3.542 \(\int \frac{1}{x^3 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{\left (a+b x^n\right ) \, _2F_1\left (3,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^3 x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

[Out]

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

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Rubi [A]  time = 0.074149, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{\left (a+b x^n\right ) \, _2F_1\left (3,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^3 x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.2043, size = 61, normalized size = 0.91 \[ - \frac{b \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}{{}_{2}F_{1}\left (\begin{matrix} 3, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{3} x^{2} \left (a b + b^{2} x^{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*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*x**2*(a*b + b**2*x**n))

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Mathematica [A]  time = 0.110042, size = 94, normalized size = 1.4 \[ \frac{\left (a+b x^n\right ) \left (a^2 n-\left (n^2+3 n+2\right ) \left (a+b x^n\right )^2 \, _2F_1\left (1,-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )+2 a (n+1) \left (a+b x^n\right )\right )}{2 a^3 n^2 x^2 \left (\left (a+b x^n\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((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*Hy
pergeometric2F1[1, -2/n, (-2 + n)/n, -((b*x^n)/a)]))/(2*a^3*n^2*x^2*((a + b*x^n)
^2)^(3/2))

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ({a}^{2}+2\,ab{x}^{n}+{b}^{2}{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/x^3/(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. \[{\left (n^{2} + 3 \, n + 2\right )} \int \frac{1}{a^{2} b n^{2} x^{3} x^{n} + a^{3} n^{2} x^{3}}\,{d x} + \frac{2 \, b{\left (n + 1\right )} x^{n} + a{\left (3 \, n + 2\right )}}{2 \,{\left (a^{2} b^{2} n^{2} x^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{2} x^{n} + a^{4} n^{2} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{3} x^{2 \, n} + 2 \, a b x^{3} x^{n} + a^{2} x^{3}\right )} \sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)*x^3),x, algorithm="giac")

[Out]

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