Integrand size = 26, antiderivative size = 707 \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=-\frac {e \left (2 b c d^2-6 a c d e+a b e^2\right ) x}{a c \left (b^2-4 a c\right ) n}-\frac {e^2 (b d-2 a e) x^{1+n}}{a \left (b^2-4 a c\right ) n}-\frac {b e x \left (d+e x^n\right )^2}{a \left (b^2-4 a c\right ) n}+\frac {x \left (b^2-2 a c+b c x^n\right ) \left (d+e x^n\right )^3}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac {\left (a b^3 e^3-b^2 \left (a \sqrt {b^2-4 a c} e^3-c^2 d^3 (1-n)+3 a c d e^2 (1-n)\right )+2 a c \left (3 c d e \left (2 a e-\sqrt {b^2-4 a c} d (1-n)\right )-2 c^2 d^3 (1-2 n)+a \sqrt {b^2-4 a c} e^3 (1+n)\right )+b c \left (c d^2 \left (\sqrt {b^2-4 a c} d (1-n)-6 a e n\right )+a e^2 \left (3 \sqrt {b^2-4 a c} d (1-n)-2 a e (2+n)\right )\right )\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{a c \left (b^2-4 a c\right ) \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) n}+\frac {\left (a b^3 e^3+b^2 \left (a \sqrt {b^2-4 a c} e^3+c^2 d^3 (1-n)-3 a c d e^2 (1-n)\right )+2 a c \left (3 c d e \left (2 a e+\sqrt {b^2-4 a c} d (1-n)\right )-2 c^2 d^3 (1-2 n)-a \sqrt {b^2-4 a c} e^3 (1+n)\right )-b c \left (c d^2 \left (\sqrt {b^2-4 a c} d (1-n)+6 a e n\right )+a e^2 \left (3 \sqrt {b^2-4 a c} d (1-n)+2 a e (2+n)\right )\right )\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a c \left (b^2-4 a c\right ) \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) n} \] Output:
-e*(a*b*e^2-6*a*c*d*e+2*b*c*d^2)*x/a/c/(-4*a*c+b^2)/n-e^2*(-2*a*e+b*d)*x^( 1+n)/a/(-4*a*c+b^2)/n-b*e*x*(d+e*x^n)^2/a/(-4*a*c+b^2)/n+x*(b^2-2*a*c+b*c* x^n)*(d+e*x^n)^3/a/(-4*a*c+b^2)/n/(a+b*x^n+c*x^(2*n))+(a*b^3*e^3-b^2*(a*(- 4*a*c+b^2)^(1/2)*e^3-c^2*d^3*(1-n)+3*a*c*d*e^2*(1-n))+2*a*c*(3*c*d*e*(2*a* e-(-4*a*c+b^2)^(1/2)*d*(1-n))-2*c^2*d^3*(1-2*n)+a*(-4*a*c+b^2)^(1/2)*e^3*( 1+n))+b*c*(c*d^2*((-4*a*c+b^2)^(1/2)*d*(1-n)-6*a*e*n)+a*e^2*(3*(-4*a*c+b^2 )^(1/2)*d*(1-n)-2*a*e*(2+n))))*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b-(- 4*a*c+b^2)^(1/2)))/a/c/(-4*a*c+b^2)/(b^2-4*a*c-b*(-4*a*c+b^2)^(1/2))/n+(a* b^3*e^3+b^2*(a*(-4*a*c+b^2)^(1/2)*e^3+c^2*d^3*(1-n)-3*a*c*d*e^2*(1-n))+2*a *c*(3*c*d*e*(2*a*e+(-4*a*c+b^2)^(1/2)*d*(1-n))-2*c^2*d^3*(1-2*n)-a*(-4*a*c +b^2)^(1/2)*e^3*(1+n))-b*c*(c*d^2*((-4*a*c+b^2)^(1/2)*d*(1-n)+6*a*e*n)+a*e ^2*(3*(-4*a*c+b^2)^(1/2)*d*(1-n)+2*a*e*(2+n))))*x*hypergeom([1, 1/n],[1+1/ n],-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))/a/c/(-4*a*c+b^2)/(b*(-4*a*c+b^2)^(1/2) -4*a*c+b^2)/n
Leaf count is larger than twice the leaf count of optimal. \(5537\) vs. \(2(707)=1414\).
