Integrand size = 35, antiderivative size = 111 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=\frac {\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac {c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{5 e^4 (d+e x)^5}-\frac {c^3 d^3}{4 e^4 (d+e x)^4} \] Output:
1/7*(-a*e^2+c*d^2)^3/e^4/(e*x+d)^7-1/2*c*d*(-a*e^2+c*d^2)^2/e^4/(e*x+d)^6+ 3/5*c^2*d^2*(-a*e^2+c*d^2)/e^4/(e*x+d)^5-1/4*c^3*d^3/e^4/(e*x+d)^4
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {20 a^3 e^6+10 a^2 c d e^4 (d+7 e x)+4 a c^2 d^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+c^3 d^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )}{140 e^4 (d+e x)^7} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^11,x]
Output:
-1/140*(20*a^3*e^6 + 10*a^2*c*d*e^4*(d + 7*e*x) + 4*a*c^2*d^2*e^2*(d^2 + 7 *d*e*x + 21*e^2*x^2) + c^3*d^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^ 3))/(e^4*(d + e*x)^7)
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, 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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3}{(d+e x)^{11}} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)^6}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^7}+\frac {\left (a e^2-c d^2\right )^3}{e^3 (d+e x)^8}+\frac {c^3 d^3}{e^3 (d+e x)^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{5 e^4 (d+e x)^5}-\frac {c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac {\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac {c^3 d^3}{4 e^4 (d+e x)^4}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^11,x]
Output:
(c*d^2 - a*e^2)^3/(7*e^4*(d + e*x)^7) - (c*d*(c*d^2 - a*e^2)^2)/(2*e^4*(d + e*x)^6) + (3*c^2*d^2*(c*d^2 - a*e^2))/(5*e^4*(d + e*x)^5) - (c^3*d^3)/(4 *e^4*(d + e*x)^4)
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 = 1.41 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.16
method | result | size |
risch | \(\frac {-\frac {c^{3} d^{3} x^{3}}{4 e}-\frac {3 d^{2} c^{2} \left (4 a \,e^{2}+c \,d^{2}\right ) x^{2}}{20 e^{2}}-\frac {d c \left (10 a^{2} e^{4}+4 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{20 e^{3}}-\frac {20 e^{6} a^{3}+10 d^{2} e^{4} a^{2} c +4 d^{4} e^{2} a \,c^{2}+d^{6} c^{3}}{140 e^{4}}}{\left (e x +d \right )^{7}}\) | \(129\) |
gosper | \(-\frac {35 c^{3} d^{3} e^{3} x^{3}+84 x^{2} a \,c^{2} d^{2} e^{4}+21 c^{3} d^{4} e^{2} x^{2}+70 x \,a^{2} c d \,e^{5}+28 x a \,c^{2} d^{3} e^{3}+7 c^{3} d^{5} e x +20 e^{6} a^{3}+10 d^{2} e^{4} a^{2} c +4 d^{4} e^{2} a \,c^{2}+d^{6} c^{3}}{140 e^{4} \left (e x +d \right )^{7}}\) | \(130\) |
parallelrisch | \(\frac {-35 d^{3} c^{3} x^{3} e^{6}-84 a \,c^{2} d^{2} e^{7} x^{2}-21 c^{3} d^{4} e^{5} x^{2}-70 a^{2} c d \,e^{8} x -28 a \,c^{2} d^{3} e^{6} x -7 c^{3} d^{5} e^{4} x -20 a^{3} e^{9}-10 a^{2} c \,d^{2} e^{7}-4 d^{4} a \,c^{2} e^{5}-c^{3} d^{6} e^{3}}{140 e^{7} \left (e x +d \right )^{7}}\) | \(136\) |
default | \(-\frac {c^{3} d^{3}}{4 e^{4} \left (e x +d \right )^{4}}-\frac {e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}}{7 e^{4} \left (e x +d \right )^{7}}-\frac {3 c^{2} d^{2} \left (a \,e^{2}-c \,d^{2}\right )}{5 e^{4} \left (e x +d \right )^{5}}-\frac {d c \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{2 e^{4} \left (e x +d \right )^{6}}\) | \(141\) |
