Integrand size = 35, antiderivative size = 179 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^4 \left (5 c d^2-4 a e^2\right ) x}{c^5 d^5}+\frac {e^5 x^2}{2 c^4 d^4}-\frac {\left (c d^2-a e^2\right )^5}{3 c^6 d^6 (a e+c d x)^3}-\frac {5 e \left (c d^2-a e^2\right )^4}{2 c^6 d^6 (a e+c d x)^2}-\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^6 d^6 (a e+c d x)}+\frac {10 e^3 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^6 d^6} \] Output:
e^4*(-4*a*e^2+5*c*d^2)*x/c^5/d^5+1/2*e^5*x^2/c^4/d^4-1/3*(-a*e^2+c*d^2)^5/ c^6/d^6/(c*d*x+a*e)^3-5/2*e*(-a*e^2+c*d^2)^4/c^6/d^6/(c*d*x+a*e)^2-10*e^2* (-a*e^2+c*d^2)^3/c^6/d^6/(c*d*x+a*e)+10*e^3*(-a*e^2+c*d^2)^2*ln(c*d*x+a*e) /c^6/d^6
Time = 0.07 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.45 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {47 a^5 e^{10}+a^4 c d e^8 (-130 d+81 e x)+a^3 c^2 d^2 e^6 \left (110 d^2-270 d e x-9 e^2 x^2\right )-a^2 c^3 d^3 e^4 \left (20 d^3-270 d^2 e x+90 d e^2 x^2+63 e^3 x^3\right )-5 a c^4 d^4 e^2 \left (d^4+12 d^3 e x-36 d^2 e^2 x^2-18 d e^3 x^3+3 e^4 x^4\right )+c^5 d^5 \left (-2 d^5-15 d^4 e x-60 d^3 e^2 x^2+30 d e^4 x^4+3 e^5 x^5\right )+60 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3 \log (a e+c d x)}{6 c^6 d^6 (a e+c d x)^3} \] Input:
Integrate[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
Output:
(47*a^5*e^10 + a^4*c*d*e^8*(-130*d + 81*e*x) + a^3*c^2*d^2*e^6*(110*d^2 - 270*d*e*x - 9*e^2*x^2) - a^2*c^3*d^3*e^4*(20*d^3 - 270*d^2*e*x + 90*d*e^2* x^2 + 63*e^3*x^3) - 5*a*c^4*d^4*e^2*(d^4 + 12*d^3*e*x - 36*d^2*e^2*x^2 - 1 8*d*e^3*x^3 + 3*e^4*x^4) + c^5*d^5*(-2*d^5 - 15*d^4*e*x - 60*d^3*e^2*x^2 + 30*d*e^4*x^4 + 3*e^5*x^5) + 60*e^3*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^3*Log[ a*e + c*d*x])/(6*c^6*d^6*(a*e + c*d*x)^3)
Time = 0.67 (sec) , antiderivative size = 179, 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.
| \(\displaystyle \int \frac {(d+e x)^9}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}+\frac {5 e \left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)^3}+\frac {\left (c d^2-a e^2\right )^5}{c^5 d^5 (a e+c d x)^4}+\frac {5 c d^2 e^4-4 a e^6}{c^5 d^5}+\frac {10 e^3 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}+\frac {e^5 x}{c^4 d^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^6 d^6 (a e+c d x)}-\frac {5 e \left (c d^2-a e^2\right )^4}{2 c^6 d^6 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^5}{3 c^6 d^6 (a e+c d x)^3}+\frac {10 e^3 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^6 d^6}+\frac {e^4 x \left (5 c d^2-4 a e^2\right )}{c^5 d^5}+\frac {e^5 x^2}{2 c^4 d^4}\) |
