3.2483 \(\int \frac{\sqrt{a+b x^n}}{x^2} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\left (a+b x^n\right )^{3/2} \, _2F_1\left (1,\frac{3}{2}-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x} \]

[Out]

-(((a + b*x^n)^(3/2)*Hypergeometric2F1[1, 3/2 - n^(-1), -((1 - n)/n), -((b*x^n)/
a)])/(a*x))

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Rubi [A]  time = 0.0673001, antiderivative size = 58, 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 \[ -\frac{\sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{x \sqrt{\frac{b x^n}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^n]/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.71517, size = 44, normalized size = 0.9 \[ - \frac{\sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{x \sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0712452, size = 73, normalized size = 1.49 \[ \frac{2 \left (a+b x^n\right )-a n \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )}{(n-2) x \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^n]/x^2,x]

[Out]

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

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}\sqrt{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^(1/2)/x^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{n} + a}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n + a)/x^2,x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^n + a)/x^2, 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(sqrt(b*x^n + a)/x^2,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((a+b*x**n)**(1/2)/x**2,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{n} + a}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x^n + a)/x^2, x)