3.538 \(\int \frac{x}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx\)

Optimal. Leaf size=64 \[ \frac{x^2 \left (a+b x^n\right ) \, _2F_1\left (3,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

[Out]

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

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Rubi [A]  time = 0.0596573, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x^2 \left (a+b x^n\right ) \, _2F_1\left (3,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.1949, size = 58, normalized size = 0.91 \[ \frac{b x^{2} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{3} \left (a b + b^{2} x^{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**2*sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n))*hyper((3, 2/n), ((n + 2)/n,), -b*
x**n/a)/(2*a**3*(a*b + b**2*x**n))

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Mathematica [A]  time = 0.0936577, size = 93, normalized size = 1.45 \[ \frac{x^2 \left (a+b x^n\right ) \left (a^2 n+\left (n^2-3 n+2\right ) \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )+2 a (n-1) \left (a+b x^n\right )\right )}{2 a^3 n^2 \left (\left (a+b x^n\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]

[Out]

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

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int{x \left ({a}^{2}+2\,ab{x}^{n}+{b}^{2}{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a^2+2*a*b*x^n+b^2*x^(2*n))^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[{\left (n^{2} - 3 \, n + 2\right )} \int \frac{x}{a^{2} b n^{2} x^{n} + a^{3} n^{2}}\,{d x} + \frac{2 \, b{\left (n - 1\right )} x^{2} x^{n} + a{\left (3 \, n - 2\right )} x^{2}}{2 \,{\left (a^{2} b^{2} n^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{n} + a^{4} n^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2),x, algorithm="maxima")

[Out]

(n^2 - 3*n + 2)*integrate(x/(a^2*b*n^2*x^n + a^3*n^2), x) + 1/2*(2*b*(n - 1)*x^2
*x^n + a*(3*n - 2)*x^2)/(a^2*b^2*n^2*x^(2*n) + 2*a^3*b*n^2*x^n + a^4*n^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (\left (a + b x^{n}\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/((a + b*x**n)**2)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2),x, algorithm="giac")

[Out]

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