Integrand size = 35, antiderivative size = 217 \[ \int \frac {(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^4 \left (15 c^2 d^4-24 a c d^2 e^2+10 a^2 e^4\right ) x}{c^6 d^6}+\frac {e^5 \left (3 c d^2-2 a e^2\right ) x^2}{c^5 d^5}+\frac {e^6 x^3}{3 c^4 d^4}-\frac {\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}-\frac {3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}+\frac {20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7} \] Output:
e^4*(10*a^2*e^4-24*a*c*d^2*e^2+15*c^2*d^4)*x/c^6/d^6+e^5*(-2*a*e^2+3*c*d^2 )*x^2/c^5/d^5+1/3*e^6*x^3/c^4/d^4-1/3*(-a*e^2+c*d^2)^6/c^7/d^7/(c*d*x+a*e) ^3-3*e*(-a*e^2+c*d^2)^5/c^7/d^7/(c*d*x+a*e)^2-15*e^2*(-a*e^2+c*d^2)^4/c^7/ d^7/(c*d*x+a*e)+20*e^3*(-a*e^2+c*d^2)^3*ln(c*d*x+a*e)/c^7/d^7
Time = 0.11 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.54 \[ \int \frac {(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {-37 a^6 e^{12}+3 a^5 c d e^{10} (47 d-17 e x)+3 a^4 c^2 d^2 e^8 \left (-65 d^2+81 d e x+13 e^2 x^2\right )+a^3 c^3 d^3 e^6 \left (110 d^3-405 d^2 e x-27 d e^2 x^2+73 e^3 x^3\right )-3 a^2 c^4 d^4 e^4 \left (5 d^4-90 d^3 e x+45 d^2 e^2 x^2+63 d e^3 x^3-5 e^4 x^4\right )-3 a c^5 d^5 e^2 \left (d^5+15 d^4 e x-60 d^3 e^2 x^2-45 d^2 e^3 x^3+15 d e^4 x^4+e^5 x^5\right )+c^6 d^6 \left (-d^6-9 d^5 e x-45 d^4 e^2 x^2+45 d^2 e^4 x^4+9 d e^5 x^5+e^6 x^6\right )-60 e^3 \left (-c d^2+a e^2\right )^3 (a e+c d x)^3 \log (a e+c d x)}{3 c^7 d^7 (a e+c d x)^3} \] Input:
Integrate[(d + e*x)^10/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
Output:
(-37*a^6*e^12 + 3*a^5*c*d*e^10*(47*d - 17*e*x) + 3*a^4*c^2*d^2*e^8*(-65*d^ 2 + 81*d*e*x + 13*e^2*x^2) + a^3*c^3*d^3*e^6*(110*d^3 - 405*d^2*e*x - 27*d *e^2*x^2 + 73*e^3*x^3) - 3*a^2*c^4*d^4*e^4*(5*d^4 - 90*d^3*e*x + 45*d^2*e^ 2*x^2 + 63*d*e^3*x^3 - 5*e^4*x^4) - 3*a*c^5*d^5*e^2*(d^5 + 15*d^4*e*x - 60 *d^3*e^2*x^2 - 45*d^2*e^3*x^3 + 15*d*e^4*x^4 + e^5*x^5) + c^6*d^6*(-d^6 - 9*d^5*e*x - 45*d^4*e^2*x^2 + 45*d^2*e^4*x^4 + 9*d*e^5*x^5 + e^6*x^6) - 60* e^3*(-(c*d^2) + a*e^2)^3*(a*e + c*d*x)^3*Log[a*e + c*d*x])/(3*c^7*d^7*(a*e + c*d*x)^3)
Time = 0.81 (sec) , antiderivative size = 217, 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.
