Optimal. Leaf size=100 \[ \frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}-\frac{b n \sqrt{d+e x^2}}{e^2}+\frac{2 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{e^2} \]
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Rubi [A] time = 0.160054, antiderivative size = 100, 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, 80, 63, 208} \[ \frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}-\frac{b n \sqrt{d+e x^2}}{e^2}+\frac{2 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{e^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2350
Rule 12
Rule 446
Rule 80
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 )^{3/2}} \, dx &=\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-(b n) \int \frac{2 d+e x^2}{e^2 x \sqrt{d+e x^2}} \, dx\\ &=\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{(b n) \int \frac{2 d+e x^2}{x \sqrt{d+e x^2}} \, dx}{e^2}\\ &=\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{(b n) \operatorname{Subst}\left (\int \frac{2 d+e x}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{2 e^2}\\ &=-\frac{b n \sqrt{d+e x^2}}{e^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{(b d n) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{e^2}\\ &=-\frac{b n \sqrt{d+e x^2}}{e^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{(2 b d n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{e^3}\\ &=-\frac{b n \sqrt{d+e x^2}}{e^2}+\frac{2 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{e^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}\\ \end{align*}
Mathematica [A] time = 0.143372, size = 118, normalized size = 1.18 \[ \frac{2 a d+a e x^2+b \left (2 d+e x^2\right ) \log \left (c x^n\right )-2 b \sqrt{d} n \log (x) \sqrt{d+e x^2}+2 b \sqrt{d} n \sqrt{d+e x^2} \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )-b d n-b e n x^2}{e^2 \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.407, 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{3}{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 = 1.44449, size = 560, normalized size = 5.6 \begin{align*} \left [\frac{{\left (b e n x^{2} + b d 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 n +{\left (b e n - a e\right )} x^{2} - 2 \, a d -{\left (b e x^{2} + 2 \, b d\right )} \log \left (c\right ) -{\left (b e n x^{2} + 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{e^{3} x^{2} + d e^{2}}, -\frac{2 \,{\left (b e n x^{2} + b d n\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d}}{\sqrt{e x^{2} + d}}\right ) +{\left (b d n +{\left (b e n - a e\right )} x^{2} - 2 \, a d -{\left (b e x^{2} + 2 \, b d\right )} \log \left (c\right ) -{\left (b e n x^{2} + 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{e^{3} x^{2} + d e^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 49.9151, size = 163, normalized size = 1.63 \begin{align*} a \left (\begin{cases} \frac{x^{4}}{4 d^{\frac{3}{2}}} & \text{for}\: e = 0 \\\frac{d}{e^{2} \sqrt{d + e x^{2}}} + \frac{\sqrt{d + e x^{2}}}{e^{2}} & \text{otherwise} \end{cases}\right ) - b n \left (\begin{cases} \frac{x^{4}}{16 d^{\frac{3}{2}}} & \text{for}\: e = 0 \\- \frac{2 \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{e} x} \right )}}{e^{2}} + \frac{d}{e^{\frac{5}{2}} x \sqrt{\frac{d}{e x^{2}} + 1}} + \frac{x}{e^{\frac{3}{2}} \sqrt{\frac{d}{e x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) + b \left (\begin{cases} \frac{x^{4}}{4 d^{\frac{3}{2}}} & \text{for}\: e = 0 \\\frac{d}{e^{2} \sqrt{d + e x^{2}}} + \frac{\sqrt{d + e x^{2}}}{e^{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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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