Input:
Int[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
Output:
(e^4*(5*c*d^2 - 4*a*e^2)*x)/(c^5*d^5) + (e^5*x^2)/(2*c^4*d^4) - (c*d^2 - a *e^2)^5/(3*c^6*d^6*(a*e + c*d*x)^3) - (5*e*(c*d^2 - a*e^2)^4)/(2*c^6*d^6*( a*e + c*d*x)^2) - (10*e^2*(c*d^2 - a*e^2)^3)/(c^6*d^6*(a*e + c*d*x)) + (10 *e^3*(c*d^2 - a*e^2)^2*Log[a*e + c*d*x])/(c^6*d^6)
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 = 1.59 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.68
| method | result | size |
| default | \(-\frac {e^{4} \left (-\frac {1}{2} c d \,x^{2} e +4 a \,e^{2} x -5 c \,d^{2} x \right )}{d^{5} c^{5}}-\frac {-a^{5} e^{10}+5 a^{4} c \,d^{2} e^{8}-10 a^{3} c^{2} d^{4} e^{6}+10 a^{2} c^{3} d^{6} e^{4}-5 a \,c^{4} d^{8} e^{2}+c^{5} d^{10}}{3 d^{6} c^{6} \left (c d x +a e \right )^{3}}+\frac {10 e^{3} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (c d x +a e \right )}{d^{6} c^{6}}-\frac {5 e \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}{2 d^{6} c^{6} \left (c d x +a e \right )^{2}}+\frac {10 e^{2} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}{c^{6} d^{6} \left (c d x +a e \right )}\) | \(300\) |
| risch | \(\frac {e^{5} x^{2}}{2 c^{4} d^{4}}-\frac {4 e^{6} a x}{d^{5} c^{5}}+\frac {5 e^{4} x}{d^{3} c^{4}}+\frac {\left (10 a^{3} c \,e^{8} d -30 a^{2} c^{2} e^{6} d^{3}+30 a \,c^{3} e^{4} d^{5}-10 c^{4} e^{2} d^{7}\right ) x^{2}+\frac {5 e \left (7 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}+18 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}-c^{4} d^{8}\right ) x}{2}+\frac {47 a^{5} e^{10}-130 a^{4} c \,d^{2} e^{8}+110 a^{3} c^{2} d^{4} e^{6}-20 a^{2} c^{3} d^{6} e^{4}-5 a \,c^{4} d^{8} e^{2}-2 c^{5} d^{10}}{6 c d}}{d^{5} c^{5} \left (c d x +a e \right )^{3}}+\frac {10 e^{7} \ln \left (c d x +a e \right ) a^{2}}{d^{6} c^{6}}-\frac {20 e^{5} \ln \left (c d x +a e \right ) a}{d^{4} c^{5}}+\frac {10 e^{3} \ln \left (c d x +a e \right )}{d^{2} c^{4}}\) | \(311\) |
| parallelrisch | \(\frac {110 a^{5} e^{10}-2 c^{5} d^{10}+270 a^{4} c d \,e^{9} x -540 a^{3} c^{2} d^{3} e^{7} x +270 a^{2} c^{3} d^{5} e^{5} x -15 c^{5} d^{9} e x -360 \ln \left (c d x +a e \right ) x \,a^{3} c^{2} d^{3} e^{7}+180 \ln \left (c d x +a e \right ) x \,a^{2} c^{3} d^{5} e^{5}-60 x^{2} c^{5} d^{8} e^{2}+3 x^{5} e^{5} d^{5} c^{5}+30 x^{4} c^{5} d^{6} e^{4}-220 a^{4} c \,d^{2} e^{8}+110 a^{3} c^{2} d^{4} e^{6}-20 a^{2} c^{3} d^{6} e^{4}-5 a \,c^{4} d^{8} e^{2}+180 \ln \left (c d x +a e \right ) x \,a^{4} c d \,e^{9}-15 x^{4} a \,c^{4} d^{4} e^{6}+180 x^{2} a^{3} c^{2} d^{2} e^{8}-360 