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

Optimal result

Integrand size = 33, antiderivative size = 144 \[ \int \frac {\left (d+e x^n\right ) \left (A+B x^n+C x^{2 n}\right )}{a+c x^{2 n}} \, dx=\frac {(C d+B e) x}{c}+\frac {C e x^{1+n}}{c (1+n)}+\frac {(A c d-a (C d+B e)) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {(B c d+A c e-a C e) x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d+e x^n\right ) \left (A+B x^n+C x^{2 n}\right )}{a+c x^{2 n}} \, dx=\frac {x \left (C d+B e+\frac {C e x^n}{1+n}+\frac {(A c d-a (C d+B e)) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a}+\frac {(B c d+A c e-a C e) x^n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (1+n)}\right )}{c} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {7293, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

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

Input:

int((d+e*x^n)*(A+B*x^n+C*x^(2*n))/(a+c*x^(2*n)),x)
 

Output:

int((d+e*x^n)*(A+B*x^n+C*x^(2*n))/(a+c*x^(2*n)),x)
 

Fricas [F]

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

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

Output:

integral((B*e*x^(2*n) + A*d + (C*e*x^n + C*d)*x^(2*n) + (B*d + A*e)*x^n)/( 
c*x^(2*n) + a), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.66 (sec) , antiderivative size = 692, normalized size of antiderivative = 4.81 \[ \int \frac {\left (d+e x^n\right ) \left (A+B x^n+C x^{2 n}\right )}{a+c x^{2 n}} \, dx =\text {Too large to display} \] Input:

integrate((d+e*x**n)*(A+B*x**n+C*x**(2*n))/(a+c*x**(2*n)),x)
 

Output:

A*a**(1/(2*n))*a**(-1 - 1/(2*n))*d*x*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a 
, 1, 1/(2*n))*gamma(1/(2*n))/(4*n**2*gamma(1 + 1/(2*n))) + A*a**(-3/2 - 1/ 
(2*n))*a**(1/2 + 1/(2*n))*e*x**(n + 1)*lerchphi(c*x**(2*n)*exp_polar(I*pi) 
/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/(4*n*gamma(3/2 + 1/(2*n))) + A* 
a**(-3/2 - 1/(2*n))*a**(1/2 + 1/(2*n))*e*x**(n + 1)*lerchphi(c*x**(2*n)*ex 
p_polar(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/(4*n**2*gamma(3/2 
+ 1/(2*n))) + B*a**(-3/2 - 1/(2*n))*a**(1/2 + 1/(2*n))*d*x**(n + 1)*lerchp 
hi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/(4 
*n*gamma(3/2 + 1/(2*n))) + B*a**(-3/2 - 1/(2*n))*a**(1/2 + 1/(2*n))*d*x**( 
n + 1)*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 
+ 1/(2*n))/(4*n**2*gamma(3/2 + 1/(2*n))) - B*a**(1 + 1/(2*n))*c**(1/(2*n)) 
*c**(-1 - 1/(2*n))*e*x*lerchphi(a*exp_polar(I*pi)/(c*x**(2*n)), 1, exp_pol 
ar(I*pi)/(2*n))*gamma(1/(2*n))/(4*a*a**(1/(2*n))*n**2*gamma(1 + 1/(2*n))) 
+ 3*C*a**(-5/2 - 1/(2*n))*a**(3/2 + 1/(2*n))*e*x**(3*n + 1)*lerchphi(c*x** 
(2*n)*exp_polar(I*pi)/a, 1, 3/2 + 1/(2*n))*gamma(3/2 + 1/(2*n))/(4*n*gamma 
(5/2 + 1/(2*n))) + C*a**(-5/2 - 1/(2*n))*a**(3/2 + 1/(2*n))*e*x**(3*n + 1) 
*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 3/2 + 1/(2*n))*gamma(3/2 + 1/(2 
*n))/(4*n**2*gamma(5/2 + 1/(2*n))) - C*a**(1 + 1/(2*n))*c**(1/(2*n))*c**(- 
1 - 1/(2*n))*d*x*lerchphi(a*exp_polar(I*pi)/(c*x**(2*n)), 1, exp_polar(I*p 
i)/(2*n))*gamma(1/(2*n))/(4*a*a**(1/(2*n))*n**2*gamma(1 + 1/(2*n)))
 

Maxima [F]

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

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

Output:

(C*e*x*x^n + (C*d*(n + 1) + B*e*(n + 1))*x)/(c*(n + 1)) + integrate(-(C*a* 
d - A*c*d + B*a*e - (B*c*d - C*a*e + A*c*e)*x^n)/(c^2*x^(2*n) + a*c), x)
 

Giac [F]

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

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

Output:

integrate((C*x^(2*n) + B*x^n + A)*(e*x^n + d)/(c*x^(2*n) + a), x)
 

Mupad [F(-1)]

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

int(((d + e*x^n)*(A + C*x^(2*n) + B*x^n))/(a + c*x^(2*n)),x)
 

Output:

int(((d + e*x^n)*(A + C*x^(2*n) + B*x^n))/(a + c*x^(2*n)), x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^n\right ) \left (A+B x^n+C x^{2 n}\right )}{a+c x^{2 n}} \, dx=\frac {x^{n} c e x +\left (\int \frac {x^{n}}{x^{2 n} c +a}d x \right ) b c d n +\left (\int \frac {x^{n}}{x^{2 n} c +a}d x \right ) b c d -\left (\int \frac {1}{x^{2 n} c +a}d x \right ) a b e n -\left (\int \frac {1}{x^{2 n} c +a}d x \right ) a b e +b e n x +b e x +c d n x +c d x}{c \left (n +1\right )} \] Input:

int((d+e*x^n)*(A+B*x^n+C*x^(2*n))/(a+c*x^(2*n)),x)
 

Output:

(x**n*c*e*x + int(x**n/(x**(2*n)*c + a),x)*b*c*d*n + int(x**n/(x**(2*n)*c 
+ a),x)*b*c*d - int(1/(x**(2*n)*c + a),x)*a*b*e*n - int(1/(x**(2*n)*c + a) 
,x)*a*b*e + b*e*n*x + b*e*x + c*d*n*x + c*d*x)/(c*(n + 1))