3.2375 \(\int \frac{1}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=32 \[ -\frac{2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

[Out]

(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.0168804, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(-3/2),x]

[Out]

(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 2.08615, size = 31, normalized size = 0.97 \[ - \frac{2 b + 4 c x}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x+a)**(3/2),x)

[Out]

-(2*b + 4*c*x)/((-4*a*c + b**2)*sqrt(a + b*x + c*x**2))

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Mathematica [A]  time = 0.0252556, size = 32, normalized size = 1. \[ -\frac{2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(-3/2),x]

[Out]

(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])

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Maple [A]  time = 0.005, size = 33, normalized size = 1. \[ 2\,{\frac{2\,cx+b}{\sqrt{c{x}^{2}+bx+a} \left ( 4\,ac-{b}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x+a)^(3/2),x)

[Out]

2/(c*x^2+b*x+a)^(1/2)*(2*c*x+b)/(4*a*c-b^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234996, size = 82, normalized size = 2.56 \[ -\frac{2 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )}}{a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-3/2),x, algorithm="fricas")

[Out]

-2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)/(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 +
(b^3 - 4*a*b*c)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x+a)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.214517, size = 55, normalized size = 1.72 \[ -\frac{2 \,{\left (\frac{2 \, c x}{b^{2} - 4 \, a c} + \frac{b}{b^{2} - 4 \, a c}\right )}}{\sqrt{c x^{2} + b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-3/2),x, algorithm="giac")

[Out]

-2*(2*c*x/(b^2 - 4*a*c) + b/(b^2 - 4*a*c))/sqrt(c*x^2 + b*x + a)