Optimal. Leaf size=75 \[ \frac{a x^{m+1} \left (d+e x^2\right )^{3/2} \text{Hypergeometric2F1}\left (1,\frac{m+4}{2},\frac{m+3}{2},-\frac{e x^2}{d}\right )}{d (m+1)}+b \text{Unintegrable}\left (x^m \tan ^{-1}(c x) \sqrt{d+e x^2},x\right ) \]
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Rubi [A] time = 0.153588, 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 = {} \[ \int x^m \sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^m \sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx &=a \int x^m \sqrt{d+e x^2} \, dx+b \int x^m \sqrt{d+e x^2} \tan ^{-1}(c x) \, dx\\ &=b \int x^m \sqrt{d+e x^2} \tan ^{-1}(c x) \, dx+\frac{\left (a \sqrt{d+e x^2}\right ) \int x^m \sqrt{1+\frac{e x^2}{d}} \, dx}{\sqrt{1+\frac{e x^2}{d}}}\\ &=\frac{a x^{1+m} \sqrt{d+e x^2} \, _2F_1\left (-\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};-\frac{e x^2}{d}\right )}{(1+m) \sqrt{1+\frac{e x^2}{d}}}+b \int x^m \sqrt{d+e x^2} \tan ^{-1}(c x) \, dx\\ \end{align*}
Mathematica [A] time = 0.0903251, size = 0, normalized size = 0. \[ \int x^m \sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.816, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{e{x}^{2}+d} \left ( a+b\arctan \left ( cx \right ) \right ) \, 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 \sqrt{e x^{2} + d}{\left (b \arctan \left (c x\right ) + a\right )} x^{m}\,{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 (\sqrt{e x^{2} + d}{\left (b \arctan \left (c x\right ) + a\right )} x^{m}, 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 \sqrt{e x^{2} + d}{\left (b \arctan \left (c x\right ) + a\right )} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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