3.2485 \(\int \frac{1}{x^2 (a+b x^n)^3} \, dx\)

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 = 0.0071909, 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, Rules used = {364} \[ -\frac{\, _2F_1\left (3,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^3 x} \]

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))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a+b x^n\right )^3} \, dx &=-\frac{\, _2F_1\left (3,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^3 x}\\ \end{align*}

Mathematica [A]  time = 0.0033527, size = 31, normalized size = 0.91 \[ -\frac{\, _2F_1\left (3,-\frac{1}{n};1-\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^n)^3),x]

[Out]

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

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b{x}^{n} \right ) ^{3}}}\, dx \end{align*}

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. \begin{align*}{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^n)^3,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. \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^n)^3,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 [C]  time = 2.11638, size = 2118, normalized size = 62.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*n**2*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n)*gamma(-1/n)/(2*a**4*n**4*x*gamma(1 - 1/n) +
6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n**4*x*x**(2*n)*gamma(1 - 1/n) + 2*a*b**3*n**4*x*x**(3*n)*g
amma(1 - 1/n)) - 3*a*n**2*gamma(-1/n)/(2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*
a**2*b**2*n**4*x*x**(2*n)*gamma(1 - 1/n) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n)) - 3*a*n*lerchphi(b*x**n*ex
p_polar(I*pi)/a, 1, exp_polar(I*pi)/n)*gamma(-1/n)/(2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(
1 - 1/n) + 6*a**2*b**2*n**4*x*x**(2*n)*gamma(1 - 1/n) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n)) - a*n*gamma(-
1/n)/(2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n**4*x*x**(2*n)*gamma(1
- 1/n) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n)) - a*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n
)*gamma(-1/n)/(2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n**4*x*x**(2*n
)*gamma(1 - 1/n) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n)) - 6*b*n**2*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a,
1, exp_polar(I*pi)/n)*gamma(-1/n)/(2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**
2*b**2*n**4*x*x**(2*n)*gamma(1 - 1/n) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n)) - 5*b*n**2*x**n*gamma(-1/n)/(
2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n**4*x*x**(2*n)*gamma(1 - 1/n
) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n)) - 9*b*n*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi
)/n)*gamma(-1/n)/(2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n**4*x*x**(
2*n)*gamma(1 - 1/n) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n)) - 2*b*n*x**n*gamma(-1/n)/(2*a**4*n**4*x*gamma(1
- 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n**4*x*x**(2*n)*gamma(1 - 1/n) + 2*a*b**3*n**4*x*x
**(3*n)*gamma(1 - 1/n)) - 3*b*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n)*gamma(-1/n)/(2*a**
4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n**4*x*x**(2*n)*gamma(1 - 1/n) + 2
*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n)) - 6*b**2*n**2*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar
(I*pi)/n)*gamma(-1/n)/(a*(2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n**
4*x*x**(2*n)*gamma(1 - 1/n) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n))) - 2*b**2*n**2*x**(2*n)*gamma(-1/n)/(a*
(2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n**4*x*x**(2*n)*gamma(1 - 1/
n) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n))) - 9*b**2*n*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_p
olar(I*pi)/n)*gamma(-1/n)/(a*(2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2
*n**4*x*x**(2*n)*gamma(1 - 1/n) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n))) - b**2*n*x**(2*n)*gamma(-1/n)/(a*(
2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n**4*x*x**(2*n)*gamma(1 - 1/n
) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n))) - 3*b**2*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_pola
r(I*pi)/n)*gamma(-1/n)/(a*(2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n*
*4*x*x**(2*n)*gamma(1 - 1/n) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n))) - 2*b**3*n**2*x**(3*n)*lerchphi(b*x**
n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n)*gamma(-1/n)/(a**2*(2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x
**n*gamma(1 - 1/n) + 6*a**2*b**2*n**4*x*x**(2*n)*gamma(1 - 1/n) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n))) -
3*b**3*n*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n)*gamma(-1/n)/(a**2*(2*a**4*n**4*x*ga
mma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n**4*x*x**(2*n)*gamma(1 - 1/n) + 2*a*b**3*n**
4*x*x**(3*n)*gamma(1 - 1/n))) - b**3*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n)*gamma(-
1/n)/(a**2*(2*a**4*n**4*x*gamma(1 - 1/n) + 6*a**3*b*n**4*x*x**n*gamma(1 - 1/n) + 6*a**2*b**2*n**4*x*x**(2*n)*g
amma(1 - 1/n) + 2*a*b**3*n**4*x*x**(3*n)*gamma(1 - 1/n)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x^n + a)^3*x^2), x)