Integrand size = 35, antiderivative size = 221 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {20 e^3 \left (c d^2-a e^2\right )^3 x}{c^6 d^6}-\frac {\left (c d^2-a e^2\right )^6}{2 c^7 d^7 (a e+c d x)^2}-\frac {6 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{2 c^7 d^7}+\frac {2 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^3}{c^7 d^7}+\frac {e^6 (a e+c d x)^4}{4 c^7 d^7}+\frac {15 e^2 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^7 d^7} \] Output:
20*e^3*(-a*e^2+c*d^2)^3*x/c^6/d^6-1/2*(-a*e^2+c*d^2)^6/c^7/d^7/(c*d*x+a*e) ^2-6*e*(-a*e^2+c*d^2)^5/c^7/d^7/(c*d*x+a*e)+15/2*e^4*(-a*e^2+c*d^2)^2*(c*d *x+a*e)^2/c^7/d^7+2*e^5*(-a*e^2+c*d^2)*(c*d*x+a*e)^3/c^7/d^7+1/4*e^6*(c*d* x+a*e)^4/c^7/d^7+15*e^2*(-a*e^2+c*d^2)^4*ln(c*d*x+a*e)/c^7/d^7
Time = 0.10 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.52 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {22 a^6 e^{12}-4 a^5 c d e^{10} (27 d+4 e x)+2 a^4 c^2 d^2 e^8 \left (105 d^2+12 d e x-34 e^2 x^2\right )-4 a^3 c^3 d^3 e^6 \left (50 d^3-15 d^2 e x-63 d e^2 x^2+5 e^3 x^3\right )+5 a^2 c^4 d^4 e^4 \left (18 d^4-32 d^3 e x-66 d^2 e^2 x^2+16 d e^3 x^3+e^4 x^4\right )-2 a c^5 d^5 e^2 \left (6 d^5-60 d^4 e x-80 d^3 e^2 x^2+60 d^2 e^3 x^3+10 d e^4 x^4+e^5 x^5\right )+c^6 d^6 \left (-2 d^6-24 d^5 e x+80 d^3 e^3 x^3+30 d^2 e^4 x^4+8 d e^5 x^5+e^6 x^6\right )+60 e^2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2 \log (a e+c d x)}{4 c^7 d^7 (a e+c d x)^2} \] Input:
Integrate[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
Output:
(22*a^6*e^12 - 4*a^5*c*d*e^10*(27*d + 4*e*x) + 2*a^4*c^2*d^2*e^8*(105*d^2 + 12*d*e*x - 34*e^2*x^2) - 4*a^3*c^3*d^3*e^6*(50*d^3 - 15*d^2*e*x - 63*d*e ^2*x^2 + 5*e^3*x^3) + 5*a^2*c^4*d^4*e^4*(18*d^4 - 32*d^3*e*x - 66*d^2*e^2* x^2 + 16*d*e^3*x^3 + e^4*x^4) - 2*a*c^5*d^5*e^2*(6*d^5 - 60*d^4*e*x - 80*d ^3*e^2*x^2 + 60*d^2*e^3*x^3 + 10*d*e^4*x^4 + e^5*x^5) + c^6*d^6*(-2*d^6 - 24*d^5*e*x + 80*d^3*e^3*x^3 + 30*d^2*e^4*x^4 + 8*d*e^5*x^5 + e^6*x^6) + 60 *e^2*(c*d^2 - a*e^2)^4*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(4*c^7*d^7*(a*e + c*d*x)^2)
Time = 0.53 (sec) , antiderivative size = 221, 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.
