Optimal. Leaf size=77 \[ \frac {x^3 \tan ^{-1}(a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{3 a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4944, 266, 43} \[ \frac {1}{3 a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \tan ^{-1}(a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 4944
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac {x^3 \tan ^{-1}(a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {1}{3} a \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {x^3 \tan ^{-1}(a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {1}{6} a \operatorname {Subst}\left (\int \frac {x}{\left (c+a^2 c x\right )^{5/2}} \, dx,x,x^2\right )\\ &=\frac {x^3 \tan ^{-1}(a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {1}{6} a \operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \left (c+a^2 c x\right )^{5/2}}+\frac {1}{a^2 c \left (c+a^2 c x\right )^{3/2}}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {1}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^3 \tan ^{-1}(a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 57, normalized size = 0.74 \[ \frac {\sqrt {a^2 c x^2+c} \left (3 a^3 x^3 \tan ^{-1}(a x)+3 a^2 x^2+2\right )}{9 a^3 c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 67, normalized size = 0.87 \[ \frac {{\left (3 \, a^{3} x^{3} \arctan \left (a x\right ) + 3 \, a^{2} x^{2} + 2\right )} \sqrt {a^{2} c x^{2} + c}}{9 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.83, size = 240, normalized size = 3.12 \[ \frac {\left (i+3 \arctan \left (a x \right )\right ) \left (a^{3} x^{3}-3 i x^{2} a^{2}-3 a x +i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{72 \left (a^{2} x^{2}+1\right )^{2} c^{3} a^{3}}+\frac {\left (i+\arctan \left (a x \right )\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 a^{3} c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )-i\right )}{8 a^{3} c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\left (-i+3 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a^{3} x^{3}+3 i x^{2} a^{2}-3 a x -i\right )}{72 \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) c^{3} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 93, normalized size = 1.21 \[ \frac {1}{9} \, a {\left (\frac {3}{\sqrt {a^{2} c x^{2} + c} a^{4} c^{2}} - \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{4} c}\right )} + \frac {1}{3} \, {\left (\frac {x}{\sqrt {a^{2} c x^{2} + c} a^{2} c^{2}} - \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c}\right )} \arctan \left (a x\right ) \]
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 \[ \int \frac {x^2\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{2} \operatorname {atan}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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