Optimal. Leaf size=201 \[ -\frac{a c^2 x^{m+2} \text{Hypergeometric2F1}\left (1,\frac{m+2}{2},\frac{m+4}{2},-a^2 x^2\right )}{m^2+3 m+2}-\frac{2 a^3 c^2 x^{m+4} \text{Hypergeometric2F1}\left (1,\frac{m+4}{2},\frac{m+6}{2},-a^2 x^2\right )}{m^2+7 m+12}-\frac{a^5 c^2 x^{m+6} \text{Hypergeometric2F1}\left (1,\frac{m+6}{2},\frac{m+8}{2},-a^2 x^2\right )}{(m+5) (m+6)}+\frac{2 a^2 c^2 x^{m+3} \tan ^{-1}(a x)}{m+3}+\frac{a^4 c^2 x^{m+5} \tan ^{-1}(a x)}{m+5}+\frac{c^2 x^{m+1} \tan ^{-1}(a x)}{m+1} \]
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Rubi [A] time = 0.156996, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4948, 4852, 364} \[ -\frac{a c^2 x^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right )}{m^2+3 m+2}-\frac{2 a^3 c^2 x^{m+4} \, _2F_1\left (1,\frac{m+4}{2};\frac{m+6}{2};-a^2 x^2\right )}{m^2+7 m+12}-\frac{a^5 c^2 x^{m+6} \, _2F_1\left (1,\frac{m+6}{2};\frac{m+8}{2};-a^2 x^2\right )}{(m+5) (m+6)}+\frac{2 a^2 c^2 x^{m+3} \tan ^{-1}(a x)}{m+3}+\frac{a^4 c^2 x^{m+5} \tan ^{-1}(a x)}{m+5}+\frac{c^2 x^{m+1} \tan ^{-1}(a x)}{m+1} \]
Antiderivative was successfully verified.
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Rule 4948
Rule 4852
Rule 364
Rubi steps
\begin{align*} \int x^m \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x) \, dx &=\int \left (c^2 x^m \tan ^{-1}(a x)+2 a^2 c^2 x^{2+m} \tan ^{-1}(a x)+a^4 c^2 x^{4+m} \tan ^{-1}(a x)\right ) \, dx\\ &=c^2 \int x^m \tan ^{-1}(a x) \, dx+\left (2 a^2 c^2\right ) \int x^{2+m} \tan ^{-1}(a x) \, dx+\left (a^4 c^2\right ) \int x^{4+m} \tan ^{-1}(a x) \, dx\\ &=\frac{c^2 x^{1+m} \tan ^{-1}(a x)}{1+m}+\frac{2 a^2 c^2 x^{3+m} \tan ^{-1}(a x)}{3+m}+\frac{a^4 c^2 x^{5+m} \tan ^{-1}(a x)}{5+m}-\frac{\left (a c^2\right ) \int \frac{x^{1+m}}{1+a^2 x^2} \, dx}{1+m}-\frac{\left (2 a^3 c^2\right ) \int \frac{x^{3+m}}{1+a^2 x^2} \, dx}{3+m}-\frac{\left (a^5 c^2\right ) \int \frac{x^{5+m}}{1+a^2 x^2} \, dx}{5+m}\\ &=\frac{c^2 x^{1+m} \tan ^{-1}(a x)}{1+m}+\frac{2 a^2 c^2 x^{3+m} \tan ^{-1}(a x)}{3+m}+\frac{a^4 c^2 x^{5+m} \tan ^{-1}(a x)}{5+m}-\frac{a c^2 x^{2+m} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};-a^2 x^2\right )}{2+3 m+m^2}-\frac{2 a^3 c^2 x^{4+m} \, _2F_1\left (1,\frac{4+m}{2};\frac{6+m}{2};-a^2 x^2\right )}{12+7 m+m^2}-\frac{a^5 c^2 x^{6+m} \, _2F_1\left (1,\frac{6+m}{2};\frac{8+m}{2};-a^2 x^2\right )}{(5+m) (6+m)}\\ \end{align*}