x^{2} a^{2} c^{3} d^{4} e^{6}+180 x^{2} a \,c^{4} d^{6} e^{4}-120 \ln \left (c d x +a e \right ) a^{4} c \,d^{2} e^{8}+60 \ln \left (c d x +a e \right ) a^{3} c^{2} d^{4} e^{6}-60 x a \,c^{4} d^{7} e^{3}+180 \ln \left (c d x +a e \right ) x^{2} a \,c^{4} d^{6} e^{4}+60 \ln \left (c d x +a e \right ) a^{5} e^{10}+60 \ln \left (c d x +a e \right ) x^{3} c^{5} d^{7} e^{3}+180 \ln \left (c d x +a e \right ) x^{2} a^{3} c^{2} d^{2} e^{8}-360 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{3} d^{4} e^{6}+60 \ln \left (c d x +a e \right ) x^{3} a^{2} c^{3} d^{3} e^{7}-120 \ln \left (c d x +a e \right ) x^{3} a \,c^{4} d^{5} e^{5}}{6 d^{6} c^{6} \left (c d x +a e \right )^{3}}\) | \(536\) |
| norman | \(\frac {\frac {110 a^{5} e^{10}-175 a^{4} c \,d^{2} e^{8}+11 a^{3} c^{2} d^{4} e^{6}-20 a^{2} c^{3} d^{6} e^{4}-5 a \,c^{4} d^{8} e^{2}-2 c^{5} d^{10}}{6 c^{6} d^{3}}+\frac {e^{8} x^{8}}{2 c d}+\frac {\left (110 a^{5} e^{16}+635 a^{4} c \,d^{2} e^{14}-664 a^{3} c^{2} d^{4} e^{12}-776 a^{2} c^{3} d^{6} e^{10}-491 a \,c^{4} d^{8} e^{8}-326 c^{5} d^{10} e^{6}\right ) x^{3}}{6 d^{6} c^{6} e^{3}}+\frac {\left (110 a^{5} e^{14}+95 a^{4} c \,d^{2} e^{12}-334 a^{3} d^{4} e^{10} c^{2}-122 a^{2} c^{3} d^{6} e^{8}-104 a \,d^{8} e^{6} c^{4}-37 c^{5} d^{10} e^{4}\right ) x^{2}}{2 d^{5} c^{6} e^{2}}+\frac {\left (90 a^{4} e^{14}+45 a^{3} c \,d^{2} e^{12}-234 a^{2} c^{2} d^{4} e^{10}-97 a \,c^{3} d^{6} e^{8}-154 c^{4} d^{8} e^{6}\right ) x^{4}}{2 d^{5} c^{5} e^{2}}+\frac {\left (110 a^{5} e^{12}-85 e^{10} d^{2} c \,a^{4}-124 a^{3} d^{4} e^{8} c^{2}-29 a^{2} c^{3} d^{6} e^{6}-25 a \,c^{4} d^{8} e^{4}-7 e^{2} d^{10} c^{5}\right ) x}{2 d^{4} c^{6} e}+\frac {\left (60 a^{3} e^{12}-75 a^{2} c \,d^{2} e^{10}-9 d^{4} a \,c^{2} e^{8}-88 d^{6} e^{6} c^{3}\right ) x^{5}}{2 d^{4} c^{4} e}-\frac {e^{7} \left (5 a \,e^{2}-13 c \,d^{2}\right ) x^{7}}{2 c^{2} d^{2}}}{\left (e x +d \right )^{3} \left (c d x +a e \right )^{3}}+\frac {10 e^{3} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (c d x +a e \right )}{d^{6} c^{6}}\) | \(570\) |
Input:
int((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^4,x,method=_RETURNVERBOSE)
Output:
-e^4/d^5/c^5*(-1/2*c*d*x^2*e+4*a*e^2*x-5*c*d^2*x)-1/3/d^6/c^6*(-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)^3+10*e^3/d^6/c^6*(a^2*e^4-2*a*c*d^2*e^2+c^2*d^4)*ln(c*d*x +a*e)-5/2/d^6*e/c^6*(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)^2+10*e^2/c^6/d^6*(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^ 2*d^4*e^2-c^3*d^6)/(c*d*x+a*e)
Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (173) = 346\).