| \(\displaystyle \int \frac {(d+e x)^9}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {e^6 (a e+c d x)^3}{c^6 d^6}+\frac {20 \left (c d^2 e-a e^3\right )^3}{c^6 d^6}+\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^6 d^6 (a e+c d x)}+\frac {6 e \left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)^2}+\frac {\left (c d^2-a e^2\right )^6}{c^6 d^6 (a e+c d x)^3}+\frac {6 \left (c d^2 e^5-a e^7\right ) (a e+c d x)^2}{c^6 d^6}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)}{c^6 d^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^6 (a e+c d x)^4}{4 c^7 d^7}-\frac {6 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^6}{2 c^7 d^7 (a e+c d x)^2}+\frac {15 e^2 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^7 d^7}+\frac {2 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^3}{c^7 d^7}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{2 c^7 d^7}+\frac {20 e^3 x \left (c d^2-a e^2\right )^3}{c^6 d^6}\) |
Input:
Int[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
Output:
(20*e^3*(c*d^2 - a*e^2)^3*x)/(c^6*d^6) - (c*d^2 - a*e^2)^6/(2*c^7*d^7*(a*e + c*d*x)^2) - (6*e*(c*d^2 - a*e^2)^5)/(c^7*d^7*(a*e + c*d*x)) + (15*e^4*( c*d^2 - a*e^2)^2*(a*e + c*d*x)^2)/(2*c^7*d^7) + (2*e^5*(c*d^2 - a*e^2)*(a* e + c*d*x)^3)/(c^7*d^7) + (e^6*(a*e + c*d*x)^4)/(4*c^7*d^7) + (15*e^2*(c*d ^2 - a*e^2)^4*Log[a*e + c*d*x])/(c^7*d^7)
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 = 1.61 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.81
| method | result | size |
| default | \(-\frac {e^{3} \left (-\frac {1}{4} x^{4} c^{3} d^{3} e^{3}+x^{3} a \,c^{2} d^{2} e^{4}-2 x^{3} c^{3} d^{4} e^{2}-3 x^{2} a^{2} c d \,e^{5}+9 x^{2} a \,c^{2} d^{3} e^{3}-\frac {15}{2} x^{2} c^{3} d^{5} e +10 e^{6} a^{3} x -36 d^{2} e^{4} a^{2} c x +45 d^{4} e^{2} a \,c^{2} x -20 d^{6} c^{3} x \right )}{d^{6} c^{6}}+\frac {15 e^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \ln \left (c d x +a e \right )}{c^{7} d^{7}}-\frac {a^{6} e^{12}-6 a^{5} d^{2} e^{10} c +15 a^{4} d^{4} e^{8} c^{2}-20 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}-6 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}}{2 d^{7} c^{7} \left (c d x +a e \right )^{2}}+\frac {6 e \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}{d^{7} c^{7} \left (c d x +a e \right )}\) | \(399\) |
| risch | \(\frac {e^{6} x^{4}}{4 d^{3} c^{3}}-\frac {e^{7} x^{3} a}{d^{4} c^{4}}+\frac {2 e^{5} x^{3}}{d^{2} c^{3}}+\frac {3 e^{8} x^{2} a^{2}}{d^{5} c^{5}}-\frac {9 e^{6} x^{2} a}{d^{3} c^{4}}+\frac {15 e^{4} x^{2}}{2 d \,c^{3}}-\frac {10 e^{9} a^{3} x}{d^{6} c^{6}}+\frac {36 e^{7} a^{2} x}{d^{4} c^{5}}-\frac {45 e^{5} a x}{d^{2} c^{4}}+\frac {20 e^{3} x}{c^{3}}+\frac {\left (6 a^{5} e^{11}-30 a^{4} d^{2} e^{9} c +60 a^{3} d^{4} e^{7} c^{2}-60 a^{2} d^{6} e^{5} c^{3}+30 a \,c^{4} d^{8} e^{3}-6 d^{10} e \,c^{5}\right ) x +\frac {11 a^{6} e^{12}-54 a^{5} d^{2} e^{10} c +105 a^{4} d^{4} e^{8} c^{2}-100 a^{3} d^{6} e^{6} c^{3}+45 a^{2} d^{8} e^{4} c^{4}-6 a \,d^{10} e^{2} c^{5}-d^{12} c^{6}}{2 c d}}{d^{6} c^{6} \left (c d x +a e \right )^{2}}+\frac {15 e^{10} \ln \left (c d x +a e \right ) a^{4}}{c^{7} d^{7}}-\frac {60 e^{8} \ln \left (c d x +a e \right ) a^{3}}{c^{6} d^{5}}+\frac {90 e^{6} \ln \left (c d x +a e \right ) a^{2}}{c^{5} d^{3}}-\frac {60 e^{4} \ln \left (c d x +a e \right ) a}{c^{4} d}+\frac {15 e^{2} d \ln \left (c d x +a e \right )}{c^{3}}\) | \(432\) |
