\(\int \frac {(a+b x^n)^2}{(c+d x^n)^2} \, dx\) [79]

Optimal result

Integrand size = 19, antiderivative size = 115 \[ \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^2} \, dx=-\frac {b (a d-b c (1+n)) x}{c d^2 n}-\frac {(b c-a d) x \left (a+b x^n\right )}{c d n \left (c+d x^n\right )}+\frac {(b c-a d) (a d (1-n)-b c (1+n)) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c^2 d^2 n} \] Output:

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^2} \, dx=\frac {x \left (\frac {c \left (-2 a b c d+a^2 d^2+b^2 c \left (c+c n+d n x^n\right )\right )}{c+d x^n}-(b c-a d) (a d (-1+n)+b c (1+n)) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )\right )}{c^2 d^2 n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {930, 913, 778}

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.

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

\(\Big \downarrow \) 930

\(\displaystyle \frac {\int \frac {a (b c-a d (1-n))-b (a d-b c (n+1)) x^n}{d x^n+c}dx}{c d n}-\frac {x (b c-a d) \left (a+b x^n\right )}{c d n \left (c+d x^n\right )}\)

\(\Big \downarrow \) 913

\(\displaystyle \frac {\frac {(b c-a d) (a d (1-n)-b c (n+1)) \int \frac {1}{d x^n+c}dx}{d}-\frac {b x (a d-b c (n+1))}{d}}{c d n}-\frac {x (b c-a d) \left (a+b x^n\right )}{c d n \left (c+d x^n\right )}\)

\(\Big \downarrow \) 778

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-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 913

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( 
p + 1) + 1))/(b*(n*(p + 1) + 1))   Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b 
, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
 

rule 930

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(a*d - c*b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 
1))), x] - Simp[1/(a*b*n*(p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q 
- 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*( 
p + q) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 
 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*x**n)**2/(c + d*x**n)**2, x)
 

Maxima [F]

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

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

Output:

-(b^2*c^2*(n + 1) - a^2*d^2*(n - 1) - 2*a*b*c*d)*integrate(1/(c*d^3*n*x^n 
+ c^2*d^2*n), x) + (b^2*c*d*n*x*x^n + (b^2*c^2*(n + 1) - 2*a*b*c*d + a^2*d 
^2)*x)/(c*d^3*n*x^n + c^2*d^2*n)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - x**n*int(x**(2*n)/(x**(2*n)*d**2*n + x**(2*n)*d**2 + 2*x**n*c*d*n + 2* 
x**n*c*d + c**2*n + c**2),x)*a**2*d**3*n**2 + x**n*int(x**(2*n)/(x**(2*n)* 
d**2*n + x**(2*n)*d**2 + 2*x**n*c*d*n + 2*x**n*c*d + c**2*n + c**2),x)*a** 
2*d**3 - 2*x**n*int(x**(2*n)/(x**(2*n)*d**2*n + x**(2*n)*d**2 + 2*x**n*c*d 
*n + 2*x**n*c*d + c**2*n + c**2),x)*a*b*c*d**2*n - 2*x**n*int(x**(2*n)/(x* 
*(2*n)*d**2*n + x**(2*n)*d**2 + 2*x**n*c*d*n + 2*x**n*c*d + c**2*n + c**2) 
,x)*a*b*c*d**2 + x**n*int(x**(2*n)/(x**(2*n)*d**2*n + x**(2*n)*d**2 + 2*x* 
*n*c*d*n + 2*x**n*c*d + c**2*n + c**2),x)*b**2*c**2*d*n**2 + 2*x**n*int(x* 
*(2*n)/(x**(2*n)*d**2*n + x**(2*n)*d**2 + 2*x**n*c*d*n + 2*x**n*c*d + c**2 
*n + c**2),x)*b**2*c**2*d*n + x**n*int(x**(2*n)/(x**(2*n)*d**2*n + x**(2*n 
)*d**2 + 2*x**n*c*d*n + 2*x**n*c*d + c**2*n + c**2),x)*b**2*c**2*d + x**n* 
a**2*d*n*x - x**n*a**2*d*x + 2*x**n*a*b*c*x - int(x**(2*n)/(x**(2*n)*d**2* 
n + x**(2*n)*d**2 + 2*x**n*c*d*n + 2*x**n*c*d + c**2*n + c**2),x)*a**2*c*d 
**2*n**2 + int(x**(2*n)/(x**(2*n)*d**2*n + x**(2*n)*d**2 + 2*x**n*c*d*n + 
2*x**n*c*d + c**2*n + c**2),x)*a**2*c*d**2 - 2*int(x**(2*n)/(x**(2*n)*d**2 
*n + x**(2*n)*d**2 + 2*x**n*c*d*n + 2*x**n*c*d + c**2*n + c**2),x)*a*b*c** 
2*d*n - 2*int(x**(2*n)/(x**(2*n)*d**2*n + x**(2*n)*d**2 + 2*x**n*c*d*n + 2 
*x**n*c*d + c**2*n + c**2),x)*a*b*c**2*d + int(x**(2*n)/(x**(2*n)*d**2*n + 
 x**(2*n)*d**2 + 2*x**n*c*d*n + 2*x**n*c*d + c**2*n + c**2),x)*b**2*c**3*n 
**2 + 2*int(x**(2*n)/(x**(2*n)*d**2*n + x**(2*n)*d**2 + 2*x**n*c*d*n + ...