Optimal. Leaf size=177 \[ -\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{2 b d^3 n \sqrt{d+e x^2}}{35 e^2}+\frac{2 b d^2 n \left (d+e x^2\right )^{3/2}}{105 e^2}-\frac{2 b d^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{35 e^2}+\frac{2 b d n \left (d+e x^2\right )^{5/2}}{175 e^2}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 e^2} \]
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Rubi [A] time = 0.204717, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {266, 43, 2350, 12, 446, 80, 50, 63, 208} \[ -\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{2 b d^3 n \sqrt{d+e x^2}}{35 e^2}+\frac{2 b d^2 n \left (d+e x^2\right )^{3/2}}{105 e^2}-\frac{2 b d^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{35 e^2}+\frac{2 b d n \left (d+e x^2\right )^{5/2}}{175 e^2}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 e^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2350
Rule 12
Rule 446
Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}-(b n) \int \frac{\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{35 e^2 x} \, dx\\ &=-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}-\frac{(b n) \int \frac{\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{x} \, dx}{35 e^2}\\ &=-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}-\frac{(b n) \operatorname{Subst}\left (\int \frac{(d+e x)^{5/2} (-2 d+5 e x)}{x} \, dx,x,x^2\right )}{70 e^2}\\ &=-\frac{b n \left (d+e x^2\right )^{7/2}}{49 e^2}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{(b d n) \operatorname{Subst}\left (\int \frac{(d+e x)^{5/2}}{x} \, dx,x,x^2\right )}{35 e^2}\\ &=\frac{2 b d n \left (d+e x^2\right )^{5/2}}{175 e^2}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 e^2}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{\left (b d^2 n\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2}}{x} \, dx,x,x^2\right )}{35 e^2}\\ &=\frac{2 b d^2 n \left (d+e x^2\right )^{3/2}}{105 e^2}+\frac{2 b d n \left (d+e x^2\right )^{5/2}}{175 e^2}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 e^2}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{\left (b d^3 n\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x}}{x} \, dx,x,x^2\right )}{35 e^2}\\ &=\frac{2 b d^3 n \sqrt{d+e x^2}}{35 e^2}+\frac{2 b d^2 n \left (d+e x^2\right )^{3/2}}{105 e^2}+\frac{2 b d n \left (d+e x^2\right )^{5/2}}{175 e^2}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 e^2}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{\left (b d^4 n\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{35 e^2}\\ &=\frac{2 b d^3 n \sqrt{d+e x^2}}{35 e^2}+\frac{2 b d^2 n \left (d+e x^2\right )^{3/2}}{105 e^2}+\frac{2 b d n \left (d+e x^2\right )^{5/2}}{175 e^2}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 e^2}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}+\frac{\left (2 b d^4 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{35 e^3}\\ &=\frac{2 b d^3 n \sqrt{d+e x^2}}{35 e^2}+\frac{2 b d^2 n \left (d+e x^2\right )^{3/2}}{105 e^2}+\frac{2 b d n \left (d+e x^2\right )^{5/2}}{175 e^2}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 e^2}-\frac{2 b d^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{35 e^2}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^2}\\ \end{align*}
Mathematica [A] time = 0.192953, size = 227, normalized size = 1.28 \[ \sqrt{d+e x^2} \left (-\frac{d^3 \left (210 a+210 b \left (\log \left (c x^n\right )-n \log (x)\right )-247 b n\right )}{3675 e^2}+\frac{d^2 x^2 \left (105 a+105 b \left (\log \left (c x^n\right )-n \log (x)\right )-71 b n\right )}{3675 e}+\frac{d x^4 \left (280 a+280 b \left (\log \left (c x^n\right )-n \log (x)\right )-61 b n\right )}{1225}+\frac{1}{49} e x^6 \left (7 a+7 b \left (\log \left (c x^n\right )-n \log (x)\right )-b n\right )\right )-\frac{2 b d^{7/2} n \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )}{35 e^2}+\frac{2 b d^{7/2} n \log (x)}{35 e^2}-\frac{b n \log (x) \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^{5/2}}{35 e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.441, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( e{x}^{2}+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 = 1.69552, size = 983, normalized size = 5.55 \begin{align*} \left [\frac{105 \, b d^{\frac{7}{2}} n \log \left (-\frac{e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{d} + 2 \, d}{x^{2}}\right ) -{\left (75 \,{\left (b e^{3} n - 7 \, a e^{3}\right )} x^{6} - 247 \, b d^{3} n + 3 \,{\left (61 \, b d e^{2} n - 280 \, a d e^{2}\right )} x^{4} + 210 \, a d^{3} +{\left (71 \, b d^{2} e n - 105 \, a d^{2} e\right )} x^{2} - 105 \,{\left (5 \, b e^{3} x^{6} + 8 \, b d e^{2} x^{4} + b d^{2} e x^{2} - 2 \, b d^{3}\right )} \log \left (c\right ) - 105 \,{\left (5 \, b e^{3} n x^{6} + 8 \, b d e^{2} n x^{4} + b d^{2} e n x^{2} - 2 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{3675 \, e^{2}}, \frac{210 \, b \sqrt{-d} d^{3} n \arctan \left (\frac{\sqrt{-d}}{\sqrt{e x^{2} + d}}\right ) -{\left (75 \,{\left (b e^{3} n - 7 \, a e^{3}\right )} x^{6} - 247 \, b d^{3} n + 3 \,{\left (61 \, b d e^{2} n - 280 \, a d e^{2}\right )} x^{4} + 210 \, a d^{3} +{\left (71 \, b d^{2} e n - 105 \, a d^{2} e\right )} x^{2} - 105 \,{\left (5 \, b e^{3} x^{6} + 8 \, b d e^{2} x^{4} + b d^{2} e x^{2} - 2 \, b d^{3}\right )} \log \left (c\right ) - 105 \,{\left (5 \, b e^{3} n x^{6} + 8 \, b d e^{2} n x^{4} + b d^{2} e n x^{2} - 2 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{3675 \, e^{2}}\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^{2} + d\right )}^{\frac{3}{2}}{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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