3.2478 \(\int \frac{1}{x^2 \left (a+b x^n\right )^3} \, dx\)

Optimal. Leaf size=34 \[ -\frac{\, _2F_1\left (3,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^3 x} \]

[Out]

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

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Rubi [A]  time = 0.029681, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\, _2F_1\left (3,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^3 x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x^n)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.66769, size = 22, normalized size = 0.65 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} 3, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-hyper((3, -1/n), ((n - 1)/n,), -b*x**n/a)/(a**3*x)

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Mathematica [B]  time = 0.0996309, size = 76, normalized size = 2.24 \[ \frac{\frac{a \left (3 a n+a+b (2 n+1) x^n\right )}{\left (a+b x^n\right )^2}-\left (2 n^2+3 n+1\right ) \, _2F_1\left (1,-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )}{2 a^3 n^2 x} \]

Antiderivative was successfully verified.

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

[Out]

((a*(a + 3*a*n + b*(1 + 2*n)*x^n))/(a + b*x^n)^2 - (1 + 3*n + 2*n^2)*Hypergeomet
ric2F1[1, -n^(-1), (-1 + n)/n, -((b*x^n)/a)])/(2*a^3*n^2*x)

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Maple [F]  time = 0.094, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( a+b{x}^{n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b*x^n)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + a)^3*x^2), x)