Integrand size = 26, antiderivative size = 1129 \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^2} \, dx=-\frac {x \left (2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )+c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n \left (a+b x^n+c x^{2 n}\right )}-\frac {2 c e^2 \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 )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}+\frac {c \left (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 )-b^2 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 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 e^2 (1-n)-b^3 e \left (2 c d-\sqrt {b^2-4 a c} e\right ) (1-n)\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^2-4 a c-b \sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n}-\frac {2 c e^2 \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 )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}+\frac {c \left (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 )-b^2 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 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 e^2 (1-n)-b^3 e \left (2 c d+\sqrt {b^2-4 a c} e\right ) (1-n)\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^2-4 a c+b \sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n}+\frac {2 e^4 (2 c d-b e) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^3}+\frac {e^4 x \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d^2 \left (c d^2-b d e+a e^2\right )^2} \]
-x*(2*b^3*c*d*e-6*a*b*c^2*d*e-b^4*e^2-b^2*c*(-4*a*e^2+c*d^2)+2*a*c^2*(-a*e ^2+c*d^2)+c*(2*b^2*c*d*e-4*a*c^2*d*e-b^3*e^2-b*c*(-3*a*e^2+c*d^2))*x^n)/a/ (-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/n/(a+b*x^n+c*x^(2*n))+2*e^4*(-b*e+2*c*d )*x*hypergeom([1, 1/n],[1+1/n],-e*x^n/d)/d/(a*e^2-b*d*e+c*d^2)^3+e^4*x*hyp ergeom([2, 1/n],[1+1/n],-e*x^n/d)/d^2/(a*e^2-b*d*e+c*d^2)^2-2*c*e^2*x*hype rgeom([1, 1/n],[1+1/n],-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))*(3*c^2*d^2+b*e^2*( b-(-4*a*c+b^2)^(1/2))-c*e*(3*b*d+a*e-2*d*(-4*a*c+b^2)^(1/2)))/(a*e^2-b*d*e +c*d^2)^3/(b^2-4*a*c+b*(-4*a*c+b^2)^(1/2))-2*c*e^2*x*hypergeom([1, 1/n],[1 +1/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))*(3*c^2*d^2+b*e^2*(b+(-4*a*c+b^2)^(1 /2))-c*e*(3*b*d+a*e+2*d*(-4*a*c+b^2)^(1/2)))/(a*e^2-b*d*e+c*d^2)^3/(b^2-4* a*c-b*(-4*a*c+b^2)^(1/2))+c*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b+(-4*a *c+b^2)^(1/2)))*(b^4*e^2*(1-n)-b^3*e*(1-n)*(2*c*d+e*(-4*a*c+b^2)^(1/2))-b^ 2*c*(-c*d^2*(1-n)+e*(a*e*(5-7*n)-2*d*(1-n)*(-4*a*c+b^2)^(1/2)))+b*c*(3*a*e ^2*(1-n)*(-4*a*c+b^2)^(1/2)+c*d*(4*a*e*(2-3*n)-d*(1-n)*(-4*a*c+b^2)^(1/2)) )+4*a*c^2*(-c*d^2*(1-2*n)+e*(a*e*(1-2*n)-d*(1-n)*(-4*a*c+b^2)^(1/2))))/a/( -4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/n/(b^2-4*a*c+b*(-4*a*c+b^2)^(1/2))+c*x*h ypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))*(b^4*e^2*(1-n)- b^3*e*(1-n)*(2*c*d-e*(-4*a*c+b^2)^(1/2))+b*c*(-3*a*e^2*(1-n)*(-4*a*c+b^2)^ (1/2)+c*d*(4*a*e*(2-3*n)+d*(1-n)*(-4*a*c+b^2)^(1/2)))+4*a*c^2*(-c*d^2*(1-2 *n)+e*(a*e*(1-2*n)+d*(1-n)*(-4*a*c+b^2)^(1/2)))-b^2*c*(-c*d^2*(1-n)+e*(...
Leaf count is larger than twice the leaf count of optimal. \(16855\) vs. \(2(1129)=2258\).
Time = 7.87 (sec) , antiderivative size = 16855, normalized size of antiderivative = 14.93 \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^2} \, dx=\text {Result too large to show} \]
Time = 2.84 (sec) , antiderivative size = 1129, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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 )}\) |
-((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)...
3.1.79.3.1 Defintions of rubi rules used
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]))
\[\int \frac {1}{\left (d +e \,x^{n}\right )^{2} \left (a +b \,x^{n}+c \,x^{2 n}\right )^{2}}d x\]
\[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2} {\left (e x^{n} + d\right )}^{2}} \,d x } \]
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)
Timed out. \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2} {\left (e x^{n} + d\right )}^{2}} \,d x } \]
(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^...
\[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2} {\left (e x^{n} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int \frac {1}{{\left (d+e\,x^n\right )}^2\,{\left (a+b\,x^n+c\,x^{2\,n}\right )}^2} \,d x \]