3.2486 \(\int \frac{1}{x^3 (a+b x^n)^3} \, dx\)

Optimal. Leaf size=36 \[ -\frac{\, _2F_1\left (3,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^3 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0070326, 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, Rules used = {364} \[ -\frac{\, _2F_1\left (3,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^n)^3),x]

[Out]

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

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

Mathematica [A]  time = 0.0031756, size = 33, normalized size = 0.92 \[ -\frac{\, _2F_1\left (3,-\frac{2}{n};1-\frac{2}{n};-\frac{b x^n}{a}\right )}{2 a^3 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x^n)^3),x]

[Out]

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

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( a+b{x}^{n} \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.77602, size = 2207, normalized size = 61.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*n**2*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**4*n**4*x**2*gamma(1 - 2/n
) + 3*a**3*b*n**4*x**2*x**n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*
x**(3*n)*gamma(1 - 2/n)) - 3*a*n**2*gamma(-2/n)/(a**4*n**4*x**2*gamma(1 - 2/n) + 3*a**3*b*n**4*x**2*x**n*gamma
(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*x**(3*n)*gamma(1 - 2/n)) - 6*a*n*
lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**4*n**4*x**2*gamma(1 - 2/n) + 3*a**3
*b*n**4*x**2*x**n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*x**(3*n)*g
amma(1 - 2/n)) - 2*a*n*gamma(-2/n)/(a**4*n**4*x**2*gamma(1 - 2/n) + 3*a**3*b*n**4*x**2*x**n*gamma(1 - 2/n) + 3
*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*x**(3*n)*gamma(1 - 2/n)) - 4*a*lerchphi(b*x**n
*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**4*n**4*x**2*gamma(1 - 2/n) + 3*a**3*b*n**4*x**2*x*
*n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*x**(3*n)*gamma(1 - 2/n))
- 6*b*n**2*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**4*n**4*x**2*gamma(1
- 2/n) + 3*a**3*b*n**4*x**2*x**n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4
*x**2*x**(3*n)*gamma(1 - 2/n)) - 5*b*n**2*x**n*gamma(-2/n)/(a**4*n**4*x**2*gamma(1 - 2/n) + 3*a**3*b*n**4*x**2
*x**n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*x**(3*n)*gamma(1 - 2/n
)) - 18*b*n*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**4*n**4*x**2*gamma(
1 - 2/n) + 3*a**3*b*n**4*x**2*x**n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**
4*x**2*x**(3*n)*gamma(1 - 2/n)) - 4*b*n*x**n*gamma(-2/n)/(a**4*n**4*x**2*gamma(1 - 2/n) + 3*a**3*b*n**4*x**2*x
**n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*x**(3*n)*gamma(1 - 2/n))
- 12*b*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**4*n**4*x**2*gamma(1 -
2/n) + 3*a**3*b*n**4*x**2*x**n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x*
*2*x**(3*n)*gamma(1 - 2/n)) - 6*b**2*n**2*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*
gamma(-2/n)/(a*(a**4*n**4*x**2*gamma(1 - 2/n) + 3*a**3*b*n**4*x**2*x**n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2
*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*x**(3*n)*gamma(1 - 2/n))) - 2*b**2*n**2*x**(2*n)*gamma(-2/n)/(a*(a
**4*n**4*x**2*gamma(1 - 2/n) + 3*a**3*b*n**4*x**2*x**n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1
- 2/n) + a*b**3*n**4*x**2*x**(3*n)*gamma(1 - 2/n))) - 18*b**2*n*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1
, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a*(a**4*n**4*x**2*gamma(1 - 2/n) + 3*a**3*b*n**4*x**2*x**n*gamma(1 - 2/n)
+ 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*x**(3*n)*gamma(1 - 2/n))) - 2*b**2*n*x**(2*
n)*gamma(-2/n)/(a*(a**4*n**4*x**2*gamma(1 - 2/n) + 3*a**3*b*n**4*x**2*x**n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x
**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*x**(3*n)*gamma(1 - 2/n))) - 12*b**2*x**(2*n)*lerchphi(b*x**n*ex
p_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a*(a**4*n**4*x**2*gamma(1 - 2/n) + 3*a**3*b*n**4*x**2*x*
*n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*x**(3*n)*gamma(1 - 2/n)))
- 2*b**3*n**2*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**2*(a**4*n**
4*x**2*gamma(1 - 2/n) + 3*a**3*b*n**4*x**2*x**n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n)
+ a*b**3*n**4*x**2*x**(3*n)*gamma(1 - 2/n))) - 6*b**3*n*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_
polar(I*pi)/n)*gamma(-2/n)/(a**2*(a**4*n**4*x**2*gamma(1 - 2/n) + 3*a**3*b*n**4*x**2*x**n*gamma(1 - 2/n) + 3*a
**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*x**(3*n)*gamma(1 - 2/n))) - 4*b**3*x**(3*n)*lerc
hphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**2*(a**4*n**4*x**2*gamma(1 - 2/n) + 3*a*
*3*b*n**4*x**2*x**n*gamma(1 - 2/n) + 3*a**2*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n) + a*b**3*n**4*x**2*x**(3*n)
*gamma(1 - 2/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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x^n + a)^3*x^3), x)