Optimal. Leaf size=33 \[ \text {Int}\left (\frac {\sqrt {f x}}{\left (c^2 d x^2+d\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.10, 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 = {} \[ \int \frac {\sqrt {f x}}{\left (d+c^2 d x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {f x}}{\left (d+c^2 d x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2} \, dx &=\int \frac {\sqrt {f x}}{\left (d+c^2 d x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 31.57, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {f x}}{\left (d+c^2 d x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {f x}}{a^{2} c^{4} d^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arctan \left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} + 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arctan \left (c x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 1.40, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {f x}}{\left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arctan \left (c x \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\frac {1}{2} \, {\left (a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arctan \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arctan \left (c x\right )\right )} \sqrt {f} \int \frac {{\left (a c^{2} x^{2} + 4 \, b c x + {\left (b c^{2} x^{2} + b\right )} \arctan \left (c x\right ) + a\right )} \sqrt {x}}{a^{3} c^{4} d^{2} x^{4} + 2 \, a^{3} c^{2} d^{2} x^{2} + a^{3} d^{2} + {\left (b^{3} c^{4} d^{2} x^{4} + 2 \, b^{3} c^{2} d^{2} x^{2} + b^{3} d^{2}\right )} \arctan \left (c x\right )^{3} + 3 \, {\left (a b^{2} c^{4} d^{2} x^{4} + 2 \, a b^{2} c^{2} d^{2} x^{2} + a b^{2} d^{2}\right )} \arctan \left (c x\right )^{2} + 3 \, {\left (a^{2} b c^{4} d^{2} x^{4} + 2 \, a^{2} b c^{2} d^{2} x^{2} + a^{2} b d^{2}\right )} \arctan \left (c x\right )}\,{d x} + \sqrt {f} x^{\frac {3}{2}}}{2 \, {\left (a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arctan \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arctan \left (c x\right )\right )}} \]
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 \[ \int \frac {\sqrt {f\,x}}{{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]
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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