| \(\displaystyle \int \frac {(d+e x)^{10}}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {10 a^2 e^8-24 a c d^2 e^6+15 c^2 d^4 e^4}{c^6 d^6}+\frac {20 \left (c d^2 e-a e^3\right )^3}{c^6 d^6 (a e+c d x)}+\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^6 d^6 (a e+c d x)^2}+\frac {6 e \left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)^3}+\frac {\left (c d^2-a e^2\right )^6}{c^6 d^6 (a e+c d x)^4}+\frac {2 e^5 x \left (3 c d^2-2 a e^2\right )}{c^5 d^5}+\frac {e^6 x^2}{c^4 d^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 x \left (10 a^2 e^4-24 a c d^2 e^2+15 c^2 d^4\right )}{c^6 d^6}-\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}-\frac {3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}+\frac {20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7}+\frac {e^5 x^2 \left (3 c d^2-2 a e^2\right )}{c^5 d^5}+\frac {e^6 x^3}{3 c^4 d^4}\) |
Input:
Int[(d + e*x)^10/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
Output:
(e^4*(15*c^2*d^4 - 24*a*c*d^2*e^2 + 10*a^2*e^4)*x)/(c^6*d^6) + (e^5*(3*c*d ^2 - 2*a*e^2)*x^2)/(c^5*d^5) + (e^6*x^3)/(3*c^4*d^4) - (c*d^2 - a*e^2)^6/( 3*c^7*d^7*(a*e + c*d*x)^3) - (3*e*(c*d^2 - a*e^2)^5)/(c^7*d^7*(a*e + c*d*x )^2) - (15*e^2*(c*d^2 - a*e^2)^4)/(c^7*d^7*(a*e + c*d*x)) + (20*e^3*(c*d^2 - a*e^2)^3*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 = 1.60 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.84
| method | result | size |
| default | \(\frac {e^{4} \left (\frac {1}{3} x^{3} c^{2} d^{2} e^{2}-2 x^{2} a c d \,e^{3}+3 x^{2} c^{2} d^{3} e +10 a^{2} e^{4} x -24 a c \,d^{2} e^{2} x +15 c^{2} d^{4} x \right )}{d^{6} c^{6}}-\frac {a^{6} e^{12}-6 a^{5} d^{2} e^{10} c +15 a^{4} d^{4} e^{8} c^{2}-20 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}-6 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}}{3 d^{7} c^{7} \left (c d x +a e \right )^{3}}-\frac {20 e^{3} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \ln \left (c d x +a e \right )}{d^{7} c^{7}}+\frac {3 e \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}{d^{7} c^{7} \left (c d x +a e \right )^{2}}-\frac {15 e^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}{c^{7} d^{7} \left (c d x +a e \right )}\) | \(399\) |
| risch | \(\frac {e^{6} x^{3}}{3 c^{4} d^{4}}-\frac {2 e^{7} x^{2} a}{d^{5} c^{5}}+\frac {3 e^{5} x^{2}}{d^{3} c^{4}}+\frac {10 e^{8} a^{2} x}{d^{6} c^{6}}-\frac {24 e^{6} a x}{d^{4} c^{5}}+\frac {15 e^{4} x}{d^{2} c^{4}}+\frac {\left (-15 a^{4} e^{10} d c +60 a^{3} e^{8} d^{3} c^{2}-90 a^{2} e^{6} d^{5} c^{3}+60 a \,c^{4} d^{7} e^{4}-15 e^{2} d^{9} c^{5}\right ) x^{2}-3 e \left (9 a^{5} e^{10}-35 a^{4} c \,d^{2} e^{8}+50 a^{3} c^{2} d^{4} e^{6}-30 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}+c^{5} d^{10}\right ) x -\frac {37 a^{6} e^{12}-141 a^{5} d^{2} e^{10} c +195 a^{4} d^{4} e^{8} c^{2}-110 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}+3 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}}{3 c d}}{d^{6} c^{6} \left (c d x +a e \right )^{3}}-\frac {20 e^{9} \ln \left (c d x +a e \right ) a^{3}}{d^{7} c^{7}}+\frac {60 e^{7} \ln \left (c d x +a e \right ) a^{2}}{d^{5} c^{6}}-\frac {60 e^{5} \ln \left (c d x +a e \right ) a}{d^{3} c^{5}}+\frac {20 e^{3} \ln \left (c d x +a e \right )}{d \,c^{4}}\) | \(418\) |
