Optimal. Leaf size=89 \[ \frac{c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} (b c-a d)^{3/2}}-\frac{x \sqrt{c+d x^2}}{2 \left (a+b x^2\right ) (b c-a d)} \]
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Rubi [A] time = 0.15905, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} (b c-a d)^{3/2}}-\frac{x \sqrt{c+d x^2}}{2 \left (a+b x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^2/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 24.4845, size = 73, normalized size = 0.82 \[ \frac{x \sqrt{c + d x^{2}}}{2 \left (a + b x^{2}\right ) \left (a d - b c\right )} - \frac{c \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 \sqrt{a} \left (a d - b c\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x**2+a)**2/(d*x**2+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.151971, size = 86, normalized size = 0.97 \[ \frac{\frac{x \sqrt{c+d x^2}}{a+b x^2}-\frac{c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} \sqrt{b c-a d}}}{2 a d-2 b c} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]
[Out]
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Maple [B] time = 0.019, size = 817, normalized size = 9.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x^2+a)^2/(d*x^2+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^2 + a)^2*sqrt(d*x^2 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.341371, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} x +{\left (b c x^{2} + a c\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} - 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x -{\left (b c x^{2} + a c\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{4 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt{a b c - a^{2} d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^2 + a)^2*sqrt(d*x^2 + c)),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x**2+a)**2/(d*x**2+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 2.00856, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^2 + a)^2*sqrt(d*x^2 + c)),x, algorithm="giac")
[Out]