Integrand size = 35, antiderivative size = 73 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=\frac {(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {c d (a e+c d x)^4}{20 \left (c d^2-a e^2\right )^2 (d+e x)^4} \] Output:
1/5*(c*d*x+a*e)^4/(-a*e^2+c*d^2)/(e*x+d)^5+1/20*c*d*(c*d*x+a*e)^4/(-a*e^2+ c*d^2)^2/(e*x+d)^4
Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=-\frac {4 a^3 e^6+3 a^2 c d e^4 (d+5 e x)+2 a c^2 d^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+c^3 d^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )}{20 e^4 (d+e x)^5} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^9,x]
Output:
-1/20*(4*a^3*e^6 + 3*a^2*c*d*e^4*(d + 5*e*x) + 2*a*c^2*d^2*e^2*(d^2 + 5*d* e*x + 10*e^2*x^2) + c^3*d^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)) /(e^4*(d + e*x)^5)
Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1121, 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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3}{(d+e x)^9} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)^4}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^5}+\frac {\left (a e^2-c d^2\right )^3}{e^3 (d+e x)^6}+\frac {c^3 d^3}{e^3 (d+e x)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c^2 d^2 \left (c d^2-a e^2\right )}{e^4 (d+e x)^3}-\frac {3 c d \left (c d^2-a e^2\right )^2}{4 e^4 (d+e x)^4}+\frac {\left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^5}-\frac {c^3 d^3}{2 e^4 (d+e x)^2}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^9,x]
Output:
(c*d^2 - a*e^2)^3/(5*e^4*(d + e*x)^5) - (3*c*d*(c*d^2 - a*e^2)^2)/(4*e^4*( d + e*x)^4) + (c^2*d^2*(c*d^2 - a*e^2))/(e^4*(d + e*x)^3) - (c^3*d^3)/(2*e ^4*(d + e*x)^2)
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 1.34 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.77
method | result | size |
risch | \(\frac {-\frac {c^{3} d^{3} x^{3}}{2 e}-\frac {d^{2} c^{2} \left (2 a \,e^{2}+c \,d^{2}\right ) x^{2}}{2 e^{2}}-\frac {d c \left (3 a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{4 e^{3}}-\frac {4 e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +2 d^{4} e^{2} a \,c^{2}+d^{6} c^{3}}{20 e^{4}}}{\left (e x +d \right )^{5}}\) | \(129\) |
gosper | \(-\frac {10 c^{3} d^{3} e^{3} x^{3}+20 x^{2} a \,c^{2} d^{2} e^{4}+10 c^{3} d^{4} e^{2} x^{2}+15 x \,a^{2} c d \,e^{5}+10 x a \,c^{2} d^{3} e^{3}+5 c^{3} d^{5} e x +4 e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +2 d^{4} e^{2} a \,c^{2}+d^{6} c^{3}}{20 e^{4} \left (e x +d \right )^{5}}\) | \(130\) |
parallelrisch | \(\frac {-10 d^{3} c^{3} x^{3} e^{4}-20 a \,c^{2} d^{2} e^{5} x^{2}-10 c^{3} d^{4} e^{3} x^{2}-15 a^{2} c d \,e^{6} x -10 a \,c^{2} d^{3} e^{4} x -5 c^{3} d^{5} e^{2} x -4 a^{3} e^{7}-3 a^{2} e^{5} c \,d^{2}-2 a \,e^{3} c^{2} d^{4}-d^{6} c^{3} e}{20 e^{5} \left (e x +d \right )^{5}}\) | \(134\) |
default | \(-\frac {c^{2} d^{2} \left (a \,e^{2}-c \,d^{2}\right )}{e^{4} \left (e x +d \right )^{3}}-\frac {3 d c \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{4 e^{4} \left (e x +d \right )^{4}}-\frac {e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}}{5 e^{4} \left (e x +d \right )^{5}}-\frac {d^{3} c^{3}}{2 e^{4} \left (e x +d \right )^{2}}\) | \(141\) |
