3.25.82 \(\int \frac {x}{(a+b x^n)^3} \, dx\) [2482]
Optimal. Leaf size=33 \[ \frac {x^2 \, _2F_1\left (3,\frac {2}{n};\frac {2+n}{n};-\frac {b x^n}{a}\right )}{2 a^3} \]
[Out]
1/2*x^2*hypergeom([3, 2/n],[(2+n)/n],-b*x^n/a)/a^3
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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 = {371}
\begin {gather*} \frac {x^2 \, _2F_1\left (3,\frac {2}{n};\frac {n+2}{n};-\frac {b x^n}{a}\right )}{2 a^3} \end {gather*}
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 371
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], 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*}
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Mathematica [A]
time = 0.02, size = 33, normalized size = 1.00 \begin {gather*} \frac {x^2 \, _2F_1\left (3,\frac {2}{n};1+\frac {2}{n};-\frac {b x^n}{a}\right )}{2 a^3} \end {gather*}
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 = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (a +b \,x^{n}\right )^{3}}\, dx\]
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
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 = 0.39, size = 39, normalized size = 1.18 \begin {gather*} {\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 {gather*}
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] Result contains complex when optimal does not.
time = 0.77, size = 1930, normalized size = 58.48 \begin {gather*} \text {Too large to display} \end {gather*}
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x}{{\left (a+b\,x^n\right )}^3} \,d x \end {gather*}
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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