Optimal. Leaf size=112 \[ -\frac{a x^{m+2} \sqrt{a^2 c x^2+c} \text{Hypergeometric2F1}\left (1,\frac{m+3}{2},\frac{m+4}{2},-a^2 x^2\right )}{(m+2)^2}+\frac{c \text{Unintegrable}\left (\frac{x^m \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}},x\right )}{m+2}+\frac{x^{m+1} \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{m+2} \]
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Rubi [A] time = 0.17032, 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 = {} \[ \int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx &=\frac{x^{1+m} \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2+m}+\frac{c \int \frac{x^m \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{2+m}-\frac{(a c) \int \frac{x^{1+m}}{\sqrt{c+a^2 c x^2}} \, dx}{2+m}\\ &=\frac{x^{1+m} \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2+m}+\frac{c \int \frac{x^m \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{2+m}-\frac{\left (a c \sqrt{1+a^2 x^2}\right ) \int \frac{x^{1+m}}{\sqrt{1+a^2 x^2}} \, dx}{(2+m) \sqrt{c+a^2 c x^2}}\\ &=\frac{x^{1+m} \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2+m}-\frac{a c x^{2+m} \sqrt{1+a^2 x^2} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};-a^2 x^2\right )}{(2+m)^2 \sqrt{c+a^2 c x^2}}+\frac{c \int \frac{x^m \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{2+m}\\ \end{align*}
Mathematica [A] time = 0.10003, size = 0, normalized size = 0. \[ \int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.596, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{{a}^{2}c{x}^{2}+c}\arctan \left ( ax \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{a^{2} c x^{2} + c} x^{m} \arctan \left (a x\right )\,{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{a^{2} c x^{2} + c} x^{m} \arctan \left (a x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sqrt{c \left (a^{2} x^{2} + 1\right )} \operatorname{atan}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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