orering | \(-\frac {\left (10 c^{3} d^{3} e^{3} x^{3}+20 x^{2} a \,c^{2} d^{2} e^{4}+10 c^{3} d^{4} e^{2} x^{2}+15 x \,a^{2} c d \,e^{5}+10 x a \,c^{2} d^{3} e^{3}+5 c^{3} d^{5} e x +4 e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +2 d^{4} e^{2} a \,c^{2}+d^{6} c^{3}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{3}}{20 e^{4} \left (c d x +a e \right )^{3} \left (e x +d \right )^{8}}\) | \(167\) |
norman | \(\frac {-\frac {d^{3} \left (4 a^{3} e^{10}+3 a^{2} c \,d^{2} e^{8}+2 d^{4} a \,c^{2} e^{6}+c^{3} d^{6} e^{4}\right )}{20 e^{8}}-\frac {\left (a^{3} e^{10}+12 a^{2} c \,d^{2} e^{8}+23 d^{4} a \,c^{2} e^{6}+14 c^{3} d^{6} e^{4}\right ) x^{3}}{5 e^{5}}-\frac {d \left (3 a^{2} c \,e^{8}+14 a \,e^{6} c^{2} d^{2}+13 d^{4} c^{3} e^{4}\right ) x^{4}}{4 e^{4}}-\frac {d \left (6 a^{3} e^{10}+27 a^{2} c \,d^{2} e^{8}+28 d^{4} a \,c^{2} e^{6}+14 c^{3} d^{6} e^{4}\right ) x^{2}}{10 e^{6}}-\frac {e^{2} d^{3} c^{3} x^{6}}{2}-\frac {d^{2} \left (a \,c^{2} e^{6}+2 c^{3} d^{2} e^{4}\right ) x^{5}}{e^{3}}-\frac {d^{2} \left (3 a^{3} e^{10}+6 a^{2} c \,d^{2} e^{8}+4 d^{4} a \,c^{2} e^{6}+2 c^{3} d^{6} e^{4}\right ) x}{5 e^{7}}}{\left (e x +d \right )^{8}}\) | \(305\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3/(e*x+d)^9,x,method=_RETURNVERBOSE)
Output:
(-1/2*c^3*d^3/e*x^3-1/2*d^2/e^2*c^2*(2*a*e^2+c*d^2)*x^2-1/4*d*c/e^3*(3*a^2 *e^4+2*a*c*d^2*e^2+c^2*d^4)*x-1/20/e^4*(4*a^3*e^6+3*a^2*c*d^2*e^4+2*a*c^2* d^4*e^2+c^3*d^6))/(e*x+d)^5
Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (69) = 138\).
Time = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=-\frac {10 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \, {\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^9,x, algorithm="fric as")
Output:
-1/20*(10*c^3*d^3*e^3*x^3 + c^3*d^6 + 2*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 4*a^3*e^6 + 10*(c^3*d^4*e^2 + 2*a*c^2*d^2*e^4)*x^2 + 5*(c^3*d^5*e + 2*a*c^ 2*d^3*e^3 + 3*a^2*c*d*e^5)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10 *d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (61) = 122\).
Time = 4.61 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.56 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=\frac {- 4 a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} - 2 a c^{2} d^{4} e^{2} - c^{3} d^{6} - 10 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 20 a c^{2} d^{2} e^{4} - 10 c^{3} d^{4} e^{2}\right ) + x \left (- 15 a^{2} c d e^{5} - 10 a c^{2} d^{3} e^{3} - 5 c^{3} d^{5} e\right )}{20 d^{5} e^{4} + 100 d^{4} e^{5} x + 200 d^{3} e^{6} x^{2} + 200 d^{2} e^{7} x^{3} + 100 d e^{8} x^{4} + 20 e^{9} x^{5}} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**9,x)
Output:
(-4*a**3*e**6 - 3*a**2*c*d**2*e**4 - 2*a*c**2*d**4*e**2 - c**3*d**6 - 10*c **3*d**3*e**3*x**3 + x**2*(-20*a*c**2*d**2*e**4 - 10*c**3*d**4*e**2) + x*( -15*a**2*c*d*e**5 - 10*a*c**2*d**3*e**3 - 5*c**3*d**5*e))/(20*d**5*e**4 + 100*d**4*e**5*x + 200*d**3*e**6*x**2 + 200*d**2*e**7*x**3 + 100*d*e**8*x** 4 + 20*e**9*x**5)
Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (69) = 138\).
