3.48 \(\int \frac{1}{x^2 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=48 \[ \frac{4 c \sqrt{b x+c x^2}}{3 b^2 x}-\frac{2 \sqrt{b x+c x^2}}{3 b x^2} \]

[Out]

(-2*Sqrt[b*x + c*x^2])/(3*b*x^2) + (4*c*Sqrt[b*x + c*x^2])/(3*b^2*x)

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Rubi [A]  time = 0.0637451, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{4 c \sqrt{b x+c x^2}}{3 b^2 x}-\frac{2 \sqrt{b x+c x^2}}{3 b x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*Sqrt[b*x + c*x^2])/(3*b*x^2) + (4*c*Sqrt[b*x + c*x^2])/(3*b^2*x)

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Rubi in Sympy [A]  time = 6.33403, size = 41, normalized size = 0.85 \[ - \frac{2 \sqrt{b x + c x^{2}}}{3 b x^{2}} + \frac{4 c \sqrt{b x + c x^{2}}}{3 b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0352768, size = 29, normalized size = 0.6 \[ \frac{2 \sqrt{x (b+c x)} (2 c x-b)}{3 b^2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*Sqrt[b*x + c*x^2]),x]

[Out]

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

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Maple [A]  time = 0.007, size = 31, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -2\,cx+b \right ) }{3\,{b}^{2}x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(c*x^2+b*x)^(1/2),x)

[Out]

-2/3*(c*x+b)*(-2*c*x+b)/x/b^2/(c*x^2+b*x)^(1/2)

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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(1/(sqrt(c*x^2 + b*x)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237992, size = 36, normalized size = 0.75 \[ \frac{2 \, \sqrt{c x^{2} + b x}{\left (2 \, c x - b\right )}}{3 \, b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*x^2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*sqrt(x*(b + c*x))), x)

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GIAC/XCAS [A]  time = 0.219003, size = 66, normalized size = 1.38 \[ \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} + b\right )}}{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*x^2),x, algorithm="giac")

[Out]

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