Integrand size = 21, antiderivative size = 210 \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^2} \, dx=-\frac {3 d e^2 x}{2 a c n}-\frac {e^3 x^{1+n}}{2 a c n}+\frac {x \left (d+e x^n\right )^3}{2 a n \left (a+c x^{2 n}\right )}+\frac {d \left (3 a e^2-c d^2 (1-2 n)\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 a^2 c n}-\frac {e \left (3 c d^2 (1-n)-a e^2 (1+n)\right ) x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 a^2 c n (1+n)} \] Output:
-3/2*d*e^2*x/a/c/n-1/2*e^3*x^(1+n)/a/c/n+1/2*x*(d+e*x^n)^3/a/n/(a+c*x^(2*n ))+1/2*d*(3*a*e^2-c*d^2*(1-2*n))*x*hypergeom([1, 1/2/n],[1+1/2/n],-c*x^(2* n)/a)/a^2/c/n-1/2*e*(3*c*d^2*(1-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^2/c/n/(1+n)
Time = 0.50 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^2} \, dx=\frac {x \left (3 a d e^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+\frac {a e^3 x^n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{1+n}+d \left (c d^2-3 a e^2\right ) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+\frac {e \left (3 c d^2-a e^2\right ) x^n \operatorname {Hypergeometric2F1}\left (2,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{1+n}\right )}{a^2 c} \] Input:
Integrate[(d + e*x^n)^3/(a + c*x^(2*n))^2,x]
Output:
(x*(3*a*d*e^2*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/ a)] + (a*e^3*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c* x^(2*n))/a)])/(1 + n) + d*(c*d^2 - 3*a*e^2)*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + (e*(3*c*d^2 - a*e^2)*x^n*Hypergeometri c2F1[2, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + n)))/(a^2*c )
Time = 0.46 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.37, 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.
| \(\displaystyle \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^2} \, dx\) |
| \(\Big \downarrow \) 1767 |
| \(\displaystyle \int \left (\frac {x^n \left (3 c d^2 e-a e^3\right )-3 a d e^2+c d^3}{c \left (a+c x^{2 n}\right )^2}+\frac {e^2 \left (3 d+e x^n\right )}{c \left (a+c x^{2 n}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {e (1-n) x^{n+1} \left (3 c d^2-a e^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 a^2 c n (n+1)}-\frac {d (1-2 n) x \left (c d^2-3 a e^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 a^2 c n}+\frac {x \left (e x^n \left (3 c d^2-a e^2\right )+d \left (c d^2-3 a e^2\right )\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac {3 d e^2 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {e^3 x^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c (n+1)}\) |
Input:
Int[(d + e*x^n)^3/(a + c*x^(2*n))^2,x]
Output:
(x*(d*(c*d^2 - 3*a*e^2) + e*(3*c*d^2 - a*e^2)*x^n))/(2*a*c*n*(a + c*x^(2*n ))) + (3*d*e^2*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n ))/a)])/(a*c) - (d*(c*d^2 - 3*a*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*x^(1 + n)*Hyper geometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c*(1 + n)) - (e*(3*c*d^2 - a*e^2)*(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))
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]) )
\[\int \frac {\left (d +e \,x^{n}\right )^{3}}{\left (a +c \,x^{2 n}\right )^{2}}d x\]
Input:
