Optimal. Leaf size=160 \[ \frac {x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\frac {x^3 \sqrt {a^2 c x^2+c}}{20 a}-\frac {2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{15 a^4}+\frac {11 \sqrt {c} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{120 a^4}+\frac {x \sqrt {a^2 c x^2+c}}{24 a^3} \]
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Rubi [A] time = 0.27, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4946, 4952, 321, 217, 206, 4930} \[ -\frac {x^3 \sqrt {a^2 c x^2+c}}{20 a}+\frac {x \sqrt {a^2 c x^2+c}}{24 a^3}+\frac {1}{5} x^4 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{15 a^2}-\frac {2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{15 a^4}+\frac {11 \sqrt {c} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{120 a^4} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 321
Rule 4930
Rule 4946
Rule 4952
Rubi steps
\begin {align*} \int x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx &=\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{5} c \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{5} (a c) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {x^3 \sqrt {c+a^2 c x^2}}{20 a}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {(2 c) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}-\frac {c \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}+\frac {(3 c) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{20 a}\\ &=\frac {x \sqrt {c+a^2 c x^2}}{24 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2}}{20 a}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{30 a^3}-\frac {(3 c) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^3}+\frac {(2 c) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}\\ &=\frac {x \sqrt {c+a^2 c x^2}}{24 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2}}{20 a}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c \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 {(3 c) \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 {(2 c) \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 {x \sqrt {c+a^2 c x^2}}{24 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2}}{20 a}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {11 \sqrt {c} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^4}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 105, normalized size = 0.66 \[ \frac {a x \left (5-6 a^2 x^2\right ) \sqrt {a^2 c x^2+c}+11 \sqrt {c} \log \left (\sqrt {c} \sqrt {a^2 c x^2+c}+a c x\right )+8 \left (3 a^4 x^4+a^2 x^2-2\right ) \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{120 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 94, normalized size = 0.59 \[ -\frac {2 \, {\left (6 \, a^{3} x^{3} - 5 \, a x - 8 \, {\left (3 \, a^{4} x^{4} + a^{2} x^{2} - 2\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c} - 11 \, \sqrt {c} \log \left (-2 \, a^{2} c x^{2} - 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right )}{240 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.72, size = 176, normalized size = 1.10 \[ \frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (24 \arctan \left (a x \right ) x^{4} a^{4}-6 a^{3} x^{3}+8 \arctan \left (a x \right ) x^{2} a^{2}+5 a x -16 \arctan \left (a x \right )\right )}{120 a^{4}}+\frac {11 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right )}{120 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {11 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right )}{120 a^{4} \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 127, normalized size = 0.79 \[ -\frac {1}{120} \, {\left (a {\left (\frac {3 \, {\left (\frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}\right )}}{a^{2}} - \frac {8 \, {\left (\sqrt {a^{2} x^{2} + 1} x + \frac {\operatorname {arsinh}\left (a x\right )}{a}\right )}}{a^{4}}\right )} - 8 \, {\left (\frac {3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{a^{2}} - \frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4}}\right )} \arctan \left (a x\right )\right )} \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\mathrm {atan}\left (a\,x\right )\,\sqrt {c\,a^2\,x^2+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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