Optimal. Leaf size=79 \[ \frac {2 x}{9 a c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{9 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {\tan ^{-1}(a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4930, 192, 191} \[ \frac {2 x}{9 a c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{9 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {\tan ^{-1}(a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 4930
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac {\tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{3 a}\\ &=\frac {x}{9 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac {\tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 a c}\\ &=\frac {x}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x}{9 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 51, normalized size = 0.65 \[ \frac {\sqrt {a^2 c x^2+c} \left (2 a^3 x^3+3 a x-3 \tan ^{-1}(a x)\right )}{9 c^3 \left (a^3 x^2+a\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 64, normalized size = 0.81 \[ \frac {{\left (2 \, a^{3} x^{3} + 3 \, a x - 3 \, \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{9 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \]
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 = 0.90, size = 244, normalized size = 3.09 \[ \frac {\left (i+3 \arctan \left (a x \right )\right ) \left (i x^{3} a^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{72 \left (a^{2} x^{2}+1\right )^{2} c^{3} a^{2}}-\frac {\left (i+\arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 a^{2} c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )-i\right )}{8 a^{2} c^{3} \left (a^{2} x^{2}+1\right )}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i x^{3} a^{3}-3 a^{2} x^{2}-3 i a x +1\right ) \left (-i+3 \arctan \left (a x \right )\right )}{72 a^{2} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 66, normalized size = 0.84 \[ \frac {{\left (2 \, a^{3} x^{3} + 3 \, a x - 3 \, \arctan \left (a x\right )\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {c}}{9 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \]
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 \[ \int \frac {x\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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