\(\int \frac {(d+e x^n)^3}{(a+c x^{2 n})^3} \, dx\) [37]

Optimal result

Integrand size = 21, antiderivative size = 259 \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^3} \, dx=\frac {x \left (d+e x^n\right )^3}{4 a n \left (a+c x^{2 n}\right )^2}-\frac {x \left (d \left (c d^2 (1-4 n)-3 a e^2 (1-2 n)\right )+e \left (3 c d^2 (1-3 n)-a e^2 (1-n)\right ) x^n\right )}{8 a^2 c n^2 \left (a+c x^{2 n}\right )}-\frac {d \left (3 a e^2-c d^2 (1-4 n)\right ) (1-2 n) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{8 a^3 c n^2}+\frac {e (1-n) \left (3 c d^2 (1-3 n)-a e^2 (1+n)\right ) x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{8 a^3 c n^2 (1+n)} \] Output:

1/4*x*(d+e*x^n)^3/a/n/(a+c*x^(2*n))^2-1/8*x*(d*(c*d^2*(1-4*n)-3*a*e^2*(1-2 
*n))+e*(3*c*d^2*(1-3*n)-a*e^2*(1-n))*x^n)/a^2/c/n^2/(a+c*x^(2*n))-1/8*d*(3 
*a*e^2-c*d^2*(1-4*n))*(1-2*n)*x*hypergeom([1, 1/2/n],[1+1/2/n],-c*x^(2*n)/ 
a)/a^3/c/n^2+1/8*e*(1-n)*(3*c*d^2*(1-3*n)-a*e^2*(1+n))*x^(1+n)*hypergeom([ 
1, 1/2*(1+n)/n],[3/2+1/2/n],-c*x^(2*n)/a)/a^3/c/n^2/(1+n)
 

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.73 \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^3} \, dx=\frac {x \left (3 a d e^2 \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+\frac {a e^3 x^n \operatorname {Hypergeometric2F1}\left (2,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{1+n}+d \left (c d^2-3 a e^2\right ) \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+\frac {e \left (3 c d^2-a e^2\right ) x^n \operatorname {Hypergeometric2F1}\left (3,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{1+n}\right )}{a^3 c} \] Input:

Integrate[(d + e*x^n)^3/(a + c*x^(2*n))^3,x]
 

Output:

(x*(3*a*d*e^2*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/ 
a)] + (a*e^3*x^n*Hypergeometric2F1[2, (1 + n)/(2*n), (3 + n^(-1))/2, -((c* 
x^(2*n))/a)])/(1 + n) + d*(c*d^2 - 3*a*e^2)*Hypergeometric2F1[3, 1/(2*n), 
(2 + n^(-1))/2, -((c*x^(2*n))/a)] + (e*(3*c*d^2 - a*e^2)*x^n*Hypergeometri 
c2F1[3, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + n)))/(a^3*c 
)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.64, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1767, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^3} \, dx\)

\(\Big \downarrow \) 1767

\(\displaystyle \int \left (\frac {x^n \left (3 c d^2 e-a e^3\right )-3 a d e^2+c d^3}{c \left (a+c x^{2 n}\right )^3}+\frac {e^2 \left (3 d+e x^n\right )}{c \left (a+c x^{2 n}\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e (1-3 n) (1-n) x^{n+1} \left (3 c d^2-a e^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{8 a^3 c n^2 (n+1)}+\frac {d (1-4 n) (1-2 n) x \left (c d^2-3 a e^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{8 a^3 c n^2}-\frac {x \left (e (1-3 n) x^n \left (3 c d^2-a e^2\right )+d (1-4 n) \left (c d^2-3 a e^2\right )\right )}{8 a^2 c n^2 \left (a+c x^{2 n}\right )}-\frac {3 d e^2 (1-2 n) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 a^2 c n}-\frac {e^3 (1-n) x^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 a^2 c n (n+1)}+\frac {x \left (e x^n \left (3 c d^2-a e^2\right )+d \left (c d^2-3 a e^2\right )\right )}{4 a c n \left (a+c x^{2 n}\right )^2}+\frac {e^2 x \left (3 d+e x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}\)

Input:

Int[(d + e*x^n)^3/(a + c*x^(2*n))^3,x]
 

Output:

(x*(d*(c*d^2 - 3*a*e^2) + e*(3*c*d^2 - a*e^2)*x^n))/(4*a*c*n*(a + c*x^(2*n 
))^2) + (e^2*x*(3*d + e*x^n))/(2*a*c*n*(a + c*x^(2*n))) - (x*(d*(c*d^2 - 3 
*a*e^2)*(1 - 4*n) + e*(3*c*d^2 - a*e^2)*(1 - 3*n)*x^n))/(8*a^2*c*n^2*(a + 
c*x^(2*n))) + (d*(c*d^2 - 3*a*e^2)*(1 - 4*n)*(1 - 2*n)*x*Hypergeometric2F1 
[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*c*n^2) - (3*d*e^2*( 
1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] 
)/(2*a^2*c*n) + (e*(3*c*d^2 - a*e^2)*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeom 
etric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*c*n^2 
*(1 + n)) - (e^3*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 
+ n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*c*n*(1 + n))
 

Defintions of rubi rules used

rule 1767

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] 
 :> Int[ExpandIntegrand[(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a 
, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && ((Integ 
ersQ[p, q] &&  !IntegerQ[n]) || IGtQ[p, 0] || (IGtQ[q, 0] &&  !IntegerQ[n]) 
)
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \frac {\left (d +e \,x^{n}\right )^{3}}{\left (a +c \,x^{2 n}\right )^{3}}d x\]

Input:

int((d+e*x^n)^3/(a+c*x^(2*n))^3,x)
 

