3.51 \(\int \frac{1}{x^5 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{256 c^4 \sqrt{b x+c x^2}}{315 b^5 x}+\frac{128 c^3 \sqrt{b x+c x^2}}{315 b^4 x^2}-\frac{32 c^2 \sqrt{b x+c x^2}}{105 b^3 x^3}+\frac{16 c \sqrt{b x+c x^2}}{63 b^2 x^4}-\frac{2 \sqrt{b x+c x^2}}{9 b x^5} \]

[Out]

(-2*Sqrt[b*x + c*x^2])/(9*b*x^5) + (16*c*Sqrt[b*x + c*x^2])/(63*b^2*x^4) - (32*c
^2*Sqrt[b*x + c*x^2])/(105*b^3*x^3) + (128*c^3*Sqrt[b*x + c*x^2])/(315*b^4*x^2)
- (256*c^4*Sqrt[b*x + c*x^2])/(315*b^5*x)

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Rubi [A]  time = 0.177386, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{256 c^4 \sqrt{b x+c x^2}}{315 b^5 x}+\frac{128 c^3 \sqrt{b x+c x^2}}{315 b^4 x^2}-\frac{32 c^2 \sqrt{b x+c x^2}}{105 b^3 x^3}+\frac{16 c \sqrt{b x+c x^2}}{63 b^2 x^4}-\frac{2 \sqrt{b x+c x^2}}{9 b x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*Sqrt[b*x + c*x^2])/(9*b*x^5) + (16*c*Sqrt[b*x + c*x^2])/(63*b^2*x^4) - (32*c
^2*Sqrt[b*x + c*x^2])/(105*b^3*x^3) + (128*c^3*Sqrt[b*x + c*x^2])/(315*b^4*x^2)
- (256*c^4*Sqrt[b*x + c*x^2])/(315*b^5*x)

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Rubi in Sympy [A]  time = 18.9127, size = 117, normalized size = 0.93 \[ - \frac{2 \sqrt{b x + c x^{2}}}{9 b x^{5}} + \frac{16 c \sqrt{b x + c x^{2}}}{63 b^{2} x^{4}} - \frac{32 c^{2} \sqrt{b x + c x^{2}}}{105 b^{3} x^{3}} + \frac{128 c^{3} \sqrt{b x + c x^{2}}}{315 b^{4} x^{2}} - \frac{256 c^{4} \sqrt{b x + c x^{2}}}{315 b^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*x + c*x**2)/(9*b*x**5) + 16*c*sqrt(b*x + c*x**2)/(63*b**2*x**4) - 32*c
**2*sqrt(b*x + c*x**2)/(105*b**3*x**3) + 128*c**3*sqrt(b*x + c*x**2)/(315*b**4*x
**2) - 256*c**4*sqrt(b*x + c*x**2)/(315*b**5*x)

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Mathematica [A]  time = 0.0414826, size = 62, normalized size = 0.49 \[ -\frac{2 \sqrt{x (b+c x)} \left (35 b^4-40 b^3 c x+48 b^2 c^2 x^2-64 b c^3 x^3+128 c^4 x^4\right )}{315 b^5 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*Sqrt[x*(b + c*x)]*(35*b^4 - 40*b^3*c*x + 48*b^2*c^2*x^2 - 64*b*c^3*x^3 + 128
*c^4*x^4))/(315*b^5*x^5)

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Maple [A]  time = 0.007, size = 66, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 128\,{c}^{4}{x}^{4}-64\,{x}^{3}{c}^{3}b+48\,{c}^{2}{x}^{2}{b}^{2}-40\,cx{b}^{3}+35\,{b}^{4} \right ) }{315\,{x}^{4}{b}^{5}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(c*x^2+b*x)^(1/2),x)

[Out]

-2/315*(c*x+b)*(128*c^4*x^4-64*b*c^3*x^3+48*b^2*c^2*x^2-40*b^3*c*x+35*b^4)/x^4/b
^5/(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^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221522, size = 81, normalized size = 0.64 \[ -\frac{2 \,{\left (128 \, c^{4} x^{4} - 64 \, b c^{3} x^{3} + 48 \, b^{2} c^{2} x^{2} - 40 \, b^{3} c x + 35 \, b^{4}\right )} \sqrt{c x^{2} + b x}}{315 \, b^{5} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*x^5),x, algorithm="fricas")

[Out]

-2/315*(128*c^4*x^4 - 64*b*c^3*x^3 + 48*b^2*c^2*x^2 - 40*b^3*c*x + 35*b^4)*sqrt(
c*x^2 + b*x)/(b^5*x^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{5} \sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**5*sqrt(x*(b + c*x))), x)

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GIAC/XCAS [A]  time = 0.218695, size = 184, normalized size = 1.46 \[ \frac{2 \,{\left (1008 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} c^{2} + 1680 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b c^{\frac{3}{2}} + 1080 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{2} c + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{3} \sqrt{c} + 35 \, b^{4}\right )}}{315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*x^5),x, algorithm="giac")

[Out]

2/315*(1008*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*c^2 + 1680*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^3*b*c^(3/2) + 1080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*c + 315*(sqrt(
c)*x - sqrt(c*x^2 + b*x))*b^3*sqrt(c) + 35*b^4)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^
9