Optimal. Leaf size=217 \[ \frac{17 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{560 a^4}-\frac{1}{42} a c x^5 \sqrt{a^2 c x^2+c}-\frac{23 c x^3 \sqrt{a^2 c x^2+c}}{840 a}+\frac{3 c x \sqrt{a^2 c x^2+c}}{112 a^3}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{8}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{35 a^2}-\frac{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{35 a^4} \]
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Rubi [A] time = 0.764624, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {4950, 4946, 4952, 321, 217, 206, 4930} \[ \frac{17 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{560 a^4}-\frac{1}{42} a c x^5 \sqrt{a^2 c x^2+c}-\frac{23 c x^3 \sqrt{a^2 c x^2+c}}{840 a}+\frac{3 c x \sqrt{a^2 c x^2+c}}{112 a^3}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{8}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{35 a^2}-\frac{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{35 a^4} \]
Antiderivative was successfully verified.
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Rule 4950
Rule 4946
Rule 4952
Rule 321
Rule 217
Rule 206
Rule 4930
Rubi steps
\begin{align*} \int x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx &=c \int x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx+\left (a^2 c\right ) \int x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{5} c^2 \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{5} \left (a c^2\right ) \int \frac{x^4}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{7} \left (a^2 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{7} \left (a^3 c^2\right ) \int \frac{x^6}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{c x^3 \sqrt{c+a^2 c x^2}}{20 a}-\frac{1}{42} a c x^5 \sqrt{c+a^2 c x^2}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^2}+\frac{8}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{35} \left (4 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{\left (2 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^2}-\frac{c^2 \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{15 a}+\frac{\left (3 c^2\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{20 a}-\frac{1}{35} \left (a c^2\right ) \int \frac{x^4}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{42} \left (5 a c^2\right ) \int \frac{x^4}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{c x \sqrt{c+a^2 c x^2}}{24 a^3}-\frac{23 c x^3 \sqrt{c+a^2 c x^2}}{840 a}-\frac{1}{42} a c x^5 \sqrt{c+a^2 c x^2}-\frac{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{35 a^2}+\frac{8}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{c^2 \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{30 a^3}-\frac{\left (3 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{40 a^3}+\frac{\left (2 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^3}+\frac{\left (8 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{105 a^2}+\frac{\left (3 c^2\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{140 a}+\frac{\left (4 c^2\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{105 a}-\frac{\left (5 c^2\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{56 a}\\ &=\frac{3 c x \sqrt{c+a^2 c x^2}}{112 a^3}-\frac{23 c x^3 \sqrt{c+a^2 c x^2}}{840 a}-\frac{1}{42} a c x^5 \sqrt{c+a^2 c x^2}-\frac{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{35 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{35 a^2}+\frac{8}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{\left (3 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{280 a^3}-\frac{\left (2 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{105 a^3}+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{30 a^3}+\frac{\left (5 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{112 a^3}-\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{40 a^3}-\frac{\left (8 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{105 a^3}+\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{15 a^3}\\ &=\frac{3 c x \sqrt{c+a^2 c x^2}}{112 a^3}-\frac{23 c x^3 \sqrt{c+a^2 c x^2}}{840 a}-\frac{1}{42} a c x^5 \sqrt{c+a^2 c x^2}-\frac{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{35 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{35 a^2}+\frac{8}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{11 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{120 a^4}-\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{280 a^3}-\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{105 a^3}+\frac{\left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{112 a^3}-\frac{\left (8 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{105 a^3}\\ &=\frac{3 c x \sqrt{c+a^2 c x^2}}{112 a^3}-\frac{23 c x^3 \sqrt{c+a^2 c x^2}}{840 a}-\frac{1}{42} a c x^5 \sqrt{c+a^2 c x^2}-\frac{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{35 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{35 a^2}+\frac{8}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{17 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{560 a^4}\\ \end{align*}
Mathematica [A] time = 0.165537, size = 119, normalized size = 0.55 \[ \frac{51 c^{3/2} \log \left (\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right )+a c x \left (-40 a^4 x^4-46 a^2 x^2+45\right ) \sqrt{a^2 c x^2+c}+48 c \left (5 a^2 x^2-2\right ) \left (a^2 x^2+1\right )^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{1680 a^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.914, size = 199, normalized size = 0.9 \begin{align*}{\frac{c \left ( 240\,\arctan \left ( ax \right ){x}^{6}{a}^{6}-40\,{a}^{5}{x}^{5}+384\,\arctan \left ( ax \right ){x}^{4}{a}^{4}-46\,{a}^{3}{x}^{3}+48\,\arctan \left ( ax \right ){a}^{2}{x}^{2}+45\,ax-96\,\arctan \left ( ax \right ) \right ) }{1680\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{17\,c}{560\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-i \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{17\,c}{560\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+i \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \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 [A] time = 1.86986, size = 285, normalized size = 1.31 \begin{align*} \frac{51 \, c^{\frac{3}{2}} \log \left (-2 \, a^{2} c x^{2} - 2 \, \sqrt{a^{2} c x^{2} + c} a \sqrt{c} x - c\right ) - 2 \,{\left (40 \, a^{5} c x^{5} + 46 \, a^{3} c x^{3} - 45 \, a c x - 48 \,{\left (5 \, a^{6} c x^{6} + 8 \, a^{4} c x^{4} + a^{2} c x^{2} - 2 \, c\right )} \arctan \left (a x\right )\right )} \sqrt{a^{2} c x^{2} + c}}{3360 \, a^{4}} \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*} \int x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15812, size = 246, normalized size = 1.13 \begin{align*} \frac{{\left (\frac{7 \,{\left (3 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} - 5 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} c\right )}}{a^{2} c} + \frac{15 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{7}{2}} - 42 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} c + 35 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} c^{2}}{a^{2} c^{2}}\right )} \arctan \left (a x\right )}{105 \, a^{2}} - \frac{\sqrt{a^{2} c x^{2} + c}{\left (2 \,{\left (20 \, a^{4} c x^{2} + 23 \, a^{2} c\right )} x^{2} - 45 \, c\right )} x + \frac{51 \, c^{\frac{3}{2}} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}}}{1680 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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