Integrand size = 33, antiderivative size = 226 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (a e+c d x)^3}+\frac {3 c^2 d^2 e}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac {6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac {e^3}{2 \left (c d^2-a e^2\right )^4 (d+e x)^2}-\frac {4 c d e^3}{\left (c d^2-a e^2\right )^5 (d+e x)}-\frac {10 c^2 d^2 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}+\frac {10 c^2 d^2 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^6} \] Output:
-1/3*c^2*d^2/(-a*e^2+c*d^2)^3/(c*d*x+a*e)^3+3/2*c^2*d^2*e/(-a*e^2+c*d^2)^4 /(c*d*x+a*e)^2-6*c^2*d^2*e^2/(-a*e^2+c*d^2)^5/(c*d*x+a*e)-1/2*e^3/(-a*e^2+ c*d^2)^4/(e*x+d)^2-4*c*d*e^3/(-a*e^2+c*d^2)^5/(e*x+d)-10*c^2*d^2*e^3*ln(c* d*x+a*e)/(-a*e^2+c*d^2)^6+10*c^2*d^2*e^3*ln(e*x+d)/(-a*e^2+c*d^2)^6
Time = 0.14 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.91 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {\frac {2 c^2 d^2 \left (-c d^2+a e^2\right )^3}{(a e+c d x)^3}+\frac {9 c^2 d^2 e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {36 c^2 d^2 e^2 \left (-c d^2+a e^2\right )}{a e+c d x}-\frac {3 e^3 \left (c d^2-a e^2\right )^2}{(d+e x)^2}+\frac {24 c d e^3 \left (-c d^2+a e^2\right )}{d+e x}-60 c^2 d^2 e^3 \log (a e+c d x)+60 c^2 d^2 e^3 \log (d+e x)}{6 \left (c d^2-a e^2\right )^6} \] Input:
Integrate[(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
Output:
((2*c^2*d^2*(-(c*d^2) + a*e^2)^3)/(a*e + c*d*x)^3 + (9*c^2*d^2*e*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (36*c^2*d^2*e^2*(-(c*d^2) + a*e^2))/(a*e + c*d *x) - (3*e^3*(c*d^2 - a*e^2)^2)/(d + e*x)^2 + (24*c*d*e^3*(-(c*d^2) + a*e^ 2))/(d + e*x) - 60*c^2*d^2*e^3*Log[a*e + c*d*x] + 60*c^2*d^2*e^3*Log[d + e *x])/(6*(c*d^2 - a*e^2)^6)
Time = 0.73 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, 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.
| \(\displaystyle \int \frac {d+e x}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {6 c^3 d^3 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)^2}-\frac {3 c^3 d^3 e}{\left (c d^2-a e^2\right )^4 (a e+c d x)^3}+\frac {c^3 d^3}{\left (c d^2-a e^2\right )^3 (a e+c d x)^4}-\frac {10 c^3 d^3 e^3}{\left (c d^2-a e^2\right )^6 (a e+c d x)}+\frac {10 c^2 d^2 e^4}{(d+e x) \left (c d^2-a e^2\right )^6}+\frac {4 c d e^4}{(d+e x)^2 \left (c d^2-a e^2\right )^5}+\frac {e^4}{(d+e x)^3 \left (c d^2-a e^2\right )^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac {3 c^2 d^2 e}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac {c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (a e+c d x)^3}-\frac {10 c^2 d^2 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}+\frac {10 c^2 d^2 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^6}-\frac {4 c d e^3}{(d+e x) \left (c d^2-a e^2\right )^5}-\frac {e^3}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4}\) |
Input:
Int[(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
Output:
