3.2499 \(\int \frac{1}{x^3 \sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{\sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2}-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a x^2} \]

[Out]

-(Sqrt[a + b*x^n]*Hypergeometric2F1[1, 1/2 - 2/n, -((2 - n)/n), -((b*x^n)/a)])/(
2*a*x^2)

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Rubi [A]  time = 0.0635509, antiderivative size = 60, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 x^2 \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[a + b*x^n]),x]

[Out]

-(Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, -2/n, -((2 - n)/n), -((b*x^n)/a)])/
(2*x^2*Sqrt[a + b*x^n])

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Rubi in Sympy [A]  time = 7.0744, size = 48, normalized size = 0.94 \[ - \frac{\sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a x^{2} \sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(a+b*x**n)**(1/2),x)

[Out]

-sqrt(a + b*x**n)*hyper((1/2, -2/n), ((n - 2)/n,), -b*x**n/a)/(2*a*x**2*sqrt(1 +
b*x**n/a))

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Mathematica [A]  time = 0.0422867, size = 58, normalized size = 1.14 \[ -\frac{\sqrt{\frac{a+b x^n}{a}} \, _2F_1\left (\frac{1}{2},-\frac{2}{n};1-\frac{2}{n};-\frac{b x^n}{a}\right )}{2 x^2 \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[a + b*x^n]),x]

[Out]

-(Sqrt[(a + b*x^n)/a]*Hypergeometric2F1[1/2, -2/n, 1 - 2/n, -((b*x^n)/a)])/(2*x^
2*Sqrt[a + b*x^n])

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a+b*x^n)^(1/2),x)

[Out]

int(1/x^3/(a+b*x^n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{n} + a} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^n + a)*x^3),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^n + a)*x^3), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^n + a)*x^3),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(a+b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{n} + a} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^n + a)*x^3),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^n + a)*x^3), x)