Integrand size = 26, antiderivative size = 750 \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\frac {x \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^n\right )}{a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac {e^2 \left (e+\frac {6 c d-3 b e}{\sqrt {b^2-4 a c}}\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{c \left (b-\sqrt {b^2-4 a c}\right )}+\frac {\left (\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) (1-n)+\frac {b^2 c d \left (3 a e^2 (1-3 n)-c d^2 (1-n)\right )-a b^3 e^3 (1-3 n)+4 a c^2 d \left (c d^2-3 a e^2\right ) (1-2 n)+2 a b c e \left (a e^2 (2-5 n)+3 c d^2 n\right )}{\sqrt {b^2-4 a c}}\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{a c \left (b^2-4 a c\right ) \left (b-\sqrt {b^2-4 a c}\right ) n}+\frac {e^2 \left (e-\frac {3 (2 c d-b e)}{\sqrt {b^2-4 a c}}\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{c \left (b+\sqrt {b^2-4 a c}\right )}+\frac {\left (\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) (1-n)-\frac {b^2 c d \left (3 a e^2 (1-3 n)-c d^2 (1-n)\right )-a b^3 e^3 (1-3 n)+4 a c^2 d \left (c d^2-3 a e^2\right ) (1-2 n)+2 a b c e \left (a e^2 (2-5 n)+3 c d^2 n\right )}{\sqrt {b^2-4 a c}}\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a c \left (b^2-4 a c\right ) \left (b+\sqrt {b^2-4 a c}\right ) n} \]
x*(b^2*c*d^3-2*a*c*d*(-3*a*e^2+c*d^2)-a*b*e*(a*e^2+3*c*d^2)-(a*b^2*e^3+2*a *c*e*(-a*e^2+3*c*d^2)-b*c*d*(3*a*e^2+c*d^2))*x^n)/a/c/(-4*a*c+b^2)/n/(a+b* x^n+c*x^(2*n))+e^2*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b-(-4*a*c+b^2)^( 1/2)))*(e+(-3*b*e+6*c*d)/(-4*a*c+b^2)^(1/2))/c/(b-(-4*a*c+b^2)^(1/2))+x*hy pergeom([1, 1/n],[1+1/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))*((a*b^2*e^3+2*a* c*e*(-a*e^2+3*c*d^2)-b*c*d*(3*a*e^2+c*d^2))*(1-n)+(b^2*c*d*(3*a*e^2*(1-3*n )-c*d^2*(1-n))-a*b^3*e^3*(1-3*n)+4*a*c^2*d*(-3*a*e^2+c*d^2)*(1-2*n)+2*a*b* c*e*(a*e^2*(2-5*n)+3*c*d^2*n))/(-4*a*c+b^2)^(1/2))/a/c/(-4*a*c+b^2)/n/(b-( -4*a*c+b^2)^(1/2))+e^2*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b+(-4*a*c+b^ 2)^(1/2)))*(e-3*(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))/c/(b+(-4*a*c+b^2)^(1/2))+ x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))*((a*b^2*e^3+ 2*a*c*e*(-a*e^2+3*c*d^2)-b*c*d*(3*a*e^2+c*d^2))*(1-n)+(-b^2*c*d*(3*a*e^2*( 1-3*n)-c*d^2*(1-n))+a*b^3*e^3*(1-3*n)-4*a*c^2*d*(-3*a*e^2+c*d^2)*(1-2*n)-2 *a*b*c*e*(a*e^2*(2-5*n)+3*c*d^2*n))/(-4*a*c+b^2)^(1/2))/a/c/(-4*a*c+b^2)/n /(b+(-4*a*c+b^2)^(1/2))
Leaf count is larger than twice the leaf count of optimal. \(5537\) vs. \(2(750)=1500\).
Time = 7.56 (sec) , antiderivative size = 5537, normalized size of antiderivative = 7.38 \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\text {Result too large to show} \]
Time = 2.33 (sec) , antiderivative size = 750, 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.
