3.3.89 \(\int \frac {a+b x^n}{(c+d x^n)^2} \, dx\) [289]

3.3.89.1 Optimal result

Integrand size = 17, antiderivative size = 73 \[ \int \frac {a+b x^n}{\left (c+d x^n\right )^2} \, dx=-\frac {(b c-a d) x}{c d n \left (c+d x^n\right )}+\frac {(b c-a d (1-n)) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c^2 d n} \]

output

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

3.3.89.2 Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \frac {a+b x^n}{\left (c+d x^n\right )^2} \, dx=\frac {x \left (\frac {b}{c+d x^n}-\frac {(b c+a d (-1+n)) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c^2}\right )}{d-d n} \]

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)
 

3.3.89.3 Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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)
 

3.3.89.3.1 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])
 

3.3.89.4 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)
 

3.3.89.5 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)
 

3.3.89.6 Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.39 (sec) , antiderivative size = 741, normalized size of antiderivative = 10.15 \[ \int \frac {a+b x^n}{\left (c+d x^n\right )^2} \, dx=a \left (\frac {c c^{\frac {1}{n}} c^{-2 - \frac {1}{n}} n x \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )} + \frac {c c^{\frac {1}{n}} c^{-2 - \frac {1}{n}} n x \Gamma \left (\frac {1}{n}\right )}{c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )} - \frac {c c^{\frac {1}{n}} c^{-2 - \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 )}{c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )} + \frac {c^{\frac {1}{n}} c^{-2 - \frac {1}{n}} d n x x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )} - \frac {c^{\frac {1}{n}} c^{-2 - \frac {1}{n}} d x x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )}\right ) + b \left (\frac {c c^{-3 - \frac {1}{n}} c^{1 + \frac {1}{n}} n^{2} x^{n + 1} \Gamma \left (1 + \frac {1}{n}\right )}{c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )} - \frac {c c^{-3 - \frac {1}{n}} c^{1 + \frac {1}{n}} n x^{n + 1} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 1 + \frac {1}{n}\right ) \Gamma \left (1 + \frac {1}{n}\right )}{c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )} + \frac {c c^{-3 - \frac {1}{n}} c^{1 + \frac {1}{n}} n x^{n + 1} \Gamma \left (1 + \frac {1}{n}\right )}{c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )} - \frac {c c^{-3 - \frac {1}{n}} c^{1 + \frac {1}{n}} x^{n + 1} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 1 + \frac {1}{n}\right ) \Gamma \left (1 + \frac {1}{n}\right )}{c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )} - \frac {c^{-3 - \frac {1}{n}} c^{1 + \frac {1}{n}} d n x^{n} x^{n + 1} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 1 + \frac {1}{n}\right ) \Gamma \left (1 + \frac {1}{n}\right )}{c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )} - \frac {c^{-3 - \frac {1}{n}} c^{1 + \frac {1}{n}} d x^{n} x^{n + 1} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 1 + \frac {1}{n}\right ) \Gamma \left (1 + \frac {1}{n}\right )}{c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )}\right ) \]

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*...
 

3.3.89.7 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)
 

3.3.89.8 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)
 

3.3.89.9 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)