Optimal. Leaf size=163 \[ -\frac{2 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{8 b d^3 n \sqrt{d+e x}}{35 e^2}+\frac{8 b d^2 n (d+e x)^{3/2}}{105 e^2}-\frac{8 b d^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{35 e^2}+\frac{8 b d n (d+e x)^{5/2}}{175 e^2}-\frac{4 b n (d+e x)^{7/2}}{49 e^2} \]
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Rubi [A] time = 0.117007, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {43, 2350, 12, 80, 50, 63, 208} \[ -\frac{2 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{8 b d^3 n \sqrt{d+e x}}{35 e^2}+\frac{8 b d^2 n (d+e x)^{3/2}}{105 e^2}-\frac{8 b d^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{35 e^2}+\frac{8 b d n (d+e x)^{5/2}}{175 e^2}-\frac{4 b n (d+e x)^{7/2}}{49 e^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2350
Rule 12
Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int x (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{2 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}-(b n) \int \frac{2 (d+e x)^{5/2} (-2 d+5 e x)}{35 e^2 x} \, dx\\ &=-\frac{2 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}-\frac{(2 b n) \int \frac{(d+e x)^{5/2} (-2 d+5 e x)}{x} \, dx}{35 e^2}\\ &=-\frac{4 b n (d+e x)^{7/2}}{49 e^2}-\frac{2 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{(4 b d n) \int \frac{(d+e x)^{5/2}}{x} \, dx}{35 e^2}\\ &=\frac{8 b d n (d+e x)^{5/2}}{175 e^2}-\frac{4 b n (d+e x)^{7/2}}{49 e^2}-\frac{2 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{\left (4 b d^2 n\right ) \int \frac{(d+e x)^{3/2}}{x} \, dx}{35 e^2}\\ &=\frac{8 b d^2 n (d+e x)^{3/2}}{105 e^2}+\frac{8 b d n (d+e x)^{5/2}}{175 e^2}-\frac{4 b n (d+e x)^{7/2}}{49 e^2}-\frac{2 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{\left (4 b d^3 n\right ) \int \frac{\sqrt{d+e x}}{x} \, dx}{35 e^2}\\ &=\frac{8 b d^3 n \sqrt{d+e x}}{35 e^2}+\frac{8 b d^2 n (d+e x)^{3/2}}{105 e^2}+\frac{8 b d n (d+e x)^{5/2}}{175 e^2}-\frac{4 b n (d+e x)^{7/2}}{49 e^2}-\frac{2 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{\left (4 b d^4 n\right ) \int \frac{1}{x \sqrt{d+e x}} \, dx}{35 e^2}\\ &=\frac{8 b d^3 n \sqrt{d+e x}}{35 e^2}+\frac{8 b d^2 n (d+e x)^{3/2}}{105 e^2}+\frac{8 b d n (d+e x)^{5/2}}{175 e^2}-\frac{4 b n (d+e x)^{7/2}}{49 e^2}-\frac{2 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{\left (8 b d^4 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{35 e^3}\\ &=\frac{8 b d^3 n \sqrt{d+e x}}{35 e^2}+\frac{8 b d^2 n (d+e x)^{3/2}}{105 e^2}+\frac{8 b d n (d+e x)^{5/2}}{175 e^2}-\frac{4 b n (d+e x)^{7/2}}{49 e^2}-\frac{8 b d^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{35 e^2}-\frac{2 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}\\ \end{align*}
Mathematica [A] time = 0.167948, size = 120, normalized size = 0.74 \[ -\frac{2 \left (\sqrt{d+e x} \left (105 a (2 d-5 e x) (d+e x)^2+105 b (2 d-5 e x) (d+e x)^2 \log \left (c x^n\right )+2 b n \left (71 d^2 e x-247 d^3+183 d e^2 x^2+75 e^3 x^3\right )\right )+420 b d^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right )}{3675 e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.557, size = 0, normalized size = 0. \begin{align*} \int x \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 = 1.5215, size = 960, normalized size = 5.89 \begin{align*} \left [\frac{2 \,{\left (210 \, b d^{\frac{7}{2}} n \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) +{\left (494 \, b d^{3} n - 210 \, a d^{3} - 75 \,{\left (2 \, b e^{3} n - 7 \, a e^{3}\right )} x^{3} - 6 \,{\left (61 \, b d e^{2} n - 140 \, a d e^{2}\right )} x^{2} -{\left (142 \, b d^{2} e n - 105 \, a d^{2} e\right )} x + 105 \,{\left (5 \, b e^{3} x^{3} + 8 \, b d e^{2} x^{2} + b d^{2} e x - 2 \, b d^{3}\right )} \log \left (c\right ) + 105 \,{\left (5 \, b e^{3} n x^{3} + 8 \, b d e^{2} n x^{2} + b d^{2} e n x - 2 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{3675 \, e^{2}}, \frac{2 \,{\left (420 \, b \sqrt{-d} d^{3} n \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) +{\left (494 \, b d^{3} n - 210 \, a d^{3} - 75 \,{\left (2 \, b e^{3} n - 7 \, a e^{3}\right )} x^{3} - 6 \,{\left (61 \, b d e^{2} n - 140 \, a d e^{2}\right )} x^{2} -{\left (142 \, b d^{2} e n - 105 \, a d^{2} e\right )} x + 105 \,{\left (5 \, b e^{3} x^{3} + 8 \, b d e^{2} x^{2} + b d^{2} e x - 2 \, b d^{3}\right )} \log \left (c\right ) + 105 \,{\left (5 \, b e^{3} n x^{3} + 8 \, b d e^{2} n x^{2} + b d^{2} e n x - 2 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{3675 \, e^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 121.169, size = 583, normalized size = 3.58 \begin{align*} \text{result too large to display} \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 )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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