Time = 7.89 (sec) , antiderivative size = 5537, normalized size of antiderivative = 7.83 \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\text {Result too large to show} \] Input:
Integrate[(d + e*x^n)^3/(a + b*x^n + c*x^(2*n))^2,x]
Output:
Result too large to show
Time = 2.45 (sec) , antiderivative size = 750, normalized size of antiderivative = 1.06, 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 |
| \(\displaystyle \frac {x \left (-\left (x^n \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )-a b e \left (a e^2+3 c d^2\right )-2 a c d \left (c d^2-3 a e^2\right )+b^2 c d^3\right )}{a c n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}+\frac {e^2 x \left (\frac {6 c d-3 b e}{\sqrt {b^2-4 a c}}+e\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{c \left (b-\sqrt {b^2-4 a c}\right )}+\frac {e^2 x \left (e-\frac {3 (2 c d-b e)}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{c \left (\sqrt {b^2-4 a c}+b\right )}+\frac {x \left ((1-n) \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )+\frac {-a b^3 e^3 (1-3 n)+b^2 c d \left (3 a e^2 (1-3 n)-c d^2 (1-n)\right )+2 a b c e \left (a e^2 (2-5 n)+3 c d^2 n\right )+4 a c^2 d (1-2 n) \left (c d^2-3 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{a c n \left (b^2-4 a c\right ) \left (b-\sqrt {b^2-4 a c}\right )}+\frac {x \left ((1-n) \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )-\frac {-a b^3 e^3 (1-3 n)+b^2 c d \left (3 a e^2 (1-3 n)-c d^2 (1-n)\right )+2 a b c e \left (a e^2 (2-5 n)+3 c d^2 n\right )+4 a c^2 d (1-2 n) \left (c d^2-3 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a c n \left (b^2-4 a c\right ) \left (\sqrt {b^2-4 a c}+b\right )}\) |
Input:
Int[(d + e*x^n)^3/(a + b*x^n + c*x^(2*n))^2,x]
Output:
(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)
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]))
\[\int \frac {\left (d +e \,x^{n}\right )^{3}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{2}}d x\]
Input:
int((d+e*x^n)^3/(a+b*x^n+c*x^(2*n))^2,x)
Output:
int((d+e*x^n)^3/(a+b*x^n+c*x^(2*n))^2,x)
\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}} \,d x } \] Input:
integrate((d+e*x^n)^3/(a+b*x^n+c*x^(2*n))^2,x, algorithm="fricas")
Output:
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. \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\text {Timed out} \] Input:
integrate((d+e*x**n)**3/(a+b*x**n+c*x**(2*n))**2,x)
Output:
Timed out
\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}} \,d x } \] Input:
integrate((d+e*x^n)^3/(a+b*x^n+c*x^(2*n))^2,x, algorithm="maxima")
Output:
((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)
\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}} \,d x } \] Input:
integrate((d+e*x^n)^3/(a+b*x^n+c*x^(2*n))^2,x, algorithm="giac")
Output:
integrate((e*x^n + d)^3/(c*x^(2*n) + b*x^n + a)^2, x)
Timed out. \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int \frac {{\left (d+e\,x^n\right )}^3}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^2} \,d x \] Input:
int((d + e*x^n)^3/(a + b*x^n + c*x^(2*n))^2,x)
Output:
int((d + e*x^n)^3/(a + b*x^n + c*x^(2*n))^2, x)
\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\text {too large to display} \] Input:
int((d+e*x^n)^3/(a+b*x^n+c*x^(2*n))^2,x)
Output:
( - 2*x**(2*n)*int(x**(2*n)/(x**(4*n)*c**2*n**2 - 2*x**(4*n)*c**2*n + x**( 4*n)*c**2 + 2*x**(3*n)*b*c*n**2 - 4*x**(3*n)*b*c*n + 2*x**(3*n)*b*c + 2*x* *(2*n)*a*c*n**2 - 4*x**(2*n)*a*c*n + 2*x**(2*n)*a*c + x**(2*n)*b**2*n**2 - 2*x**(2*n)*b**2*n + x**(2*n)*b**2 + 2*x**n*a*b*n**2 - 4*x**n*a*b*n + 2*x* *n*a*b + a**2*n**2 - 2*a**2*n + a**2),x)*a*c**2*e**3*n**4 + 3*x**(2*n)*int (x**(2*n)/(x**(4*n)*c**2*n**2 - 2*x**(4*n)*c**2*n + x**(4*n)*c**2 + 2*x**( 3*n)*b*c*n**2 - 4*x**(3*n)*b*c*n + 2*x**(3*n)*b*c + 2*x**(2*n)*a*c*n**2 - 4*x**(2*n)*a*c*n + 2*x**(2*n)*a*c + x**(2*n)*b**2*n**2 - 2*x**(2*n)*b**2*n + x**(2*n)*b**2 + 2*x**n*a*b*n**2 - 4*x**n*a*b*n + 2*x**n*a*b + a**2*n**2 - 2*a**2*n + a**2),x)*a*c**2*e**3*n**3 + x**(2*n)*int(x**(2*n)/(x**(4*n)* c**2*n**2 - 2*x**(4*n)*c**2*n + x**(4*n)*c**2 + 2*x**(3*n)*b*c*n**2 - 4*x* *(3*n)*b*c*n + 2*x**(3*n)*b*c + 2*x**(2*n)*a*c*n**2 - 4*x**(2*n)*a*c*n + 2 *x**(2*n)*a*c + x**(2*n)*b**2*n**2 - 2*x**(2*n)*b**2*n + x**(2*n)*b**2 + 2 *x**n*a*b*n**2 - 4*x**n*a*b*n + 2*x**n*a*b + a**2*n**2 - 2*a**2*n + a**2), x)*a*c**2*e**3*n**2 - 3*x**(2*n)*int(x**(2*n)/(x**(4*n)*c**2*n**2 - 2*x**( 4*n)*c**2*n + x**(4*n)*c**2 + 2*x**(3*n)*b*c*n**2 - 4*x**(3*n)*b*c*n + 2*x **(3*n)*b*c + 2*x**(2*n)*a*c*n**2 - 4*x**(2*n)*a*c*n + 2*x**(2*n)*a*c + x* *(2*n)*b**2*n**2 - 2*x**(2*n)*b**2*n + x**(2*n)*b**2 + 2*x**n*a*b*n**2 - 4 *x**n*a*b*n + 2*x**n*a*b + a**2*n**2 - 2*a**2*n + a**2),x)*a*c**2*e**3*n + x**(2*n)*int(x**(2*n)/(x**(4*n)*c**2*n**2 - 2*x**(4*n)*c**2*n + x**(4*...