3.49 \(\int \frac{1}{x^3 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{16 c^2 \sqrt{b x+c x^2}}{15 b^3 x}+\frac{8 c \sqrt{b x+c x^2}}{15 b^2 x^2}-\frac{2 \sqrt{b x+c x^2}}{5 b x^3} \]

[Out]

(-2*Sqrt[b*x + c*x^2])/(5*b*x^3) + (8*c*Sqrt[b*x + c*x^2])/(15*b^2*x^2) - (16*c^
2*Sqrt[b*x + c*x^2])/(15*b^3*x)

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Rubi [A]  time = 0.0956, antiderivative size = 74, 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 \[ -\frac{16 c^2 \sqrt{b x+c x^2}}{15 b^3 x}+\frac{8 c \sqrt{b x+c x^2}}{15 b^2 x^2}-\frac{2 \sqrt{b x+c x^2}}{5 b x^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[b*x + c*x^2])/(5*b*x^3) + (8*c*Sqrt[b*x + c*x^2])/(15*b^2*x^2) - (16*c^
2*Sqrt[b*x + c*x^2])/(15*b^3*x)

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Rubi in Sympy [A]  time = 9.85392, size = 66, normalized size = 0.89 \[ - \frac{2 \sqrt{b x + c x^{2}}}{5 b x^{3}} + \frac{8 c \sqrt{b x + c x^{2}}}{15 b^{2} x^{2}} - \frac{16 c^{2} \sqrt{b x + c x^{2}}}{15 b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*x + c*x**2)/(5*b*x**3) + 8*c*sqrt(b*x + c*x**2)/(15*b**2*x**2) - 16*c*
*2*sqrt(b*x + c*x**2)/(15*b**3*x)

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Mathematica [A]  time = 0.031278, size = 40, normalized size = 0.54 \[ -\frac{2 \sqrt{x (b+c x)} \left (3 b^2-4 b c x+8 c^2 x^2\right )}{15 b^3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[x*(b + c*x)]*(3*b^2 - 4*b*c*x + 8*c^2*x^2))/(15*b^3*x^3)

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Maple [A]  time = 0.006, size = 44, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 8\,{c}^{2}{x}^{2}-4\,bcx+3\,{b}^{2} \right ) }{15\,{x}^{2}{b}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(c*x^2+b*x)^(1/2),x)

[Out]

-2/15*(c*x+b)*(8*c^2*x^2-4*b*c*x+3*b^2)/x^2/b^3/(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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223293, size = 51, normalized size = 0.69 \[ -\frac{2 \,{\left (8 \, c^{2} x^{2} - 4 \, b c x + 3 \, b^{2}\right )} \sqrt{c x^{2} + b x}}{15 \, b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(8*c^2*x^2 - 4*b*c*x + 3*b^2)*sqrt(c*x^2 + b*x)/(b^3*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219465, size = 105, normalized size = 1.42 \[ \frac{2 \,{\left (20 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} c + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b \sqrt{c} + 3 \, b^{2}\right )}}{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(20*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*c + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)*b*sqrt(c) + 3*b^2)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^5