\(\int \frac {1}{(d+e x^n)^2 (a+b x^n+c x^{2 n})^2} \, dx\) [73]

Optimal result

Integrand size = 26, antiderivative size = 1302 \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^2} \, dx =\text {Too large to display} \] Output:

e*(b^3*d*e^2+b*c*d*(-3*a*e^2+c*d^2)+4*a*c*e*(-a*e^2+c*d^2)-b^2*(-a*e^3+2*c 
*d^2*e))*x/a/(-4*a*c+b^2)/d/(a*e^2-b*d*e+c*d^2)^2/n/(d+e*x^n)+x*(c*(-2*a*c 
+b^2)*d-b*(-3*a*c+b^2)*e+c*(2*a*c*e-b^2*e+b*c*d)*x^n)/a/(-4*a*c+b^2)/(a*e^ 
2-b*d*e+c*d^2)/n/(d+e*x^n)/(a+b*x^n+c*x^(2*n))+c*(b*c*(2*a*c*d*e^2*(2*a*e* 
(1-7*n)-3*(-4*a*c+b^2)^(1/2)*d*(1-n))+c^2*d^3*(4*a*e*(3-5*n)+(-4*a*c+b^2)^ 
(1/2)*d*(1-n))-a^2*(-4*a*c+b^2)^(1/2)*e^4*(3-11*n))+b^4*e^2*(e*(a*e*(1-3*n 
)-(-4*a*c+b^2)^(1/2)*d*(1-n))+3*c*d^2*(1-n))-b^5*d*e^3*(1-n)+4*a*c^2*(a*e^ 
3*((-4*a*c+b^2)^(1/2)*d*(1-5*n)+a*e*(1-4*n))-c^2*d^4*(1-2*n)+c*d^2*e*((-4* 
a*c+b^2)^(1/2)*d*(1-n)+6*a*e*n))+b^3*e*(a*(-4*a*c+b^2)^(1/2)*e^3*(1-3*n)-3 
*c^2*d^3*(1-n)+c*d*e*(3*(-4*a*c+b^2)^(1/2)*d*(1-n)+a*e*(3+n)))+b^2*c*(c^2* 
d^4*(1-n)-3*c*d^2*e*((-4*a*c+b^2)^(1/2)*d+4*a*e)*(1-n)-a*e^3*(a*e*(5-17*n) 
-(-4*a*c+b^2)^(1/2)*d*(1+3*n))))*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b- 
(-4*a*c+b^2)^(1/2)))/a/(-4*a*c+b^2)/(b^2-4*a*c-b*(-4*a*c+b^2)^(1/2))/(a*e^ 
2-b*d*e+c*d^2)^3/n+c*(b*c*(c^2*d^3*(4*a*e*(3-5*n)-(-4*a*c+b^2)^(1/2)*d*(1- 
n))+2*a*c*d*e^2*(2*a*e*(1-7*n)+3*(-4*a*c+b^2)^(1/2)*d*(1-n))+a^2*(-4*a*c+b 
^2)^(1/2)*e^4*(3-11*n))+b^4*e^2*(e*(a*e*(1-3*n)+(-4*a*c+b^2)^(1/2)*d*(1-n) 
)+3*c*d^2*(1-n))-b^5*d*e^3*(1-n)-4*a*c^2*(a*e^3*((-4*a*c+b^2)^(1/2)*d*(1-5 
*n)-a*e*(1-4*n))+c^2*d^4*(1-2*n)+c*d^2*e*((-4*a*c+b^2)^(1/2)*d*(1-n)-6*a*e 
*n))-b^3*e*(a*(-4*a*c+b^2)^(1/2)*e^3*(1-3*n)+3*c^2*d^3*(1-n)+c*d*e*(3*(-4* 
a*c+b^2)^(1/2)*d*(1-n)-a*e*(3+n)))+b^2*c*(c^2*d^4*(1-n)+3*c*d^2*e*((-4*...
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(16855\) vs. \(2(1302)=2604\).

Time = 8.68 (sec) , antiderivative size = 16855, normalized size of antiderivative = 12.95 \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.85 (sec) , antiderivative size = 1129, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1766, 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.