| norman | \(\frac {\frac {e^{7} \left (5 a^{2} e^{4}-18 a c \,d^{2} e^{2}+25 c^{2} d^{4}\right ) x^{7}}{d^{3} c^{3}}-\frac {110 a^{6} e^{12}-285 a^{5} d^{2} e^{10} c +186 a^{4} d^{4} e^{8} c^{2}+53 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}+3 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}}{3 d^{4} c^{7}}+\frac {e^{9} x^{9}}{3 c d}-\frac {\left (110 a^{6} e^{18}+525 c \,d^{2} a^{5} e^{16}-1299 a^{4} c^{2} d^{4} e^{14}-28 a^{3} d^{6} c^{3} e^{12}+1041 a^{2} c^{4} d^{8} e^{10}+921 a \,c^{5} d^{10} e^{8}+326 d^{12} c^{6} e^{6}\right ) x^{3}}{3 d^{7} c^{7} e^{3}}-\frac {\left (110 a^{6} e^{16}-15 a^{5} c \,d^{2} e^{14}-429 a^{4} c^{2} d^{4} e^{12}+296 a^{3} d^{6} c^{3} e^{10}+270 a^{2} c^{4} d^{8} e^{8}+151 a \,c^{5} d^{10} e^{6}+25 d^{12} c^{6} e^{4}\right ) x^{2}}{d^{6} c^{7} e^{2}}-\frac {\left (90 a^{5} e^{16}-45 a^{4} c \,d^{2} e^{14}-279 a^{3} c^{2} d^{4} e^{12}+221 a^{2} c^{3} d^{6} e^{10}+195 a \,c^{4} d^{8} e^{8}+196 c^{5} d^{10} e^{6}\right ) x^{4}}{d^{6} c^{6} e^{2}}-\frac {\left (110 a^{6} e^{14}-195 c \,d^{2} e^{12} a^{5}-39 a^{4} d^{4} e^{10} c^{2}+179 a^{3} c^{3} d^{6} e^{8}+88 a^{2} d^{8} e^{6} c^{4}+18 a \,d^{10} e^{4} c^{5}+4 d^{12} c^{6} e^{2}\right ) x}{d^{5} c^{7} e}-\frac {\left (60 a^{4} e^{14}-135 a^{3} c \,d^{2} e^{12}+66 a^{2} c^{2} d^{4} e^{10}+5 a \,c^{3} d^{6} e^{8}+130 c^{4} d^{8} e^{6}\right ) x^{5}}{d^{5} c^{5} e}-\frac {e^{8} \left (a \,e^{2}-4 c \,d^{2}\right ) x^{8}}{c^{2} d^{2}}}{\left (e x +d \right )^{3} \left (c d x +a e \right )^{3}}-\frac {20 e^{3} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \ln \left (c d x +a e \right )}{d^{7} c^{7}}\) | \(707\) |
| parallelrisch | \(-\frac {270 a^{5} c d \,e^{11} x -810 a^{4} c^{2} d^{3} e^{9} x +810 a^{3} c^{3} d^{5} e^{7} x -270 a^{2} c^{4} d^{7} e^{5} x +3 x^{5} a \,c^{5} d^{5} e^{7}-15 x^{4} a^{2} c^{4} d^{4} e^{8}+45 x^{4} a \,c^{5} d^{6} e^{6}+110 a^{6} e^{12}+d^{12} c^{6}+180 \ln \left (c d x +a e \right ) x \,a^{5} c d \,e^{11}-540 \ln \left (c d x +a e \right ) x \,a^{4} c^{2} d^{3} e^{9}+540 \ln \left (c d x +a e \right ) x \,a^{3} c^{3} d^{5} e^{7}-180 \ln \left (c d x +a e \right ) x \,a^{2} c^{4} d^{7} e^{5}+15 a^{2} d^{8} e^{4} c^{4}-110 a^{3} d^{6} e^{6} c^{3}+330 a^{4} d^{4} e^{8} c^{2}-330 a^{5} d^{2} e^{10} c -x^{6} e^{6} c^{6} d^{6}-9 x^{5} c^{6} d^{7} e^{5}-45 x^{4} c^{6} d^{8} e^{4}+45 x^{2} c^{6} d^{10} e^{2}+9 c^{6} d^{11} e x +3 a \,d^{10} e^{2} c^{5}+180 x^{2} a^{4} c^{2} d^{2} e^{10}-540 x^{2} a^{3} c^{3} d^{4} e^{8}+540 x^{2} a^{2} c^{4} d^{6} e^{6}-180 x^{2} a \,c^{5} d^{8} e^{4}-180 \ln \left (c d x +a e \right ) a^{5} c \,d^{2} e^{10}+180 \ln \left (c d x +a e \right ) a^{4} c^{2} d^{4} e^{8}-60 \ln \left (c d x +a e \right ) a^{3} c^{3} d^{6} e^{6}+60 \ln \left (c d x +a e \right ) a^{6} e^{12}+60 \ln \left (c d x +a e \right ) x^{3} a^{3} c^{3} d^{3} e^{9}-180 \ln \left (c d x +a e \right ) x^{3} a^{2} c^{4} d^{5} e^{7}+180 \ln \left (c d x +a e \right ) x^{3} a \,c^{5} d^{7} e^{5}+180 \ln \left (c d x +a e \right ) x^{2} a^{4} c^{2} d^{2} e^{10}-540 \ln \left (c d x +a e \right ) x^{2} a^{3} c^{3} d^{4} e^{8}+540 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{4} d^{6} e^{6}-180 \ln \left (c d x +a e \right ) x^{2} a \,c^{5} d^{8} e^{4}+45 x a \,c^{5} d^{9} e^{3}-60 \ln \left (c d x +a e \right ) x^{3} c^{6} d^{9} e^{3}}{3 d^{7} c^{7} \left (c d x +a e \right )^{3}}\) | \(726\) |
Input:
int((e*x+d)^10/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^4,x,method=_RETURNVERBOSE )
Output:
e^4/d^6/c^6*(1/3*x^3*c^2*d^2*e^2-2*x^2*a*c*d*e^3+3*x^2*c^2*d^3*e+10*a^2*e^ 4*x-24*a*c*d^2*e^2*x+15*c^2*d^4*x)-1/3/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)^3-20*e^3/d^7/c^7*(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^ 4*e^2-c^3*d^6)*ln(c*d*x+a*e)+3/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)^2-15* 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)/(c*d*x+a*e)
Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (213) = 426\).