| norman | \(\frac {\frac {90 a^{6} e^{12}-320 a^{5} d^{2} e^{10} c +375 a^{4} d^{4} e^{8} c^{2}-100 a^{3} d^{6} e^{6} c^{3}-100 a^{2} d^{8} e^{4} c^{4}-12 a \,d^{10} e^{2} c^{5}-2 d^{12} c^{6}}{4 c^{7} d^{5}}+\frac {e^{8} x^{8}}{4 c d}+\frac {\left (90 a^{6} e^{16}-80 c \,d^{2} a^{5} e^{14}-425 a^{4} c^{2} d^{4} e^{12}+720 a^{3} d^{6} c^{3} e^{10}-185 a^{2} c^{4} d^{8} e^{8}-272 a \,c^{5} d^{10} e^{6}-240 d^{12} c^{6} e^{4}\right ) x^{2}}{4 d^{7} c^{7} e^{2}}+\frac {\left (60 a^{5} e^{14}-200 a^{4} c \,d^{2} e^{12}+225 a^{3} c^{2} d^{4} e^{10}-105 a^{2} c^{3} d^{6} e^{8}+70 a \,c^{4} d^{8} e^{6}-162 c^{5} d^{10} e^{4}\right ) x^{3}}{2 d^{6} c^{6} e}-\frac {e^{5} \left (10 e^{6} a^{3}-45 d^{2} e^{4} a^{2} c +81 d^{4} e^{2} a \,c^{2}-74 d^{6} c^{3}\right ) x^{5}}{2 d^{4} c^{4}}+\frac {e^{6} \left (5 a^{2} e^{4}-24 a c \,d^{2} e^{2}+47 c^{2} d^{4}\right ) x^{6}}{4 d^{3} c^{3}}-\frac {e^{7} \left (a \,e^{2}-5 c \,d^{2}\right ) x^{7}}{2 c^{2} d^{2}}+\frac {\left (90 a^{6} e^{14}-260 a^{5} c \,d^{2} e^{12}+175 a^{4} c^{2} d^{4} e^{10}+95 a^{3} c^{3} d^{6} e^{8}-80 a^{2} c^{4} d^{8} e^{6}-142 a \,c^{5} d^{10} e^{4}-14 d^{12} c^{6} e^{2}\right ) x}{2 c^{7} d^{6} e}}{\left (e x +d \right )^{2} \left (c d x +a e \right )^{2}}+\frac {15 e^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \ln \left (c d x +a e \right )}{c^{7} d^{7}}\) | \(605\) |
| parallelrisch | \(\frac {120 a^{5} c d \,e^{11} x -480 a^{4} c^{2} d^{3} e^{9} x +720 a^{3} c^{3} d^{5} e^{7} x -480 a^{2} c^{4} d^{7} e^{5} x -2 x^{5} a \,c^{5} d^{5} e^{7}+5 x^{4} a^{2} c^{4} d^{4} e^{8}-20 x^{4} a \,c^{5} d^{6} e^{6}-20 x^{3} a^{3} c^{3} d^{3} e^{9}+80 x^{3} a^{2} c^{4} d^{5} e^{7}-120 x^{3} a \,c^{5} d^{7} e^{5}+90 a^{6} e^{12}-2 d^{12} c^{6}+120 \ln \left (c d x +a e \right ) x \,a^{5} c d \,e^{11}-480 \ln \left (c d x +a e \right ) x \,a^{4} c^{2} d^{3} e^{9}+720 \ln \left (c d x +a e \right ) x \,a^{3} c^{3} d^{5} e^{7}-480 \ln \left (c d x +a e \right ) x \,a^{2} c^{4} d^{7} e^{5}+120 \ln \left (c d x +a e \right ) x a \,c^{5} d^{9} e^{3}+90 a^{2} d^{8} e^{4} c^{4}-360 a^{3} d^{6} e^{6} c^{3}+540 a^{4} d^{4} e^{8} c^{2}-360 a^{5} d^{2} e^{10} c +x^{6} e^{6} c^{6} d^{6}+8 x^{5} c^{6} d^{7} e^{5}+30 x^{4} c^{6} d^{8} e^{4}+80 x^{3} c^{6} d^{9} e^{3}+60 \ln \left (c d x +a e \right ) a^{2} c^{4} d^{8} e^{4}-24 c^{6} d^{11} e x -12 a \,d^{10} e^{2} c^{5}-240 \ln \left (c d x +a e \right ) a^{5} c \,d^{2} e^{10}+360 \ln \left (c d x +a e \right ) a^{4} c^{2} d^{4} e^{8}-240 \ln \left (c d x +a e \right ) a^{3} c^{3} d^{6} e^{6}+60 \ln \left (c d x +a e \right ) a^{6} e^{12}+60 \ln \left (c d x +a e \right ) x^{2} a^{4} c^{2} d^{2} e^{10}-240 \ln \left (c d x +a e \right ) x^{2} a^{3} c^{3} d^{4} e^{8}+360 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{4} d^{6} e^{6}-240 \ln \left (c d x +a e \right ) x^{2} a \,c^{5} d^{8} e^{4}+60 \ln \left (c d x +a e \right ) x^{2} c^{6} d^{10} e^{2}+120 x a \,c^{5} d^{9} e^{3}}{4 d^{7} c^{7} \left (c d x +a e \right )^{2}}\) | \(678\) |
Input:
int((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3,x,method=_RETURNVERBOSE)
Output:
-e^3/d^6/c^6*(-1/4*x^4*c^3*d^3*e^3+x^3*a*c^2*d^2*e^4-2*x^3*c^3*d^4*e^2-3*x ^2*a^2*c*d*e^5+9*x^2*a*c^2*d^3*e^3-15/2*x^2*c^3*d^5*e+10*e^6*a^3*x-36*d^2* e^4*a^2*c*x+45*d^4*e^2*a*c^2*x-20*d^6*c^3*x)+15*e^2/c^7/d^7*(a^4*e^8-4*a^3 *c*d^2*e^6+6*a^2*c^2*d^4*e^4-4*a*c^3*d^6*e^2+c^4*d^8)*ln(c*d*x+a*e)-1/2/d^ 7/c^7*(a^6*e^12-6*a^5*c*d^2*e^10+15*a^4*c^2*d^4*e^8-20*a^3*c^3*d^6*e^6+15* a^2*c^4*d^8*e^4-6*a*c^5*d^10*e^2+c^6*d^12)/(c*d*x+a*e)^2+6/d^7*e/c^7*(a^5* e^10-5*a^4*c*d^2*e^8+10*a^3*c^2*d^4*e^6-10*a^2*c^3*d^6*e^4+5*a*c^4*d^8*e^2 -c^5*d^10)/(c*d*x+a*e)
Leaf count of result is larger than twice the leaf count of optimal. 606 vs. \(2 (215) = 430\).
