Optimal. Leaf size=51 \[ \frac{x \sqrt{c+\frac{d}{x^2}} (3 b c-2 a d)}{3 c^2}+\frac{a x^3 \sqrt{c+\frac{d}{x^2}}}{3 c} \]
[Out]
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Rubi [A] time = 0.0847891, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x \sqrt{c+\frac{d}{x^2}} (3 b c-2 a d)}{3 c^2}+\frac{a x^3 \sqrt{c+\frac{d}{x^2}}}{3 c} \]
Antiderivative was successfully verified.
[In] Int[((a + b/x^2)*x^2)/Sqrt[c + d/x^2],x]
[Out]
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Rubi in Sympy [A] time = 7.11603, size = 44, normalized size = 0.86 \[ \frac{a x^{3} \sqrt{c + \frac{d}{x^{2}}}}{3 c} - \frac{x \sqrt{c + \frac{d}{x^{2}}} \left (2 a d - 3 b c\right )}{3 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)*x**2/(c+d/x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.042313, size = 34, normalized size = 0.67 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (a c x^2-2 a d+3 b c\right )}{3 c^2} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b/x^2)*x^2)/Sqrt[c + d/x^2],x]
[Out]
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Maple [A] time = 0.009, size = 44, normalized size = 0.9 \[{\frac{ \left ( a{x}^{2}c-2\,ad+3\,bc \right ) \left ( c{x}^{2}+d \right ) }{3\,x{c}^{2}}{\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)*x^2/(c+d/x^2)^(1/2),x)
[Out]
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Maxima [A] time = 1.39092, size = 66, normalized size = 1.29 \[ \frac{b \sqrt{c + \frac{d}{x^{2}}} x}{c} + \frac{{\left ({\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{3} - 3 \, \sqrt{c + \frac{d}{x^{2}}} d x\right )} a}{3 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x^2/sqrt(c + d/x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215878, size = 49, normalized size = 0.96 \[ \frac{{\left (a c x^{3} +{\left (3 \, b c - 2 \, a d\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x^2/sqrt(c + d/x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.79483, size = 70, normalized size = 1.37 \[ \frac{a \sqrt{d} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c} - \frac{2 a d^{\frac{3}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{2}} + \frac{b \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)*x**2/(c+d/x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{2}}\right )} x^{2}}{\sqrt{c + \frac{d}{x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x^2/sqrt(c + d/x^2),x, algorithm="giac")
[Out]