-1/3*(c^2*d^2)/((c*d^2 - a*e^2)^3*(a*e + c*d*x)^3) + (3*c^2*d^2*e)/(2*(c*d ^2 - a*e^2)^4*(a*e + c*d*x)^2) - (6*c^2*d^2*e^2)/((c*d^2 - a*e^2)^5*(a*e + c*d*x)) - e^3/(2*(c*d^2 - a*e^2)^4*(d + e*x)^2) - (4*c*d*e^3)/((c*d^2 - a *e^2)^5*(d + e*x)) - (10*c^2*d^2*e^3*Log[a*e + c*d*x])/(c*d^2 - a*e^2)^6 + (10*c^2*d^2*e^3*Log[d + e*x])/(c*d^2 - a*e^2)^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 = 1.67 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(\frac {c^{2} d^{2}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c d x +a e \right )^{3}}-\frac {10 c^{2} d^{2} e^{3} \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{6}}+\frac {6 c^{2} d^{2} e^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{5} \left (c d x +a e \right )}+\frac {3 c^{2} d^{2} e}{2 \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (c d x +a e \right )^{2}}-\frac {e^{3}}{2 \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (e x +d \right )^{2}}+\frac {10 c^{2} d^{2} e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{6}}+\frac {4 e^{3} c d}{\left (a \,e^{2}-c \,d^{2}\right )^{5} \left (e x +d \right )}\) | \(221\) |
| risch | \(\frac {\frac {10 c^{4} d^{4} e^{4} x^{4}}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}+\frac {5 d^{3} e^{3} c^{3} \left (5 a \,e^{2}+3 c \,d^{2}\right ) x^{3}}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}+\frac {5 \left (11 a^{2} e^{4}+23 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right ) c^{2} d^{2} e^{2} x^{2}}{3 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}+\frac {5 \left (3 e^{6} a^{3}+35 d^{2} e^{4} a^{2} c +11 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) c d e x}{6 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}-\frac {3 a^{4} e^{8}-27 a^{3} c \,d^{2} e^{6}-47 a^{2} c^{2} d^{4} e^{4}+13 a \,c^{3} d^{6} e^{2}-2 c^{4} d^{8}}{6 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}}{\left (c d x +a e \right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{2}}+\frac {10 c^{2} d^{2} e^{3} \ln \left (-e x -d \right )}{a^{6} e^{12}-6 a^{5} d^{2} e^{10} c +15 a^{4} d^{4} e^{8} c^{2}-20 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}-6 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}}-\frac {10 c^{2} d^{2} e^{3} \ln \left (c d x +a e \right )}{a^{6} e^{12}-6 a^{5} d^{2} e^{10} c +15 a^{4} d^{4} e^{8} c^{2}-20 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}-6 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}}\) | \(782\) |
| norman | \(\frac {\frac {10 d^{4} c^{4} e^{5} x^{5}}{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}}+\frac {\left (25 a \,c^{5} d^{5} e^{8}+25 c^{6} d^{7} e^{6}\right ) x^{4}}{d^{2} c^{2} e^{2} \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 )}+\frac {-3 a^{4} c^{3} d \,e^{8}+27 a^{3} c^{4} d^{3} e^{6}+47 a^{2} c^{5} d^{5} e^{4}-13 a \,c^{6} d^{7} e^{2}+2 c^{7} d^{9}}{6 c^{3} \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 )}+\frac {\left (-a^{4} c^{3} d \,e^{10}+14 a^{3} c^{4} d^{3} e^{8}+74 a^{2} c^{5} d^{5} e^{6}+14 a \,c^{6} d^{7} e^{4}-c^{7} d^{9} e^{2}\right ) x}{2 e d \,c^{3} \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 )}+\frac {\left (55 a^{2} c^{5} d^{5} e^{10}+190 a \,c^{6} d^{7} e^{8}+55 c^{7} d^{9} e^{6}\right ) x^{3}}{3 c^{3} d^{3} e^{3} \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 )}+\frac {\left (5 a^{3} c^{4} d^{3} e^{10}+95 a^{2} c^{5} d^{5} e^{8}+95 a \,c^{6} d^{7} e^{6}+5 c^{7} d^{9} e^{4}\right ) x^{2}}{2 e^{2} d^{2} c^{3} \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 )}}{\left (e x +d \right )^{3} \left (c d x +a e \right )^{3}}+\frac {10 c^{2} d^{2} e^{3} \ln \left (e x +d \right )}{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}}-\frac {10 c^{2} d^{2} e^{3} \ln \left (c d x +a e \right )}{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}}\) | \(955\) |
