Optimal. Leaf size=21 \[ \frac{c x \sqrt{\frac{c}{a+b x^2}}}{a} \]
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Rubi [A] time = 0.0312931, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{c x \sqrt{\frac{c}{a+b x^2}}}{a} \]
Antiderivative was successfully verified.
[In] Int[(c/(a + b*x^2))^(3/2),x]
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Rubi in Sympy [A] time = 1.95366, size = 15, normalized size = 0.71 \[ \frac{c x \sqrt{\frac{c}{a + b x^{2}}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c/(b*x**2+a))**(3/2),x)
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Mathematica [A] time = 0.0130355, size = 21, normalized size = 1. \[ \frac{c x \sqrt{\frac{c}{a+b x^2}}}{a} \]
Antiderivative was successfully verified.
[In] Integrate[(c/(a + b*x^2))^(3/2),x]
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Maple [A] time = 0.003, size = 26, normalized size = 1.2 \[{\frac{x \left ( b{x}^{2}+a \right ) }{a} \left ({\frac{c}{b{x}^{2}+a}} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c/(b*x^2+a))^(3/2),x)
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Maxima [A] time = 0.700135, size = 23, normalized size = 1.1 \[ \frac{c^{\frac{3}{2}} x}{\sqrt{b x^{2} + a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x^2 + a))^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.303143, size = 26, normalized size = 1.24 \[ \frac{c x \sqrt{\frac{c}{b x^{2} + a}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x^2 + a))^(3/2),x, algorithm="fricas")
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Sympy [A] time = 4.69849, size = 66, normalized size = 3.14 \[ \begin{cases} c^{\frac{3}{2}} x \left (\frac{1}{a + b x^{2}}\right )^{\frac{3}{2}} + \frac{b c^{\frac{3}{2}} x^{3} \left (\frac{1}{a + b x^{2}}\right )^{\frac{3}{2}}}{a} & \text{for}\: a \neq 0 \\- \frac{c^{\frac{3}{2}} x \left (\frac{1}{b}\right )^{\frac{3}{2}} \left (\frac{1}{x^{2}}\right )^{\frac{3}{2}}}{2} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x**2+a))**(3/2),x)
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GIAC/XCAS [A] time = 0.272286, size = 38, normalized size = 1.81 \[ \frac{c^{2} x{\rm sign}\left (b x^{2} + a\right )}{\sqrt{b c x^{2} + a c} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x^2 + a))^(3/2),x, algorithm="giac")
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