Output:

int((d+e*x^n)^3/(a+c*x^(2*n))^3,x)
 

Fricas [F]

\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + a\right )}^{3}} \,d x } \] Input:

integrate((d+e*x^n)^3/(a+c*x^(2*n))^3,x, algorithm="fricas")
 

Output:

integral((e^3*x^(3*n) + 3*d*e^2*x^(2*n) + 3*d^2*e*x^n + d^3)/(c^3*x^(6*n) 
+ 3*a*c^2*x^(4*n) + 3*a^2*c*x^(2*n) + a^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^3} \, dx=\text {Timed out} \] Input:

integrate((d+e*x**n)**3/(a+c*x**(2*n))**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + a\right )}^{3}} \,d x } \] Input:

integrate((d+e*x^n)^3/(a+c*x^(2*n))^3,x, algorithm="maxima")
 

Output:

1/8*((3*c^2*d^2*e*(3*n - 1) + a*c*e^3*(n + 1))*x*x^(3*n) + (c^2*d^3*(4*n - 
 1) + 3*a*c*d*e^2)*x*x^(2*n) + (3*a*c*d^2*e*(5*n - 1) - a^2*e^3*(n - 1))*x 
*x^n + (a*c*d^3*(6*n - 1) - 3*a^2*d*e^2*(2*n - 1))*x)/(a^2*c^3*n^2*x^(4*n) 
 + 2*a^3*c^2*n^2*x^(2*n) + a^4*c*n^2) + integrate(1/8*((8*n^2 - 6*n + 1)*c 
*d^3 + 3*a*d*e^2*(2*n - 1) + (3*(3*n^2 - 4*n + 1)*c*d^2*e + (n^2 - 1)*a*e^ 
3)*x^n)/(a^2*c^2*n^2*x^(2*n) + a^3*c*n^2), x)
 

Giac [F]

\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + a\right )}^{3}} \,d x } \] Input:

integrate((d+e*x^n)^3/(a+c*x^(2*n))^3,x, algorithm="giac")
 

Output:

integrate((e*x^n + d)^3/(c*x^(2*n) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^3} \, dx=\int \frac {{\left (d+e\,x^n\right )}^3}{{\left (a+c\,x^{2\,n}\right )}^3} \,d x \] Input:

int((d + e*x^n)^3/(a + c*x^(2*n))^3,x)
 

Output:

int((d + e*x^n)^3/(a + c*x^(2*n))^3, x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^3} \, dx=\text {too large to display} \] Input:

int((d+e*x^n)^3/(a+c*x^(2*n))^3,x)
 

Output:

(x**(4*n)*int(x**(3*n)/(x**(6*n)*c**3*n + x**(6*n)*c**3 + 3*x**(4*n)*a*c** 
2*n + 3*x**(4*n)*a*c**2 + 3*x**(2*n)*a**2*c*n + 3*x**(2*n)*a**2*c + a**3*n 
 + a**3),x)*a*c**2*e**3*n**2 + 2*x**(4*n)*int(x**(3*n)/(x**(6*n)*c**3*n + 
x**(6*n)*c**3 + 3*x**(4*n)*a*c**2*n + 3*x**(4*n)*a*c**2 + 3*x**(2*n)*a**2* 
c*n + 3*x**(2*n)*a**2*c + a**3*n + a**3),x)*a*c**2*e**3*n + x**(4*n)*int(x 
**(3*n)/(x**(6*n)*c**3*n + x**(6*n)*c**3 + 3*x**(4*n)*a*c**2*n + 3*x**(4*n 
)*a*c**2 + 3*x**(2*n)*a**2*c*n + 3*x**(2*n)*a**2*c + a**3*n + a**3),x)*a*c 
**2*e**3 + 9*x**(4*n)*int(x**(3*n)/(x**(6*n)*c**3*n + x**(6*n)*c**3 + 3*x* 
*(4*n)*a*c**2*n + 3*x**(4*n)*a*c**2 + 3*x**(2*n)*a**2*c*n + 3*x**(2*n)*a** 
2*c + a**3*n + a**3),x)*c**3*d**2*e*n**2 + 6*x**(4*n)*int(x**(3*n)/(x**(6* 
n)*c**3*n + x**(6*n)*c**3 + 3*x**(4*n)*a*c**2*n + 3*x**(4*n)*a*c**2 + 3*x* 
*(2*n)*a**2*c*n + 3*x**(2*n)*a**2*c + a**3*n + a**3),x)*c**3*d**2*e*n - 3* 
x**(4*n)*int(x**(3*n)/(x**(6*n)*c**3*n + x**(6*n)*c**3 + 3*x**(4*n)*a*c**2 
*n + 3*x**(4*n)*a*c**2 + 3*x**(2*n)*a**2*c*n + 3*x**(2*n)*a**2*c + a**3*n 
+ a**3),x)*c**3*d**2*e + 3*x**(4*n)*int(x**(2*n)/(x**(6*n)*c**3*n + x**(6* 
n)*c**3 + 3*x**(4*n)*a*c**2*n + 3*x**(4*n)*a*c**2 + 3*x**(2*n)*a**2*c*n + 
3*x**(2*n)*a**2*c + a**3*n + a**3),x)*a*c**2*d*e**2*n**2 + 6*x**(4*n)*int( 
x**(2*n)/(x**(6*n)*c**3*n + x**(6*n)*c**3 + 3*x**(4*n)*a*c**2*n + 3*x**(4* 
n)*a*c**2 + 3*x**(2*n)*a**2*c*n + 3*x**(2*n)*a**2*c + a**3*n + a**3),x)*a* 
c**2*d*e**2*n + 3*x**(4*n)*int(x**(2*n)/(x**(6*n)*c**3*n + x**(6*n)*c**...