Optimal. Leaf size=213 \[ \frac{2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac{4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac{2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac{32 b d^4 n \sqrt{d+e x}}{315 e^3}-\frac{32 b d^3 n (d+e x)^{3/2}}{945 e^3}-\frac{32 b d^2 n (d+e x)^{5/2}}{1575 e^3}+\frac{32 b d^{9/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{315 e^3}+\frac{44 b d n (d+e x)^{7/2}}{441 e^3}-\frac{4 b n (d+e x)^{9/2}}{81 e^3} \]
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Rubi [A] time = 0.197397, antiderivative size = 213, 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, 1261, 208} \[ \frac{2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac{4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac{2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac{32 b d^4 n \sqrt{d+e x}}{315 e^3}-\frac{32 b d^3 n (d+e x)^{3/2}}{945 e^3}-\frac{32 b d^2 n (d+e x)^{5/2}}{1575 e^3}+\frac{32 b d^{9/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{315 e^3}+\frac{44 b d n (d+e x)^{7/2}}{441 e^3}-\frac{4 b n (d+e x)^{9/2}}{81 e^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2350
Rule 12
Rule 897
Rule 1261
Rule 208
Rubi steps
\begin{align*} \int x^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac{4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac{2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-(b n) \int \frac{2 (d+e x)^{5/2} \left (8 d^2-20 d e x+35 e^2 x^2\right )}{315 e^3 x} \, dx\\ &=\frac{2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac{4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac{2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac{(2 b n) \int \frac{(d+e x)^{5/2} \left (8 d^2-20 d e x+35 e^2 x^2\right )}{x} \, dx}{315 e^3}\\ &=\frac{2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac{4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac{2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac{(4 b n) \operatorname{Subst}\left (\int \frac{x^6 \left (63 d^2-90 d x^2+35 x^4\right )}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{315 e^4}\\ &=\frac{2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac{4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac{2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac{(4 b n) \operatorname{Subst}\left (\int \left (8 d^4 e+8 d^3 e x^2+8 d^2 e x^4-55 d e x^6+35 e x^8+\frac{8 d^5}{-\frac{d}{e}+\frac{x^2}{e}}\right ) \, dx,x,\sqrt{d+e x}\right )}{315 e^4}\\ &=-\frac{32 b d^4 n \sqrt{d+e x}}{315 e^3}-\frac{32 b d^3 n (d+e x)^{3/2}}{945 e^3}-\frac{32 b d^2 n (d+e x)^{5/2}}{1575 e^3}+\frac{44 b d n (d+e x)^{7/2}}{441 e^3}-\frac{4 b n (d+e x)^{9/2}}{81 e^3}+\frac{2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac{4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac{2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac{\left (32 b d^5 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{315 e^4}\\ &=-\frac{32 b d^4 n \sqrt{d+e x}}{315 e^3}-\frac{32 b d^3 n (d+e x)^{3/2}}{945 e^3}-\frac{32 b d^2 n (d+e x)^{5/2}}{1575 e^3}+\frac{44 b d n (d+e x)^{7/2}}{441 e^3}-\frac{4 b n (d+e x)^{9/2}}{81 e^3}+\frac{32 b d^{9/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{315 e^3}+\frac{2 d^2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac{4 d (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac{2 (d+e x)^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}\\ \end{align*}
Mathematica [A] time = 0.229311, size = 153, normalized size = 0.72 \[ \frac{2 \left (\sqrt{d+e x} \left (315 a \left (8 d^2-20 d e x+35 e^2 x^2\right ) (d+e x)^2+315 b \left (8 d^2-20 d e x+35 e^2 x^2\right ) (d+e x)^2 \log \left (c x^n\right )-2 b n \left (429 d^2 e^2 x^2-677 d^3 e x+2614 d^4+2425 d e^3 x^3+1225 e^4 x^4\right )\right )+5040 b d^{9/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right )}{99225 e^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.561, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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 = 1.47006, size = 1223, normalized size = 5.74 \begin{align*} \left [\frac{2 \,{\left (2520 \, b d^{\frac{9}{2}} n \log \left (\frac{e x + 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) -{\left (5228 \, b d^{4} n - 2520 \, a d^{4} + 1225 \,{\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{4} + 50 \,{\left (97 \, b d e^{3} n - 315 \, a d e^{3}\right )} x^{3} + 3 \,{\left (286 \, b d^{2} e^{2} n - 315 \, a d^{2} e^{2}\right )} x^{2} - 2 \,{\left (677 \, b d^{3} e n - 630 \, a d^{3} e\right )} x - 315 \,{\left (35 \, b e^{4} x^{4} + 50 \, b d e^{3} x^{3} + 3 \, b d^{2} e^{2} x^{2} - 4 \, b d^{3} e x + 8 \, b d^{4}\right )} \log \left (c\right ) - 315 \,{\left (35 \, b e^{4} n x^{4} + 50 \, b d e^{3} n x^{3} + 3 \, b d^{2} e^{2} n x^{2} - 4 \, b d^{3} e n x + 8 \, b d^{4} n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{99225 \, e^{3}}, -\frac{2 \,{\left (5040 \, b \sqrt{-d} d^{4} n \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) +{\left (5228 \, b d^{4} n - 2520 \, a d^{4} + 1225 \,{\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{4} + 50 \,{\left (97 \, b d e^{3} n - 315 \, a d e^{3}\right )} x^{3} + 3 \,{\left (286 \, b d^{2} e^{2} n - 315 \, a d^{2} e^{2}\right )} x^{2} - 2 \,{\left (677 \, b d^{3} e n - 630 \, a d^{3} e\right )} x - 315 \,{\left (35 \, b e^{4} x^{4} + 50 \, b d e^{3} x^{3} + 3 \, b d^{2} e^{2} x^{2} - 4 \, b d^{3} e x + 8 \, b d^{4}\right )} \log \left (c\right ) - 315 \,{\left (35 \, b e^{4} n x^{4} + 50 \, b d e^{3} n x^{3} + 3 \, b d^{2} e^{2} n x^{2} - 4 \, b d^{3} e n x + 8 \, b d^{4} n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{99225 \, e^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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