3.25.83 \(\int \frac {1}{(a+b x^n)^3} \, dx\) [2483]
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*hypergeom([3, 1/n],[1+1/n],-b*x^n/a)/a^3
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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {251}
\begin {gather*} \frac {x \, _2F_1\left (3,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^3} \end {gather*}
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 251
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*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} \frac {x \, _2F_1\left (3,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^3} \end {gather*}
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
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \,x^{n}\right )^{3}}\, dx\]
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)
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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(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)
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Fricas [F]
time = 0.39, size = 37, normalized size = 1.54 \begin {gather*} {\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 {gather*}
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)
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Sympy [C] Result contains complex when optimal does not.
time = 0.74, size = 1953, normalized size = 81.38 \begin {gather*} \text {Too large to display} \end {gather*}
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)))
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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(1/(a+b*x^n)^3,x, algorithm="giac")
[Out]
integrate((b*x^n + a)^(-3), x)
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Mupad [B]
time = 1.18, size = 25, normalized size = 1.04 \begin {gather*} \frac {x\,{{}}_2{\mathrm {F}}_1\left (3,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a + b*x^n)^3,x)
[Out]
(x*hypergeom([3, 1/n], 1/n + 1, -(b*x^n)/a))/a^3
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