3.9 \(\int \frac{\sqrt{b x+c x^2}}{x^5} \, dx\)

Optimal. Leaf size=74 \[ -\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^3}+\frac{8 c \left (b x+c x^2\right )^{3/2}}{35 b^2 x^4}-\frac{2 \left (b x+c x^2\right )^{3/2}}{7 b x^5} \]

[Out]

(-2*(b*x + c*x^2)^(3/2))/(7*b*x^5) + (8*c*(b*x + c*x^2)^(3/2))/(35*b^2*x^4) - (1
6*c^2*(b*x + c*x^2)^(3/2))/(105*b^3*x^3)

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Rubi [A]  time = 0.0939739, 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 \[ -\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^3}+\frac{8 c \left (b x+c x^2\right )^{3/2}}{35 b^2 x^4}-\frac{2 \left (b x+c x^2\right )^{3/2}}{7 b x^5} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/x^5,x]

[Out]

(-2*(b*x + c*x^2)^(3/2))/(7*b*x^5) + (8*c*(b*x + c*x^2)^(3/2))/(35*b^2*x^4) - (1
6*c^2*(b*x + c*x^2)^(3/2))/(105*b^3*x^3)

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Rubi in Sympy [A]  time = 9.65861, size = 68, normalized size = 0.92 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{7 b x^{5}} + \frac{8 c \left (b x + c x^{2}\right )^{\frac{3}{2}}}{35 b^{2} x^{4}} - \frac{16 c^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{105 b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(1/2)/x**5,x)

[Out]

-2*(b*x + c*x**2)**(3/2)/(7*b*x**5) + 8*c*(b*x + c*x**2)**(3/2)/(35*b**2*x**4) -
16*c**2*(b*x + c*x**2)**(3/2)/(105*b**3*x**3)

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Mathematica [A]  time = 0.0231776, size = 51, normalized size = 0.69 \[ -\frac{2 \sqrt{x (b+c x)} \left (15 b^3+3 b^2 c x-4 b c^2 x^2+8 c^3 x^3\right )}{105 b^3 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x + c*x^2]/x^5,x]

[Out]

(-2*Sqrt[x*(b + c*x)]*(15*b^3 + 3*b^2*c*x - 4*b*c^2*x^2 + 8*c^3*x^3))/(105*b^3*x
^4)

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Maple [A]  time = 0.007, size = 44, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 8\,{c}^{2}{x}^{2}-12\,bcx+15\,{b}^{2} \right ) }{105\,{b}^{3}{x}^{4}}\sqrt{c{x}^{2}+bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/x^5,x)

[Out]

-2/105*(c*x+b)*(8*c^2*x^2-12*b*c*x+15*b^2)*(c*x^2+b*x)^(1/2)/b^3/x^4

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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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.214396, size = 66, normalized size = 0.89 \[ -\frac{2 \,{\left (8 \, c^{3} x^{3} - 4 \, b c^{2} x^{2} + 3 \, b^{2} c x + 15 \, b^{3}\right )} \sqrt{c x^{2} + b x}}{105 \, b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^5,x, algorithm="fricas")

[Out]

-2/105*(8*c^3*x^3 - 4*b*c^2*x^2 + 3*b^2*c*x + 15*b^3)*sqrt(c*x^2 + b*x)/(b^3*x^4
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(1/2)/x**5,x)

[Out]

Integral(sqrt(x*(b + c*x))/x**5, x)

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GIAC/XCAS [A]  time = 0.214396, size = 184, normalized size = 2.49 \[ \frac{2 \,{\left (140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} c^{2} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b c^{\frac{3}{2}} + 273 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{2} c + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{3} \sqrt{c} + 15 \, b^{4}\right )}}{105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^5,x, algorithm="giac")

[Out]

2/105*(140*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*c^2 + 315*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^3*b*c^(3/2) + 273*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*c + 105*(sqrt(c)*
x - sqrt(c*x^2 + b*x))*b^3*sqrt(c) + 15*b^4)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^7