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

Optimal. Leaf size=51 \[ \frac{8 c (b+2 c x)}{3 b^3 \sqrt{b x+c x^2}}-\frac{2}{3 b x \sqrt{b x+c x^2}} \]

[Out]

-2/(3*b*x*Sqrt[b*x + c*x^2]) + (8*c*(b + 2*c*x))/(3*b^3*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.0498431, antiderivative size = 51, 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 \[ \frac{8 c (b+2 c x)}{3 b^3 \sqrt{b x+c x^2}}-\frac{2}{3 b x \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-2/(3*b*x*Sqrt[b*x + c*x^2]) + (8*c*(b + 2*c*x))/(3*b^3*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 5.32063, size = 46, normalized size = 0.9 \[ - \frac{2}{3 b x \sqrt{b x + c x^{2}}} + \frac{4 c \left (2 b + 4 c x\right )}{3 b^{3} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0321132, size = 40, normalized size = 0.78 \[ \frac{2 \left (-b^2+4 b c x+8 c^2 x^2\right )}{3 b^3 x \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-b^2 + 4*b*c*x + 8*c^2*x^2))/(3*b^3*x*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.007, size = 39, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -8\,{c}^{2}{x}^{2}-4\,bcx+{b}^{2} \right ) }{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^2+b*x)^(3/2),x)

[Out]

-2/3*(c*x+b)*(-8*c^2*x^2-4*b*c*x+b^2)/b^3/(c*x^2+b*x)^(3/2)

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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/((c*x^2 + b*x)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21999, size = 51, normalized size = 1. \[ \frac{2 \,{\left (8 \, c^{2} x^{2} + 4 \, b c x - b^{2}\right )}}{3 \, \sqrt{c x^{2} + b x} b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(3/2)*x),x, algorithm="fricas")

[Out]

2/3*(8*c^2*x^2 + 4*b*c*x - b^2)/(sqrt(c*x^2 + b*x)*b^3*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(x*(b + c*x))**(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(3/2)*x),x, algorithm="giac")

[Out]

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