Time = 0.09 (sec) , antiderivative size = 606, normalized size of antiderivative = 2.74 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {c^{6} d^{6} e^{6} x^{6} - 2 \, c^{6} d^{12} - 12 \, a c^{5} d^{10} e^{2} + 90 \, a^{2} c^{4} d^{8} e^{4} - 200 \, a^{3} c^{3} d^{6} e^{6} + 210 \, a^{4} c^{2} d^{4} e^{8} - 108 \, a^{5} c d^{2} e^{10} + 22 \, a^{6} e^{12} + 2 \, {\left (4 \, c^{6} d^{7} e^{5} - a c^{5} d^{5} e^{7}\right )} x^{5} + 5 \, {\left (6 \, c^{6} d^{8} e^{4} - 4 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} + 20 \, {\left (4 \, c^{6} d^{9} e^{3} - 6 \, a c^{5} d^{7} e^{5} + 4 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 2 \, {\left (80 \, a c^{5} d^{8} e^{4} - 165 \, a^{2} c^{4} d^{6} e^{6} + 126 \, a^{3} c^{3} d^{4} e^{8} - 34 \, a^{4} c^{2} d^{2} e^{10}\right )} x^{2} - 4 \, {\left (6 \, c^{6} d^{11} e - 30 \, a c^{5} d^{9} e^{3} + 40 \, a^{2} c^{4} d^{7} e^{5} - 15 \, a^{3} c^{3} d^{5} e^{7} - 6 \, a^{4} c^{2} d^{3} e^{9} + 4 \, a^{5} c d e^{11}\right )} x + 60 \, {\left (a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} + 6 \, a^{4} c^{2} d^{4} e^{8} - 4 \, a^{5} c d^{2} e^{10} + a^{6} e^{12} + {\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 2 \, {\left (a c^{5} d^{9} e^{3} - 4 \, a^{2} c^{4} d^{7} e^{5} + 6 \, a^{3} c^{3} d^{5} e^{7} - 4 \, a^{4} c^{2} d^{3} e^{9} + a^{5} c d e^{11}\right )} x\right )} \log \left (c d x + a e\right )}{4 \, {\left (c^{9} d^{9} x^{2} + 2 \, a c^{8} d^{8} e x + a^{2} c^{7} d^{7} e^{2}\right )}} \] Input:
integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fric as")
Output:
1/4*(c^6*d^6*e^6*x^6 - 2*c^6*d^12 - 12*a*c^5*d^10*e^2 + 90*a^2*c^4*d^8*e^4 - 200*a^3*c^3*d^6*e^6 + 210*a^4*c^2*d^4*e^8 - 108*a^5*c*d^2*e^10 + 22*a^6 *e^12 + 2*(4*c^6*d^7*e^5 - a*c^5*d^5*e^7)*x^5 + 5*(6*c^6*d^8*e^4 - 4*a*c^5 *d^6*e^6 + a^2*c^4*d^4*e^8)*x^4 + 20*(4*c^6*d^9*e^3 - 6*a*c^5*d^7*e^5 + 4* a^2*c^4*d^5*e^7 - a^3*c^3*d^3*e^9)*x^3 + 2*(80*a*c^5*d^8*e^4 - 165*a^2*c^4 *d^6*e^6 + 126*a^3*c^3*d^4*e^8 - 34*a^4*c^2*d^2*e^10)*x^2 - 4*(6*c^6*d^11* e - 30*a*c^5*d^9*e^3 + 40*a^2*c^4*d^7*e^5 - 15*a^3*c^3*d^5*e^7 - 6*a^4*c^2 *d^3*e^9 + 4*a^5*c*d*e^11)*x + 60*(a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 + 6 *a^4*c^2*d^4*e^8 - 4*a^5*c*d^2*e^10 + a^6*e^12 + (c^6*d^10*e^2 - 4*a*c^5*d ^8*e^4 + 6*a^2*c^4*d^6*e^6 - 4*a^3*c^3*d^4*e^8 + a^4*c^2*d^2*e^10)*x^2 + 2 *(a*c^5*d^9*e^3 - 4*a^2*c^4*d^7*e^5 + 6*a^3*c^3*d^5*e^7 - 4*a^4*c^2*d^3*e^ 9 + a^5*c*d*e^11)*x)*log(c*d*x + a*e))/(c^9*d^9*x^2 + 2*a*c^8*d^8*e*x + a^ 2*c^7*d^7*e^2)
