Optimal. Leaf size=79 \[ \frac{2 x \sqrt{c+\frac{d}{x^2}} (3 b c-4 a d)}{3 c^3}-\frac{x (3 b c-4 a d)}{3 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^3}{3 c \sqrt{c+\frac{d}{x^2}}} \]
[Out]
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Rubi [A] time = 0.11222, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{2 x \sqrt{c+\frac{d}{x^2}} (3 b c-4 a d)}{3 c^3}-\frac{x (3 b c-4 a d)}{3 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^3}{3 c \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Int[((a + b/x^2)*x^2)/(c + d/x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.76509, size = 73, normalized size = 0.92 \[ \frac{a x^{3}}{3 c \sqrt{c + \frac{d}{x^{2}}}} + \frac{x \left (4 a d - 3 b c\right )}{3 c^{2} \sqrt{c + \frac{d}{x^{2}}}} - \frac{2 x \sqrt{c + \frac{d}{x^{2}}} \left (4 a d - 3 b c\right )}{3 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)*x**2/(c+d/x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.070274, size = 57, normalized size = 0.72 \[ \frac{a \left (c^2 x^4-4 c d x^2-8 d^2\right )+3 b c \left (c x^2+2 d\right )}{3 c^3 x \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b/x^2)*x^2)/(c + d/x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 66, normalized size = 0.8 \[{\frac{ \left ( a{x}^{4}{c}^{2}-4\,acd{x}^{2}+3\,b{c}^{2}{x}^{2}-8\,a{d}^{2}+6\,bcd \right ) \left ( c{x}^{2}+d \right ) }{3\,{x}^{3}{c}^{3}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)*x^2/(c+d/x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.38796, size = 122, normalized size = 1.54 \[ b{\left (\frac{\sqrt{c + \frac{d}{x^{2}}} x}{c^{2}} + \frac{d}{\sqrt{c + \frac{d}{x^{2}}} c^{2} x}\right )} + \frac{1}{3} \, a{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{3} - 6 \, \sqrt{c + \frac{d}{x^{2}}} d x}{c^{3}} - \frac{3 \, d^{2}}{\sqrt{c + \frac{d}{x^{2}}} c^{3} x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x^2/(c + d/x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228403, size = 95, normalized size = 1.2 \[ \frac{{\left (a c^{2} x^{5} +{\left (3 \, b c^{2} - 4 \, a c d\right )} x^{3} + 2 \,{\left (3 \, b c d - 4 \, a d^{2}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3 \,{\left (c^{4} x^{2} + c^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x^2/(c + d/x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 11.4922, size = 267, normalized size = 3.38 \[ a \left (\frac{c^{3} d^{\frac{9}{2}} x^{6} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{5} d^{4} x^{4} + 6 c^{4} d^{5} x^{2} + 3 c^{3} d^{6}} - \frac{3 c^{2} d^{\frac{11}{2}} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{5} d^{4} x^{4} + 6 c^{4} d^{5} x^{2} + 3 c^{3} d^{6}} - \frac{12 c d^{\frac{13}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{5} d^{4} x^{4} + 6 c^{4} d^{5} x^{2} + 3 c^{3} d^{6}} - \frac{8 d^{\frac{15}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{5} d^{4} x^{4} + 6 c^{4} d^{5} x^{2} + 3 c^{3} d^{6}}\right ) + b \left (\frac{x^{2}}{c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{2 \sqrt{d}}{c^{2} \sqrt{\frac{c x^{2}}{d} + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)*x**2/(c+d/x**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{2}}\right )} x^{2}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x^2/(c + d/x^2)^(3/2),x, algorithm="giac")
[Out]