\(\int \frac {1}{(a+b x^n)^2 (c+d x^n)^3} \, dx\) [94]

Optimal result

Integrand size = 19, antiderivative size = 299 \[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^3} \, dx=\frac {d (2 b c+a d) x}{2 a c (b c-a d)^2 n \left (c+d x^n\right )^2}+\frac {b x}{a (b c-a d) n \left (a+b x^n\right ) \left (c+d x^n\right )^2}-\frac {d \left (a b c d (1-6 n)-a^2 d^2 (1-2 n)-2 b^2 c^2 n\right ) x}{2 a c^2 (b c-a d)^3 n^2 \left (c+d x^n\right )}+\frac {b^3 (a d (1-4 n)-b c (1-n)) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^4 n}+\frac {d^2 \left (a^2 d^2 \left (1-3 n+2 n^2\right )-2 a b c d \left (1-5 n+4 n^2\right )+b^2 c^2 \left (1-7 n+12 n^2\right )\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{2 c^3 (b c-a d)^4 n^2} \] Output:

1/2*d*(a*d+2*b*c)*x/a/c/(-a*d+b*c)^2/n/(c+d*x^n)^2+b*x/a/(-a*d+b*c)/n/(a+b 
*x^n)/(c+d*x^n)^2-1/2*d*(a*b*c*d*(1-6*n)-a^2*d^2*(1-2*n)-2*b^2*c^2*n)*x/a/ 
c^2/(-a*d+b*c)^3/n^2/(c+d*x^n)+b^3*(a*d*(1-4*n)-b*c*(1-n))*x*hypergeom([1, 
 1/n],[1+1/n],-b*x^n/a)/a^2/(-a*d+b*c)^4/n+1/2*d^2*(a^2*d^2*(2*n^2-3*n+1)- 
2*a*b*c*d*(4*n^2-5*n+1)+b^2*c^2*(12*n^2-7*n+1))*x*hypergeom([1, 1/n],[1+1/ 
n],-d*x^n/c)/c^3/(-a*d+b*c)^4/n^2
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^3} \, dx=\frac {x \left (\frac {2 b^3 (b c-a d) n}{a \left (a+b x^n\right )}+\frac {d^2 (b c-a d)^2 n}{c \left (c+d x^n\right )^2}+\frac {d^2 (-b c+a d) (a d (-1+2 n)+b (c-6 c n))}{c^2 \left (c+d x^n\right )}+\frac {2 b^3 (a d (1-4 n)+b c (-1+n)) n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2}+\frac {d^2 \left (a^2 d^2 \left (1-3 n+2 n^2\right )-2 a b c d \left (1-5 n+4 n^2\right )+b^2 c^2 \left (1-7 n+12 n^2\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c^3}\right )}{2 (b c-a d)^4 n^2} \] Input:

Integrate[1/((a + b*x^n)^2*(c + d*x^n)^3),x]
 

Output:

(x*((2*b^3*(b*c - a*d)*n)/(a*(a + b*x^n)) + (d^2*(b*c - a*d)^2*n)/(c*(c + 
d*x^n)^2) + (d^2*(-(b*c) + a*d)*(a*d*(-1 + 2*n) + b*(c - 6*c*n)))/(c^2*(c 
+ d*x^n)) + (2*b^3*(a*d*(1 - 4*n) + b*c*(-1 + n))*n*Hypergeometric2F1[1, n 
^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^2 + (d^2*(a^2*d^2*(1 - 3*n + 2*n^2) - 
2*a*b*c*d*(1 - 5*n + 4*n^2) + b^2*c^2*(1 - 7*n + 12*n^2))*Hypergeometric2F 
1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/c^3))/(2*(b*c - a*d)^4*n^2)
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {931, 1024, 1024, 1020, 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.

\(\displaystyle \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^3} \, dx\)

\(\Big \downarrow \) 931

\(\displaystyle \frac {b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )^2}-\frac {\int \frac {b d (1-3 n) x^n+a d n+b (c-c n)}{\left (b x^n+a\right ) \left (d x^n+c\right )^3}dx}{a n (b c-a d)}\)

\(\Big \downarrow \) 1024

\(\displaystyle \frac {b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )^2}-\frac {\frac {\int \frac {b d (2 b c+a d) (1-2 n) n x^n+n \left (2 b^2 (1-n) c^2+4 a b d n c+a^2 d^2 (1-2 n)\right )}{\left (b x^n+a\right ) \left (d x^n+c\right )^2}dx}{2 c n (b c-a d)}-\frac {d x (a d+2 b c)}{2 c (b c-a d) \left (c+d x^n\right )^2}}{a n (b c-a d)}\)