Time = 4.72 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.75 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=x^{3} \left (- \frac {a e^{7}}{c^{4} d^{4}} + \frac {2 e^{5}}{c^{3} d^{2}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} e^{8}}{c^{5} d^{5}} - \frac {9 a e^{6}}{c^{4} d^{3}} + \frac {15 e^{4}}{2 c^{3} d}\right ) + x \left (- \frac {10 a^{3} e^{9}}{c^{6} d^{6}} + \frac {36 a^{2} e^{7}}{c^{5} d^{4}} - \frac {45 a e^{5}}{c^{4} d^{2}} + \frac {20 e^{3}}{c^{3}}\right ) + \frac {11 a^{6} e^{12} - 54 a^{5} c d^{2} e^{10} + 105 a^{4} c^{2} d^{4} e^{8} - 100 a^{3} c^{3} d^{6} e^{6} + 45 a^{2} c^{4} d^{8} e^{4} - 6 a c^{5} d^{10} e^{2} - c^{6} d^{12} + x \left (12 a^{5} c d e^{11} - 60 a^{4} c^{2} d^{3} e^{9} + 120 a^{3} c^{3} d^{5} e^{7} - 120 a^{2} c^{4} d^{7} e^{5} + 60 a c^{5} d^{9} e^{3} - 12 c^{6} d^{11} e\right )}{2 a^{2} c^{7} d^{7} e^{2} + 4 a c^{8} d^{8} e x + 2 c^{9} d^{9} x^{2}} + \frac {e^{6} x^{4}}{4 c^{3} d^{3}} + \frac {15 e^{2} \left (a e^{2} - c d^{2}\right )^{4} \log {\left (a e + c d x \right )}}{c^{7} d^{7}} \] Input:
integrate((e*x+d)**9/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
Output:
x**3*(-a*e**7/(c**4*d**4) + 2*e**5/(c**3*d**2)) + x**2*(3*a**2*e**8/(c**5* d**5) - 9*a*e**6/(c**4*d**3) + 15*e**4/(2*c**3*d)) + x*(-10*a**3*e**9/(c** 6*d**6) + 36*a**2*e**7/(c**5*d**4) - 45*a*e**5/(c**4*d**2) + 20*e**3/c**3) + (11*a**6*e**12 - 54*a**5*c*d**2*e**10 + 105*a**4*c**2*d**4*e**8 - 100*a **3*c**3*d**6*e**6 + 45*a**2*c**4*d**8*e**4 - 6*a*c**5*d**10*e**2 - c**6*d **12 + x*(12*a**5*c*d*e**11 - 60*a**4*c**2*d**3*e**9 + 120*a**3*c**3*d**5* e**7 - 120*a**2*c**4*d**7*e**5 + 60*a*c**5*d**9*e**3 - 12*c**6*d**11*e))/( 2*a**2*c**7*d**7*e**2 + 4*a*c**8*d**8*e*x + 2*c**9*d**9*x**2) + e**6*x**4/ (4*c**3*d**3) + 15*e**2*(a*e**2 - c*d**2)**4*log(a*e + c*d*x)/(c**7*d**7)
Time = 0.04 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {c^{6} d^{12} + 6 \, a c^{5} d^{10} e^{2} - 45 \, a^{2} c^{4} d^{8} e^{4} + 100 \, a^{3} c^{3} d^{6} e^{6} - 105 \, a^{4} c^{2} d^{4} e^{8} + 54 \, a^{5} c d^{2} e^{10} - 11 \, a^{6} e^{12} + 12 \, {\left (c^{6} d^{11} e - 5 \, a c^{5} d^{9} e^{3} + 10 \, a^{2} c^{4} d^{7} e^{5} - 10 \, a^{3} c^{3} d^{5} e^{7} + 5 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x}{2 \, {\left (c^{9} d^{9} x^{2} + 2 \, a c^{8} d^{8} e x + a^{2} c^{7} d^{7} e^{2}\right )}} + \frac {c^{3} d^{3} e^{6} x^{4} + 4 \, {\left (2 \, c^{3} d^{4} e^{5} - a c^{2} d^{2} e^{7}\right )} x^{3} + 6 \, {\left (5 \, c^{3} d^{5} e^{4} - 6 \, a c^{2} d^{3} e^{6} + 2 \, a^{2} c d e^{8}\right )} x^{2} + 4 \, {\left (20 \, c^{3} d^{6} e^{3} - 45 \, a c^{2} d^{4} e^{5} + 36 \, a^{2} c d^{2} e^{7} - 10 \, a^{3} e^{9}\right )} x}{4 \, c^{6} d^{6}} + \frac {15 \, {\left (c^{4} d^{8} e^{2} - 4 \, a c^{3} d^{6} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{6} - 4 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}\right )} \log \left (c d x + a e\right )}{c^{7} d^{7}} \] Input:
integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxi ma")
Output:
-1/2*(c^6*d^12 + 6*a*c^5*d^10*e^2 - 45*a^2*c^4*d^8*e^4 + 100*a^3*c^3*d^6*e ^6 - 105*a^4*c^2*d^4*e^8 + 54*a^5*c*d^2*e^10 - 11*a^6*e^12 + 12*(c^6*d^11* e - 5*a*c^5*d^9*e^3 + 10*a^2*c^4*d^7*e^5 - 10*a^3*c^3*d^5*e^7 + 5*a^4*c^2* d^3*e^9 - a^5*c*d*e^11)*x)/(c^9*d^9*x^2 + 2*a*c^8*d^8*e*x + a^2*c^7*d^7*e^ 2) + 1/4*(c^3*d^3*e^6*x^4 + 4*(2*c^3*d^4*e^5 - a*c^2*d^2*e^7)*x^3 + 6*(5*c ^3*d^5*e^4 - 6*a*c^2*d^3*e^6 + 2*a^2*c*d*e^8)*x^2 + 4*(20*c^3*d^6*e^3 - 45 *a*c^2*d^4*e^5 + 36*a^2*c*d^2*e^7 - 10*a^3*e^9)*x)/(c^6*d^6) + 15*(c^4*d^8 *e^2 - 4*a*c^3*d^6*e^4 + 6*a^2*c^2*d^4*e^6 - 4*a^3*c*d^2*e^8 + a^4*e^10)*l og(c*d*x + a*e)/(c^7*d^7)
Time = 0.15 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.82 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {15 \, {\left (c^{4} d^{8} e^{2} - 4 \, a c^{3} d^{6} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{6} - 4 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{7} d^{7}} - \frac {c^{6} d^{12} + 6 \, a c^{5} d^{10} e^{2} - 45 \, a^{2} c^{4} d^{8} e^{4} + 100 \, a^{3} c^{3} d^{6} e^{6} - 105 \, a^{4} c^{2} d^{4} e^{8} + 54 \, a^{5} c d^{2} e^{10} - 11 \, a^{6} e^{12} + 12 \, {\left (c^{6} d^{11} e - 5 \, a c^{5} d^{9} e^{3} + 10 \, a^{2} c^{4} d^{7} e^{5} - 10 \, a^{3} c^{3} d^{5} e^{7} + 5 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x}{2 \, {\left (c d x + a e\right )}^{2} c^{7} d^{7}} + \frac {c^{9} d^{9} e^{6} x^{4} + 8 \, c^{9} d^{10} e^{5} x^{3} - 4 \, a c^{8} d^{8} e^{7} x^{3} + 30 \, c^{9} d^{11} e^{4} x^{2} - 36 \, a c^{8} d^{9} e^{6} x^{2} + 12 \, a^{2} c^{7} d^{7} e^{8} x^{2} + 80 \, c^{9} d^{12} e^{3} x - 180 \, a c^{8} d^{10} e^{5} x + 144 \, a^{2} c^{7} d^{8} e^{7} x - 40 \, a^{3} c^{6} d^{6} e^{9} x}{4 \, c^{12} d^{12}} \] Input:
integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac ")
Output:
15*(c^4*d^8*e^2 - 4*a*c^3*d^6*e^4 + 6*a^2*c^2*d^4*e^6 - 4*a^3*c*d^2*e^8 + a^4*e^10)*log(abs(c*d*x + a*e))/(c^7*d^7) - 1/2*(c^6*d^12 + 6*a*c^5*d^10*e ^2 - 45*a^2*c^4*d^8*e^4 + 100*a^3*c^3*d^6*e^6 - 105*a^4*c^2*d^4*e^8 + 54*a ^5*c*d^2*e^10 - 11*a^6*e^12 + 12*(c^6*d^11*e - 5*a*c^5*d^9*e^3 + 10*a^2*c^ 4*d^7*e^5 - 10*a^3*c^3*d^5*e^7 + 5*a^4*c^2*d^3*e^9 - a^5*c*d*e^11)*x)/((c* d*x + a*e)^2*c^7*d^7) + 1/4*(c^9*d^9*e^6*x^4 + 8*c^9*d^10*e^5*x^3 - 4*a*c^ 8*d^8*e^7*x^3 + 30*c^9*d^11*e^4*x^2 - 36*a*c^8*d^9*e^6*x^2 + 12*a^2*c^7*d^ 7*e^8*x^2 + 80*c^9*d^12*e^3*x - 180*a*c^8*d^10*e^5*x + 144*a^2*c^7*d^8*e^7 *x - 40*a^3*c^6*d^6*e^9*x)/(c^12*d^12)
