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

Optimal. Leaf size=24 \[ -\frac{2 (b+2 c x)}{b^2 \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0145355, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 (b+2 c x)}{b^2 \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(-3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.54375, size = 24, normalized size = 1. \[ - \frac{2 b + 4 c x}{b^{2} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0218273, size = 22, normalized size = 0.92 \[ -\frac{2 (b+2 c x)}{b^2 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 29, normalized size = 1.2 \[ -2\,{\frac{ \left ( 2\,cx+b \right ) x \left ( cx+b \right ) }{{b}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.713439, size = 47, normalized size = 1.96 \[ -\frac{4 \, c x}{\sqrt{c x^{2} + b x} b^{2}} - \frac{2}{\sqrt{c x^{2} + b x} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216463, size = 30, normalized size = 1.25 \[ -\frac{2 \,{\left (2 \, c x + b\right )}}{\sqrt{c x^{2} + b x} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((b*x + c*x**2)**(-3/2), x)

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GIAC/XCAS [A]  time = 0.221073, size = 32, normalized size = 1.33 \[ -\frac{2 \,{\left (\frac{2 \, c x}{b^{2}} + \frac{1}{b}\right )}}{\sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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