Time = 0.08 (sec) , antiderivative size = 644, normalized size of antiderivative = 2.97 \[ \int \frac {(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {c^{6} d^{6} e^{6} x^{6} - c^{6} d^{12} - 3 \, a c^{5} d^{10} e^{2} - 15 \, a^{2} c^{4} d^{8} e^{4} + 110 \, a^{3} c^{3} d^{6} e^{6} - 195 \, a^{4} c^{2} d^{4} e^{8} + 141 \, a^{5} c d^{2} e^{10} - 37 \, a^{6} e^{12} + 3 \, {\left (3 \, c^{6} d^{7} e^{5} - a c^{5} d^{5} e^{7}\right )} x^{5} + 15 \, {\left (3 \, c^{6} d^{8} e^{4} - 3 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} + {\left (135 \, a c^{5} d^{7} e^{5} - 189 \, a^{2} c^{4} d^{5} e^{7} + 73 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{3} - 3 \, {\left (15 \, c^{6} d^{10} e^{2} - 60 \, a c^{5} d^{8} e^{4} + 45 \, a^{2} c^{4} d^{6} e^{6} + 9 \, a^{3} c^{3} d^{4} e^{8} - 13 \, a^{4} c^{2} d^{2} e^{10}\right )} x^{2} - 3 \, {\left (3 \, c^{6} d^{11} e + 15 \, a c^{5} d^{9} e^{3} - 90 \, a^{2} c^{4} d^{7} e^{5} + 135 \, a^{3} c^{3} d^{5} e^{7} - 81 \, a^{4} c^{2} d^{3} e^{9} + 17 \, a^{5} c d e^{11}\right )} x + 60 \, {\left (a^{3} c^{3} d^{6} e^{6} - 3 \, a^{4} c^{2} d^{4} e^{8} + 3 \, a^{5} c d^{2} e^{10} - a^{6} e^{12} + {\left (c^{6} d^{9} e^{3} - 3 \, a c^{5} d^{7} e^{5} + 3 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 3 \, {\left (a c^{5} d^{8} e^{4} - 3 \, a^{2} c^{4} d^{6} e^{6} + 3 \, a^{3} c^{3} d^{4} e^{8} - a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 3 \, {\left (a^{2} c^{4} d^{7} e^{5} - 3 \, a^{3} c^{3} d^{5} e^{7} + 3 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x\right )} \log \left (c d x + a e\right )}{3 \, {\left (c^{10} d^{10} x^{3} + 3 \, a c^{9} d^{9} e x^{2} + 3 \, a^{2} c^{8} d^{8} e^{2} x + a^{3} c^{7} d^{7} e^{3}\right )}} \] Input:
integrate((e*x+d)^10/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fri cas")
Output:
1/3*(c^6*d^6*e^6*x^6 - c^6*d^12 - 3*a*c^5*d^10*e^2 - 15*a^2*c^4*d^8*e^4 + 110*a^3*c^3*d^6*e^6 - 195*a^4*c^2*d^4*e^8 + 141*a^5*c*d^2*e^10 - 37*a^6*e^ 12 + 3*(3*c^6*d^7*e^5 - a*c^5*d^5*e^7)*x^5 + 15*(3*c^6*d^8*e^4 - 3*a*c^5*d ^6*e^6 + a^2*c^4*d^4*e^8)*x^4 + (135*a*c^5*d^7*e^5 - 189*a^2*c^4*d^5*e^7 + 73*a^3*c^3*d^3*e^9)*x^3 - 3*(15*c^6*d^10*e^2 - 60*a*c^5*d^8*e^4 + 45*a^2* c^4*d^6*e^6 + 9*a^3*c^3*d^4*e^8 - 13*a^4*c^2*d^2*e^10)*x^2 - 3*(3*c^6*d^11 *e + 15*a*c^5*d^9*e^3 - 90*a^2*c^4*d^7*e^5 + 135*a^3*c^3*d^5*e^7 - 81*a^4* c^2*d^3*e^9 + 17*a^5*c*d*e^11)*x + 60*(a^3*c^3*d^6*e^6 - 3*a^4*c^2*d^4*e^8 + 3*a^5*c*d^2*e^10 - a^6*e^12 + (c^6*d^9*e^3 - 3*a*c^5*d^7*e^5 + 3*a^2*c^ 4*d^5*e^7 - a^3*c^3*d^3*e^9)*x^3 + 3*(a*c^5*d^8*e^4 - 3*a^2*c^4*d^6*e^6 + 3*a^3*c^3*d^4*e^8 - a^4*c^2*d^2*e^10)*x^2 + 3*(a^2*c^4*d^7*e^5 - 3*a^3*c^3 *d^5*e^7 + 3*a^4*c^2*d^3*e^9 - a^5*c*d*e^11)*x)*log(c*d*x + a*e))/(c^10*d^ 10*x^3 + 3*a*c^9*d^9*e*x^2 + 3*a^2*c^8*d^8*e^2*x + a^3*c^7*d^7*e^3)
Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (211) = 422\).
