}}{25 e}+\frac{2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac{\left (2 b d^3 n\right ) \int \frac{1}{x \sqrt{d+e x}} \, dx}{5 e}\\ &=-\frac{4 b d^2 n \sqrt{d+e x}}{5 e}-\frac{4 b d n (d+e x)^{3/2}}{15 e}-\frac{4 b n (d+e x)^{5/2}}{25 e}+\frac{2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac{\left (4 b d^3 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{5 e^2}\\ &=-\frac{4 b d^2 n \sqrt{d+e x}}{5 e}-\frac{4 b d n (d+e x)^{3/2}}{15 e}-\frac{4 b n (d+e x)^{5/2}}{25 e}+\frac{4 b d^{5/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{5 e}+\frac{2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}\\ \end{align*}
Mathematica [A] time = 0.0828026, size = 87, normalized size = 0.76 \[ \frac{2 \left ((d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )-\frac{2}{15} b n \sqrt{d+e x} \left (23 d^2+11 d e x+3 e^2 x^2\right )+2 b d^{5/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right )}{5 e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.563, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \, 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 = 1.41295, size = 710, normalized size = 6.17 \begin{align*} \left [\frac{2 \,{\left (15 \, b d^{\frac{5}{2}} n \log \left (\frac{e x + 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) -{\left (46 \, b d^{2} n - 15 \, a d^{2} + 3 \,{\left (2 \, b e^{2} n - 5 \, a e^{2}\right )} x^{2} + 2 \,{\left (11 \, b d e n - 15 \, a d e\right )} x - 15 \,{\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \log \left (c\right ) - 15 \,{\left (b e^{2} n x^{2} + 2 \, b d e n x + b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{75 \, e}, -\frac{2 \,{\left (30 \, b \sqrt{-d} d^{2} n \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) +{\left (46 \, b d^{2} n - 15 \, a d^{2} + 3 \,{\left (2 \, b e^{2} n - 5 \, a e^{2}\right )} x^{2} + 2 \,{\left (11 \, b d e n - 15 \, a d e\right )} x - 15 \,{\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \log \left (c\right ) - 15 \,{\left (b e^{2} n x^{2} + 2 \, b d e n x + b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{75 \, e}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 51.687, size = 333, normalized size = 2.9 \begin{align*} a d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 a \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{2 b d \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} + \frac{2 b \left (- d \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 ) + \frac{\left (d + e x\right )^{\frac{5}{2}} \log{\left (c \left (- \frac{d}{e} + \frac{d + e x}{e}\right )^{n} \right )}}{5} - \frac{2 n \left (\frac{d^{3} e \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{\sqrt{- d}} + d^{2} e \sqrt{d + e x} + \frac{d e \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{e \left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{5 e}\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{\left (e x + d\right )}^{\frac{3}{2}}{\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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