Integrand size = 35, antiderivative size = 139 \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac {c d}{2 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac {c^2 d^2}{\left (c d^2-a e^2\right )^3 (d+e x)}+\frac {c^3 d^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac {c^3 d^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \] Output:
1/3/(-a*e^2+c*d^2)/(e*x+d)^3+1/2*c*d/(-a*e^2+c*d^2)^2/(e*x+d)^2+c^2*d^2/(- a*e^2+c*d^2)^3/(e*x+d)+c^3*d^3*ln(c*d*x+a*e)/(-a*e^2+c*d^2)^4-c^3*d^3*ln(e *x+d)/(-a*e^2+c*d^2)^4
Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {\left (c d^2-a e^2\right ) \left (2 a^2 e^4-a c d e^2 (7 d+3 e x)+c^2 d^2 \left (11 d^2+15 d e x+6 e^2 x^2\right )\right )+6 c^3 d^3 (d+e x)^3 \log (a e+c d x)-6 c^3 d^3 (d+e x)^3 \log (d+e x)}{6 \left (c d^2-a e^2\right )^4 (d+e x)^3} \] Input:
Integrate[1/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]
Output:
((c*d^2 - a*e^2)*(2*a^2*e^4 - a*c*d*e^2*(7*d + 3*e*x) + c^2*d^2*(11*d^2 + 15*d*e*x + 6*e^2*x^2)) + 6*c^3*d^3*(d + e*x)^3*Log[a*e + c*d*x] - 6*c^3*d^ 3*(d + e*x)^3*Log[d + e*x])/(6*(c*d^2 - a*e^2)^4*(d + e*x)^3)
Time = 0.35 (sec) , antiderivative size = 139, 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.
| \(\displaystyle \int \frac {1}{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {c^4 d^4}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac {c^3 d^3 e}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac {c^2 d^2 e}{(d+e x)^2 \left (c d^2-a e^2\right )^3}-\frac {c d e}{(d+e x)^3 \left (c d^2-a e^2\right )^2}-\frac {e}{(d+e x)^4 \left (c d^2-a e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^3 d^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac {c^3 d^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4}+\frac {c^2 d^2}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac {c d}{2 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac {1}{3 (d+e x)^3 \left (c d^2-a e^2\right )}\) |
Input:
Int[1/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]
Output:
1/(3*(c*d^2 - a*e^2)*(d + e*x)^3) + (c*d)/(2*(c*d^2 - a*e^2)^2*(d + e*x)^2 ) + (c^2*d^2)/((c*d^2 - a*e^2)^3*(d + e*x)) + (c^3*d^3*Log[a*e + c*d*x])/( c*d^2 - a*e^2)^4 - (c^3*d^3*Log[d + e*x])/(c*d^2 - a*e^2)^4
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 1.83 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(\frac {c^{3} d^{3} \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{4}}-\frac {1}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{3}}-\frac {c^{2} d^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \left (e x +d \right )}+\frac {c d}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e x +d \right )^{2}}-\frac {c^{3} d^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{4}}\) | \(137\) |
| risch | \(\frac {-\frac {e^{2} c^{2} d^{2} x^{2}}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}}+\frac {\left (a \,e^{2}-5 c \,d^{2}\right ) e c d x}{2 e^{6} a^{3}-6 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} a \,c^{2}-2 d^{6} c^{3}}-\frac {2 a^{2} e^{4}-7 a c \,d^{2} e^{2}+11 c^{2} d^{4}}{6 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}}{\left (e x +d \right )^{3}}-\frac {d^{3} c^{3} \ln \left (e x +d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {d^{3} c^{3} \ln \left (-c d x -a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\) | \(340\) |