| parallelrisch | \(\frac {120 \ln \left (e x +d \right ) x^{4} c^{8} d^{9} e^{6}-120 \ln \left (c d x +a e \right ) x^{4} c^{8} d^{9} e^{6}+60 \ln \left (e x +d \right ) x^{3} c^{8} d^{10} e^{5}-60 \ln \left (c d x +a e \right ) x^{3} c^{8} d^{10} e^{5}+60 \ln \left (e x +d \right ) a^{3} c^{5} d^{7} e^{8}-60 \ln \left (c d x +a e \right ) a^{3} c^{5} d^{7} e^{8}+150 x^{3} a^{2} c^{6} d^{6} e^{9}-60 x^{3} a \,c^{7} d^{8} e^{7}+110 x^{2} a^{3} c^{5} d^{5} e^{10}+120 x^{2} a^{2} c^{6} d^{7} e^{8}+360 \ln \left (e x +d \right ) x^{2} a^{2} c^{6} d^{7} e^{8}+180 \ln \left (e x +d \right ) x^{2} a \,c^{7} d^{9} e^{6}-60 \ln \left (c d x +a e \right ) x^{2} a^{3} c^{5} d^{5} e^{10}-360 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{6} d^{7} e^{8}-180 \ln \left (c d x +a e \right ) x^{2} a \,c^{7} d^{9} e^{6}+120 \ln \left (e x +d \right ) x \,a^{3} c^{5} d^{6} e^{9}+180 \ln \left (e x +d \right ) x \,a^{2} c^{6} d^{8} e^{7}-120 \ln \left (c d x +a e \right ) x \,a^{3} c^{5} d^{6} e^{9}-180 \ln \left (c d x +a e \right ) x \,a^{2} c^{6} d^{8} e^{7}-3 a^{5} c^{3} d^{3} e^{12}+30 a^{4} c^{4} d^{5} e^{10}+20 a^{3} c^{5} d^{7} e^{8}-60 a^{2} c^{6} d^{9} e^{6}+15 a \,c^{7} d^{11} e^{4}-2 c^{8} d^{13} e^{2}-60 x^{4} c^{8} d^{9} e^{6}-90 x^{3} c^{8} d^{10} e^{5}-20 x^{2} c^{8} d^{11} e^{4}+5 x \,c^{8} d^{12} e^{3}-210 x^{2} a \,c^{7} d^{9} e^{6}+15 x \,a^{4} c^{4} d^{4} e^{11}+160 x \,a^{3} c^{5} d^{6} e^{9}-120 x \,a^{2} c^{6} d^{8} e^{7}-60 x a \,c^{7} d^{10} e^{5}+60 \ln \left (e x +d \right ) x^{5} c^{8} d^{8} e^{7}-60 \ln \left (c d x +a e \right ) x^{5} c^{8} d^{8} e^{7}+60 x^{4} a \,c^{7} d^{7} e^{8}+180 \ln \left (e x +d \right ) x^{4} a \,c^{7} d^{7} e^{8}-180 \ln \left (c d x +a e \right ) x^{4} a \,c^{7} d^{7} e^{8}+180 \ln \left (e x +d \right ) x^{3} a^{2} c^{6} d^{6} e^{9}+360 \ln \left (e x +d \right ) x^{3} a \,c^{7} d^{8} e^{7}-180 \ln \left (c d x +a e \right ) x^{3} a^{2} c^{6} d^{6} e^{9}-360 \ln \left (c d x +a e \right ) x^{3} a \,c^{7} d^{8} e^{7}+60 \ln \left (e x +d \right ) x^{2} a^{3} c^{5} d^{5} e^{10}}{6 \left (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}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{2} \left (c d x +a e \right ) e^{2} d^{3} c^{3}}\) | \(963\) |
Input:
int((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^4,x,method=_RETURNVERBOSE)
Output:
1/3*c^2*d^2/(a*e^2-c*d^2)^3/(c*d*x+a*e)^3-10*c^2*d^2/(a*e^2-c*d^2)^6*e^3*l n(c*d*x+a*e)+6*c^2*d^2/(a*e^2-c*d^2)^5*e^2/(c*d*x+a*e)+3/2*c^2*d^2/(a*e^2- c*d^2)^4*e/(c*d*x+a*e)^2-1/2*e^3/(a*e^2-c*d^2)^4/(e*x+d)^2+10*c^2*d^2/(a*e ^2-c*d^2)^6*e^3*ln(e*x+d)+4*e^3/(a*e^2-c*d^2)^5*c*d/(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 1242 vs. \(2 (220) = 440\).
