\(\int \frac {c+d x^n}{(a+b x^n)^2} \, dx\) [91]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.79 \[ \int \frac {c+d x^n}{\left (a+b x^n\right )^2} \, dx=\frac {x \left (\frac {d}{a+b x^n}-\frac {(a d+b c (-1+n)) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2}\right )}{b-b n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01, 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.

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

\(\Big \downarrow \) 910

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

\(\Big \downarrow \) 778

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

Input:

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

Output:

((b*c - a*d)*x)/(a*b*n*(a + b*x^n)) + ((a*d - b*c*(1 - n))*x*Hypergeometri 
c2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a^2*b*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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.36 (sec) , antiderivative size = 741, normalized size of antiderivative = 10.44 \[ \int \frac {c+d x^n}{\left (a+b x^n\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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