\(\Big \downarrow \) 1024

\(\displaystyle \frac {b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )^2}-\frac {\frac {\frac {\int \frac {n \left (2 b^3 (1-n) n c^3+6 a b^2 d n^2 c^2-a^2 b d^2 \left (6 n^2-7 n+1\right ) c+a^3 d^3 \left (2 n^2-3 n+1\right )\right )-b d (1-n) n \left (-2 b^2 n c^2+a b d (1-6 n) c-a^2 d^2 (1-2 n)\right ) x^n}{\left (b x^n+a\right ) \left (d x^n+c\right )}dx}{c n (b c-a d)}+\frac {d x \left (-a^2 d^2 (1-2 n)+a b c d (1-6 n)-2 b^2 c^2 n\right )}{c (b c-a d) \left (c+d x^n\right )}}{2 c n (b c-a d)}-\frac {d x (a d+2 b c)}{2 c (b c-a d) \left (c+d x^n\right )^2}}{a n (b c-a d)}\)

\(\Big \downarrow \) 1020

\(\displaystyle \frac {b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )^2}-\frac {\frac {\frac {-\frac {a d^2 n \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (4 n^2-5 n+1\right )+b^2 c^2 \left (12 n^2-7 n+1\right )\right ) \int \frac {1}{d x^n+c}dx}{b c-a d}-\frac {2 b^3 c^2 n^2 (a d (1-4 n)-b c (1-n)) \int \frac {1}{b x^n+a}dx}{b c-a d}}{c n (b c-a d)}+\frac {d x \left (-a^2 d^2 (1-2 n)+a b c d (1-6 n)-2 b^2 c^2 n\right )}{c (b c-a d) \left (c+d x^n\right )}}{2 c n (b c-a d)}-\frac {d x (a d+2 b c)}{2 c (b c-a d) \left (c+d x^n\right )^2}}{a n (b c-a d)}\)

\(\Big \downarrow \) 778

\(\displaystyle \frac {b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )^2}-\frac {\frac {\frac {d x \left (-a^2 d^2 (1-2 n)+a b c d (1-6 n)-2 b^2 c^2 n\right )}{c (b c-a d) \left (c+d x^n\right )}+\frac {-\frac {a d^2 n x \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (4 n^2-5 n+1\right )+b^2 c^2 \left (12 n^2-7 n+1\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c (b c-a d)}-\frac {2 b^3 c^2 n^2 x (a d (1-4 n)-b c (1-n)) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a (b c-a d)}}{c n (b c-a d)}}{2 c n (b c-a d)}-\frac {d x (a d+2 b c)}{2 c (b c-a d) \left (c+d x^n\right )^2}}{a n (b c-a d)}\)

Input:

Int[1/((a + b*x^n)^2*(c + d*x^n)^3),x]
 

Output:

(b*x)/(a*(b*c - a*d)*n*(a + b*x^n)*(c + d*x^n)^2) - (-1/2*(d*(2*b*c + a*d) 
*x)/(c*(b*c - a*d)*(c + d*x^n)^2) + ((d*(a*b*c*d*(1 - 6*n) - a^2*d^2*(1 - 
2*n) - 2*b^2*c^2*n)*x)/(c*(b*c - a*d)*(c + d*x^n)) + ((-2*b^3*c^2*(a*d*(1 
- 4*n) - b*c*(1 - n))*n^2*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b* 
x^n)/a)])/(a*(b*c - a*d)) - (a*d^2*n*(a^2*d^2*(1 - 3*n + 2*n^2) - 2*a*b*c* 
d*(1 - 5*n + 4*n^2) + b^2*c^2*(1 - 7*n + 12*n^2))*x*Hypergeometric2F1[1, n 
^(-1), 1 + n^(-1), -((d*x^n)/c)])/(c*(b*c - a*d)))/(c*(b*c - a*d)*n))/(2*c 
*(b*c - a*d)*n))/(a*(b*c - a*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 931

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

rule 1020

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^( 
n_))), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^n), x], x 
] - Simp[(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b 
, c, d, e, f, n}, x]
 

rule 1024

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f 
_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c 
+ d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( 
p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b 
*c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr 
eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
 

Maple [F]

\[\int \frac {1}{\left (a +b \,x^{n}\right )^{2} \left (c +d \,x^{n}\right )^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^3} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:

integrate(1/(a+b*x^n)^2/(c+d*x^n)^3,x, algorithm="fricas")
 

Output:

integral(1/(b^2*d^3*x^(5*n) + a^2*c^3 + (3*b^2*c*d^2 + 2*a*b*d^3)*x^(4*n) 
+ (3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^(3*n) + (b^2*c^3 + 6*a*b*c^2*d + 
 3*a^2*c*d^2)*x^(2*n) + (2*a*b*c^3 + 3*a^2*c^2*d)*x^n), x)
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^3} \, dx=\int \frac {1}{\left (a + b x^{n}\right )^{2} \left (c + d x^{n}\right )^{3}}\, dx \] Input:

