Optimal. Leaf size=122 \[ \frac{b x (2 b c-5 a d)}{3 a^2 \sqrt{a+b x^2} (b c-a d)^2}+\frac{d^2 \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} (b c-a d)^{5/2}}+\frac{b x}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.345895, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{b x (2 b c-5 a d)}{3 a^2 \sqrt{a+b x^2} (b c-a d)^2}+\frac{d^2 \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} (b c-a d)^{5/2}}+\frac{b x}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 55.144, size = 107, normalized size = 0.88 \[ \frac{d^{2} \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{\sqrt{c} \left (a d - b c\right )^{\frac{5}{2}}} - \frac{b x}{3 a \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{b x \left (5 a d - 2 b c\right )}{3 a^{2} \sqrt{a + b x^{2}} \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.360047, size = 112, normalized size = 0.92 \[ \frac{b x \left (-6 a^2 d+a b \left (3 c-5 d x^2\right )+2 b^2 c x^2\right )}{3 a^2 \left (a+b x^2\right )^{3/2} (b c-a d)^2}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} (a d-b c)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)),x]
[Out]
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Maple [B] time = 0.027, size = 1070, normalized size = 8.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2+a)^(5/2)/(d*x^2+c),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(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.546142, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (2 \, b^{3} c - 5 \, a b^{2} d\right )} x^{3} + 3 \,{\left (a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt{b c^{2} - a c d} \sqrt{b x^{2} + a} + 3 \,{\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )} \log \left (\frac{{\left ({\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt{b c^{2} - a c d} + 4 \,{\left ({\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{3} +{\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt{b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{12 \,{\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} +{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} x^{2}\right )} \sqrt{b c^{2} - a c d}}, \frac{2 \,{\left ({\left (2 \, b^{3} c - 5 \, a b^{2} d\right )} x^{3} + 3 \,{\left (a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt{-b c^{2} + a c d} \sqrt{b x^{2} + a} + 3 \,{\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )} \arctan \left (\frac{\sqrt{-b c^{2} + a c d}{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )}}{2 \,{\left (b c^{2} - a c d\right )} \sqrt{b x^{2} + a} x}\right )}{6 \,{\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} +{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} x^{2}\right )} \sqrt{-b c^{2} + a c d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (c + d x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.229209, size = 432, normalized size = 3.54 \[ -\frac{\sqrt{b} d^{2} \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c^{2} + a b c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-b^{2} c^{2} + a b c d}} + \frac{{\left (\frac{{\left (2 \, b^{6} c^{3} - 9 \, a b^{5} c^{2} d + 12 \, a^{2} b^{4} c d^{2} - 5 \, a^{3} b^{3} d^{3}\right )} x^{2}}{a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}} + \frac{3 \,{\left (a b^{5} c^{3} - 4 \, a^{2} b^{4} c^{2} d + 5 \, a^{3} b^{3} c d^{2} - 2 \, a^{4} b^{2} d^{3}\right )}}{a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}}\right )} x}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)),x, algorithm="giac")
[Out]