3.2483 \(\int \frac{1}{(a+b x^n)^3} \, dx\)
Optimal. Leaf size=24 \[ \frac{x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3} \]
[Out]
(x*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^3
________________________________________________________________________________________
Rubi [A] time = 0.0045994, antiderivative size = 24, normalized size of antiderivative = 1.,
number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used =
{245} \[ \frac{x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x^n)^(-3),x]
[Out]
(x*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^3
Rule 245
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p, 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^n\right )^3} \, dx &=\frac{x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0019963, size = 24, normalized size = 1. \[ \frac{x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x^n)^(-3),x]
[Out]
(x*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^3
________________________________________________________________________________________
Maple [F] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{-3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*x^n)^3,x)
[Out]
int(1/(a+b*x^n)^3,x)
________________________________________________________________________________________
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^{n} + a^{3} n^{2}\right )}}\,{d x} + \frac{b{\left (2 \, n - 1\right )} x x^{n} + a{\left (3 \, n - 1\right )} x}{2 \,{\left (a^{2} b^{2} n^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{n} + a^{4} n^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*x^n)^3,x, algorithm="maxima")
[Out]
(2*n^2 - 3*n + 1)*integrate(1/2/(a^2*b*n^2*x^n + a^3*n^2), x) + 1/2*(b*(2*n - 1)*x*x^n + a*(3*n - 1)*x)/(a^2*b
^2*n^2*x^(2*n) + 2*a^3*b*n^2*x^n + a^4*n^2)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*x^n)^3,x, algorithm="fricas")
[Out]
integral(1/(b^3*x^(3*n) + 3*a*b^2*x^(2*n) + 3*a^2*b*x^n + a^3), x)
________________________________________________________________________________________
Sympy [C] time = 1.89139, size = 1953, normalized size = 81.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*x**n)**3,x)
[Out]
2*a*n**2*x*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x
**n*gamma(1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n)) + 3*a*n
**2*x*gamma(1/n)/(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*g
amma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n)) - 3*a*n*x*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*ga
mma(1/n)/(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(1 +
1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n)) - a*n*x*gamma(1/n)/(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*
x**n*gamma(1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n)) + a*x*
lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1
+ 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n)) + 6*b*n**2*x*x**n*
lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1
+ 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n)) + 5*b*n**2*x*x**n*
gamma(1/n)/(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(1
+ 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n)) - 9*b*n*x*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gam
ma(1/n)/(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(1 +
1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n)) - 2*b*n*x*x**n*gamma(1/n)/(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b
*n**4*x**n*gamma(1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n))
+ 3*b*x*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4
*x**n*gamma(1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n)) + 6*b
**2*n**2*x*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a*(2*a**4*n**4*gamma(1 + 1/n) + 6*a
**3*b*n**4*x**n*gamma(1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1
/n))) + 2*b**2*n**2*x*x**(2*n)*gamma(1/n)/(a*(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1 + 1/n) +
6*a**2*b**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n))) - 9*b**2*n*x*x**(2*n)*lerc
hphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a*(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1
+ 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n))) - b**2*n*x*x**(2*n
)*gamma(1/n)/(a*(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*ga
mma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n))) + 3*b**2*x*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a,
1, 1/n)*gamma(1/n)/(a*(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1 + 1/n) + 6*a**2*b**2*n**4*x**(2
*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n))) + 2*b**3*n**2*x*x**(3*n)*lerchphi(b*x**n*exp_pola
r(I*pi)/a, 1, 1/n)*gamma(1/n)/(a**2*(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1 + 1/n) + 6*a**2*b
**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n))) - 3*b**3*n*x*x**(3*n)*lerchphi(b*x*
*n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a**2*(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1 + 1/n)
+ 6*a**2*b**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n))) + b**3*x*x**(3*n)*lerchp
hi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a**2*(2*a**4*n**4*gamma(1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(1
+ 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(1 + 1/n)))
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*x^n)^3,x, algorithm="giac")
[Out]
integrate((b*x^n + a)^(-3), x)