Time = 5.28 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.33 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=x^3\,\left (\frac {2\,e^5}{c^3\,d^2}-\frac {a\,e^7}{c^4\,d^4}\right )-x^2\,\left (\frac {3\,a^2\,e^8}{2\,c^5\,d^5}-\frac {15\,e^4}{2\,c^3\,d}+\frac {3\,a\,e\,\left (\frac {6\,e^5}{c^3\,d^2}-\frac {3\,a\,e^7}{c^4\,d^4}\right )}{2\,c\,d}\right )+\frac {x\,\left (6\,a^5\,e^{11}-30\,a^4\,c\,d^2\,e^9+60\,a^3\,c^2\,d^4\,e^7-60\,a^2\,c^3\,d^6\,e^5+30\,a\,c^4\,d^8\,e^3-6\,c^5\,d^{10}\,e\right )-\frac {-11\,a^6\,e^{12}+54\,a^5\,c\,d^2\,e^{10}-105\,a^4\,c^2\,d^4\,e^8+100\,a^3\,c^3\,d^6\,e^6-45\,a^2\,c^4\,d^8\,e^4+6\,a\,c^5\,d^{10}\,e^2+c^6\,d^{12}}{2\,c\,d}}{a^2\,c^6\,d^6\,e^2+2\,a\,c^7\,d^7\,e\,x+c^8\,d^8\,x^2}+x\,\left (\frac {20\,e^3}{c^3}-\frac {a^3\,e^9}{c^6\,d^6}-\frac {3\,a^2\,e^2\,\left (\frac {6\,e^5}{c^3\,d^2}-\frac {3\,a\,e^7}{c^4\,d^4}\right )}{c^2\,d^2}+\frac {3\,a\,e\,\left (\frac {3\,a^2\,e^8}{c^5\,d^5}-\frac {15\,e^4}{c^3\,d}+\frac {3\,a\,e\,\left (\frac {6\,e^5}{c^3\,d^2}-\frac {3\,a\,e^7}{c^4\,d^4}\right )}{c\,d}\right )}{c\,d}\right )+\frac {e^6\,x^4}{4\,c^3\,d^3}+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (15\,a^4\,e^{10}-60\,a^3\,c\,d^2\,e^8+90\,a^2\,c^2\,d^4\,e^6-60\,a\,c^3\,d^6\,e^4+15\,c^4\,d^8\,e^2\right )}{c^7\,d^7} \] Input:
int((d + e*x)^9/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)
Output:
x^3*((2*e^5)/(c^3*d^2) - (a*e^7)/(c^4*d^4)) - x^2*((3*a^2*e^8)/(2*c^5*d^5) - (15*e^4)/(2*c^3*d) + (3*a*e*((6*e^5)/(c^3*d^2) - (3*a*e^7)/(c^4*d^4)))/ (2*c*d)) + (x*(6*a^5*e^11 - 6*c^5*d^10*e + 30*a*c^4*d^8*e^3 - 30*a^4*c*d^2 *e^9 - 60*a^2*c^3*d^6*e^5 + 60*a^3*c^2*d^4*e^7) - (c^6*d^12 - 11*a^6*e^12 + 6*a*c^5*d^10*e^2 + 54*a^5*c*d^2*e^10 - 45*a^2*c^4*d^8*e^4 + 100*a^3*c^3* d^6*e^6 - 105*a^4*c^2*d^4*e^8)/(2*c*d))/(c^8*d^8*x^2 + a^2*c^6*d^6*e^2 + 2 *a*c^7*d^7*e*x) + x*((20*e^3)/c^3 - (a^3*e^9)/(c^6*d^6) - (3*a^2*e^2*((6*e ^5)/(c^3*d^2) - (3*a*e^7)/(c^4*d^4)))/(c^2*d^2) + (3*a*e*((3*a^2*e^8)/(c^5 *d^5) - (15*e^4)/(c^3*d) + (3*a*e*((6*e^5)/(c^3*d^2) - (3*a*e^7)/(c^4*d^4) ))/(c*d)))/(c*d)) + (e^6*x^4)/(4*c^3*d^3) + (log(a*e + c*d*x)*(15*a^4*e^10 + 15*c^4*d^8*e^2 - 60*a*c^3*d^6*e^4 - 60*a^3*c*d^2*e^8 + 90*a^2*c^2*d^4*e ^6))/(c^7*d^7)
Time = 0.23 (sec) , antiderivative size = 716, normalized size of antiderivative = 3.24 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {-480 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{4} d^{7} e^{5} x +360 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{4} d^{6} e^{6} x^{2}+120 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{5} d^{9} e^{3} x -240 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{5} d^{8} e^{4} x^{2}+60 \,\mathrm {log}\left (c d x +a e \right ) a \,c^{6} d^{10} e^{2} x^{2}+30 a^{7} e^{12}+a \,c^{6} d^{6} e^{6} x^{6}+120 \,\mathrm {log}\left (c d x +a e \right ) a^{6} c d \,e^{11} x -480 \,\mathrm {log}\left (c d x +a e \right ) a^{5} c^{2} d^{3} e^{9} x +60 \,\mathrm {log}\left (c d x +a e \right ) a^{5} c^{2} d^{2} e^{10} x^{2}+720 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c^{3} d^{5} e^{7} x -240 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c^{3} d^{4} e^{8} x^{2}-240 \,\mathrm {log}\left (c d x +a e \right ) a^{6} c \,d^{2} e^{10}+360 \,\mathrm {log}\left (c d x +a e \right ) a^{5} c^{2} d^{4} e^{8}-240 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c^{3} d^{6} e^{6}+60 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{4} d^{8} e^{4}-60 a^{5} c^{2} d^{2} e^{10} x^{2}+240 a^{4} c^{3} d^{4} e^{8} x^{2}-20 a^{4} c^{3} d^{3} e^{9} x^{3}-360 a^{3} c^{4} d^{6} e^{6} x^{2}+80 a^{3} c^{4} d^{5} e^{7} x^{3}+5 a^{3} c^{4} d^{4} e^{8} x^{4}+240 a^{2} c^{5} d^{8} e^{4} x^{2}-120 a^{2} c^{5} d^{7} e^{5} x^{3}-20 a^{2} c^{5} d^{6} e^{6} x^{4}-2 a^{2} c^{5} d^{5} e^{7} x^{5}-60 a \,c^{6} d^{10} e^{2} x^{2}+80 a \,c^{6} d^{9} e^{3} x^{3}+30 a \,c^{6} d^{8} e^{4} x^{4}+8 a \,c^{6} d^{7} e^{5} x^{5}-120 a^{6} c \,d^{2} e^{10}+180 a^{5} c^{2} d^{4} e^{8}-120 a^{4} c^{3} d^{6} e^{6}+30 a^{3} c^{4} d^{8} e^{4}+60 \,\mathrm {log}\left (c d x +a e \right ) a^{7} e^{12}-2 a \,c^{6} d^{12}+12 c^{7} d^{12} x^{2}}{4 a \,c^{7} d^{7} \left (c^{2} d^{2} x^{2}+2 a c d e x +a^{2} e^{2}\right )} \] Input:
int((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)
Output:
(60*log(a*e + c*d*x)*a**7*e**12 - 240*log(a*e + c*d*x)*a**6*c*d**2*e**10 + 120*log(a*e + c*d*x)*a**6*c*d*e**11*x + 360*log(a*e + c*d*x)*a**5*c**2*d* *4*e**8 - 480*log(a*e + c*d*x)*a**5*c**2*d**3*e**9*x + 60*log(a*e + c*d*x) *a**5*c**2*d**2*e**10*x**2 - 240*log(a*e + c*d*x)*a**4*c**3*d**6*e**6 + 72 0*log(a*e + c*d*x)*a**4*c**3*d**5*e**7*x - 240*log(a*e + c*d*x)*a**4*c**3* d**4*e**8*x**2 + 60*log(a*e + c*d*x)*a**3*c**4*d**8*e**4 - 480*log(a*e + c *d*x)*a**3*c**4*d**7*e**5*x + 360*log(a*e + c*d*x)*a**3*c**4*d**6*e**6*x** 2 + 120*log(a*e + c*d*x)*a**2*c**5*d**9*e**3*x - 240*log(a*e + c*d*x)*a**2 *c**5*d**8*e**4*x**2 + 60*log(a*e + c*d*x)*a*c**6*d**10*e**2*x**2 + 30*a** 7*e**12 - 120*a**6*c*d**2*e**10 + 180*a**5*c**2*d**4*e**8 - 60*a**5*c**2*d **2*e**10*x**2 - 120*a**4*c**3*d**6*e**6 + 240*a**4*c**3*d**4*e**8*x**2 - 20*a**4*c**3*d**3*e**9*x**3 + 30*a**3*c**4*d**8*e**4 - 360*a**3*c**4*d**6* e**6*x**2 + 80*a**3*c**4*d**5*e**7*x**3 + 5*a**3*c**4*d**4*e**8*x**4 + 240 *a**2*c**5*d**8*e**4*x**2 - 120*a**2*c**5*d**7*e**5*x**3 - 20*a**2*c**5*d* *6*e**6*x**4 - 2*a**2*c**5*d**5*e**7*x**5 - 2*a*c**6*d**12 - 60*a*c**6*d** 10*e**2*x**2 + 80*a*c**6*d**9*e**3*x**3 + 30*a*c**6*d**8*e**4*x**4 + 8*a*c **6*d**7*e**5*x**5 + a*c**6*d**6*e**6*x**6 + 12*c**7*d**12*x**2)/(4*a*c**7 *d**7*(a**2*e**2 + 2*a*c*d*e*x + c**2*d**2*x**2))