3.95 \(\int \frac{1}{x^5 \sqrt{a x+b x^4}} \, dx\)

Optimal. Leaf size=48 \[ \frac{4 b \sqrt{a x+b x^4}}{9 a^2 x^2}-\frac{2 \sqrt{a x+b x^4}}{9 a x^5} \]

[Out]

(-2*Sqrt[a*x + b*x^4])/(9*a*x^5) + (4*b*Sqrt[a*x + b*x^4])/(9*a^2*x^2)

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Rubi [A]  time = 0.117729, 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 \[ \frac{4 b \sqrt{a x+b x^4}}{9 a^2 x^2}-\frac{2 \sqrt{a x+b x^4}}{9 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[a*x + b*x^4]),x]

[Out]

(-2*Sqrt[a*x + b*x^4])/(9*a*x^5) + (4*b*Sqrt[a*x + b*x^4])/(9*a^2*x^2)

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Rubi in Sympy [A]  time = 12.2708, size = 42, normalized size = 0.88 \[ - \frac{2 \sqrt{a x + b x^{4}}}{9 a x^{5}} + \frac{4 b \sqrt{a x + b x^{4}}}{9 a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x**4+a*x)**(1/2),x)

[Out]

-2*sqrt(a*x + b*x**4)/(9*a*x**5) + 4*b*sqrt(a*x + b*x**4)/(9*a**2*x**2)

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Mathematica [A]  time = 0.0349873, size = 31, normalized size = 0.65 \[ -\frac{2 \left (a-2 b x^3\right ) \sqrt{x \left (a+b x^3\right )}}{9 a^2 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*Sqrt[a*x + b*x^4]),x]

[Out]

(-2*(a - 2*b*x^3)*Sqrt[x*(a + b*x^3)])/(9*a^2*x^5)

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Maple [A]  time = 0.007, size = 35, normalized size = 0.7 \[ -{\frac{ \left ( 2\,b{x}^{3}+2\,a \right ) \left ( -2\,b{x}^{3}+a \right ) }{9\,{a}^{2}{x}^{4}}{\frac{1}{\sqrt{b{x}^{4}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^4+a*x)^(1/2),x)

[Out]

-2/9*(b*x^3+a)*(-2*b*x^3+a)/x^4/a^2/(b*x^4+a*x)^(1/2)

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Maxima [A]  time = 1.41109, size = 51, normalized size = 1.06 \[ \frac{2 \,{\left (2 \, b^{2} x^{7} + a b x^{4} - a^{2} x\right )}}{9 \, \sqrt{b x^{3} + a} a^{2} x^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a*x)*x^5),x, algorithm="maxima")

[Out]

2/9*(2*b^2*x^7 + a*b*x^4 - a^2*x)/(sqrt(b*x^3 + a)*a^2*x^(11/2))

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Fricas [A]  time = 0.215991, size = 39, normalized size = 0.81 \[ \frac{2 \, \sqrt{b x^{4} + a x}{\left (2 \, b x^{3} - a\right )}}{9 \, a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a*x)*x^5),x, algorithm="fricas")

[Out]

2/9*sqrt(b*x^4 + a*x)*(2*b*x^3 - a)/(a^2*x^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{5} \sqrt{x \left (a + b x^{3}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(b*x**4+a*x)**(1/2),x)

[Out]

Integral(1/(x**5*sqrt(x*(a + b*x**3))), x)

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GIAC/XCAS [A]  time = 0.223347, size = 36, normalized size = 0.75 \[ -\frac{2 \,{\left ({\left (b + \frac{a}{x^{3}}\right )}^{\frac{3}{2}} - 3 \, \sqrt{b + \frac{a}{x^{3}}} b\right )}}{9 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a*x)*x^5),x, algorithm="giac")

[Out]

-2/9*((b + a/x^3)^(3/2) - 3*sqrt(b + a/x^3)*b)/a^2