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

Optimal. Leaf size=77 \[ -\frac{16 c^2 (b+2 c x)}{5 b^4 \sqrt{b x+c x^2}}+\frac{4 c}{5 b^2 x \sqrt{b x+c x^2}}-\frac{2}{5 b x^2 \sqrt{b x+c x^2}} \]

[Out]

-2/(5*b*x^2*Sqrt[b*x + c*x^2]) + (4*c)/(5*b^2*x*Sqrt[b*x + c*x^2]) - (16*c^2*(b
+ 2*c*x))/(5*b^4*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.0853996, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{16 c^2 (b+2 c x)}{5 b^4 \sqrt{b x+c x^2}}+\frac{4 c}{5 b^2 x \sqrt{b x+c x^2}}-\frac{2}{5 b x^2 \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-2/(5*b*x^2*Sqrt[b*x + c*x^2]) + (4*c)/(5*b^2*x*Sqrt[b*x + c*x^2]) - (16*c^2*(b
+ 2*c*x))/(5*b^4*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 8.8254, size = 71, normalized size = 0.92 \[ - \frac{2}{5 b x^{2} \sqrt{b x + c x^{2}}} + \frac{4 c}{5 b^{2} x \sqrt{b x + c x^{2}}} - \frac{8 c^{2} \left (2 b + 4 c x\right )}{5 b^{4} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(5*b*x**2*sqrt(b*x + c*x**2)) + 4*c/(5*b**2*x*sqrt(b*x + c*x**2)) - 8*c**2*(2
*b + 4*c*x)/(5*b**4*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.0361456, size = 49, normalized size = 0.64 \[ -\frac{2 \left (b^3-2 b^2 c x+8 b c^2 x^2+16 c^3 x^3\right )}{5 b^4 x^2 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b^3 - 2*b^2*c*x + 8*b*c^2*x^2 + 16*c^3*x^3))/(5*b^4*x^2*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.006, size = 53, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 16\,{x}^{3}{c}^{3}+8\,b{x}^{2}{c}^{2}-2\,{b}^{2}xc+{b}^{3} \right ) }{5\,x{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(c*x^2+b*x)^(3/2),x)

[Out]

-2/5*(c*x+b)*(16*c^3*x^3+8*b*c^2*x^2-2*b^2*c*x+b^3)/x/b^4/(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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219423, size = 63, normalized size = 0.82 \[ -\frac{2 \,{\left (16 \, c^{3} x^{3} + 8 \, b c^{2} x^{2} - 2 \, b^{2} c x + b^{3}\right )}}{5 \, \sqrt{c x^{2} + b x} b^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*(16*c^3*x^3 + 8*b*c^2*x^2 - 2*b^2*c*x + b^3)/(sqrt(c*x^2 + b*x)*b^4*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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