3.2482 \(\int \frac{x}{(a+b x^n)^3} \, dx\)

Optimal. Leaf size=33 \[ \frac{x^2 \, _2F_1\left (3,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a^3} \]

[Out]

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

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Rubi [A]  time = 0.0060812, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {364} \[ \frac{x^2 \, _2F_1\left (3,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^n)^3,x]

[Out]

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

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

Mathematica [A]  time = 0.0033113, size = 33, normalized size = 1. \[ \frac{x^2 \, _2F_1\left (3,\frac{2}{n};1+\frac{2}{n};-\frac{b x^n}{a}\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^n)^3,x]

[Out]

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

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ( a+b{x}^{n} \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x/(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{x}{a^{2} b n^{2} x^{n} + a^{3} n^{2}}\,{d x} + \frac{2 \, b{\left (n - 1\right )} x^{2} x^{n} + a{\left (3 \, n - 2\right )} x^{2}}{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(x/(a+b*x^n)^3,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{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(x/(a+b*x^n)^3,x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 1.6393, size = 1930, normalized size = 58.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*n**2*x**2*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*
x**n*gamma(1 + 2/n) + 3*a**2*b**2*n**4*x**(2*n)*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n)) + 3*a*n*
*2*x**2*gamma(2/n)/(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(1 + 2/n) + 3*a**2*b**2*n**4*x**(2*n)*g
amma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n)) - 6*a*n*x**2*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*g
amma(2/n)/(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(1 + 2/n) + 3*a**2*b**2*n**4*x**(2*n)*gamma(1 +
2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n)) - 2*a*n*x**2*gamma(2/n)/(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*
x**n*gamma(1 + 2/n) + 3*a**2*b**2*n**4*x**(2*n)*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n)) + 4*a*x*
*2*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(
1 + 2/n) + 3*a**2*b**2*n**4*x**(2*n)*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n)) + 6*b*n**2*x**2*x**
n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(1
+ 2/n) + 3*a**2*b**2*n**4*x**(2*n)*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n)) + 5*b*n**2*x**2*x**n
*gamma(2/n)/(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(1 + 2/n) + 3*a**2*b**2*n**4*x**(2*n)*gamma(1
+ 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n)) - 18*b*n*x**2*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*ga
mma(2/n)/(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(1 + 2/n) + 3*a**2*b**2*n**4*x**(2*n)*gamma(1 + 2
/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n)) - 4*b*n*x**2*x**n*gamma(2/n)/(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n
**4*x**n*gamma(1 + 2/n) + 3*a**2*b**2*n**4*x**(2*n)*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n)) + 12
*b*x**2*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x
**n*gamma(1 + 2/n) + 3*a**2*b**2*n**4*x**(2*n)*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n)) + 6*b**2*
n**2*x**2*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a*(a**4*n**4*gamma(1 + 2/n) + 3*a**3
*b*n**4*x**n*gamma(1 + 2/n) + 3*a**2*b**2*n**4*x**(2*n)*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n)))
+ 2*b**2*n**2*x**2*x**(2*n)*gamma(2/n)/(a*(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(1 + 2/n) + 3*a
**2*b**2*n**4*x**(2*n)*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n))) - 18*b**2*n*x**2*x**(2*n)*lerchp
hi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a*(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(1 + 2/
n) + 3*a**2*b**2*n**4*x**(2*n)*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n))) - 2*b**2*n*x**2*x**(2*n)
*gamma(2/n)/(a*(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(1 + 2/n) + 3*a**2*b**2*n**4*x**(2*n)*gamma
(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n))) + 12*b**2*x**2*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1
, 2/n)*gamma(2/n)/(a*(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(1 + 2/n) + 3*a**2*b**2*n**4*x**(2*n)
*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n))) + 2*b**3*n**2*x**2*x**(3*n)*lerchphi(b*x**n*exp_polar(
I*pi)/a, 1, 2/n)*gamma(2/n)/(a**2*(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(1 + 2/n) + 3*a**2*b**2*
n**4*x**(2*n)*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n))) - 6*b**3*n*x**2*x**(3*n)*lerchphi(b*x**n*
exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a**2*(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(1 + 2/n) + 3*
a**2*b**2*n**4*x**(2*n)*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n))) + 4*b**3*x**2*x**(3*n)*lerchphi
(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a**2*(a**4*n**4*gamma(1 + 2/n) + 3*a**3*b*n**4*x**n*gamma(1 + 2
/n) + 3*a**2*b**2*n**4*x**(2*n)*gamma(1 + 2/n) + a*b**3*n**4*x**(3*n)*gamma(1 + 2/n)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(b*x^n + a)^3, x)