3.290 \(\int \frac {a+b x^n}{(c+d x^n)^3} \, dx\)
Optimal. Leaf size=78 \[ \frac {x (b c-a d (1-2 n)) \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{2 c^3 d n}-\frac {x (b c-a d)}{2 c d n \left (c+d x^n\right )^2} \]
[Out]
-1/2*(-a*d+b*c)*x/c/d/n/(c+d*x^n)^2+1/2*(b*c-a*d*(1-2*n))*x*hypergeom([2, 1/n],[1+1/n],-d*x^n/c)/c^3/d/n
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Rubi [A] time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used =
{385, 245} \[ \frac {x (b c-a d (1-2 n)) \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{2 c^3 d n}-\frac {x (b c-a d)}{2 c d n \left (c+d x^n\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x^n)/(c + d*x^n)^3,x]
[Out]
-((b*c - a*d)*x)/(2*c*d*n*(c + d*x^n)^2) + ((b*c - a*d*(1 - 2*n))*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -
((d*x^n)/c)])/(2*c^3*d*n)
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])
Rule 385
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])
Rubi steps
\begin {align*} \int \frac {a+b x^n}{\left (c+d x^n\right )^3} \, dx &=-\frac {(b c-a d) x}{2 c d n \left (c+d x^n\right )^2}+\frac {(b c-a d (1-2 n)) \int \frac {1}{\left (c+d x^n\right )^2} \, dx}{2 c d n}\\ &=-\frac {(b c-a d) x}{2 c d n \left (c+d x^n\right )^2}+\frac {(b c-a d (1-2 n)) x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{2 c^3 d n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 58, normalized size = 0.74 \[ \frac {x \left (\frac {b}{\left (c+d x^n\right )^2}-\frac {(a d (2 n-1)+b c) \, _2F_1\left (3,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^3}\right )}{d-2 d n} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x^n)/(c + d*x^n)^3,x]
[Out]
(x*(b/(c + d*x^n)^2 - ((b*c + a*d*(-1 + 2*n))*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/c^3))/(d
- 2*d*n)
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{n} + a}{d^{3} x^{3 \, n} + 3 \, c d^{2} x^{2 \, n} + 3 \, c^{2} d x^{n} + c^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*x^n)/(c+d*x^n)^3,x, algorithm="fricas")
[Out]
integral((b*x^n + a)/(d^3*x^(3*n) + 3*c*d^2*x^(2*n) + 3*c^2*d*x^n + c^3), x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b x^{n} + a}{{\left (d x^{n} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*x^n)/(c+d*x^n)^3,x, algorithm="giac")
[Out]
integrate((b*x^n + a)/(d*x^n + c)^3, x)
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int \frac {b \,x^{n}+a}{\left (d \,x^{n}+c \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x^n+a)/(d*x^n+c)^3,x)
[Out]
int((b*x^n+a)/(d*x^n+c)^3,x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left ({\left (2 \, n^{2} - 3 \, n + 1\right )} a d + b c {\left (n - 1\right )}\right )} \int \frac {1}{2 \, {\left (c^{2} d^{2} n^{2} x^{n} + c^{3} d n^{2}\right )}}\,{d x} + \frac {{\left (a d^{2} {\left (2 \, n - 1\right )} + b c d\right )} x x^{n} + {\left (a c d {\left (3 \, n - 1\right )} - b c^{2} {\left (n - 1\right )}\right )} x}{2 \, {\left (c^{2} d^{3} n^{2} x^{2 \, n} + 2 \, c^{3} d^{2} n^{2} x^{n} + c^{4} d n^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*x^n)/(c+d*x^n)^3,x, algorithm="maxima")
[Out]
((2*n^2 - 3*n + 1)*a*d + b*c*(n - 1))*integrate(1/2/(c^2*d^2*n^2*x^n + c^3*d*n^2), x) + 1/2*((a*d^2*(2*n - 1)
+ b*c*d)*x*x^n + (a*c*d*(3*n - 1) - b*c^2*(n - 1))*x)/(c^2*d^3*n^2*x^(2*n) + 2*c^3*d^2*n^2*x^n + c^4*d*n^2)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,x^n}{{\left (c+d\,x^n\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x^n)/(c + d*x^n)^3,x)
[Out]
int((a + b*x^n)/(c + d*x^n)^3, x)
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sympy [C] time = 64.99, size = 3706, normalized size = 47.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*x**n)/(c+d*x**n)**3,x)
[Out]
a*(2*c*n**2*x*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*gamma(1/n)/(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**
4*x**n*gamma(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n)) + 3*
c*n**2*x*gamma(1/n)/(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamma(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n
)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n)) - 3*c*n*x*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)
*gamma(1/n)/(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamma(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(
1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n)) - c*n*x*gamma(1/n)/(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n*
*4*x**n*gamma(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n)) + c
*x*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*gamma(1/n)/(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamm
a(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n)) + 6*d*n**2*x*x*
*n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*gamma(1/n)/(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamm
a(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n)) + 5*d*n**2*x*x*
*n*gamma(1/n)/(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamma(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamm
a(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n)) - 9*d*n*x*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*
gamma(1/n)/(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamma(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(1
+ 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n)) - 2*d*n*x*x**n*gamma(1/n)/(2*c**4*n**4*gamma(1 + 1/n) + 6*c**
