3.532 \(\int \frac{1}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \, dx\)

Optimal. Leaf size=55 \[ \frac{x \left (a+b x^n\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

[Out]

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

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Rubi [A]  time = 0.0383224, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{x \left (a+b x^n\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.94657, size = 53, normalized size = 0.96 \[ \frac{b x \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \left (a b + b^{2} x^{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0202204, size = 44, normalized size = 0.8 \[ \frac{x \left (a+b x^n\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a \sqrt{\left (a+b x^n\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt{{a}^{2}+2\,ab{x}^{n}+{b}^{2}{x}^{2\,n}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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