Time = 0.09 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.73 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {3 \, c^{5} d^{5} e^{5} x^{5} - 2 \, c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} - 20 \, a^{2} c^{3} d^{6} e^{4} + 110 \, a^{3} c^{2} d^{4} e^{6} - 130 \, a^{4} c d^{2} e^{8} + 47 \, a^{5} e^{10} + 15 \, {\left (2 \, c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} + 9 \, {\left (10 \, a c^{4} d^{5} e^{5} - 7 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{3} - 3 \, {\left (20 \, c^{5} d^{8} e^{2} - 60 \, a c^{4} d^{6} e^{4} + 30 \, a^{2} c^{3} d^{4} e^{6} + 3 \, a^{3} c^{2} d^{2} e^{8}\right )} x^{2} - 3 \, {\left (5 \, c^{5} d^{9} e + 20 \, a c^{4} d^{7} e^{3} - 90 \, a^{2} c^{3} d^{5} e^{5} + 90 \, a^{3} c^{2} d^{3} e^{7} - 27 \, a^{4} c d e^{9}\right )} x + 60 \, {\left (a^{3} c^{2} d^{4} e^{6} - 2 \, a^{4} c d^{2} e^{8} + a^{5} e^{10} + {\left (c^{5} d^{7} e^{3} - 2 \, a c^{4} d^{5} e^{5} + a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \, {\left (a c^{4} d^{6} e^{4} - 2 \, a^{2} c^{3} d^{4} e^{6} + a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 3 \, {\left (a^{2} c^{3} d^{5} e^{5} - 2 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x\right )} \log \left (c d x + a e\right )}{6 \, {\left (c^{9} d^{9} x^{3} + 3 \, a c^{8} d^{8} e x^{2} + 3 \, a^{2} c^{7} d^{7} e^{2} x + a^{3} c^{6} d^{6} e^{3}\right )}} \] Input:
integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fric as")
Output:
1/6*(3*c^5*d^5*e^5*x^5 - 2*c^5*d^10 - 5*a*c^4*d^8*e^2 - 20*a^2*c^3*d^6*e^4 + 110*a^3*c^2*d^4*e^6 - 130*a^4*c*d^2*e^8 + 47*a^5*e^10 + 15*(2*c^5*d^6*e ^4 - a*c^4*d^4*e^6)*x^4 + 9*(10*a*c^4*d^5*e^5 - 7*a^2*c^3*d^3*e^7)*x^3 - 3 *(20*c^5*d^8*e^2 - 60*a*c^4*d^6*e^4 + 30*a^2*c^3*d^4*e^6 + 3*a^3*c^2*d^2*e ^8)*x^2 - 3*(5*c^5*d^9*e + 20*a*c^4*d^7*e^3 - 90*a^2*c^3*d^5*e^5 + 90*a^3* c^2*d^3*e^7 - 27*a^4*c*d*e^9)*x + 60*(a^3*c^2*d^4*e^6 - 2*a^4*c*d^2*e^8 + a^5*e^10 + (c^5*d^7*e^3 - 2*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^3 + 3*(a*c^ 4*d^6*e^4 - 2*a^2*c^3*d^4*e^6 + a^3*c^2*d^2*e^8)*x^2 + 3*(a^2*c^3*d^5*e^5 - 2*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)*x)*log(c*d*x + a*e))/(c^9*d^9*x^3 + 3*a *c^8*d^8*e*x^2 + 3*a^2*c^7*d^7*e^2*x + a^3*c^6*d^6*e^3)
Time = 25.01 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.88 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=x \left (- \frac {4 a e^{6}}{c^{5} d^{5}} + \frac {5 e^{4}}{c^{4} d^{3}}\right ) + \frac {47 a^{5} e^{10} - 130 a^{4} c d^{2} e^{8} + 110 a^{3} c^{2} d^{4} e^{6} - 20 a^{2} c^{3} d^{6} e^{4} - 5 a c^{4} d^{8} e^{2} - 2 c^{5} d^{10} + x^{2} \cdot \left (60 a^{3} c^{2} d^{2} e^{8} - 180 a^{2} c^{3} d^{4} e^{6} + 