int((d+e*x^n)^3/(a+c*x^(2*n))^2,x)
Output:
int((d+e*x^n)^3/(a+c*x^(2*n))^2,x)
\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^2} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + a\right )}^{2}} \,d x } \] Input:
integrate((d+e*x^n)^3/(a+c*x^(2*n))^2,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^2*x^(4*n) + 2*a*c*x^(2*n) + a^2), x)
\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^2} \, dx=\int \frac {\left (d + e x^{n}\right )^{3}}{\left (a + c x^{2 n}\right )^{2}}\, dx \] Input:
integrate((d+e*x**n)**3/(a+c*x**(2*n))**2,x)
Output:
Integral((d + e*x**n)**3/(a + c*x**(2*n))**2, x)
\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^2} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + a\right )}^{2}} \,d x } \] Input:
integrate((d+e*x^n)^3/(a+c*x^(2*n))^2,x, algorithm="maxima")
Output:
1/2*((3*c*d^2*e - a*e^3)*x*x^n + (c*d^3 - 3*a*d*e^2)*x)/(a*c^2*n*x^(2*n) + a^2*c*n) + integrate(1/2*(c*d^3*(2*n - 1) + 3*a*d*e^2 + (a*e^3*(n + 1) + 3*c*d^2*e*(n - 1))*x^n)/(a*c^2*n*x^(2*n) + a^2*c*n), x)
\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^2} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + a\right )}^{2}} \,d x } \] Input:
integrate((d+e*x^n)^3/(a+c*x^(2*n))^2,x, algorithm="giac")
Output:
integrate((e*x^n + d)^3/(c*x^(2*n) + a)^2, x)
Timed out. \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^2} \, dx=\int \frac {{\left (d+e\,x^n\right )}^3}{{\left (a+c\,x^{2\,n}\right )}^2} \,d x \] Input:
int((d + e*x^n)^3/(a + c*x^(2*n))^2,x)
Output:
int((d + e*x^n)^3/(a + c*x^(2*n))^2, x)
\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^2} \, dx =\text {Too large to display} \] Input:
int((d+e*x^n)^3/(a+c*x^(2*n))^2,x)
Output:
(x**(2*n)*int(x**(3*n)/(x**(4*n)*c**2*n + x**(4*n)*c**2 + 2*x**(2*n)*a*c*n + 2*x**(2*n)*a*c + a**2*n + a**2),x)*a*c*e**3*n**2 + 2*x**(2*n)*int(x**(3 *n)/(x**(4*n)*c**2*n + x**(4*n)*c**2 + 2*x**(2*n)*a*c*n + 2*x**(2*n)*a*c + a**2*n + a**2),x)*a*c*e**3*n + x**(2*n)*int(x**(3*n)/(x**(4*n)*c**2*n + x **(4*n)*c**2 + 2*x**(2*n)*a*c*n + 2*x**(2*n)*a*c + a**2*n + a**2),x)*a*c*e **3 + 3*x**(2*n)*int(x**(3*n)/(x**(4*n)*c**2*n + x**(4*n)*c**2 + 2*x**(2*n )*a*c*n + 2*x**(2*n)*a*c + a**2*n + a**2),x)*c**2*d**2*e*n**2 - 3*x**(2*n) *int(x**(3*n)/(x**(4*n)*c**2*n + x**(4*n)*c**2 + 2*x**(2*n)*a*c*n + 2*x**( 2*n)*a*c + a**2*n + a**2),x)*c**2*d**2*e + 3*x**(2*n)*int(x**(2*n)/(x**(4* n)*c**2*n + x**(4*n)*c**2 + 2*x**(2*n)*a*c*n + 2*x**(2*n)*a*c + a**2*n + a **2),x)*a*c*d*e**2*n**2 + 6*x**(2*n)*int(x**(2*n)/(x**(4*n)*c**2*n + x**(4 *n)*c**2 + 2*x**(2*n)*a*c*n + 2*x**(2*n)*a*c + a**2*n + a**2),x)*a*c*d*e** 2*n + 3*x**(2*n)*int(x**(2*n)/(x**(4*n)*c**2*n + x**(4*n)*c**2 + 2*x**(2*n )*a*c*n + 2*x**(2*n)*a*c + a**2*n + a**2),x)*a*c*d*e**2 + 2*x**(2*n)*int(x **(2*n)/(x**(4*n)*c**2*n + x**(4*n)*c**2 + 2*x**(2*n)*a*c*n + 2*x**(2*n)*a *c + a**2*n + a**2),x)*c**2*d**3*n**3 + 3*x**(2*n)*int(x**(2*n)/(x**(4*n)* c**2*n + x**(4*n)*c**2 + 2*x**(2*n)*a*c*n + 2*x**(2*n)*a*c + a**2*n + a**2 ),x)*c**2*d**3*n**2 - x**(2*n)*int(x**(2*n)/(x**(4*n)*c**2*n + x**(4*n)*c* *2 + 2*x**(2*n)*a*c*n + 2*x**(2*n)*a*c + a**2*n + a**2),x)*c**2*d**3 + 3*x **n*d**2*e*x + int(x**(3*n)/(x**(4*n)*c**2*n + x**(4*n)*c**2 + 2*x**(2*...