Optimal. Leaf size=109 \[ \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{b c}{3 e \left (c^2 d-e\right ) \sqrt{d+e x^2}}-\frac{b \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{3 d \left (c^2 d-e\right )^{3/2}} \]
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Rubi [A] time = 0.199304, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {264, 4976, 446, 78, 63, 208} \[ \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{b c}{3 e \left (c^2 d-e\right ) \sqrt{d+e x^2}}-\frac{b \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{3 d \left (c^2 d-e\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 4976
Rule 446
Rule 78
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-(b c) \int \frac{x^3}{\left (3 d+3 c^2 d x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{x}{\left (3 d+3 c^2 d x\right ) (d+e x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac{b c}{3 \left (c^2 d-e\right ) e \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{\left (3 d+3 c^2 d x\right ) \sqrt{d+e x}} \, dx,x,x^2\right )}{2 \left (c^2 d-e\right )}\\ &=\frac{b c}{3 \left (c^2 d-e\right ) e \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{3 d-\frac{3 c^2 d^2}{e}+\frac{3 c^2 d x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{\left (c^2 d-e\right ) e}\\ &=\frac{b c}{3 \left (c^2 d-e\right ) e \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{b \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{3 d \left (c^2 d-e\right )^{3/2}}\\ \end{align*}
Mathematica [C] time = 1.06281, size = 252, normalized size = 2.31 \[ -\frac{-\frac{2 \left (a x \left (c^2 d-e\right )+b c d\right )}{e \left (c^2 d-e\right ) \sqrt{d+e x^2}}+\frac{2 a d x}{e \left (d+e x^2\right )^{3/2}}+\frac{b \log \left (\frac{12 c d \sqrt{c^2 d-e} \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right )}{b (c x+i)}\right )}{\left (c^2 d-e\right )^{3/2}}+\frac{b \log \left (\frac{12 c d \sqrt{c^2 d-e} \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right )}{b (c x-i)}\right )}{\left (c^2 d-e\right )^{3/2}}-\frac{2 b x^3 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}}}{6 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.604, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b\arctan \left ( cx \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] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{3} \, a{\left (\frac{x}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} e} - \frac{x}{\sqrt{e x^{2} + d} d e}\right )} + 2 \, b \int \frac{x^{2} \arctan \left (c x\right )}{2 \,{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt{e x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.3693, size = 1386, normalized size = 12.72 \begin{align*} \left [-\frac{{\left (b e^{3} x^{4} + 2 \, b d e^{2} x^{2} + b d^{2} e\right )} \sqrt{c^{2} d - e} \log \left (\frac{c^{4} e^{2} x^{4} + 8 \, c^{4} d^{2} - 8 \, c^{2} d e + 2 \,{\left (4 \, c^{4} d e - 3 \, c^{2} e^{2}\right )} x^{2} + 4 \,{\left (c^{3} e x^{2} + 2 \, c^{3} d - c e\right )} \sqrt{c^{2} d - e} \sqrt{e x^{2} + d} + e^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) - 4 \,{\left (b c^{3} d^{3} - b c d^{2} e +{\left (b c^{4} d^{2} e - 2 \, b c^{2} d e^{2} + b e^{3}\right )} x^{3} \arctan \left (c x\right ) +{\left (a c^{4} d^{2} e - 2 \, a c^{2} d e^{2} + a e^{3}\right )} x^{3} +{\left (b c^{3} d^{2} e - b c d e^{2}\right )} x^{2}\right )} \sqrt{e x^{2} + d}}{12 \,{\left (c^{4} d^{5} e - 2 \, c^{2} d^{4} e^{2} + d^{3} e^{3} +{\left (c^{4} d^{3} e^{3} - 2 \, c^{2} d^{2} e^{4} + d e^{5}\right )} x^{4} + 2 \,{\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{2}\right )}}, -\frac{{\left (b e^{3} x^{4} + 2 \, b d e^{2} x^{2} + b d^{2} e\right )} \sqrt{-c^{2} d + e} \arctan \left (-\frac{{\left (c^{2} e x^{2} + 2 \, c^{2} d - e\right )} \sqrt{-c^{2} d + e} \sqrt{e x^{2} + d}}{2 \,{\left (c^{3} d^{2} - c d e +{\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) - 2 \,{\left (b c^{3} d^{3} - b c d^{2} e +{\left (b c^{4} d^{2} e - 2 \, b c^{2} d e^{2} + b e^{3}\right )} x^{3} \arctan \left (c x\right ) +{\left (a c^{4} d^{2} e - 2 \, a c^{2} d e^{2} + a e^{3}\right )} x^{3} +{\left (b c^{3} d^{2} e - b c d e^{2}\right )} x^{2}\right )} \sqrt{e x^{2} + d}}{6 \,{\left (c^{4} d^{5} e - 2 \, c^{2} d^{4} e^{2} + d^{3} e^{3} +{\left (c^{4} d^{3} e^{3} - 2 \, c^{2} d^{2} e^{4} + d e^{5}\right )} x^{4} + 2 \,{\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{2}\right )}}\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 \frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\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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