\(\int \frac {(a+b x^n)^2}{c+d x^n} \, dx\) [78]

Optimal result

Integrand size = 19, antiderivative size = 84 \[ \int \frac {\left (a+b x^n\right )^2}{c+d x^n} \, dx=-\frac {b (b c (1+n)-a d (1+2 n)) x}{d^2 (1+n)}+\frac {b x \left (a+b x^n\right )}{d (1+n)}+\frac {(b c-a d)^2 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^n\right )^2}{c+d x^n} \, dx=\frac {a^2 x}{c}-\frac {(b c-a d)^2 x}{c d^2}+\frac {b^2 x^{1+n}}{d (1+n)}+\frac {(-b c+a d)^2 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {933, 25, 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.

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

\(\Big \downarrow \) 933

\(\displaystyle \frac {\int -\frac {b (b c (n+1)-a d (2 n+1)) x^n+a (b c-a d (n+1))}{d x^n+c}dx}{d (n+1)}+\frac {b x \left (a+b x^n\right )}{d (n+1)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b x \left (a+b x^n\right )}{d (n+1)}-\frac {\int \frac {b (b c (n+1)-a d (2 n+1)) x^n+a (b c-a d (n+1))}{d x^n+c}dx}{d (n+1)}\)

\(\Big \downarrow \) 913

\(\displaystyle \frac {b x \left (a+b x^n\right )}{d (n+1)}-\frac {\frac {b x (b c (n+1)-a d (2 n+1))}{d}-\frac {(n+1) (b c-a d)^2 \int \frac {1}{d x^n+c}dx}{d}}{d (n+1)}\)

\(\Big \downarrow \) 778

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 933

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.76 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.80 \[ \int \frac {\left (a+b x^n\right )^2}{c+d x^n} \, dx=\frac {a^{2} c^{\frac {1}{n}} c^{-1 - \frac {1}{n}} x \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{n^{2} \Gamma \left (1 + \frac {1}{n}\right )} - \frac {2 a b c^{- \frac {1}{n}} c^{1 + \frac {1}{n}} d^{\frac {1}{n}} d^{-1 - \frac {1}{n}} x \Phi \left (\frac {c x^{- n} e^{i \pi }}{d}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c n^{2} \Gamma \left (1 + \frac {1}{n}\right )} + \frac {2 b^{2} c^{-3 - \frac {1}{n}} c^{2 + \frac {1}{n}} x^{2 n + 1} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 2 + \frac {1}{n}\right ) \Gamma \left (2 + \frac {1}{n}\right )}{n \Gamma \left (3 + \frac {1}{n}\right )} + \frac {b^{2} c^{-3 - \frac {1}{n}} c^{2 + \frac {1}{n}} x^{2 n + 1} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 2 + \frac {1}{n}\right ) \Gamma \left (2 + \frac {1}{n}\right )}{n^{2} \Gamma \left (3 + \frac {1}{n}\right )} \] Input:

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

Output:

a**2*c**(1/n)*c**(-1 - 1/n)*x*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*g 
amma(1/n)/(n**2*gamma(1 + 1/n)) - 2*a*b*c**(1 + 1/n)*d**(1/n)*d**(-1 - 1/n 
)*x*lerchphi(c*exp_polar(I*pi)/(d*x**n), 1, exp_polar(I*pi)/n)*gamma(1/n)/ 
(c*c**(1/n)*n**2*gamma(1 + 1/n)) + 2*b**2*c**(-3 - 1/n)*c**(2 + 1/n)*x**(2 
*n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 2 + 1/n)*gamma(2 + 1/n)/(n*g 
amma(3 + 1/n)) + b**2*c**(-3 - 1/n)*c**(2 + 1/n)*x**(2*n + 1)*lerchphi(d*x 
**n*exp_polar(I*pi)/c, 1, 2 + 1/n)*gamma(2 + 1/n)/(n**2*gamma(3 + 1/n))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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