orering | \(-\frac {\left (35 c^{3} d^{3} e^{3} x^{3}+84 x^{2} a \,c^{2} d^{2} e^{4}+21 c^{3} d^{4} e^{2} x^{2}+70 x \,a^{2} c d \,e^{5}+28 x a \,c^{2} d^{3} e^{3}+7 c^{3} d^{5} e x +20 e^{6} a^{3}+10 d^{2} e^{4} a^{2} c +4 d^{4} e^{2} a \,c^{2}+d^{6} c^{3}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{3}}{140 e^{4} \left (c d x +a e \right )^{3} \left (e x +d \right )^{10}}\) | \(167\) |
norman | \(\frac {-\frac {d^{3} \left (20 a^{3} e^{12}+10 a^{2} c \,d^{2} e^{10}+4 d^{4} a \,c^{2} e^{8}+c^{3} d^{6} e^{6}\right )}{140 e^{10}}-\frac {\left (a^{3} e^{12}+11 a^{2} c \,d^{2} e^{10}+17 d^{4} a \,c^{2} e^{8}+6 c^{3} d^{6} e^{6}\right ) x^{3}}{7 e^{7}}-\frac {d \left (2 a^{2} c \,e^{10}+8 a \,c^{2} d^{2} e^{8}+5 d^{4} c^{3} e^{6}\right ) x^{4}}{4 e^{6}}-\frac {3 d \left (4 a^{3} e^{12}+16 a^{2} c \,d^{2} e^{10}+12 d^{4} a \,c^{2} e^{8}+3 c^{3} d^{6} e^{6}\right ) x^{2}}{28 e^{8}}-\frac {3 d^{2} \left (2 a \,c^{2} e^{8}+3 c^{3} d^{2} e^{6}\right ) x^{5}}{10 e^{5}}-\frac {d^{2} \left (6 a^{3} e^{12}+10 a^{2} c \,d^{2} e^{10}+4 d^{4} a \,c^{2} e^{8}+c^{3} d^{6} e^{6}\right ) x}{14 e^{9}}-\frac {e^{2} d^{3} c^{3} x^{6}}{4}}{\left (e x +d \right )^{10}}\) | \(305\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3/(e*x+d)^11,x,method=_RETURNVERBOSE )
Output:
(-1/4*c^3*d^3/e*x^3-3/20*d^2*c^2/e^2*(4*a*e^2+c*d^2)*x^2-1/20*d*c/e^3*(10* a^2*e^4+4*a*c*d^2*e^2+c^2*d^4)*x-1/140/e^4*(20*a^3*e^6+10*a^2*c*d^2*e^4+4* a*c^2*d^4*e^2+c^3*d^6))/(e*x+d)^7
Time = 0.09 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {35 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 4 \, a c^{2} d^{4} e^{2} + 10 \, a^{2} c d^{2} e^{4} + 20 \, a^{3} e^{6} + 21 \, {\left (c^{3} d^{4} e^{2} + 4 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \, {\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + 10 \, a^{2} c d e^{5}\right )} x}{140 \, {\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^11,x, algorithm="fri cas")
Output:
-1/140*(35*c^3*d^3*e^3*x^3 + c^3*d^6 + 4*a*c^2*d^4*e^2 + 10*a^2*c*d^2*e^4 + 20*a^3*e^6 + 21*(c^3*d^4*e^2 + 4*a*c^2*d^2*e^4)*x^2 + 7*(c^3*d^5*e + 4*a *c^2*d^3*e^3 + 10*a^2*c*d*e^5)*x)/(e^11*x^7 + 7*d*e^10*x^6 + 21*d^2*e^9*x^ 5 + 35*d^3*e^8*x^4 + 35*d^4*e^7*x^3 + 21*d^5*e^6*x^2 + 7*d^6*e^5*x + d^7*e ^4)
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (99) = 198\).
Time = 50.32 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=\frac {- 20 a^{3} e^{6} - 10 a^{2} c d^{2} e^{4} - 4 a c^{2} d^{4} e^{2} - c^{3} d^{6} - 35 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 84 a c^{2} d^{2} e^{4} - 21 c^{3} d^{4} e^{2}\right ) + x \left (- 70 a^{2} c d e^{5} - 28 a c^{2} d^{3} e^{3} - 7 c^{3} d^{5} e\right )}{140 d^{7} e^{4} + 980 d^{6} e^{5} x + 2940 d^{5} e^{6} x^{2} + 4900 d^{4} e^{7} x^{3} + 4900 d^{3} e^{8} x^{4} + 2940 d^{2} e^{9} x^{5} + 980 d e^{10} x^{6} + 140 e^{11} x^{7}} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**11,x)
Output:
(-20*a**3*e**6 - 10*a**2*c*d**2*e**4 - 4*a*c**2*d**4*e**2 - c**3*d**6 - 35 *c**3*d**3*e**3*x**3 + x**2*(-84*a*c**2*d**2*e**4 - 21*c**3*d**4*e**2) + x *(-70*a**2*c*d*e**5 - 28*a*c**2*d**3*e**3 - 7*c**3*d**5*e))/(140*d**7*e**4 + 980*d**6*e**5*x + 2940*d**5*e**6*x**2 + 4900*d**4*e**7*x**3 + 4900*d**3 *e**8*x**4 + 2940*d**2*e**9*x**5 + 980*d*e**10*x**6 + 140*e**11*x**7)
