Optimal. Leaf size=33 \[ \text {Int}\left (\frac {\sqrt {f x}}{\left (d+c^2 d x^2\right )^2 (a+b \text {ArcTan}(c x))^2},x\right ) \]
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Rubi [A]
time = 0.07, 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 = {}
\begin {gather*} \int \frac {\sqrt {f x}}{\left (d+c^2 d x^2\right )^2 (a+b \text {ArcTan}(c x))^2} \, dx \end {gather*}
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 = 19.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {f x}}{\left (d+c^2 d x^2\right )^2 (a+b \text {ArcTan}(c x))^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.21, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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