Optimal. Leaf size=73 \[ \frac{a x^{m+1} \text{Hypergeometric2F1}\left (1,\frac{m}{2},\frac{m+3}{2},-\frac{e x^2}{d}\right )}{d (m+1) \sqrt{d+e x^2}}+b \text{Unintegrable}\left (\frac{x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}},x\right ) \]
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Rubi [A] time = 0.173692, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^m \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x^m \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=a \int \frac{x^m}{\left (d+e x^2\right )^{3/2}} \, dx+b \int \frac{x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx\\ &=b \int \frac{x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx+\frac{\left (a \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{x^m}{\left (1+\frac{e x^2}{d}\right )^{3/2}} \, dx}{d \sqrt{d+e x^2}}\\ &=\frac{a x^{1+m} \sqrt{1+\frac{e x^2}{d}} \, _2F_1\left (\frac{3}{2},\frac{1+m}{2};\frac{3+m}{2};-\frac{e x^2}{d}\right )}{d (1+m) \sqrt{d+e x^2}}+b \int \frac{x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx\\ \end{align*}
Mathematica [A] time = 4.37088, size = 0, normalized size = 0. \[ \int \frac{x^m \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.619, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( a+b\arctan \left ( cx \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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{{\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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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x^{2} + d}{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{{\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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