Time = 0.11 (sec) , antiderivative size = 1242, normalized size of antiderivative = 5.50 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas ")
Output:
-1/6*(2*c^5*d^10 - 15*a*c^4*d^8*e^2 + 60*a^2*c^3*d^6*e^4 - 20*a^3*c^2*d^4* e^6 - 30*a^4*c*d^2*e^8 + 3*a^5*e^10 + 60*(c^5*d^6*e^4 - a*c^4*d^4*e^6)*x^4 + 30*(3*c^5*d^7*e^3 + 2*a*c^4*d^5*e^5 - 5*a^2*c^3*d^3*e^7)*x^3 + 10*(2*c^ 5*d^8*e^2 + 21*a*c^4*d^6*e^4 - 12*a^2*c^3*d^4*e^6 - 11*a^3*c^2*d^2*e^8)*x^ 2 - 5*(c^5*d^9*e - 12*a*c^4*d^7*e^3 - 24*a^2*c^3*d^5*e^5 + 32*a^3*c^2*d^3* e^7 + 3*a^4*c*d*e^9)*x + 60*(c^5*d^5*e^5*x^5 + a^3*c^2*d^4*e^6 + (2*c^5*d^ 6*e^4 + 3*a*c^4*d^4*e^6)*x^4 + (c^5*d^7*e^3 + 6*a*c^4*d^5*e^5 + 3*a^2*c^3* d^3*e^7)*x^3 + (3*a*c^4*d^6*e^4 + 6*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)*x)*log(c*d*x + a*e) - 60*(c^5*d ^5*e^5*x^5 + a^3*c^2*d^4*e^6 + (2*c^5*d^6*e^4 + 3*a*c^4*d^4*e^6)*x^4 + (c^ 5*d^7*e^3 + 6*a*c^4*d^5*e^5 + 3*a^2*c^3*d^3*e^7)*x^3 + (3*a*c^4*d^6*e^4 + 6*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)*x)*log(e*x + d))/(a^3*c^6*d^14*e^3 - 6*a^4*c^5*d^12*e^5 + 15*a^5* c^4*d^10*e^7 - 20*a^6*c^3*d^8*e^9 + 15*a^7*c^2*d^6*e^11 - 6*a^8*c*d^4*e^13 + a^9*d^2*e^15 + (c^9*d^15*e^2 - 6*a*c^8*d^13*e^4 + 15*a^2*c^7*d^11*e^6 - 20*a^3*c^6*d^9*e^8 + 15*a^4*c^5*d^7*e^10 - 6*a^5*c^4*d^5*e^12 + a^6*c^3*d ^3*e^14)*x^5 + (2*c^9*d^16*e - 9*a*c^8*d^14*e^3 + 12*a^2*c^7*d^12*e^5 + 5* a^3*c^6*d^10*e^7 - 30*a^4*c^5*d^8*e^9 + 33*a^5*c^4*d^6*e^11 - 16*a^6*c^3*d ^4*e^13 + 3*a^7*c^2*d^2*e^15)*x^4 + (c^9*d^17 - 18*a^2*c^7*d^13*e^4 + 52*a ^3*c^6*d^11*e^6 - 60*a^4*c^5*d^9*e^8 + 24*a^5*c^4*d^7*e^10 + 10*a^6*c^3...
Leaf count of result is larger than twice the leaf count of optimal. 1363 vs. \(2 (209) = 418\).
