\(\int \frac {a+b x^n}{(c+d x^n)^2} \, dx\) [72]

Optimal result

Integrand size = 17, antiderivative size = 71 \[ \int \frac {a+b x^n}{\left (c+d x^n\right )^2} \, dx=\frac {b x}{d (1-n) \left (c+d x^n\right )}-\frac {(b c-a d (1-n)) x \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c^2 d (1-n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.79 \[ \int \frac {a+b x^n}{\left (c+d x^n\right )^2} \, dx=\frac {x \left (\frac {b}{c+d x^n}-\frac {(b c+a d (-1+n)) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c^2}\right )}{d-d n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {910, 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.

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

\(\Big \downarrow \) 910

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

\(\Big \downarrow \) 778

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

Input:

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

Output:

-(((b*c - a*d)*x)/(c*d*n*(c + d*x^n))) + ((b*c - a*d*(1 - n))*x*Hypergeome 
tric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/(c^2*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 910

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.50 (sec) , antiderivative size = 741, normalized size of antiderivative = 10.44 \[ \int \frac {a+b x^n}{\left (c+d x^n\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

a*(c*c**(1/n)*c**(-2 - 1/n)*n*x*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n) 
*gamma(1/n)/(c*n**3*gamma(1 + 1/n) + d*n**3*x**n*gamma(1 + 1/n)) + c*c**(1 
/n)*c**(-2 - 1/n)*n*x*gamma(1/n)/(c*n**3*gamma(1 + 1/n) + d*n**3*x**n*gamm 
a(1 + 1/n)) - c*c**(1/n)*c**(-2 - 1/n)*x*lerchphi(d*x**n*exp_polar(I*pi)/c 
, 1, 1/n)*gamma(1/n)/(c*n**3*gamma(1 + 1/n) + d*n**3*x**n*gamma(1 + 1/n)) 
+ c**(1/n)*c**(-2 - 1/n)*d*n*x*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 
1/n)*gamma(1/n)/(c*n**3*gamma(1 + 1/n) + d*n**3*x**n*gamma(1 + 1/n)) - c** 
(1/n)*c**(-2 - 1/n)*d*x*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*ga 
mma(1/n)/(c*n**3*gamma(1 + 1/n) + d*n**3*x**n*gamma(1 + 1/n))) + b*(c*c**( 
-3 - 1/n)*c**(1 + 1/n)*n**2*x**(n + 1)*gamma(1 + 1/n)/(c*n**3*gamma(2 + 1/ 
n) + d*n**3*x**n*gamma(2 + 1/n)) - c*c**(-3 - 1/n)*c**(1 + 1/n)*n*x**(n + 
1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(c*n**3*g 
amma(2 + 1/n) + d*n**3*x**n*gamma(2 + 1/n)) + c*c**(-3 - 1/n)*c**(1 + 1/n) 
*n*x**(n + 1)*gamma(1 + 1/n)/(c*n**3*gamma(2 + 1/n) + d*n**3*x**n*gamma(2 
+ 1/n)) - c*c**(-3 - 1/n)*c**(1 + 1/n)*x**(n + 1)*lerchphi(d*x**n*exp_pola 
r(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(c*n**3*gamma(2 + 1/n) + d*n**3*x**n 
*gamma(2 + 1/n)) - c**(-3 - 1/n)*c**(1 + 1/n)*d*n*x**n*x**(n + 1)*lerchphi 
(d*x**n*exp_polar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(c*n**3*gamma(2 + 1/ 
n) + d*n**3*x**n*gamma(2 + 1/n)) - c**(-3 - 1/n)*c**(1 + 1/n)*d*x**n*x**(n 
 + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(c*...
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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