3.50 \(\int \frac{1}{x^4 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=100 \[ \frac{32 c^3 \sqrt{b x+c x^2}}{35 b^4 x}-\frac{16 c^2 \sqrt{b x+c x^2}}{35 b^3 x^2}+\frac{12 c \sqrt{b x+c x^2}}{35 b^2 x^3}-\frac{2 \sqrt{b x+c x^2}}{7 b x^4} \]

[Out]

(-2*Sqrt[b*x + c*x^2])/(7*b*x^4) + (12*c*Sqrt[b*x + c*x^2])/(35*b^2*x^3) - (16*c
^2*Sqrt[b*x + c*x^2])/(35*b^3*x^2) + (32*c^3*Sqrt[b*x + c*x^2])/(35*b^4*x)

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Rubi [A]  time = 0.133149, 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 \[ \frac{32 c^3 \sqrt{b x+c x^2}}{35 b^4 x}-\frac{16 c^2 \sqrt{b x+c x^2}}{35 b^3 x^2}+\frac{12 c \sqrt{b x+c x^2}}{35 b^2 x^3}-\frac{2 \sqrt{b x+c x^2}}{7 b x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*Sqrt[b*x + c*x^2])/(7*b*x^4) + (12*c*Sqrt[b*x + c*x^2])/(35*b^2*x^3) - (16*c
^2*Sqrt[b*x + c*x^2])/(35*b^3*x^2) + (32*c^3*Sqrt[b*x + c*x^2])/(35*b^4*x)

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Rubi in Sympy [A]  time = 14.0888, size = 92, normalized size = 0.92 \[ - \frac{2 \sqrt{b x + c x^{2}}}{7 b x^{4}} + \frac{12 c \sqrt{b x + c x^{2}}}{35 b^{2} x^{3}} - \frac{16 c^{2} \sqrt{b x + c x^{2}}}{35 b^{3} x^{2}} + \frac{32 c^{3} \sqrt{b x + c x^{2}}}{35 b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*x + c*x**2)/(7*b*x**4) + 12*c*sqrt(b*x + c*x**2)/(35*b**2*x**3) - 16*c
**2*sqrt(b*x + c*x**2)/(35*b**3*x**2) + 32*c**3*sqrt(b*x + c*x**2)/(35*b**4*x)

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Mathematica [A]  time = 0.0374799, size = 51, normalized size = 0.51 \[ \frac{2 \sqrt{x (b+c x)} \left (-5 b^3+6 b^2 c x-8 b c^2 x^2+16 c^3 x^3\right )}{35 b^4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[b*x + c*x^2]),x]

[Out]

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

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Maple [A]  time = 0.008, size = 55, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -16\,{x}^{3}{c}^{3}+8\,b{x}^{2}{c}^{2}-6\,{b}^{2}xc+5\,{b}^{3} \right ) }{35\,{x}^{3}{b}^{4}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(c*x^2+b*x)^(1/2),x)

[Out]

-2/35*(c*x+b)*(-16*c^3*x^3+8*b*c^2*x^2-6*b^2*c*x+5*b^3)/x^3/b^4/(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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.216808, size = 66, normalized size = 0.66 \[ \frac{2 \,{\left (16 \, c^{3} x^{3} - 8 \, b c^{2} x^{2} + 6 \, b^{2} c x - 5 \, b^{3}\right )} \sqrt{c x^{2} + b x}}{35 \, b^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**4*sqrt(x*(b + c*x))), x)

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GIAC/XCAS [A]  time = 0.219524, size = 144, normalized size = 1.44 \[ \frac{2 \,{\left (70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} c^{\frac{3}{2}} + 84 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b c + 35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{2} \sqrt{c} + 5 \, b^{3}\right )}}{35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*x^4),x, algorithm="giac")

[Out]

2/35*(70*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*c^(3/2) + 84*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^2*b*c + 35*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^2*sqrt(c) + 5*b^3)/(sqrt(c)
*x - sqrt(c*x^2 + b*x))^7