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

\(\Big \downarrow \) 1766

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((x*(2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 2* 
a*c^2*(c*d^2 - a*e^2) + c*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^ 
2 - 3*a*e^2))*x^n))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*n*(a + b*x^ 
n + c*x^(2*n)))) - (2*c*e^2*(3*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - c 
*e*(3*b*d + 2*Sqrt[b^2 - 4*a*c]*d + a*e))*x*Hypergeometric2F1[1, n^(-1), 1 
 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt[b^2 
 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^3) + (c*(4*a*c^2*(e*(a*e*(1 - 2*n) + Sq 
rt[b^2 - 4*a*c]*d*(1 - n)) - c*d^2*(1 - 2*n)) - b^2*c*(e*(a*e*(5 - 7*n) + 
2*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - c*d^2*(1 - n)) + b*c*(c*d*(4*a*e*(2 - 3*n 
) + Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 3*a*Sqrt[b^2 - 4*a*c]*e^2*(1 - n)) + b^ 
4*e^2*(1 - n) - b^3*e*(2*c*d - Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeomet 
ric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 
 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2*n) 
 - (2*c*e^2*(3*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - c*e*(3*b*d - 2*Sq 
rt[b^2 - 4*a*c]*d + a*e))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c 
*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^ 
2 - b*d*e + a*e^2)^3) + (c*(4*a*c^2*(e*(a*e*(1 - 2*n) - Sqrt[b^2 - 4*a*c]* 
d*(1 - n)) - c*d^2*(1 - 2*n)) - b^2*c*(e*(a*e*(5 - 7*n) - 2*Sqrt[b^2 - 4*a 
*c]*d*(1 - n)) - c*d^2*(1 - n)) + b*c*(c*d*(4*a*e*(2 - 3*n) - Sqrt[b^2 - 4 
*a*c]*d*(1 - n)) + 3*a*Sqrt[b^2 - 4*a*c]*e^2*(1 - n)) + b^4*e^2*(1 - n)...
 

Defintions of rubi rules used

rule 1766

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ 
))^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)^q*(a + b*x^n + c*x^(2 
*n))^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ 
[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ((IntegersQ[p, q] && 
!IntegerQ[n]) || IGtQ[p, 0] || (IGtQ[q, 0] &&  !IntegerQ[n]))
 

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

(c*d^2*e^4*(5*n - 1) - b*d*e^5*(3*n - 1) + a*e^6*(n - 1))*integrate(1/(c^3 
*d^8*n - 3*b*c^2*d^7*e*n + 3*b^2*c*d^6*e^2*n - b^3*d^5*e^3*n + a^3*d^2*e^6 
*n + 3*(c*d^4*e^4*n - b*d^3*e^5*n)*a^2 + 3*(c^2*d^6*e^2*n - 2*b*c*d^5*e^3* 
n + b^2*d^4*e^4*n)*a + (c^3*d^7*e*n - 3*b*c^2*d^6*e^2*n + 3*b^2*c*d^5*e^3* 
n - b^3*d^4*e^4*n + a^3*d*e^7*n + 3*(c*d^3*e^5*n - b*d^2*e^6*n)*a^2 + 3*(c 
^2*d^5*e^3*n - 2*b*c*d^4*e^4*n + b^2*d^3*e^5*n)*a)*x^n), x) - ((b*c^3*d^3* 
e - 2*b^2*c^2*d^2*e^2 + b^3*c*d*e^3 - 4*a^2*c^2*e^4 + (4*c^3*d^2*e^2 - 3*b 
*c^2*d*e^3 + b^2*c*e^4)*a)*x*x^(2*n) + (b*c^3*d^4 - b^2*c^2*d^3*e - b^3*c* 
d^2*e^2 + b^4*d*e^3 + 2*(c^2*d*e^3 - 2*b*c*e^4)*a^2 + (2*c^3*d^3*e + 3*b*c 
^2*d^2*e^2 - 4*b^2*c*d*e^3 + b^3*e^4)*a)*x*x^n + (b^2*c^2*d^4 - 2*b^3*c*d^ 
3*e + b^4*d^2*e^2 - 4*a^3*c*e^4 + (2*c^2*d^2*e^2 + b^2*e^4)*a^2 - 2*(c^3*d 
^4 - 3*b*c^2*d^3*e + 2*b^2*c*d^2*e^2)*a)*x)/(4*a^5*c*d^2*e^4*n + (8*c^2*d^ 
4*e^2*n - 8*b*c*d^3*e^3*n - b^2*d^2*e^4*n)*a^4 + 2*(2*c^3*d^6*n - 4*b*c^2* 
d^5*e*n + b^2*c*d^4*e^2*n + b^3*d^3*e^3*n)*a^3 - (b^2*c^2*d^6*n - 2*b^3*c* 
d^5*e*n + b^4*d^4*e^2*n)*a^2 + (4*a^4*c^2*d*e^5*n + (8*c^3*d^3*e^3*n - 8*b 
*c^2*d^2*e^4*n - b^2*c*d*e^5*n)*a^3 + 2*(2*c^4*d^5*e*n - 4*b*c^3*d^4*e^2*n 
 + b^2*c^2*d^3*e^3*n + b^3*c*d^2*e^4*n)*a^2 - (b^2*c^3*d^5*e*n - 2*b^3*c^2 
*d^4*e^2*n + b^4*c*d^3*e^3*n)*a)*x^(3*n) + (4*(c^2*d^2*e^4*n + b*c*d*e^5*n 
)*a^4 + (8*c^3*d^4*e^2*n - 9*b^2*c*d^2*e^4*n - b^3*d*e^5*n)*a^3 + 2*(2*c^4 
*d^6*n - 2*b*c^3*d^5*e*n - 3*b^2*c^2*d^4*e^2*n + 2*b^3*c*d^3*e^3*n + b^...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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