180 a c^{4} d^{6} e^{4} - 60 c^{5} d^{8} e^{2}\right ) + x \left (105 a^{4} c d e^{9} - 300 a^{3} c^{2} d^{3} e^{7} + 270 a^{2} c^{3} d^{5} e^{5} - 60 a c^{4} d^{7} e^{3} - 15 c^{5} d^{9} e\right )}{6 a^{3} c^{6} d^{6} e^{3} + 18 a^{2} c^{7} d^{7} e^{2} x + 18 a c^{8} d^{8} e x^{2} + 6 c^{9} d^{9} x^{3}} + \frac {e^{5} x^{2}}{2 c^{4} d^{4}} + \frac {10 e^{3} \left (a e^{2} - c d^{2}\right )^{2} \log {\left (a e + c d x \right )}}{c^{6} d^{6}} \] Input:
integrate((e*x+d)**9/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)
Output:
x*(-4*a*e**6/(c**5*d**5) + 5*e**4/(c**4*d**3)) + (47*a**5*e**10 - 130*a**4 *c*d**2*e**8 + 110*a**3*c**2*d**4*e**6 - 20*a**2*c**3*d**6*e**4 - 5*a*c**4 *d**8*e**2 - 2*c**5*d**10 + x**2*(60*a**3*c**2*d**2*e**8 - 180*a**2*c**3*d **4*e**6 + 180*a*c**4*d**6*e**4 - 60*c**5*d**8*e**2) + x*(105*a**4*c*d*e** 9 - 300*a**3*c**2*d**3*e**7 + 270*a**2*c**3*d**5*e**5 - 60*a*c**4*d**7*e** 3 - 15*c**5*d**9*e))/(6*a**3*c**6*d**6*e**3 + 18*a**2*c**7*d**7*e**2*x + 1 8*a*c**8*d**8*e*x**2 + 6*c**9*d**9*x**3) + e**5*x**2/(2*c**4*d**4) + 10*e* *3*(a*e**2 - c*d**2)**2*log(a*e + c*d*x)/(c**6*d**6)
Time = 0.04 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.82 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {2 \, c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} + 20 \, a^{2} c^{3} d^{6} e^{4} - 110 \, a^{3} c^{2} d^{4} e^{6} + 130 \, a^{4} c d^{2} e^{8} - 47 \, a^{5} e^{10} + 60 \, {\left (c^{5} d^{8} e^{2} - 3 \, a c^{4} d^{6} e^{4} + 3 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 15 \, {\left (c^{5} d^{9} e + 4 \, a c^{4} d^{7} e^{3} - 18 \, a^{2} c^{3} d^{5} e^{5} + 20 \, a^{3} c^{2} d^{3} e^{7} - 7 \, a^{4} c d e^{9}\right )} x}{6 \, {\left (c^{9} d^{9} x^{3} + 3 \, a c^{8} d^{8} e x^{2} + 3 \, a^{2} c^{7} d^{7} e^{2} x + a^{3} c^{6} d^{6} e^{3}\right )}} + \frac {c d e^{5} x^{2} + 2 \, {\left (5 \, c d^{2} e^{4} - 4 \, a e^{6}\right )} x}{2 \, c^{5} d^{5}} + \frac {10 \, {\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} \log \left (c d x + a e\right )}{c^{6} d^{6}} \] Input:
integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxi ma")
Output:
-1/6*(2*c^5*d^10 + 5*a*c^4*d^8*e^2 + 20*a^2*c^3*d^6*e^4 - 110*a^3*c^2*d^4* e^6 + 130*a^4*c*d^2*e^8 - 47*a^5*e^10 + 60*(c^5*d^8*e^2 - 3*a*c^4*d^6*e^4 + 3*a^2*c^3*d^4*e^6 - a^3*c^2*d^2*e^8)*x^2 + 15*(c^5*d^9*e + 4*a*c^4*d^7*e ^3 - 18*a^2*c^3*d^5*e^5 + 20*a^3*c^2*d^3*e^7 - 7*a^4*c*d*e^9)*x)/(c^9*d^9* x^3 + 3*a*c^8*d^8*e*x^2 + 3*a^2*c^7*d^7*e^2*x + a^3*c^6*d^6*e^3) + 1/2*(c* d*e^5*x^2 + 2*(5*c*d^2*e^4 - 4*a*e^6)*x)/(c^5*d^5) + 10*(c^2*d^4*e^3 - 2*a *c*d^2*e^5 + a^2*e^7)*log(c*d*x + a*e)/(c^6*d^6)
