Optimal. Leaf size=108 \[ -\frac{a+b \log \left (c x^n\right )}{e^2 \sqrt{d+e x^2}}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{b n}{3 e^2 \sqrt{d+e x^2}}-\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 \sqrt{d} e^2} \]
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Rubi [A] time = 0.160831, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {266, 43, 2350, 12, 446, 78, 63, 208} \[ -\frac{a+b \log \left (c x^n\right )}{e^2 \sqrt{d+e x^2}}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{b n}{3 e^2 \sqrt{d+e x^2}}-\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 \sqrt{d} e^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2350
Rule 12
Rule 446
Rule 78
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \log \left (c x^n\right )}{e^2 \sqrt{d+e x^2}}-(b n) \int \frac{-2 d-3 e x^2}{3 e^2 x \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \log \left (c x^n\right )}{e^2 \sqrt{d+e x^2}}-\frac{(b n) \int \frac{-2 d-3 e x^2}{x \left (d+e x^2\right )^{3/2}} \, dx}{3 e^2}\\ &=\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \log \left (c x^n\right )}{e^2 \sqrt{d+e x^2}}-\frac{(b n) \operatorname{Subst}\left (\int \frac{-2 d-3 e x}{x (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^2}\\ &=-\frac{b n}{3 e^2 \sqrt{d+e x^2}}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \log \left (c x^n\right )}{e^2 \sqrt{d+e x^2}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{3 e^2}\\ &=-\frac{b n}{3 e^2 \sqrt{d+e x^2}}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \log \left (c x^n\right )}{e^2 \sqrt{d+e x^2}}+\frac{(2 b n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{3 e^3}\\ &=-\frac{b n}{3 e^2 \sqrt{d+e x^2}}-\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 \sqrt{d} e^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \log \left (c x^n\right )}{e^2 \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [A] time = 0.268131, size = 137, normalized size = 1.27 \[ \frac{\frac{d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-\left (d+e x^2\right ) \left (3 a+3 b \log \left (c x^n\right )-3 b n \log (x)+b n\right )}{\left (d+e x^2\right )^{3/2}}-\frac{b n \log (x) \left (2 d+3 e x^2\right )}{\left (d+e x^2\right )^{3/2}}-\frac{2 b n \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )}{\sqrt{d}}+\frac{2 b n \log (x)}{\sqrt{d}}}{3 e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.402, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, 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 = 1.6605, size = 737, normalized size = 6.82 \begin{align*} \left [\frac{{\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt{d} \log \left (-\frac{e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{d} + 2 \, d}{x^{2}}\right ) -{\left (b d^{2} n + 2 \, a d^{2} +{\left (b d e n + 3 \, a d e\right )} x^{2} +{\left (3 \, b d e x^{2} + 2 \, b d^{2}\right )} \log \left (c\right ) +{\left (3 \, b d e n x^{2} + 2 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{3 \,{\left (d e^{4} x^{4} + 2 \, d^{2} e^{3} x^{2} + d^{3} e^{2}\right )}}, \frac{2 \,{\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d}}{\sqrt{e x^{2} + d}}\right ) -{\left (b d^{2} n + 2 \, a d^{2} +{\left (b d e n + 3 \, a d e\right )} x^{2} +{\left (3 \, b d e x^{2} + 2 \, b d^{2}\right )} \log \left (c\right ) +{\left (3 \, b d e n x^{2} + 2 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{3 \,{\left (d e^{4} x^{4} + 2 \, d^{2} e^{3} x^{2} + d^{3} e^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 80.4278, size = 333, normalized size = 3.08 \begin{align*} a \left (\begin{cases} \frac{x^{4}}{4 d^{\frac{5}{2}}} & \text{for}\: e = 0 \\\frac{d}{3 e^{2} \left (d + e x^{2}\right )^{\frac{3}{2}}} - \frac{1}{e^{2} \sqrt{d + e x^{2}}} & \text{otherwise} \end{cases}\right ) - b n \left (\begin{cases} \frac{x^{4}}{16 d^{\frac{5}{2}}} & \text{for}\: e = 0 \\\frac{2 d^{4} \sqrt{1 + \frac{e x^{2}}{d}}}{6 d^{\frac{9}{2}} e^{2} + 6 d^{\frac{7}{2}} e^{3} x^{2}} + \frac{d^{4} \log{\left (\frac{e x^{2}}{d} \right )}}{6 d^{\frac{9}{2}} e^{2} + 6 d^{\frac{7}{2}} e^{3} x^{2}} - \frac{2 d^{4} \log{\left (\sqrt{1 + \frac{e x^{2}}{d}} + 1 \right )}}{6 d^{\frac{9}{2}} e^{2} + 6 d^{\frac{7}{2}} e^{3} x^{2}} + \frac{d^{3} x^{2} \log{\left (\frac{e x^{2}}{d} \right )}}{6 d^{\frac{9}{2}} e + 6 d^{\frac{7}{2}} e^{2} x^{2}} - \frac{2 d^{3} x^{2} \log{\left (\sqrt{1 + \frac{e x^{2}}{d}} + 1 \right )}}{6 d^{\frac{9}{2}} e + 6 d^{\frac{7}{2}} e^{2} x^{2}} + \frac{\operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{e} x} \right )}}{\sqrt{d} e^{2}} & \text{otherwise} \end{cases}\right ) + b \left (\begin{cases} \frac{x^{4}}{4 d^{\frac{5}{2}}} & \text{for}\: e = 0 \\\frac{d}{3 e^{2} \left (d + e x^{2}\right )^{\frac{3}{2}}} - \frac{1}{e^{2} \sqrt{d + e x^{2}}} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )} \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 \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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