Mathematica [A] time = 0.138472, size = 175, normalized size = 0.87 \[ c^2 x^{m+1} \left (-\frac{2 a^3 x^3 \text{Hypergeometric2F1}\left (1,\frac{m+4}{2},\frac{m+6}{2},-a^2 x^2\right )}{m^2+7 m+12}-\frac{a x \text{Hypergeometric2F1}\left (1,\frac{m+2}{2},\frac{m+4}{2},-a^2 x^2\right )}{m^2+3 m+2}-\frac{a^5 x^5 \text{Hypergeometric2F1}\left (1,\frac{m+6}{2},\frac{m+8}{2},-a^2 x^2\right )}{(m+5) (m+6)}+\frac{a^4 x^4 \tan ^{-1}(a x)}{m+5}+\frac{2 a^2 x^2 \tan ^{-1}(a x)}{m+3}+\frac{\tan ^{-1}(a x)}{m+1}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.577, size = 376, normalized size = 1.9 \begin{align*}{\frac{{a}^{-1-m}{c}^{2}}{4} \left ( -4\,{\frac{{x}^{m}{a}^{m} \left ({a}^{4}{m}^{2}{x}^{4}+2\,m{x}^{4}{a}^{4}-{a}^{2}{m}^{2}{x}^{2}-4\,m{x}^{2}{a}^{2}+{m}^{2}+6\,m+8 \right ) }{ \left ( 5+m \right ) m \left ( 2+m \right ) \left ( 4+m \right ) }}+8\,{\frac{{x}^{6+m}{a}^{6+m}\arctan \left ( \sqrt{{a}^{2}{x}^{2}} \right ) }{ \left ( 10+2\,m \right ) \sqrt{{a}^{2}{x}^{2}}}}+2\,{\frac{{x}^{m}{a}^{m}{\it LerchPhi} \left ( -{a}^{2}{x}^{2},1,m/2 \right ) }{5+m}} \right ) }+{\frac{{a}^{-1-m}{c}^{2}}{2} \left ( -4\,{\frac{{x}^{m}{a}^{m} \left ( m{x}^{2}{a}^{2}-m-2 \right ) }{ \left ( 3+m \right ) m \left ( 2+m \right ) }}+8\,{\frac{{x}^{4+m}{a}^{4+m}\arctan \left ( \sqrt{{a}^{2}{x}^{2}} \right ) }{ \left ( 6+2\,m \right ) \sqrt{{a}^{2}{x}^{2}}}}+2\,{\frac{{x}^{m}{a}^{m} \left ( -4-m \right ){\it LerchPhi} \left ( -{a}^{2}{x}^{2},1,m/2 \right ) }{ \left ( 4+m \right ) \left ( 3+m \right ) }} \right ) }+{\frac{{a}^{-1-m}{c}^{2}}{4} \left ( 4\,{\frac{{x}^{m}{a}^{m} \left ( -m-2 \right ) }{ \left ( 2+m \right ) \left ( 1+m \right ) m}}+8\,{\frac{{x}^{2+m}{a}^{2+m}\arctan \left ( \sqrt{{a}^{2}{x}^{2}} \right ) }{ \left ( 2+2\,m \right ) \sqrt{{a}^{2}{x}^{2}}}}+2\,{\frac{{x}^{m}{a}^{m}{\it LerchPhi} \left ( -{a}^{2}{x}^{2},1,m/2 \right ) }{1+m}} \right ) } \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \left (\int x^{m} \operatorname{atan}{\left (a x \right )}\, dx + \int 2 a^{2} x^{2} x^{m} \operatorname{atan}{\left (a x \right )}\, dx + \int a^{4} x^{4} x^{m} \operatorname{atan}{\left (a x \right )}\, dx\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{\left (a^{2} c x^{2} + c\right )}^{2} 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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