3.10 \(\int \frac{\sqrt{b x+c x^2}}{x^6} \, dx\)

Optimal. Leaf size=100 \[ \frac{32 c^3 \left (b x+c x^2\right )^{3/2}}{315 b^4 x^3}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}+\frac{4 c \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6} \]

[Out]

(-2*(b*x + c*x^2)^(3/2))/(9*b*x^6) + (4*c*(b*x + c*x^2)^(3/2))/(21*b^2*x^5) - (1
6*c^2*(b*x + c*x^2)^(3/2))/(105*b^3*x^4) + (32*c^3*(b*x + c*x^2)^(3/2))/(315*b^4
*x^3)

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Rubi [A]  time = 0.131598, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{32 c^3 \left (b x+c x^2\right )^{3/2}}{315 b^4 x^3}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}+\frac{4 c \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/x^6,x]

[Out]

(-2*(b*x + c*x^2)^(3/2))/(9*b*x^6) + (4*c*(b*x + c*x^2)^(3/2))/(21*b^2*x^5) - (1
6*c^2*(b*x + c*x^2)^(3/2))/(105*b^3*x^4) + (32*c^3*(b*x + c*x^2)^(3/2))/(315*b^4
*x^3)

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Rubi in Sympy [A]  time = 14.1529, size = 94, normalized size = 0.94 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{9 b x^{6}} + \frac{4 c \left (b x + c x^{2}\right )^{\frac{3}{2}}}{21 b^{2} x^{5}} - \frac{16 c^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{105 b^{3} x^{4}} + \frac{32 c^{3} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{315 b^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(1/2)/x**6,x)

[Out]

-2*(b*x + c*x**2)**(3/2)/(9*b*x**6) + 4*c*(b*x + c*x**2)**(3/2)/(21*b**2*x**5) -
16*c**2*(b*x + c*x**2)**(3/2)/(105*b**3*x**4) + 32*c**3*(b*x + c*x**2)**(3/2)/(
315*b**4*x**3)

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Mathematica [A]  time = 0.0275995, size = 62, normalized size = 0.62 \[ \frac{2 \sqrt{x (b+c x)} \left (-35 b^4-5 b^3 c x+6 b^2 c^2 x^2-8 b c^3 x^3+16 c^4 x^4\right )}{315 b^4 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x + c*x^2]/x^6,x]

[Out]

(2*Sqrt[x*(b + c*x)]*(-35*b^4 - 5*b^3*c*x + 6*b^2*c^2*x^2 - 8*b*c^3*x^3 + 16*c^4
*x^4))/(315*b^4*x^5)

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Maple [A]  time = 0.007, size = 55, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -16\,{x}^{3}{c}^{3}+24\,b{x}^{2}{c}^{2}-30\,{b}^{2}xc+35\,{b}^{3} \right ) }{315\,{x}^{5}{b}^{4}}\sqrt{c{x}^{2}+bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/x^6,x)

[Out]

-2/315*(c*x+b)*(-16*c^3*x^3+24*b*c^2*x^2-30*b^2*c*x+35*b^3)*(c*x^2+b*x)^(1/2)/x^
5/b^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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220525, size = 81, normalized size = 0.81 \[ \frac{2 \,{\left (16 \, c^{4} x^{4} - 8 \, b c^{3} x^{3} + 6 \, b^{2} c^{2} x^{2} - 5 \, b^{3} c x - 35 \, b^{4}\right )} \sqrt{c x^{2} + b x}}{315 \, b^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^6,x, algorithm="fricas")

[Out]

2/315*(16*c^4*x^4 - 8*b*c^3*x^3 + 6*b^2*c^2*x^2 - 5*b^3*c*x - 35*b^4)*sqrt(c*x^2
+ b*x)/(b^4*x^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(1/2)/x**6,x)

[Out]

Integral(sqrt(x*(b + c*x))/x**6, x)

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GIAC/XCAS [A]  time = 0.21415, size = 223, normalized size = 2.23 \[ \frac{2 \,{\left (630 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} c^{\frac{5}{2}} + 1764 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} b c^{2} + 1995 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{2} c^{\frac{3}{2}} + 1125 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{3} c + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{4} \sqrt{c} + 35 \, b^{5}\right )}}{315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^6,x, algorithm="giac")

[Out]

2/315*(630*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*c^(5/2) + 1764*(sqrt(c)*x - sqrt(c*
x^2 + b*x))^4*b*c^2 + 1995*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*c^(3/2) + 1125*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^3*c + 315*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^
4*sqrt(c) + 35*b^5)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^9