Optimal. Leaf size=84 \[ -\frac{a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 d^{3/2} e}+\frac{b n}{3 d e \sqrt{d+e x^2}} \]
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Rubi [A] time = 0.0879107, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2338, 266, 51, 63, 208} \[ -\frac{a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 d^{3/2} e}+\frac{b n}{3 d e \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 2338
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=-\frac{a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}+\frac{(b n) \int \frac{1}{x \left (d+e x^2\right )^{3/2}} \, dx}{3 e}\\ &=-\frac{a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{1}{x (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e}\\ &=\frac{b n}{3 d e \sqrt{d+e x^2}}-\frac{a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{6 d e}\\ &=\frac{b n}{3 d e \sqrt{d+e x^2}}-\frac{a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{3 d e^2}\\ &=\frac{b n}{3 d e \sqrt{d+e x^2}}-\frac{b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 d^{3/2} e}-\frac{a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.230545, size = 97, normalized size = 1.15 \[ -\frac{\frac{a}{\left (d+e x^2\right )^{3/2}}+\frac{b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}}+\frac{b n \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )}{d^{3/2}}-\frac{b n \log (x)}{d^{3/2}}-\frac{b n}{d \sqrt{d+e x^2}}}{3 e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.436, size = 0, normalized size = 0. \begin{align*} \int{x \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 = 1.5408, size = 597, normalized size = 7.11 \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 ) + 2 \,{\left (b d e n x^{2} - b d^{2} n \log \left (x\right ) + b d^{2} n - b d^{2} \log \left (c\right ) - a d^{2}\right )} \sqrt{e x^{2} + d}}{6 \,{\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}}, \frac{{\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 e n x^{2} - b d^{2} n \log \left (x\right ) + b d^{2} n - b d^{2} \log \left (c\right ) - a d^{2}\right )} \sqrt{e x^{2} + d}}{3 \,{\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 54.2549, size = 245, normalized size = 2.92 \begin{align*} - \frac{a}{3 e \left (d + e x^{2}\right )^{\frac{3}{2}}} + \frac{2 b d^{3} n \sqrt{1 + \frac{e x^{2}}{d}}}{6 d^{\frac{9}{2}} e + 6 d^{\frac{7}{2}} e^{2} x^{2}} + \frac{b d^{3} n \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 b d^{3} n \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{b d^{2} n x^{2} \log{\left (\frac{e x^{2}}{d} \right )}}{3 \left (2 d^{\frac{9}{2}} + 2 d^{\frac{7}{2}} e x^{2}\right )} - \frac{2 b d^{2} n x^{2} \log{\left (\sqrt{1 + \frac{e x^{2}}{d}} + 1 \right )}}{3 \left (2 d^{\frac{9}{2}} + 2 d^{\frac{7}{2}} e x^{2}\right )} - \frac{b \log{\left (c x^{n} \right )}}{3 e \left (d + e x^{2}\right )^{\frac{3}{2}}} \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}{{\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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