\(\int \frac {(d+e x^n)^2}{a+c x^{2 n}} \, dx\) [27]

Optimal result

Integrand size = 21, antiderivative size = 107 \[ \int \frac {\left (d+e x^n\right )^2}{a+c x^{2 n}} \, dx=\frac {e^2 x}{c}+\frac {\left (c d^2-a e^2\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {2 d e x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^n\right )^2}{a+c x^{2 n}} \, dx=\frac {e^2 x}{c}+\frac {\left (c d^2-a e^2\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {2 d e x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (1+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1755, 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.

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

\(\Big \downarrow \) 1755

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1755

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

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.10 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.77 \[ \int \frac {\left (d+e x^n\right )^2}{a+c x^{2 n}} \, dx=\frac {a^{\frac {1}{2 n}} a^{-1 - \frac {1}{2 n}} d^{2} x \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} d e x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{2 n \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} + \frac {a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} d e x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{2 n^{2} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} - \frac {a^{- \frac {1}{2 n}} a^{1 + \frac {1}{2 n}} c^{\frac {1}{2 n}} c^{-1 - \frac {1}{2 n}} e^{2} x \Phi \left (\frac {a x^{- 2 n} e^{i \pi }}{c}, 1, \frac {e^{i \pi }}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 a n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} \] Input:

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

Output:

a**(1/(2*n))*a**(-1 - 1/(2*n))*d**2*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**(-3/2 - 1/( 
2*n))*a**(1/2 + 1/(2*n))*d*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))/(2*n*gamma(3/2 + 1/(2*n))) + a 
**(-3/2 - 1/(2*n))*a**(1/2 + 1/(2*n))*d*e*x**(n + 1)*lerchphi(c*x**(2*n)*e 
xp_polar(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/(2*n**2*gamma(3/2 
 + 1/(2*n))) - a**(1 + 1/(2*n))*c**(1/(2*n))*c**(-1 - 1/(2*n))*e**2*x*lerc 
hphi(a*exp_polar(I*pi)/(c*x**(2*n)), 1, exp_polar(I*pi)/(2*n))*gamma(1/(2* 
n))/(4*a*a**(1/(2*n))*n**2*gamma(1 + 1/(2*n)))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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