Optimal. Leaf size=115 \[ \frac{x (a d+2 b c)}{3 c \sqrt{c+d x^2} (b c-a d)^2}+\frac{x}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{\sqrt{a} b \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{5/2}} \]
[Out]
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Rubi [A] time = 0.261409, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{x (a d+2 b c)}{3 c \sqrt{c+d x^2} (b c-a d)^2}+\frac{x}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{\sqrt{a} b \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[x^2/((a + b*x^2)*(c + d*x^2)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 48.3616, size = 97, normalized size = 0.84 \[ - \frac{\sqrt{a} b \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{\left (a d - b c\right )^{\frac{5}{2}}} - \frac{x}{3 \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{x \left (a d + 2 b c\right )}{3 c \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x**2+a)/(d*x**2+c)**(5/2),x)
[Out]
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Mathematica [A] time = 0.333755, size = 103, normalized size = 0.9 \[ \frac{a d^2 x^3+b c x \left (3 c+2 d x^2\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac{\sqrt{a} b \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((a + b*x^2)*(c + d*x^2)^(5/2)),x]
[Out]
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Maple [B] time = 0.02, size = 1134, normalized size = 9.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x^2+a)/(d*x^2+c)^(5/2),x)
[Out]
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Maxima [F] 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)*(d*x^2 + c)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.502157, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b c d^{2} x^{4} + 2 \, b c^{2} d x^{2} + b c^{3}\right )} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} -{\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left (3 \, b c^{2} x +{\left (2 \, b c d + a d^{2}\right )} x^{3}\right )} \sqrt{d x^{2} + c}}{12 \,{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} +{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}}, -\frac{3 \,{\left (b c d^{2} x^{4} + 2 \, b c^{2} d x^{2} + b c^{3}\right )} \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} x \sqrt{\frac{a}{b c - a d}}}\right ) - 2 \,{\left (3 \, b c^{2} x +{\left (2 \, b c d + a d^{2}\right )} x^{3}\right )} \sqrt{d x^{2} + c}}{6 \,{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} +{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^2 + a)*(d*x^2 + c)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x**2+a)/(d*x**2+c)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.239251, size = 393, normalized size = 3.42 \[ \frac{a b \sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b c d - a^{2} d^{2}}} + \frac{{\left (\frac{{\left (2 \, b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + a^{3} d^{5}\right )} x^{2}}{b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3} - 4 \, a^{3} b c^{2} d^{4} + a^{4} c d^{5}} + \frac{3 \,{\left (b^{3} c^{4} d - 2 \, a b^{2} c^{3} d^{2} + a^{2} b c^{2} d^{3}\right )}}{b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3} - 4 \, a^{3} b c^{2} d^{4} + a^{4} c d^{5}}\right )} x}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^2 + a)*(d*x^2 + c)^(5/2)),x, algorithm="giac")
[Out]