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

Output:

Integral(1/((a + b*x**n)**2*(c + d*x**n)**3), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^3} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:

integrate(1/(a+b*x^n)^2/(c+d*x^n)^3,x, algorithm="maxima")
 

Output:

((12*n^2 - 7*n + 1)*b^2*c^2*d^2 - 2*(4*n^2 - 5*n + 1)*a*b*c*d^3 + (2*n^2 - 
 3*n + 1)*a^2*d^4)*integrate(1/2/(b^4*c^7*n^2 - 4*a*b^3*c^6*d*n^2 + 6*a^2* 
b^2*c^5*d^2*n^2 - 4*a^3*b*c^4*d^3*n^2 + a^4*c^3*d^4*n^2 + (b^4*c^6*d*n^2 - 
 4*a*b^3*c^5*d^2*n^2 + 6*a^2*b^2*c^4*d^3*n^2 - 4*a^3*b*c^3*d^4*n^2 + a^4*c 
^2*d^5*n^2)*x^n), x) - (a*b^3*d*(4*n - 1) - b^4*c*(n - 1))*integrate(1/(a^ 
2*b^4*c^4*n - 4*a^3*b^3*c^3*d*n + 6*a^4*b^2*c^2*d^2*n - 4*a^5*b*c*d^3*n + 
a^6*d^4*n + (a*b^5*c^4*n - 4*a^2*b^4*c^3*d*n + 6*a^3*b^3*c^2*d^2*n - 4*a^4 
*b^2*c*d^3*n + a^5*b*d^4*n)*x^n), x) + 1/2*((a*b^2*c*d^3*(6*n - 1) - a^2*b 
*d^4*(2*n - 1) + 2*b^3*c^2*d^2*n)*x*x^(2*n) + (a*b^2*c^2*d^2*(7*n - 1) - a 
^3*d^4*(2*n - 1) + 4*b^3*c^3*d*n + 3*a^2*b*c*d^3*n)*x*x^n + (a^2*b*c^2*d^2 
*(7*n - 1) - a^3*c*d^3*(3*n - 1) + 2*b^3*c^4*n)*x)/(a^2*b^3*c^7*n^2 - 3*a^ 
3*b^2*c^6*d*n^2 + 3*a^4*b*c^5*d^2*n^2 - a^5*c^4*d^3*n^2 + (a*b^4*c^5*d^2*n 
^2 - 3*a^2*b^3*c^4*d^3*n^2 + 3*a^3*b^2*c^3*d^4*n^2 - a^4*b*c^2*d^5*n^2)*x^ 
(3*n) + (2*a*b^4*c^6*d*n^2 - 5*a^2*b^3*c^5*d^2*n^2 + 3*a^3*b^2*c^4*d^3*n^2 
 + a^4*b*c^3*d^4*n^2 - a^5*c^2*d^5*n^2)*x^(2*n) + (a*b^4*c^7*n^2 - a^2*b^3 
*c^6*d*n^2 - 3*a^3*b^2*c^5*d^2*n^2 + 5*a^4*b*c^4*d^3*n^2 - 2*a^5*c^3*d^4*n 
^2)*x^n)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^3} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:

integrate(1/(a+b*x^n)^2/(c+d*x^n)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^3} \, dx=\int \frac {1}{{\left (a+b\,x^n\right )}^2\,{\left (c+d\,x^n\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^3} \, dx=\int \frac {1}{x^{5 n} b^{2} d^{3}+2 x^{4 n} a b \,d^{3}+3 x^{4 n} b^{2} c \,d^{2}+x^{3 n} a^{2} d^{3}+6 x^{3 n} a b c \,d^{2}+3 x^{3 n} b^{2} c^{2} d +3 x^{2 n} a^{2} c \,d^{2}+6 x^{2 n} a b \,c^{2} d +x^{2 n} b^{2} c^{3}+3 x^{n} a^{2} c^{2} d +2 x^{n} a b \,c^{3}+a^{2} c^{3}}d x \] Input:

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

Output:

int(1/(x**(5*n)*b**2*d**3 + 2*x**(4*n)*a*b*d**3 + 3*x**(4*n)*b**2*c*d**2 + 
 x**(3*n)*a**2*d**3 + 6*x**(3*n)*a*b*c*d**2 + 3*x**(3*n)*b**2*c**2*d + 3*x 
**(2*n)*a**2*c*d**2 + 6*x**(2*n)*a*b*c**2*d + x**(2*n)*b**2*c**3 + 3*x**n* 
a**2*c**2*d + 2*x**n*a*b*c**3 + a**2*c**3),x)