Optimal. Leaf size=138 \[ \frac {15}{8} a^2 \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{x \sqrt {a^2 c x^2+c}},x\right )-\frac {1}{2} a^2 \text {Int}\left (\frac {\tan ^{-1}(a x)^{5/2}}{x \sqrt {a^2 c x^2+c}},x\right )-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{2 c x^2}-\frac {5 a \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{4 c x} \]
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Rubi [A] time = 0.42, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\tan ^{-1}(a x)^{5/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^{5/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{2 c x^2}+\frac {1}{4} (5 a) \int \frac {\tan ^{-1}(a x)^{3/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} a^2 \int \frac {\tan ^{-1}(a x)^{5/2}}{x \sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {5 a \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{4 c x}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{2 c x^2}-\frac {1}{2} a^2 \int \frac {\tan ^{-1}(a x)^{5/2}}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (15 a^2\right ) \int \frac {\sqrt {\tan ^{-1}(a x)}}{x \sqrt {c+a^2 c x^2}} \, dx\\ \end {align*}
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Mathematica [A] time = 4.77, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{-1}(a x)^{5/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
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fricas [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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giac [A] 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 [A] time = 4.43, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (a x \right )^{\frac {5}{2}}}{x^{3} \sqrt {a^{2} c \,x^{2}+c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{x^3\,\sqrt {c\,a^2\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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