Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\left (a^2 c x^2+c\right )^{3/2}}{x \sqrt {\tan ^{-1}(a x)}},x\right ) \]
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Rubi [A] time = 0.11, 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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \sqrt {\tan ^{-1}(a x)}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \sqrt {\tan ^{-1}(a x)}} \, dx &=\int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}
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Mathematica [A] time = 2.50, size = 0, normalized size = 0.00 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \sqrt {\tan ^{-1}(a x)}} \, dx \]
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 = 1.61, size = 0, normalized size = 0.00 \[ \int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{x \sqrt {\arctan \left (a x \right )}}\, dx \]
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 = -0.03 \[ \int \frac {{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x\,\sqrt {\mathrm {atan}\left (a\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{x \sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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