Time = 2.49 (sec) , antiderivative size = 1363, normalized size of antiderivative = 6.03 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)
Output:
10*c**2*d**2*e**3*log(x + (-10*a**7*c**2*d**2*e**17/(a*e**2 - c*d**2)**6 + 70*a**6*c**3*d**4*e**15/(a*e**2 - c*d**2)**6 - 210*a**5*c**4*d**6*e**13/( a*e**2 - c*d**2)**6 + 350*a**4*c**5*d**8*e**11/(a*e**2 - c*d**2)**6 - 350* a**3*c**6*d**10*e**9/(a*e**2 - c*d**2)**6 + 210*a**2*c**7*d**12*e**7/(a*e* *2 - c*d**2)**6 - 70*a*c**8*d**14*e**5/(a*e**2 - c*d**2)**6 + 10*a*c**2*d* *2*e**5 + 10*c**9*d**16*e**3/(a*e**2 - c*d**2)**6 + 10*c**3*d**4*e**3)/(20 *c**3*d**3*e**4))/(a*e**2 - c*d**2)**6 - 10*c**2*d**2*e**3*log(x + (10*a** 7*c**2*d**2*e**17/(a*e**2 - c*d**2)**6 - 70*a**6*c**3*d**4*e**15/(a*e**2 - c*d**2)**6 + 210*a**5*c**4*d**6*e**13/(a*e**2 - c*d**2)**6 - 350*a**4*c** 5*d**8*e**11/(a*e**2 - c*d**2)**6 + 350*a**3*c**6*d**10*e**9/(a*e**2 - c*d **2)**6 - 210*a**2*c**7*d**12*e**7/(a*e**2 - c*d**2)**6 + 70*a*c**8*d**14* e**5/(a*e**2 - c*d**2)**6 + 10*a*c**2*d**2*e**5 - 10*c**9*d**16*e**3/(a*e* *2 - c*d**2)**6 + 10*c**3*d**4*e**3)/(20*c**3*d**3*e**4))/(a*e**2 - c*d**2 )**6 + (-3*a**4*e**8 + 27*a**3*c*d**2*e**6 + 47*a**2*c**2*d**4*e**4 - 13*a *c**3*d**6*e**2 + 2*c**4*d**8 + 60*c**4*d**4*e**4*x**4 + x**3*(150*a*c**3* d**3*e**5 + 90*c**4*d**5*e**3) + x**2*(110*a**2*c**2*d**2*e**6 + 230*a*c** 3*d**4*e**4 + 20*c**4*d**6*e**2) + x*(15*a**3*c*d*e**7 + 175*a**2*c**2*d** 3*e**5 + 55*a*c**3*d**5*e**3 - 5*c**4*d**7*e))/(6*a**8*d**2*e**13 - 30*a** 7*c*d**4*e**11 + 60*a**6*c**2*d**6*e**9 - 60*a**5*c**3*d**8*e**7 + 30*a**4 *c**4*d**10*e**5 - 6*a**3*c**5*d**12*e**3 + x**5*(6*a**5*c**3*d**3*e**1...
Leaf count of result is larger than twice the leaf count of optimal. 956 vs. \(2 (220) = 440\).
Time = 0.08 (sec) , antiderivative size = 956, normalized size of antiderivative = 4.23 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima ")
Output:
-10*c^2*d^2*e^3*log(c*d*x + a*e)/(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4 *d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^ 6*e^12) + 10*c^2*d^2*e^3*log(e*x + d)/(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^ 2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12) - 1/6*(60*c^4*d^4*e^4*x^4 + 2*c^4*d^8 - 13*a*c^3*d^6*e^2 + 47 *a^2*c^2*d^4*e^4 + 27*a^3*c*d^2*e^6 - 3*a^4*e^8 + 30*(3*c^4*d^5*e^3 + 5*a* c^3*d^3*e^5)*x^3 + 10*(2*c^4*d^6*e^2 + 23*a*c^3*d^4*e^4 + 11*a^2*c^2*d^2*e ^6)*x^2 - 5*(c^4*d^7*e - 11*a*c^3*d^5*e^3 - 35*a^2*c^2*d^3*e^5 - 3*a^3*c*d *e^7)*x)/(a^3*c^5*d^12*e^3 - 5*a^4*c^4*d^10*e^5 + 10*a^5*c^3*d^8*e^7 - 10* a^6*c^2*d^6*e^9 + 5*a^7*c*d^4*e^11 - a^8*d^2*e^13 + (c^8*d^13*e^2 - 5*a*c^ 7*d^11*e^4 + 10*a^2*c^6*d^9*e^6 - 10*a^3*c^5*d^7*e^8 + 5*a^4*c^4*d^5*e^10 - a^5*c^3*d^3*e^12)*x^5 + (2*c^8*d^14*e - 7*a*c^7*d^12*e^3 + 5*a^2*c^6*d^1 0*e^5 + 10*a^3*c^5*d^8*e^7 - 20*a^4*c^4*d^6*e^9 + 13*a^5*c^3*d^4*e^11 - 3* a^6*c^2*d^2*e^13)*x^4 + (c^8*d^15 + a*c^7*d^13*e^2 - 17*a^2*c^6*d^11*e^4 + 35*a^3*c^5*d^9*e^6 - 25*a^4*c^4*d^7*e^8 - a^5*c^3*d^5*e^10 + 9*a^6*c^2*d^ 3*e^12 - 3*a^7*c*d*e^14)*x^3 + (3*a*c^7*d^14*e - 9*a^2*c^6*d^12*e^3 + a^3* c^5*d^10*e^5 + 25*a^4*c^4*d^8*e^7 - 35*a^5*c^3*d^6*e^9 + 17*a^6*c^2*d^4*e^ 11 - a^7*c*d^2*e^13 - a^8*e^15)*x^2 + (3*a^2*c^6*d^13*e^2 - 13*a^3*c^5*d^1 1*e^4 + 20*a^4*c^4*d^9*e^6 - 10*a^5*c^3*d^7*e^8 - 5*a^6*c^2*d^5*e^10 + 7*a ^7*c*d^3*e^12 - 2*a^8*d*e^14)*x)
Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (220) = 440\).