Time = 71.72 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.96 \[ \int \frac {(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=x^{2} \left (- \frac {2 a e^{7}}{c^{5} d^{5}} + \frac {3 e^{5}}{c^{4} d^{3}}\right ) + x \left (\frac {10 a^{2} e^{8}}{c^{6} d^{6}} - \frac {24 a e^{6}}{c^{5} d^{4}} + \frac {15 e^{4}}{c^{4} d^{2}}\right ) + \frac {- 37 a^{6} e^{12} + 141 a^{5} c d^{2} e^{10} - 195 a^{4} c^{2} d^{4} e^{8} + 110 a^{3} c^{3} d^{6} e^{6} - 15 a^{2} c^{4} d^{8} e^{4} - 3 a c^{5} d^{10} e^{2} - c^{6} d^{12} + x^{2} \left (- 45 a^{4} c^{2} d^{2} e^{10} + 180 a^{3} c^{3} d^{4} e^{8} - 270 a^{2} c^{4} d^{6} e^{6} + 180 a c^{5} d^{8} e^{4} - 45 c^{6} d^{10} e^{2}\right ) + x \left (- 81 a^{5} c d e^{11} + 315 a^{4} c^{2} d^{3} e^{9} - 450 a^{3} c^{3} d^{5} e^{7} + 270 a^{2} c^{4} d^{7} e^{5} - 45 a c^{5} d^{9} e^{3} - 9 c^{6} d^{11} e\right )}{3 a^{3} c^{7} d^{7} e^{3} + 9 a^{2} c^{8} d^{8} e^{2} x + 9 a c^{9} d^{9} e x^{2} + 3 c^{10} d^{10} x^{3}} + \frac {e^{6} x^{3}}{3 c^{4} d^{4}} - \frac {20 e^{3} \left (a e^{2} - c d^{2}\right )^{3} \log {\left (a e + c d x \right )}}{c^{7} d^{7}} \] Input:
integrate((e*x+d)**10/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)
Output:
x**2*(-2*a*e**7/(c**5*d**5) + 3*e**5/(c**4*d**3)) + x*(10*a**2*e**8/(c**6* d**6) - 24*a*e**6/(c**5*d**4) + 15*e**4/(c**4*d**2)) + (-37*a**6*e**12 + 1 41*a**5*c*d**2*e**10 - 195*a**4*c**2*d**4*e**8 + 110*a**3*c**3*d**6*e**6 - 15*a**2*c**4*d**8*e**4 - 3*a*c**5*d**10*e**2 - c**6*d**12 + x**2*(-45*a** 4*c**2*d**2*e**10 + 180*a**3*c**3*d**4*e**8 - 270*a**2*c**4*d**6*e**6 + 18 0*a*c**5*d**8*e**4 - 45*c**6*d**10*e**2) + x*(-81*a**5*c*d*e**11 + 315*a** 4*c**2*d**3*e**9 - 450*a**3*c**3*d**5*e**7 + 270*a**2*c**4*d**7*e**5 - 45* a*c**5*d**9*e**3 - 9*c**6*d**11*e))/(3*a**3*c**7*d**7*e**3 + 9*a**2*c**8*d **8*e**2*x + 9*a*c**9*d**9*e*x**2 + 3*c**10*d**10*x**3) + e**6*x**3/(3*c** 4*d**4) - 20*e**3*(a*e**2 - c*d**2)**3*log(a*e + c*d*x)/(c**7*d**7)
Time = 0.05 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.95 \[ \int \frac {(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {c^{6} d^{12} + 3 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 110 \, a^{3} c^{3} d^{6} e^{6} + 195 \, a^{4} c^{2} d^{4} e^{8} - 141 \, a^{5} c d^{2} e^{10} + 37 \, a^{6} e^{12} + 45 \, {\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 9 \, {\left (c^{6} d^{11} e + 5 \, a c^{5} d^{9} e^{3} - 30 \, a^{2} c^{4} d^{7} e^{5} + 50 \, a^{3} c^{3} d^{5} e^{7} - 35 \, a^{4} c^{2} d^{3} e^{9} + 9 \, a^{5} c d e^{11}\right )} x}{3 \, {\left (c^{10} d^{10} x^{3} + 3 \, a c^{9} d^{9} e x^{2} + 3 \, a^{2} c^{8} d^{8} e^{2} x + a^{3} c^{7} d^{7} e^{3}\right )}} + \frac {c^{2} d^{2} e^{6} x^{3} + 3 \, {\left (3 \, c^{2} d^{3} e^{5} - 2 \, a c d e^{7}\right )} x^{2} + 3 \, {\left (15 \, c^{2} d^{4} e^{4} - 24 \, a c d^{2} e^{6} + 10 \, a^{2} e^{8}\right )} x}{3 \, c^{6} d^{6}} + \frac {20 \, {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} \log \left (c d x + a e\right )}{c^{7} d^{7}} \] Input:
