Optimal. Leaf size=112 \[ -\frac {x^2 \tan ^{-1}(a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3}{9 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 x}{3 a^3 c^2 \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.14, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4938, 4930, 191} \[ \frac {2 x}{3 a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {x^3}{9 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {x^2 \tan ^{-1}(a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 4930
Rule 4938
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac {x^3}{9 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x^2 \tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^2 c}\\ &=\frac {x^3}{9 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x^2 \tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^3 c}\\ &=\frac {x^3}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 65, normalized size = 0.58 \[ \frac {\sqrt {a^2 c x^2+c} \left (a x \left (7 a^2 x^2+6\right )-3 \left (3 a^2 x^2+2\right ) \tan ^{-1}(a x)\right )}{9 a^4 c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 74, normalized size = 0.66 \[ \frac {{\left (7 \, a^{3} x^{3} + 6 \, a x - 3 \, {\left (3 \, a^{2} x^{2} + 2\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{9 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \]
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 = 3.06, size = 244, normalized size = 2.18 \[ -\frac {\left (i+3 \arctan \left (a x \right )\right ) \left (i x^{3} a^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{72 \left (a^{2} x^{2}+1\right )^{2} c^{3} a^{4}}-\frac {3 \left (i+\arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 a^{4} c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )-i\right )}{8 a^{4} c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\left (-i+3 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i x^{3} a^{3}-3 a^{2} x^{2}-3 i a x +1\right )}{72 \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) c^{3} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 65, normalized size = 0.58 \[ \frac {7 \, a^{3} x^{3} + 6 \, a x - 3 \, {\left (3 \, a^{2} x^{2} + 2\right )} \arctan \left (a x\right )}{9 \, {\left (a^{6} c^{2} x^{2} + a^{4} c^{2}\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {c}} \]
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 \[ \int \frac {x^3\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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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