Time = 0.14 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.23 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {10 \, c^{3} d^{3} e^{3} \log \left ({\left | c d x + a e \right |}\right )}{c^{7} d^{13} - 6 \, a c^{6} d^{11} e^{2} + 15 \, a^{2} c^{5} d^{9} e^{4} - 20 \, a^{3} c^{4} d^{7} e^{6} + 15 \, a^{4} c^{3} d^{5} e^{8} - 6 \, a^{5} c^{2} d^{3} e^{10} + a^{6} c d e^{12}} + \frac {10 \, c^{2} d^{2} e^{4} \log \left ({\left | e x + d \right |}\right )}{c^{6} d^{12} e - 6 \, a c^{5} d^{10} e^{3} + 15 \, a^{2} c^{4} d^{8} e^{5} - 20 \, a^{3} c^{3} d^{6} e^{7} + 15 \, a^{4} c^{2} d^{4} e^{9} - 6 \, a^{5} c d^{2} e^{11} + a^{6} e^{13}} - \frac {2 \, c^{5} d^{10} - 15 \, a c^{4} d^{8} e^{2} + 60 \, a^{2} c^{3} d^{6} e^{4} - 20 \, a^{3} c^{2} d^{4} e^{6} - 30 \, a^{4} c d^{2} e^{8} + 3 \, a^{5} e^{10} + 60 \, {\left (c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} + 30 \, {\left (3 \, c^{5} d^{7} e^{3} + 2 \, a c^{4} d^{5} e^{5} - 5 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 10 \, {\left (2 \, c^{5} d^{8} e^{2} + 21 \, a c^{4} d^{6} e^{4} - 12 \, a^{2} c^{3} d^{4} e^{6} - 11 \, a^{3} c^{2} d^{2} e^{8}\right )} x^{2} - 5 \, {\left (c^{5} d^{9} e - 12 \, a c^{4} d^{7} e^{3} - 24 \, a^{2} c^{3} d^{5} e^{5} + 32 \, a^{3} c^{2} d^{3} e^{7} + 3 \, a^{4} c d e^{9}\right )} x}{6 \, {\left (c d^{2} - a e^{2}\right )}^{6} {\left (c d x + a e\right )}^{3} {\left (e x + d\right )}^{2}} \] Input:
integrate((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")
Output:
-10*c^3*d^3*e^3*log(abs(c*d*x + a*e))/(c^7*d^13 - 6*a*c^6*d^11*e^2 + 15*a^ 2*c^5*d^9*e^4 - 20*a^3*c^4*d^7*e^6 + 15*a^4*c^3*d^5*e^8 - 6*a^5*c^2*d^3*e^ 10 + a^6*c*d*e^12) + 10*c^2*d^2*e^4*log(abs(e*x + d))/(c^6*d^12*e - 6*a*c^ 5*d^10*e^3 + 15*a^2*c^4*d^8*e^5 - 20*a^3*c^3*d^6*e^7 + 15*a^4*c^2*d^4*e^9 - 6*a^5*c*d^2*e^11 + a^6*e^13) - 1/6*(2*c^5*d^10 - 15*a*c^4*d^8*e^2 + 60*a ^2*c^3*d^6*e^4 - 20*a^3*c^2*d^4*e^6 - 30*a^4*c*d^2*e^8 + 3*a^5*e^10 + 60*( c^5*d^6*e^4 - a*c^4*d^4*e^6)*x^4 + 30*(3*c^5*d^7*e^3 + 2*a*c^4*d^5*e^5 - 5 *a^2*c^3*d^3*e^7)*x^3 + 10*(2*c^5*d^8*e^2 + 21*a*c^4*d^6*e^4 - 12*a^2*c^3* d^4*e^6 - 11*a^3*c^2*d^2*e^8)*x^2 - 5*(c^5*d^9*e - 12*a*c^4*d^7*e^3 - 24*a ^2*c^3*d^5*e^5 + 32*a^3*c^2*d^3*e^7 + 3*a^4*c*d*e^9)*x)/((c*d^2 - a*e^2)^6 *(c*d*x + a*e)^3*(e*x + d)^2)