3*d*n**4*x**n*gamma(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n
)) + 3*d*x*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*gamma(1/n)/(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n
**4*x**n*gamma(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n)) +
6*d**2*n**2*x*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*gamma(1/n)/(c*(2*c**4*n**4*gamma(1 + 1/n) +
6*c**3*d*n**4*x**n*gamma(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1
+ 1/n))) + 2*d**2*n**2*x*x**(2*n)*gamma(1/n)/(c*(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamma(1 + 1/n
) + 6*c**2*d**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n))) - 9*d**2*n*x*x**(2*n)*l
erchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*gamma(1/n)/(c*(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamma
(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n))) - d**2*n*x*x**(
2*n)*gamma(1/n)/(c*(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamma(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)
*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n))) + 3*d**2*x*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/
c, 1, 1/n)*gamma(1/n)/(c*(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamma(1 + 1/n) + 6*c**2*d**2*n**4*x*
*(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n))) + 2*d**3*n**2*x*x**(3*n)*lerchphi(d*x**n*exp_p
olar(I*pi)/c, 1, 1/n)*gamma(1/n)/(c**2*(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamma(1 + 1/n) + 6*c**
2*d**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n))) - 3*d**3*n*x*x**(3*n)*lerchphi(d
*x**n*exp_polar(I*pi)/c, 1, 1/n)*gamma(1/n)/(c**2*(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamma(1 + 1
/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n))) + d**3*x*x**(3*n)*ler
chphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*gamma(1/n)/(c**2*(2*c**4*n**4*gamma(1 + 1/n) + 6*c**3*d*n**4*x**n*gamm
a(1 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(1 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(1 + 1/n)))) + b*(2*c*n**3
*x*x**n*gamma(1 + 1/n)/(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4*x**n*gamma(2 + 1/n) + 6*c**2*d**2*n**4*x**(
2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n)) - c*n**2*x*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c
, 1, 1 + 1/n)*gamma(1 + 1/n)/(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4*x**n*gamma(2 + 1/n) + 6*c**2*d**2*n**
4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n)) + c*n**2*x*x**n*gamma(1 + 1/n)/(2*c**4*n**4
*gamma(2 + 1/n) + 6*c**3*d*n**4*x**n*gamma(2 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4
*x**(3*n)*gamma(2 + 1/n)) - c*n*x*x**n*gamma(1 + 1/n)/(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4*x**n*gamma(2
+ 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n)) + c*x*x**n*lerchph
i(d*x**n*exp_polar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4*x**n*gamma(
2 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n)) + 3*d*n**3*x*x**(
2*n)*gamma(1 + 1/n)/(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4*x**n*gamma(2 + 1/n) + 6*c**2*d**2*n**4*x**(2*n
)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n)) - 3*d*n**2*x*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi
)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4*x**n*gamma(2 + 1/n) + 6*c**2*d**2*
n**4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n)) + d*n**2*x*x**(2*n)*gamma(1 + 1/n)/(2*c*
*4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4*x**n*gamma(2 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(2 + 1/n) + 2*c*d*
*3*n**4*x**(3*n)*gamma(2 + 1/n)) - 2*d*n*x*x**(2*n)*gamma(1 + 1/n)/(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4
*x**n*gamma(2 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n)) + 3*d
*x*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3
*d*n**4*x**n*gamma(2 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n)
) + d**2*n**3*x*x**(3*n)*gamma(1 + 1/n)/(c*(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4*x**n*gamma(2 + 1/n) + 6
*c**2*d**2*n**4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n))) - 3*d**2*n**2*x*x**(3*n)*ler
chphi(d*x**n*exp_polar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(c*(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4*x**n
*gamma(2 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n))) - d**2*n*
x*x**(3*n)*gamma(1 + 1/n)/(c*(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4*x**n*gamma(2 + 1/n) + 6*c**2*d**2*n**
4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n))) + 3*d**2*x*x**(3*n)*lerchphi(d*x**n*exp_po
lar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(c*(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4*x**n*gamma(2 + 1/n) + 6
*c**2*d**2*n**4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n))) - d**3*n**2*x*x**(4*n)*lerch
phi(d*x**n*exp_polar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(c**2*(2*c**4*n**4*gamma(2 + 1/n) + 6*c**3*d*n**4*x**
n*gamma(2 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 + 1/n))) + d**3*x
*x**(4*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(c**2*(2*c**4*n**4*gamma(2 + 1/n) + 6*
c**3*d*n**4*x**n*gamma(2 + 1/n) + 6*c**2*d**2*n**4*x**(2*n)*gamma(2 + 1/n) + 2*c*d**3*n**4*x**(3*n)*gamma(2 +
1/n))))
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