\(\int \frac {x}{(a+b x^n)^2 (c+d x^n)} \, dx\) [443]

Optimal result

Integrand size = 20, antiderivative size = 143 \[ \int \frac {x}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\frac {b x^2}{a (b c-a d) n \left (a+b x^n\right )}+\frac {b (2 a d (1-n)-b c (2-n)) x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{n},\frac {2+n}{n},-\frac {b x^n}{a}\right )}{2 a^2 (b c-a d)^2 n}+\frac {d^2 x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{n},\frac {2+n}{n},-\frac {d x^n}{c}\right )}{2 c (b c-a d)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.94 \[ \int \frac {x}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\frac {x^2 \left (b c (b c (-2+n)-2 a d (-1+n)) \left (a+b x^n\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2}{n},\frac {2+n}{n},-\frac {b x^n}{a}\right )+a \left (2 b c (b c-a d)+a d^2 n \left (a+b x^n\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2}{n},\frac {2+n}{n},-\frac {d x^n}{c}\right )\right )\right )}{2 a^2 c (b c-a d)^2 n \left (a+b x^n\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1006, 1067, 2009}

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.

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

\(\Big \downarrow \) 1006

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

\(\Big \downarrow \) 1067

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1006

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x 
^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 
 1))   Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( 
b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ 
a, b, c, d, e, m, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomi 
alQ[a, b, c, d, e, m, n, p, q, x]
 

rule 1067

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n 
_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a 
+ b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, n, p}, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(x/(b^2*d*x^(3*n) + a^2*c + (b^2*c + 2*a*b*d)*x^(2*n) + (2*a*b*c + 
 a^2*d)*x^n), x)
 

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

d^2*integrate(x/(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c* 
d^2 + a^2*d^3)*x^n), x) + b*x^2/(a^2*b*c*n - a^3*d*n + (a*b^2*c*n - a^2*b* 
d*n)*x^n) - (2*a*b*d*(n - 1) - b^2*c*(n - 2))*integrate(x/(a^2*b^2*c^2*n - 
 2*a^3*b*c*d*n + a^4*d^2*n + (a*b^3*c^2*n - 2*a^2*b^2*c*d*n + a^3*b*d^2*n) 
*x^n), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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