Time = 0.04 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=-\frac {10 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \, {\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^9,x, algorithm="maxi ma")
Output:
-1/20*(10*c^3*d^3*e^3*x^3 + c^3*d^6 + 2*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 4*a^3*e^6 + 10*(c^3*d^4*e^2 + 2*a*c^2*d^2*e^4)*x^2 + 5*(c^3*d^5*e + 2*a*c^ 2*d^3*e^3 + 3*a^2*c*d*e^5)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10 *d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)
Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=-\frac {10 \, c^{3} d^{3} e^{3} x^{3} + 10 \, c^{3} d^{4} e^{2} x^{2} + 20 \, a c^{2} d^{2} e^{4} x^{2} + 5 \, c^{3} d^{5} e x + 10 \, a c^{2} d^{3} e^{3} x + 15 \, a^{2} c d e^{5} x + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6}}{20 \, {\left (e x + d\right )}^{5} e^{4}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^9,x, algorithm="giac ")
Output:
-1/20*(10*c^3*d^3*e^3*x^3 + 10*c^3*d^4*e^2*x^2 + 20*a*c^2*d^2*e^4*x^2 + 5* c^3*d^5*e*x + 10*a*c^2*d^3*e^3*x + 15*a^2*c*d*e^5*x + c^3*d^6 + 2*a*c^2*d^ 4*e^2 + 3*a^2*c*d^2*e^4 + 4*a^3*e^6)/((e*x + d)^5*e^4)
Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=-\frac {d^2\,\left (\frac {3\,a^2\,c}{20}+a\,c^2\,x^2-\frac {c^3\,x^4}{4}\right )-d\,\left (\frac {c^3\,e\,x^5}{20}-\frac {3\,a^2\,c\,e\,x}{4}\right )+\frac {a^3\,e^2}{5}+\frac {a\,c^2\,d^4}{10\,e^2}+\frac {a\,c^2\,d^3\,x}{2\,e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3/(d + e*x)^9,x)
Output:
-(d^2*((3*a^2*c)/20 - (c^3*x^4)/4 + a*c^2*x^2) - d*((c^3*e*x^5)/20 - (3*a^ 2*c*e*x)/4) + (a^3*e^2)/5 + (a*c^2*d^4)/(10*e^2) + (a*c^2*d^3*x)/(2*e))/(d ^5 + e^5*x^5 + 5*d*e^4*x^4 + 10*d^3*e^2*x^2 + 10*d^2*e^3*x^3 + 5*d^4*e*x)
Time = 0.22 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.38 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=\frac {-10 c^{3} d^{3} e^{3} x^{3}-20 a \,c^{2} d^{2} e^{4} x^{2}-10 c^{3} d^{4} e^{2} x^{2}-15 a^{2} c d \,e^{5} x -10 a \,c^{2} d^{3} e^{3} x -5 c^{3} d^{5} e x -4 a^{3} e^{6}-3 a^{2} c \,d^{2} e^{4}-2 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}}{20 e^{4} \left (e^{5} x^{5}+5 d \,e^{4} x^{4}+10 d^{2} e^{3} x^{3}+10 d^{3} e^{2} x^{2}+5 d^{4} e x +d^{5}\right )} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^9,x)
Output:
( - 4*a**3*e**6 - 3*a**2*c*d**2*e**4 - 15*a**2*c*d*e**5*x - 2*a*c**2*d**4* e**2 - 10*a*c**2*d**3*e**3*x - 20*a*c**2*d**2*e**4*x**2 - c**3*d**6 - 5*c* *3*d**5*e*x - 10*c**3*d**4*e**2*x**2 - 10*c**3*d**3*e**3*x**3)/(20*e**4*(d **5 + 5*d**4*e*x + 10*d**3*e**2*x**2 + 10*d**2*e**3*x**3 + 5*d*e**4*x**4 + e**5*x**5))