integrate((e*x+d)^10/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="max ima")
Output:
-1/3*(c^6*d^12 + 3*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 110*a^3*c^3*d^6*e ^6 + 195*a^4*c^2*d^4*e^8 - 141*a^5*c*d^2*e^10 + 37*a^6*e^12 + 45*(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 + 9*(c^6*d^11*e + 5*a*c^5*d^9*e^3 - 30*a^2*c^4*d^7*e^5 + 50*a ^3*c^3*d^5*e^7 - 35*a^4*c^2*d^3*e^9 + 9*a^5*c*d*e^11)*x)/(c^10*d^10*x^3 + 3*a*c^9*d^9*e*x^2 + 3*a^2*c^8*d^8*e^2*x + a^3*c^7*d^7*e^3) + 1/3*(c^2*d^2* e^6*x^3 + 3*(3*c^2*d^3*e^5 - 2*a*c*d*e^7)*x^2 + 3*(15*c^2*d^4*e^4 - 24*a*c *d^2*e^6 + 10*a^2*e^8)*x)/(c^6*d^6) + 20*(c^3*d^6*e^3 - 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 - a^3*e^9)*log(c*d*x + a*e)/(c^7*d^7)
Time = 0.11 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.83 \[ \int \frac {(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {20 \, {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{7} d^{7}} - \frac {c^{6} d^{12} + 3 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 110 \, a^{3} c^{3} d^{6} e^{6} + 195 \, a^{4} c^{2} d^{4} e^{8} - 141 \, a^{5} c d^{2} e^{10} + 37 \, a^{6} e^{12} + 45 \, {\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 9 \, {\left (c^{6} d^{11} e + 5 \, a c^{5} d^{9} e^{3} - 30 \, a^{2} c^{4} d^{7} e^{5} + 50 \, a^{3} c^{3} d^{5} e^{7} - 35 \, a^{4} c^{2} d^{3} e^{9} + 9 \, a^{5} c d e^{11}\right )} x}{3 \, {\left (c d x + a e\right )}^{3} c^{7} d^{7}} + \frac {c^{8} d^{8} e^{6} x^{3} + 9 \, c^{8} d^{9} e^{5} x^{2} - 6 \, a c^{7} d^{7} e^{7} x^{2} + 45 \, c^{8} d^{10} e^{4} x - 72 \, a c^{7} d^{8} e^{6} x + 30 \, a^{2} c^{6} d^{6} e^{8} x}{3 \, c^{12} d^{12}} \] Input:
integrate((e*x+d)^10/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="gia c")
Output:
20*(c^3*d^6*e^3 - 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 - a^3*e^9)*log(abs(c*d *x + a*e))/(c^7*d^7) - 1/3*(c^6*d^12 + 3*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e ^4 - 110*a^3*c^3*d^6*e^6 + 195*a^4*c^2*d^4*e^8 - 141*a^5*c*d^2*e^10 + 37*a ^6*e^12 + 45*(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 + 9*(c^6*d^11*e + 5*a*c^5*d^9*e^3 - 30* a^2*c^4*d^7*e^5 + 50*a^3*c^3*d^5*e^7 - 35*a^4*c^2*d^3*e^9 + 9*a^5*c*d*e^11 )*x)/((c*d*x + a*e)^3*c^7*d^7) + 1/3*(c^8*d^8*e^6*x^3 + 9*c^8*d^9*e^5*x^2 - 6*a*c^7*d^7*e^7*x^2 + 45*c^8*d^10*e^4*x - 72*a*c^7*d^8*e^6*x + 30*a^2*c^ 6*d^6*e^8*x)/(c^12*d^12)
