Optimal. Leaf size=21 \[ -\frac{c \sqrt{\frac{c}{a+b x^2}}}{b} \]
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Rubi [A] time = 0.0214014, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{c \sqrt{\frac{c}{a+b x^2}}}{b} \]
Antiderivative was successfully verified.
[In] Int[x*(c/(a + b*x^2))^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 2.13074, size = 15, normalized size = 0.71 \[ - \frac{c \sqrt{\frac{c}{a + b x^{2}}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(c/(b*x**2+a))**(3/2),x)
[Out]
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Mathematica [A] time = 0.0082162, size = 21, normalized size = 1. \[ -\frac{c \sqrt{\frac{c}{a+b x^2}}}{b} \]
Antiderivative was successfully verified.
[In] Integrate[x*(c/(a + b*x^2))^(3/2),x]
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Maple [A] time = 0.005, size = 26, normalized size = 1.2 \[ -{\frac{b{x}^{2}+a}{b} \left ({\frac{c}{b{x}^{2}+a}} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(c/(b*x^2+a))^(3/2),x)
[Out]
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Maxima [A] time = 0.686796, size = 26, normalized size = 1.24 \[ -\frac{c \sqrt{\frac{c}{b x^{2} + a}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c/(b*x^2 + a))^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.273092, size = 26, normalized size = 1.24 \[ -\frac{c \sqrt{\frac{c}{b x^{2} + a}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c/(b*x^2 + a))^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.62353, size = 53, normalized size = 2.52 \[ \begin{cases} - \frac{a c^{\frac{3}{2}} \left (\frac{1}{a + b x^{2}}\right )^{\frac{3}{2}}}{b} - c^{\frac{3}{2}} x^{2} \left (\frac{1}{a + b x^{2}}\right )^{\frac{3}{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \left (\frac{c}{a}\right )^{\frac{3}{2}}}{2} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c/(b*x**2+a))**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.263225, size = 38, normalized size = 1.81 \[ -\frac{c^{2}{\rm sign}\left (b x^{2} + a\right )}{\sqrt{b c x^{2} + a c} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c/(b*x^2 + a))^(3/2),x, algorithm="giac")
[Out]