| \(\displaystyle \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx\) |
| \(\Big \downarrow \) 1766 |
| \(\displaystyle \int \left (\frac {x^n \left (-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right )+a b e^3-3 a c d e^2+c^2 d^3}{c^2 \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 \left (-b e+3 c d+c e x^n\right )}{c^2 \left (a+b x^n+c x^{2 n}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x \left (-\left (x^n \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )-a b e \left (a e^2+3 c d^2\right )-2 a c d \left (c d^2-3 a e^2\right )+b^2 c d^3\right )}{a c n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}+\frac {e^2 x \left (\frac {6 c d-3 b e}{\sqrt {b^2-4 a c}}+e\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{c \left (b-\sqrt {b^2-4 a c}\right )}+\frac {e^2 x \left (e-\frac {3 (2 c d-b e)}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{c \left (\sqrt {b^2-4 a c}+b\right )}+\frac {x \left ((1-n) \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )+\frac {-a b^3 e^3 (1-3 n)+b^2 c d \left (3 a e^2 (1-3 n)-c d^2 (1-n)\right )+2 a b c e \left (a e^2 (2-5 n)+3 c d^2 n\right )+4 a c^2 d (1-2 n) \left (c d^2-3 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{a c n \left (b^2-4 a c\right ) \left (b-\sqrt {b^2-4 a c}\right )}+\frac {x \left ((1-n) \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )-\frac {-a b^3 e^3 (1-3 n)+b^2 c d \left (3 a e^2 (1-3 n)-c d^2 (1-n)\right )+2 a b c e \left (a e^2 (2-5 n)+3 c d^2 n\right )+4 a c^2 d (1-2 n) \left (c d^2-3 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a c n \left (b^2-4 a c\right ) \left (\sqrt {b^2-4 a c}+b\right )}\) |
(x*(b^2*c*d^3 - 2*a*c*d*(c*d^2 - 3*a*e^2) - a*b*e*(3*c*d^2 + a*e^2) - (a*b ^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2))*x^n))/(a*c*( b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (e^2*(e + (6*c*d - 3*b*e)/Sqrt[b ^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sq rt[b^2 - 4*a*c])])/(c*(b - Sqrt[b^2 - 4*a*c])) + (((a*b^2*e^3 + 2*a*c*e*(3 *c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2))*(1 - n) + (b^2*c*d*(3*a*e^2*(1 - 3*n) - c*d^2*(1 - n)) - a*b^3*e^3*(1 - 3*n) + 4*a*c^2*d*(c*d^2 - 3*a*e^2 )*(1 - 2*n) + 2*a*b*c*e*(a*e^2*(2 - 5*n) + 3*c*d^2*n))/Sqrt[b^2 - 4*a*c])* x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a* c])])/(a*c*(b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*n) + (e^2*(e - (3*(2*c*d - b*e))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2* c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(c*(b + Sqrt[b^2 - 4*a*c])) + (((a*b^2*e^ 3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2))*(1 - n) - (b^2*c* d*(3*a*e^2*(1 - 3*n) - c*d^2*(1 - n)) - a*b^3*e^3*(1 - 3*n) + 4*a*c^2*d*(c *d^2 - 3*a*e^2)*(1 - 2*n) + 2*a*b*c*e*(a*e^2*(2 - 5*n) + 3*c*d^2*n))/Sqrt[ b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + S qrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b + Sqrt[b^2 - 4*a*c])*n)
3.1.75.3.1 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]))
\[\int \frac {\left (d +e \,x^{n}\right )^{3}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{2}}d x\]
\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}} \,d x } \]
integral((e^3*x^(3*n) + 3*d*e^2*x^(2*n) + 3*d^2*e*x^n + d^3)/(c^2*x^(4*n) + b^2*x^(2*n) + 2*a*b*x^n + a^2 + 2*(b*c*x^n + a*c)*x^(2*n)), x)
Timed out. \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}} \,d x } \]
((b*c^2*d^3 + 2*a^2*c*e^3 - (6*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)*a)*x*x^n + (b^2*c*d^3 + (6*c*d*e^2 - b*e^3)*a^2 - (2*c^2*d^3 + 3*b*c*d^2*e)*a)*x)/ (a^2*b^2*c*n - 4*a^3*c^2*n + (a*b^2*c^2*n - 4*a^2*c^3*n)*x^(2*n) + (a*b^3* c*n - 4*a^2*b*c^2*n)*x^n) + integrate((b^2*c*d^3*(n - 1) - (6*c*d*e^2 - b* e^3)*a^2 - (2*c^2*d^3*(2*n - 1) - 3*b*c*d^2*e)*a - (2*a^2*c*e^3*(n + 1) - b*c^2*d^3*(n - 1) + (6*c^2*d^2*e*(n - 1) - 3*b*c*d*e^2*(n - 1) - b^2*e^3)* a)*x^n)/(a^2*b^2*c*n - 4*a^3*c^2*n + (a*b^2*c^2*n - 4*a^2*c^3*n)*x^(2*n) + (a*b^3*c*n - 4*a^2*b*c^2*n)*x^n), x)
\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int \frac {{\left (d+e\,x^n\right )}^3}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^2} \,d x \]