Time = 0.04 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {35 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 4 \, a c^{2} d^{4} e^{2} + 10 \, a^{2} c d^{2} e^{4} + 20 \, a^{3} e^{6} + 21 \, {\left (c^{3} d^{4} e^{2} + 4 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \, {\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + 10 \, a^{2} c d e^{5}\right )} x}{140 \, {\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^11,x, algorithm="max ima")
Output:
-1/140*(35*c^3*d^3*e^3*x^3 + c^3*d^6 + 4*a*c^2*d^4*e^2 + 10*a^2*c*d^2*e^4 + 20*a^3*e^6 + 21*(c^3*d^4*e^2 + 4*a*c^2*d^2*e^4)*x^2 + 7*(c^3*d^5*e + 4*a *c^2*d^3*e^3 + 10*a^2*c*d*e^5)*x)/(e^11*x^7 + 7*d*e^10*x^6 + 21*d^2*e^9*x^ 5 + 35*d^3*e^8*x^4 + 35*d^4*e^7*x^3 + 21*d^5*e^6*x^2 + 7*d^6*e^5*x + d^7*e ^4)
Time = 0.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {35 \, c^{3} d^{3} e^{3} x^{3} + 21 \, c^{3} d^{4} e^{2} x^{2} + 84 \, a c^{2} d^{2} e^{4} x^{2} + 7 \, c^{3} d^{5} e x + 28 \, a c^{2} d^{3} e^{3} x + 70 \, a^{2} c d e^{5} x + c^{3} d^{6} + 4 \, a c^{2} d^{4} e^{2} + 10 \, a^{2} c d^{2} e^{4} + 20 \, a^{3} e^{6}}{140 \, {\left (e x + d\right )}^{7} e^{4}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^11,x, algorithm="gia c")
Output:
-1/140*(35*c^3*d^3*e^3*x^3 + 21*c^3*d^4*e^2*x^2 + 84*a*c^2*d^2*e^4*x^2 + 7 *c^3*d^5*e*x + 28*a*c^2*d^3*e^3*x + 70*a^2*c*d*e^5*x + c^3*d^6 + 4*a*c^2*d ^4*e^2 + 10*a^2*c*d^2*e^4 + 20*a^3*e^6)/((e*x + d)^7*e^4)
Time = 0.05 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {\frac {20\,a^3\,e^6+10\,a^2\,c\,d^2\,e^4+4\,a\,c^2\,d^4\,e^2+c^3\,d^6}{140\,e^4}+\frac {c^3\,d^3\,x^3}{4\,e}+\frac {c\,d\,x\,\left (10\,a^2\,e^4+4\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{20\,e^3}+\frac {3\,c^2\,d^2\,x^2\,\left (c\,d^2+4\,a\,e^2\right )}{20\,e^2}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3/(d + e*x)^11,x)
Output:
-((20*a^3*e^6 + c^3*d^6 + 4*a*c^2*d^4*e^2 + 10*a^2*c*d^2*e^4)/(140*e^4) + (c^3*d^3*x^3)/(4*e) + (c*d*x*(10*a^2*e^4 + c^2*d^4 + 4*a*c*d^2*e^2))/(20*e ^3) + (3*c^2*d^2*x^2*(4*a*e^2 + c*d^2))/(20*e^2))/(d^7 + e^7*x^7 + 7*d*e^6 *x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6*e*x)
Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=\frac {-35 c^{3} d^{3} e^{3} x^{3}-84 a \,c^{2} d^{2} e^{4} x^{2}-21 c^{3} d^{4} e^{2} x^{2}-70 a^{2} c d \,e^{5} x -28 a \,c^{2} d^{3} e^{3} x -7 c^{3} d^{5} e x -20 a^{3} e^{6}-10 a^{2} c \,d^{2} e^{4}-4 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}}{140 e^{4} \left (e^{7} x^{7}+7 d \,e^{6} x^{6}+21 d^{2} e^{5} x^{5}+35 d^{3} e^{4} x^{4}+35 d^{4} e^{3} x^{3}+21 d^{5} e^{2} x^{2}+7 d^{6} e x +d^{7}\right )} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^11,x)
Output:
( - 20*a**3*e**6 - 10*a**2*c*d**2*e**4 - 70*a**2*c*d*e**5*x - 4*a*c**2*d** 4*e**2 - 28*a*c**2*d**3*e**3*x - 84*a*c**2*d**2*e**4*x**2 - c**3*d**6 - 7* c**3*d**5*e*x - 21*c**3*d**4*e**2*x**2 - 35*c**3*d**3*e**3*x**3)/(140*e**4 *(d**7 + 7*d**6*e*x + 21*d**5*e**2*x**2 + 35*d**4*e**3*x**3 + 35*d**3*e**4 *x**4 + 21*d**2*e**5*x**5 + 7*d*e**6*x**6 + e**7*x**7))