Optimal. Leaf size=32 \[ \text{Unintegrable}\left (\frac{\sqrt{f x}}{\left (c^2 d x^2+d\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.100492, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [A] time = 28.9607, size = 0, normalized size = 0. \[ \int \frac{\sqrt{f x}}{\left (d+c^2 d x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 1.223, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({c}^{2}d{x}^{2}+d \right ) ^{2} \left ( a+b\arctan \left ( cx \right ) \right ) ^{2}}\sqrt{fx}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{f x}}{{\left (c^{2} d x^{2} + d\right )}^{2}{\left (b \arctan \left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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