Time = 5.64 (sec) , antiderivative size = 891, normalized size of antiderivative = 3.94 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {\frac {-3\,a^4\,e^8+27\,a^3\,c\,d^2\,e^6+47\,a^2\,c^2\,d^4\,e^4-13\,a\,c^3\,d^6\,e^2+2\,c^4\,d^8}{6\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}+\frac {5\,e^2\,x^2\,\left (11\,a^2\,c^2\,d^2\,e^4+23\,a\,c^3\,d^4\,e^2+2\,c^4\,d^6\right )}{3\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}+\frac {5\,d\,e\,x\,\left (3\,a^3\,c\,e^6+35\,a^2\,c^2\,d^2\,e^4+11\,a\,c^3\,d^4\,e^2-c^4\,d^6\right )}{6\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}+\frac {5\,c\,e\,x^3\,\left (3\,c^3\,d^5\,e^2+5\,a\,c^2\,d^3\,e^4\right )}{a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}}+\frac {10\,c^4\,d^4\,e^4\,x^4}{a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}}}{x^2\,\left (a^3\,e^5+6\,a^2\,c\,d^2\,e^3+3\,a\,c^2\,d^4\,e\right )+x^3\,\left (3\,a^2\,c\,d\,e^4+6\,a\,c^2\,d^3\,e^2+c^3\,d^5\right )+x\,\left (2\,a^3\,d\,e^4+3\,c\,a^2\,d^3\,e^2\right )+x^4\,\left (2\,c^3\,d^4\,e+3\,a\,c^2\,d^2\,e^3\right )+a^3\,d^2\,e^3+c^3\,d^3\,e^2\,x^5}-\frac {20\,c^2\,d^2\,e^3\,\mathrm {atanh}\left (\frac {a^6\,e^{12}-4\,a^5\,c\,d^2\,e^{10}+5\,a^4\,c^2\,d^4\,e^8-5\,a^2\,c^4\,d^8\,e^4+4\,a\,c^5\,d^{10}\,e^2-c^6\,d^{12}}{{\left (a\,e^2-c\,d^2\right )}^6}+\frac {2\,c\,d\,e\,x\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}{{\left (a\,e^2-c\,d^2\right )}^6}\right )}{{\left (a\,e^2-c\,d^2\right )}^6} \] Input:
int((d + e*x)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)
Output:
((2*c^4*d^8 - 3*a^4*e^8 - 13*a*c^3*d^6*e^2 + 27*a^3*c*d^2*e^6 + 47*a^2*c^2 *d^4*e^4)/(6*(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10 *a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6)) + (5*e^2*x^2*(2*c^4*d^6 + 23*a*c^3 *d^4*e^2 + 11*a^2*c^2*d^2*e^4))/(3*(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6)) + (5*d*e*x*( 3*a^3*c*e^6 - c^4*d^6 + 11*a*c^3*d^4*e^2 + 35*a^2*c^2*d^2*e^4))/(6*(a^5*e^ 10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*e^4 + 1 0*a^3*c^2*d^4*e^6)) + (5*c*e*x^3*(3*c^3*d^5*e^2 + 5*a*c^2*d^3*e^4))/(a^5*e ^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6) + (10*c^4*d^4*e^4*x^4)/(a^5*e^10 - c^5*d^10 + 5*a*c^4* d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6))/(x^2 *(a^3*e^5 + 6*a^2*c*d^2*e^3 + 3*a*c^2*d^4*e) + x^3*(c^3*d^5 + 6*a*c^2*d^3* e^2 + 3*a^2*c*d*e^4) + x*(2*a^3*d*e^4 + 3*a^2*c*d^3*e^2) + x^4*(2*c^3*d^4* e + 3*a*c^2*d^2*e^3) + a^3*d^2*e^3 + c^3*d^3*e^2*x^5) - (20*c^2*d^2*e^3*at anh((a^6*e^12 - c^6*d^12 + 4*a*c^5*d^10*e^2 - 4*a^5*c*d^2*e^10 - 5*a^2*c^4 *d^8*e^4 + 5*a^4*c^2*d^4*e^8)/(a*e^2 - c*d^2)^6 + (2*c*d*e*x*(a^5*e^10 - c ^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*e^4 + 10*a^3* c^2*d^4*e^6))/(a*e^2 - c*d^2)^6))/(a*e^2 - c*d^2)^6
Time = 0.25 (sec) , antiderivative size = 1990, normalized size of antiderivative = 8.81 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx =\text {Too large to display} \] Input:
int((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)
Output:
( - 180*log(a*e + c*d*x)*a**4*c**2*d**4*e**8 - 360*log(a*e + c*d*x)*a**4*c **2*d**3*e**9*x - 180*log(a*e + c*d*x)*a**4*c**2*d**2*e**10*x**2 - 120*log (a*e + c*d*x)*a**3*c**3*d**6*e**6 - 780*log(a*e + c*d*x)*a**3*c**3*d**5*e* *7*x - 1200*log(a*e + c*d*x)*a**3*c**3*d**4*e**8*x**2 - 540*log(a*e + c*d* x)*a**3*c**3*d**3*e**9*x**3 - 360*log(a*e + c*d*x)*a**2*c**4*d**7*e**5*x - 1260*log(a*e + c*d*x)*a**2*c**4*d**6*e**6*x**2 - 1440*log(a*e + c*d*x)*a* *2*c**4*d**5*e**7*x**3 - 540*log(a*e + c*d*x)*a**2*c**4*d**4*e**8*x**4 - 3 60*log(a*e + c*d*x)*a*c**5*d**8*e**4*x**2 - 900*log(a*e + c*d*x)*a*c**5*d* *7*e**5*x**3 - 720*log(a*e + c*d*x)*a*c**5*d**6*e**6*x**4 - 180*log(a*e + c*d*x)*a*c**5*d**5*e**7*x**5 - 120*log(a*e + c*d*x)*c**6*d**9*e**3*x**3 - 240*log(a*e + c*d*x)*c**6*d**8*e**4*x**4 - 120*log(a*e + c*d*x)*c**6*d**7* e**5*x**5 + 180*log(d + e*x)*a**4*c**2*d**4*e**8 + 360*log(d + e*x)*a**4*c **2*d**3*e**9*x + 180*log(d + e*x)*a**4*c**2*d**2*e**10*x**2 + 120*log(d + e*x)*a**3*c**3*d**6*e**6 + 780*log(d + e*x)*a**3*c**3*d**5*e**7*x + 1200* log(d + e*x)*a**3*c**3*d**4*e**8*x**2 + 540*log(d + e*x)*a**3*c**3*d**3*e* *9*x**3 + 360*log(d + e*x)*a**2*c**4*d**7*e**5*x + 1260*log(d + e*x)*a**2* c**4*d**6*e**6*x**2 + 1440*log(d + e*x)*a**2*c**4*d**5*e**7*x**3 + 540*log (d + e*x)*a**2*c**4*d**4*e**8*x**4 + 360*log(d + e*x)*a*c**5*d**8*e**4*x** 2 + 900*log(d + e*x)*a*c**5*d**7*e**5*x**3 + 720*log(d + e*x)*a*c**5*d**6* e**6*x**4 + 180*log(d + e*x)*a*c**5*d**5*e**7*x**5 + 120*log(d + e*x)*c...