Optimal. Leaf size=79 \[ \frac {x}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x}{9 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\text {ArcTan}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}} \]
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Rubi [A]
time = 0.05, 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 = {5050, 198, 197}
\begin {gather*} -\frac {\text {ArcTan}(a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 5050
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 = 0.04, size = 51, normalized size = 0.65 \begin {gather*} \frac {\sqrt {c+a^2 c x^2} \left (3 a x+2 a^3 x^3-3 \text {ArcTan}(a x)\right )}{9 c^3 \left (a+a^3 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.40, size = 244, normalized size = 3.09
method | result | size |
default | \(\frac {\left (3 \arctan \left (a x \right )+i\right ) \left (i a^{3} x^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{72 \left (a^{2} x^{2}+1\right )^{2} a^{2} c^{3}}-\frac {\left (\arctan \left (a x \right )+i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} a^{2} \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )-i\right )}{8 c^{3} a^{2} \left (a^{2} x^{2}+1\right )}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{3} x^{3}-3 a^{2} x^{2}-3 i a x +1\right ) \left (-i+3 \arctan \left (a x \right )\right )}{72 c^{3} a^{2} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\) | \(244\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 66, normalized size = 0.84 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.29, size = 64, normalized size = 0.81 \begin {gather*} \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 )}} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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