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

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

[Out]

(c*x*Sqrt[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]

[Out]

(c*x*Sqrt[c/(a + b*x^2)])/a

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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)

[Out]

c*x*sqrt(c/(a + b*x**2))/a

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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]

[Out]

(c*x*Sqrt[c/(a + b*x^2)])/a

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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)

[Out]

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

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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")

[Out]

c^(3/2)*x/(sqrt(b*x^2 + a)*a)

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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")

[Out]

c*x*sqrt(c/(b*x^2 + a))/a

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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)

[Out]

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

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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")

[Out]

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