Time = 5.26 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.08 \[ \int \frac {(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=x^2\,\left (\frac {3\,e^5}{c^4\,d^3}-\frac {2\,a\,e^7}{c^5\,d^5}\right )-x\,\left (\frac {6\,a^2\,e^8}{c^6\,d^6}-\frac {15\,e^4}{c^4\,d^2}+\frac {4\,a\,e\,\left (\frac {6\,e^5}{c^4\,d^3}-\frac {4\,a\,e^7}{c^5\,d^5}\right )}{c\,d}\right )-\frac {x\,\left (27\,a^5\,e^{11}-105\,a^4\,c\,d^2\,e^9+150\,a^3\,c^2\,d^4\,e^7-90\,a^2\,c^3\,d^6\,e^5+15\,a\,c^4\,d^8\,e^3+3\,c^5\,d^{10}\,e\right )+x^2\,\left (15\,a^4\,c\,d\,e^{10}-60\,a^3\,c^2\,d^3\,e^8+90\,a^2\,c^3\,d^5\,e^6-60\,a\,c^4\,d^7\,e^4+15\,c^5\,d^9\,e^2\right )+\frac {37\,a^6\,e^{12}-141\,a^5\,c\,d^2\,e^{10}+195\,a^4\,c^2\,d^4\,e^8-110\,a^3\,c^3\,d^6\,e^6+15\,a^2\,c^4\,d^8\,e^4+3\,a\,c^5\,d^{10}\,e^2+c^6\,d^{12}}{3\,c\,d}}{a^3\,c^6\,d^6\,e^3+3\,a^2\,c^7\,d^7\,e^2\,x+3\,a\,c^8\,d^8\,e\,x^2+c^9\,d^9\,x^3}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (20\,a^3\,e^9-60\,a^2\,c\,d^2\,e^7+60\,a\,c^2\,d^4\,e^5-20\,c^3\,d^6\,e^3\right )}{c^7\,d^7}+\frac {e^6\,x^3}{3\,c^4\,d^4} \] Input:
int((d + e*x)^10/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)
Output:
x^2*((3*e^5)/(c^4*d^3) - (2*a*e^7)/(c^5*d^5)) - x*((6*a^2*e^8)/(c^6*d^6) - (15*e^4)/(c^4*d^2) + (4*a*e*((6*e^5)/(c^4*d^3) - (4*a*e^7)/(c^5*d^5)))/(c *d)) - (x*(27*a^5*e^11 + 3*c^5*d^10*e + 15*a*c^4*d^8*e^3 - 105*a^4*c*d^2*e ^9 - 90*a^2*c^3*d^6*e^5 + 150*a^3*c^2*d^4*e^7) + x^2*(15*c^5*d^9*e^2 - 60* a*c^4*d^7*e^4 + 90*a^2*c^3*d^5*e^6 - 60*a^3*c^2*d^3*e^8 + 15*a^4*c*d*e^10) + (37*a^6*e^12 + c^6*d^12 + 3*a*c^5*d^10*e^2 - 141*a^5*c*d^2*e^10 + 15*a^ 2*c^4*d^8*e^4 - 110*a^3*c^3*d^6*e^6 + 195*a^4*c^2*d^4*e^8)/(3*c*d))/(c^9*d ^9*x^3 + a^3*c^6*d^6*e^3 + 3*a*c^8*d^8*e*x^2 + 3*a^2*c^7*d^7*e^2*x) - (log (a*e + c*d*x)*(20*a^3*e^9 - 20*c^3*d^6*e^3 + 60*a*c^2*d^4*e^5 - 60*a^2*c*d ^2*e^7))/(c^7*d^7) + (e^6*x^3)/(3*c^4*d^4)
Time = 0.22 (sec) , antiderivative size = 749, normalized size of antiderivative = 3.45 \[ \int \frac {(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {-60 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c^{3} d^{3} e^{9} x^{3}+180 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{4} d^{5} e^{7} x^{3}-180 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{5} d^{7} e^{5} x^{3}+60 \,\mathrm {log}\left (c d x +a e \right ) a \,c^{6} d^{9} e^{3} x^{3}-3 a^{2} c^{5} d^{10} e^{2}+15 c^{7} d^{11} e \,x^{3}+180 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{4} d^{7} e^{5} x -540 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{4} d^{6} e^{6} x^{2}+180 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{5} d^{8} e^{4} x^{2}-90 a^{6} c d \,e^{11} x +270 a^{5} c^{2} d^{3} e^{9} x -270 a^{4} c^{3} d^{5} e^{7} x +90 