\(\int \frac {c+d x^{-1+n}}{(a+b x^n)^2} \, dx\) [59]

Optimal result

Integrand size = 19, antiderivative size = 44 \[ \int \frac {c+d x^{-1+n}}{\left (a+b x^n\right )^2} \, dx=-\frac {d}{b n \left (a+b x^n\right )}+\frac {c x \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^{-1+n}}{\left (a+b x^n\right )^2} \, dx=-\frac {d}{a b n+b^2 n x^n}+\frac {c x \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2430, 778, 793}

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.

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

\(\Big \downarrow \) 2430

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

\(\Big \downarrow \) 778

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

\(\Big \downarrow \) 793

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

Input:

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

Output:

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

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 793

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n) 
^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && 
 NeQ[p, -1]
 

rule 2430

Int[((A_) + (B_.)*(x_)^(m_.))*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> 
Simp[A   Int[(a + b*x^n)^p, x], x] + Simp[B   Int[x^m*(a + b*x^n)^p, x], x] 
 /; FreeQ[{a, b, A, B, m, n, p}, x] && EqQ[m - n + 1, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((d*x^(n - 1) + 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 = 7.76 (sec) , antiderivative size = 388, normalized size of antiderivative = 8.82 \[ \int \frac {c+d x^{-1+n}}{\left (a+b x^n\right )^2} \, dx=c \left (\frac {a a^{\frac {1}{n}} a^{-2 - \frac {1}{n}} n x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )} + \frac {a a^{\frac {1}{n}} a^{-2 - \frac {1}{n}} n x \Gamma \left (\frac {1}{n}\right )}{a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )} - \frac {a a^{\frac {1}{n}} a^{-2 - \frac {1}{n}} x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )} + \frac {a^{\frac {1}{n}} a^{-2 - \frac {1}{n}} b n x x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )} - \frac {a^{\frac {1}{n}} a^{-2 - \frac {1}{n}} b x x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )}\right ) + d \left (\begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac {x x^{- 2 n} x^{n - 1}}{b^{2} n} & \text {for}\: a = 0 \\\frac {\tilde {\infty } x x^{n - 1}}{n} & \text {for}\: b = - a x^{- n} \\\frac {\log {\left (x \right )}}{\left (a + b\right )^{2}} & \text {for}\: n = 0 \\\frac {x x^{n - 1}}{a^{2} n + a b n x^{n}} & \text {otherwise} \end {cases}\right ) \] Input:

integrate((c+d*x**(-1+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*Piecewi 
se((zoo*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (-x*x**(n - 1)/(b**2*n*x* 
*(2*n)), Eq(a, 0)), (zoo*x*x**(n - 1)/n, Eq(b, -a/x**n)), (log(x)/(a + b)* 
*2, Eq(n, 0)), (x*x**(n - 1)/(a**2*n + a*b*n*x**n), True))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 6.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.11 \[ \int \frac {c+d x^{-1+n}}{\left (a+b x^n\right )^2} \, dx=\frac {c\,x\,{{}}_2{\mathrm {F}}_1\left (2,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{a^2}-\frac {a\,d}{b\,\left (a^2\,n+a\,b\,n\,x^n\right )} \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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