3.2474 \(\int \frac{1}{x^3 \left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=36 \[ -\frac{\, _2F_1\left (2,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.030105, antiderivative size = 36, 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 \[ -\frac{\, _2F_1\left (2,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^n)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.78401, size = 26, normalized size = 0.72 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} 2, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0502815, size = 61, normalized size = 1.69 \[ \frac{2 a-(n+2) \left (a+b x^n\right ) \, _2F_1\left (1,-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )}{2 a^2 n x^2 \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a+b*x^n)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + a)^2*x^3), x)