a^{3} c^{4} d^{7} e^{5} x -9 a \,c^{6} d^{11} e x -50 a^{7} e^{12}+a \,c^{6} d^{6} e^{6} x^{6}-180 \,\mathrm {log}\left (c d x +a e \right ) a^{6} c d \,e^{11} x +540 \,\mathrm {log}\left (c d x +a e \right ) a^{5} c^{2} d^{3} e^{9} x -180 \,\mathrm {log}\left (c d x +a e \right ) a^{5} c^{2} d^{2} e^{10} x^{2}-540 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c^{3} d^{5} e^{7} x +540 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c^{3} d^{4} e^{8} x^{2}+180 \,\mathrm {log}\left (c d x +a e \right ) a^{6} c \,d^{2} e^{10}-180 \,\mathrm {log}\left (c d x +a e \right ) a^{5} c^{2} d^{4} e^{8}+60 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c^{3} d^{6} e^{6}+60 a^{4} c^{3} d^{3} e^{9} x^{3}-180 a^{3} c^{4} d^{5} e^{7} x^{3}+15 a^{3} c^{4} d^{4} e^{8} x^{4}+180 a^{2} c^{5} d^{7} e^{5} x^{3}-45 a^{2} c^{5} d^{6} e^{6} x^{4}-3 a^{2} c^{5} d^{5} e^{7} x^{5}-60 a \,c^{6} d^{9} e^{3} x^{3}+45 a \,c^{6} d^{8} e^{4} x^{4}+9 a \,c^{6} d^{7} e^{5} x^{5}+150 a^{6} c \,d^{2} e^{10}-150 a^{5} c^{2} d^{4} e^{8}+50 a^{4} c^{3} d^{6} e^{6}-60 \,\mathrm {log}\left (c d x +a e \right ) a^{7} e^{12}-a \,c^{6} d^{12}}{3 a \,c^{7} d^{7} \left (c^{3} d^{3} x^{3}+3 a \,c^{2} d^{2} e \,x^{2}+3 a^{2} c d \,e^{2} x +a^{3} e^{3}\right )} \] Input:
int((e*x+d)^10/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)
Output:
( - 60*log(a*e + c*d*x)*a**7*e**12 + 180*log(a*e + c*d*x)*a**6*c*d**2*e**1 0 - 180*log(a*e + c*d*x)*a**6*c*d*e**11*x - 180*log(a*e + c*d*x)*a**5*c**2 *d**4*e**8 + 540*log(a*e + c*d*x)*a**5*c**2*d**3*e**9*x - 180*log(a*e + c* d*x)*a**5*c**2*d**2*e**10*x**2 + 60*log(a*e + c*d*x)*a**4*c**3*d**6*e**6 - 540*log(a*e + c*d*x)*a**4*c**3*d**5*e**7*x + 540*log(a*e + c*d*x)*a**4*c* *3*d**4*e**8*x**2 - 60*log(a*e + c*d*x)*a**4*c**3*d**3*e**9*x**3 + 180*log (a*e + c*d*x)*a**3*c**4*d**7*e**5*x - 540*log(a*e + c*d*x)*a**3*c**4*d**6* e**6*x**2 + 180*log(a*e + c*d*x)*a**3*c**4*d**5*e**7*x**3 + 180*log(a*e + c*d*x)*a**2*c**5*d**8*e**4*x**2 - 180*log(a*e + c*d*x)*a**2*c**5*d**7*e**5 *x**3 + 60*log(a*e + c*d*x)*a*c**6*d**9*e**3*x**3 - 50*a**7*e**12 + 150*a* *6*c*d**2*e**10 - 90*a**6*c*d*e**11*x - 150*a**5*c**2*d**4*e**8 + 270*a**5 *c**2*d**3*e**9*x + 50*a**4*c**3*d**6*e**6 - 270*a**4*c**3*d**5*e**7*x + 6 0*a**4*c**3*d**3*e**9*x**3 + 90*a**3*c**4*d**7*e**5*x - 180*a**3*c**4*d**5 *e**7*x**3 + 15*a**3*c**4*d**4*e**8*x**4 - 3*a**2*c**5*d**10*e**2 + 180*a* *2*c**5*d**7*e**5*x**3 - 45*a**2*c**5*d**6*e**6*x**4 - 3*a**2*c**5*d**5*e* *7*x**5 - a*c**6*d**12 - 9*a*c**6*d**11*e*x - 60*a*c**6*d**9*e**3*x**3 + 4 5*a*c**6*d**8*e**4*x**4 + 9*a*c**6*d**7*e**5*x**5 + a*c**6*d**6*e**6*x**6 + 15*c**7*d**11*e*x**3)/(3*a*c**7*d**7*(a**3*e**3 + 3*a**2*c*d*e**2*x + 3* a*c**2*d**2*e*x**2 + c**3*d**3*x**3))