3.748 \(\int x \left (\frac{c}{a+b x^2}\right )^{3/2} \, dx\)

Optimal. Leaf size=21 \[ -\frac{c \sqrt{\frac{c}{a+b x^2}}}{b} \]

[Out]

-((c*Sqrt[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]

-((c*Sqrt[c/(a + b*x^2)])/b)

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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]

-c*sqrt(c/(a + b*x**2))/b

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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]

[Out]

-((c*Sqrt[c/(a + b*x^2)])/b)

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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]

-(b*x^2+a)/b*(c/(b*x^2+a))^(3/2)

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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]

-c*sqrt(c/(b*x^2 + a))/b

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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]

-c*sqrt(c/(b*x^2 + a))/b

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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]

Piecewise((-a*c**(3/2)*(1/(a + b*x**2))**(3/2)/b - c**(3/2)*x**2*(1/(a + b*x**2)
)**(3/2), Ne(b, 0)), (x**2*(c/a)**(3/2)/2, True))

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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]

-c^2*sign(b*x^2 + a)/(sqrt(b*c*x^2 + a*c)*b)