3.8 \(\int \frac{\sqrt{b x+c x^2}}{x^4} \, dx\)

Optimal. Leaf size=48 \[ \frac{4 c \left (b x+c x^2\right )^{3/2}}{15 b^2 x^3}-\frac{2 \left (b x+c x^2\right )^{3/2}}{5 b x^4} \]

[Out]

(-2*(b*x + c*x^2)^(3/2))/(5*b*x^4) + (4*c*(b*x + c*x^2)^(3/2))/(15*b^2*x^3)

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Rubi [A]  time = 0.0602688, 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 \[ \frac{4 c \left (b x+c x^2\right )^{3/2}}{15 b^2 x^3}-\frac{2 \left (b x+c x^2\right )^{3/2}}{5 b x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/x^4,x]

[Out]

(-2*(b*x + c*x^2)^(3/2))/(5*b*x^4) + (4*c*(b*x + c*x^2)^(3/2))/(15*b^2*x^3)

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Rubi in Sympy [A]  time = 6.23142, size = 42, normalized size = 0.88 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{5 b x^{4}} + \frac{4 c \left (b x + c x^{2}\right )^{\frac{3}{2}}}{15 b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(1/2)/x**4,x)

[Out]

-2*(b*x + c*x**2)**(3/2)/(5*b*x**4) + 4*c*(b*x + c*x**2)**(3/2)/(15*b**2*x**3)

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Mathematica [A]  time = 0.0198937, size = 40, normalized size = 0.83 \[ \frac{2 \sqrt{x (b+c x)} \left (-3 b^2-b c x+2 c^2 x^2\right )}{15 b^2 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x + c*x^2]/x^4,x]

[Out]

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

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Maple [A]  time = 0.006, size = 33, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -2\,cx+3\,b \right ) }{15\,{b}^{2}{x}^{3}}\sqrt{c{x}^{2}+bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/x^4,x)

[Out]

-2/15*(c*x+b)*(-2*c*x+3*b)*(c*x^2+b*x)^(1/2)/b^2/x^3

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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(sqrt(c*x^2 + b*x)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217756, size = 51, normalized size = 1.06 \[ \frac{2 \,{\left (2 \, c^{2} x^{2} - b c x - 3 \, b^{2}\right )} \sqrt{c x^{2} + b x}}{15 \, b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(1/2)/x**4,x)

[Out]

Integral(sqrt(x*(b + c*x))/x**4, x)

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GIAC/XCAS [A]  time = 0.212456, size = 144, normalized size = 3. \[ \frac{2 \,{\left (15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} c^{\frac{3}{2}} + 25 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b c + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{2} \sqrt{c} + 3 \, b^{3}\right )}}{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^4,x, algorithm="giac")

[Out]

2/15*(15*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*c^(3/2) + 25*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^2*b*c + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^2*sqrt(c) + 3*b^3)/(sqrt(c)
*x - sqrt(c*x^2 + b*x))^5