Time = 0.14 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.67 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {10 \, {\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{6} d^{6}} - \frac {2 \, c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} + 20 \, a^{2} c^{3} d^{6} e^{4} - 110 \, a^{3} c^{2} d^{4} e^{6} + 130 \, a^{4} c d^{2} e^{8} - 47 \, a^{5} e^{10} + 60 \, {\left (c^{5} d^{8} e^{2} - 3 \, a c^{4} d^{6} e^{4} + 3 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 15 \, {\left (c^{5} d^{9} e + 4 \, a c^{4} d^{7} e^{3} - 18 \, a^{2} c^{3} d^{5} e^{5} + 20 \, a^{3} c^{2} d^{3} e^{7} - 7 \, a^{4} c d e^{9}\right )} x}{6 \, {\left (c d x + a e\right )}^{3} c^{6} d^{6}} + \frac {c^{4} d^{4} e^{5} x^{2} + 10 \, c^{4} d^{5} e^{4} x - 8 \, a c^{3} d^{3} e^{6} x}{2 \, c^{8} d^{8}} \] Input:
integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac ")
Output:
10*(c^2*d^4*e^3 - 2*a*c*d^2*e^5 + a^2*e^7)*log(abs(c*d*x + a*e))/(c^6*d^6) - 1/6*(2*c^5*d^10 + 5*a*c^4*d^8*e^2 + 20*a^2*c^3*d^6*e^4 - 110*a^3*c^2*d^ 4*e^6 + 130*a^4*c*d^2*e^8 - 47*a^5*e^10 + 60*(c^5*d^8*e^2 - 3*a*c^4*d^6*e^ 4 + 3*a^2*c^3*d^4*e^6 - a^3*c^2*d^2*e^8)*x^2 + 15*(c^5*d^9*e + 4*a*c^4*d^7 *e^3 - 18*a^2*c^3*d^5*e^5 + 20*a^3*c^2*d^3*e^7 - 7*a^4*c*d*e^9)*x)/((c*d*x + a*e)^3*c^6*d^6) + 1/2*(c^4*d^4*e^5*x^2 + 10*c^4*d^5*e^4*x - 8*a*c^3*d^3 *e^6*x)/(c^8*d^8)
Time = 0.08 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=x\,\left (\frac {5\,e^4}{c^4\,d^3}-\frac {4\,a\,e^6}{c^5\,d^5}\right )-\frac {x^2\,\left (-10\,a^3\,c\,d\,e^8+30\,a^2\,c^2\,d^3\,e^6-30\,a\,c^3\,d^5\,e^4+10\,c^4\,d^7\,e^2\right )+x\,\left (-\frac {35\,a^4\,e^9}{2}+50\,a^3\,c\,d^2\,e^7-45\,a^2\,c^2\,d^4\,e^5+10\,a\,c^3\,d^6\,e^3+\frac {5\,c^4\,d^8\,e}{2}\right )+\frac {-47\,a^5\,e^{10}+130\,a^4\,c\,d^2\,e^8-110\,a^3\,c^2\,d^4\,e^6+20\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2+2\,c^5\,d^{10}}{6\,c\,d}}{a^3\,c^5\,d^5\,e^3+3\,a^2\,c^6\,d^6\,e^2\,x+3\,a\,c^7\,d^7\,e\,x^2+c^8\,d^8\,x^3}+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (10\,a^2\,e^7-20\,a\,c\,d^2\,e^5+10\,c^2\,d^4\,e^3\right )}{c^6\,d^6}+\frac {e^5\,x^2}{2\,c^4\,d^4} \] Input:
int((d + e*x)^9/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)
Output:
x*((5*e^4)/(c^4*d^3) - (4*a*e^6)/(c^5*d^5)) - (x^2*(10*c^4*d^7*e^2 - 30*a* c^3*d^5*e^4 + 30*a^2*c^2*d^3*e^6 - 10*a^3*c*d*e^8) + x*((5*c^4*d^8*e)/2 - (35*a^4*e^9)/2 + 10*a*c^3*d^6*e^3 + 50*a^3*c*d^2*e^7 - 45*a^2*c^2*d^4*e^5) + (2*c^5*d^10 - 47*a^5*e^10 + 5*a*c^4*d^8*e^2 + 130*a^4*c*d^2*e^8 + 20*a^ 2*c^3*d^6*e^4 - 110*a^3*c^2*d^4*e^6)/(6*c*d))/(c^8*d^8*x^3 + a^3*c^5*d^5*e ^3 + 3*a*c^7*d^7*e*x^2 + 3*a^2*c^6*d^6*e^2*x) + (log(a*e + c*d*x)*(10*a^2* e^7 + 10*c^2*d^4*e^3 - 20*a*c*d^2*e^5))/(c^6*d^6) + (e^5*x^2)/(2*c^4*d^4)
Time = 0.26 (sec) , antiderivative size = 556, normalized size of antiderivative = 3.11 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {180 \,\mathrm {log}\left (c d x +a e \right ) a^{5} c d \,e^{9} x +60 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{3} d^{3} e^{7} x^{3}-120 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{4} d^{5} e^{5} x^{3}+60 \,\mathrm {log}\left (c d x +a e \right ) a \,c^{5} d^{7} e^{3} x^{3}-100 a^{5} c \,d^{2} e^{8}+50 a^{4} c^{2} d^{4} e^{6}+90 a^{5} c d \,e^{9} x -180 a^{4} c^{2} d^{3} e^{7} x +90 a^{3} c^{3} d^{5} e^{5} x -15 a \,c^{5} d^{9} e x -360 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c^{2} d^{3} e^{7} x +180 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c^{2} d^{2} e^{8} x^{2}+180 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{3} d^{5} e^{5} x -360 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{3} d^{4} e^{6} x^{2}+180 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{4} d^{6} e^{4} x^{2}-120 \,\mathrm {log}\left (c d x +a e \right ) a^{5} c \,d^{2} e^{8}+60 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c^{2} d^{4} e^{6}-60 a^{3} c^{3} d^{3} e^{7} x^{3}+120 a^{2} c^{4} d^{5} e^{5} x^{3}-15 a^{2} c^{4} d^{4} e^{6} x^{4}-60 a \,c^{5} d^{7} e^{3} x^{3}+30 a \,c^{5} d^{6} e^{4} x^{4}+3 a \,c^{5} d^{5} e^{5} x^{5}-5 a^{2} c^{4} d^{8} e^{2}+20 c^{6} d^{9} e \,x^{3}+60 \,\mathrm {log}\left (c d x +a e \right ) a^{6} e^{10}-2 a \,c^{5} d^{10}+50 a^{6} e^{10}}{6 a \,c^{6} d^{6} \left (c^{3} d^{3} x^{3}+3 a \,c^{2} d^{2} e \,x^{2}+3 a^{2} c d \,e^{2} x +a^{3} e^{3}\right )} \] Input:
int((e*x+d)^9/(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**6*e**10 - 120*log(a*e + c*d*x)*a**5*c*d**2*e**8 + 180*log(a*e + c*d*x)*a**5*c*d*e**9*x + 60*log(a*e + c*d*x)*a**4*c**2*d**4* e**6 - 360*log(a*e + c*d*x)*a**4*c**2*d**3*e**7*x + 180*log(a*e + c*d*x)*a **4*c**2*d**2*e**8*x**2 + 180*log(a*e + c*d*x)*a**3*c**3*d**5*e**5*x - 360 *log(a*e + c*d*x)*a**3*c**3*d**4*e**6*x**2 + 60*log(a*e + c*d*x)*a**3*c**3 *d**3*e**7*x**3 + 180*log(a*e + c*d*x)*a**2*c**4*d**6*e**4*x**2 - 120*log( a*e + c*d*x)*a**2*c**4*d**5*e**5*x**3 + 60*log(a*e + c*d*x)*a*c**5*d**7*e* *3*x**3 + 50*a**6*e**10 - 100*a**5*c*d**2*e**8 + 90*a**5*c*d*e**9*x + 50*a **4*c**2*d**4*e**6 - 180*a**4*c**2*d**3*e**7*x + 90*a**3*c**3*d**5*e**5*x - 60*a**3*c**3*d**3*e**7*x**3 - 5*a**2*c**4*d**8*e**2 + 120*a**2*c**4*d**5 *e**5*x**3 - 15*a**2*c**4*d**4*e**6*x**4 - 2*a*c**5*d**10 - 15*a*c**5*d**9 *e*x - 60*a*c**5*d**7*e**3*x**3 + 30*a*c**5*d**6*e**4*x**4 + 3*a*c**5*d**5 *e**5*x**5 + 20*c**6*d**9*e*x**3)/(6*a*c**6*d**6*(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))