| parallelrisch | \(-\frac {2 a^{3} d^{2} e^{7}-6 c^{3} d^{8} e +x^{3} a^{2} c d \,e^{8}-6 x^{3} a \,c^{2} d^{3} e^{6}+3 x^{2} a^{2} c \,d^{2} e^{7}-12 x^{2} a \,c^{2} d^{4} e^{5}+6 \ln \left (e x +d \right ) x^{3} c^{3} d^{5} e^{4}-6 \ln \left (c d x +a e \right ) x^{3} c^{3} d^{5} e^{4}+18 \ln \left (e x +d \right ) x^{2} c^{3} d^{6} e^{3}-18 \ln \left (c d x +a e \right ) x^{2} c^{3} d^{6} e^{3}+18 \ln \left (e x +d \right ) x \,c^{3} d^{7} e^{2}-18 \ln \left (c d x +a e \right ) x \,c^{3} d^{7} e^{2}+6 \ln \left (e x +d \right ) c^{3} d^{8} e -6 \ln \left (c d x +a e \right ) c^{3} d^{8} e +5 x^{3} c^{3} d^{5} e^{4}+9 x^{2} c^{3} d^{6} e^{3}-8 a^{2} c \,d^{4} e^{5}+12 a \,c^{2} d^{6} e^{3}}{6 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{3} d^{2} e}\) | \(359\) |
| norman | \(\frac {\frac {-a^{2} e^{5}+3 a c \,d^{2} e^{3}-3 c^{2} d^{4} e}{3 e \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}+\frac {e^{2} \left (-d \,e^{3} a c +5 d^{3} e \,c^{2}\right ) x^{3}}{6 d^{2} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}+\frac {\left (-d \,e^{3} a c +3 d^{3} e \,c^{2}\right ) e \,x^{2}}{2 d \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}}{\left (e x +d \right )^{3}}+\frac {d^{3} c^{3} \ln \left (c d x +a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {d^{3} c^{3} \ln \left (e x +d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\) | \(366\) |
Input:
int(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e),x,method=_RETURNVERBOSE)
Output:
c^3*d^3/(a*e^2-c*d^2)^4*ln(c*d*x+a*e)-1/3/(a*e^2-c*d^2)/(e*x+d)^3-c^2*d^2/ (a*e^2-c*d^2)^3/(e*x+d)+1/2*c*d/(a*e^2-c*d^2)^2/(e*x+d)^2-c^3*d^3/(a*e^2-c *d^2)^4*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (135) = 270\).
Time = 0.09 (sec) , antiderivative size = 454, normalized size of antiderivative = 3.27 \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {11 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (5 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x + 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{4} e^{2} x^{2} + 3 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (c d x + a e\right ) - 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{4} e^{2} x^{2} + 3 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{6 \, {\left (c^{4} d^{11} - 4 \, a c^{3} d^{9} e^{2} + 6 \, a^{2} c^{2} d^{7} e^{4} - 4 \, a^{3} c d^{5} e^{6} + a^{4} d^{3} e^{8} + {\left (c^{4} d^{8} e^{3} - 4 \, a c^{3} d^{6} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{7} - 4 \, a^{3} c d^{2} e^{9} + a^{4} e^{11}\right )} x^{3} + 3 \, {\left (c^{4} d^{9} e^{2} - 4 \, a c^{3} d^{7} e^{4} + 6 \, a^{2} c^{2} d^{5} e^{6} - 4 \, a^{3} c d^{3} e^{8} + a^{4} d e^{10}\right )} x^{2} + 3 \, {\left (c^{4} d^{10} e - 4 \, a c^{3} d^{8} e^{3} + 6 \, a^{2} c^{2} d^{6} e^{5} - 4 \, a^{3} c d^{4} e^{7} + a^{4} d^{2} e^{9}\right )} x\right )}} \] Input:
integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fric as")
Output:
1/6*(11*c^3*d^6 - 18*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 2*a^3*e^6 + 6*(c^3* d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 3*(5*c^3*d^5*e - 6*a*c^2*d^3*e^3 + a^2*c*d* e^5)*x + 6*(c^3*d^3*e^3*x^3 + 3*c^3*d^4*e^2*x^2 + 3*c^3*d^5*e*x + c^3*d^6) *log(c*d*x + a*e) - 6*(c^3*d^3*e^3*x^3 + 3*c^3*d^4*e^2*x^2 + 3*c^3*d^5*e*x + c^3*d^6)*log(e*x + d))/(c^4*d^11 - 4*a*c^3*d^9*e^2 + 6*a^2*c^2*d^7*e^4 - 4*a^3*c*d^5*e^6 + a^4*d^3*e^8 + (c^4*d^8*e^3 - 4*a*c^3*d^6*e^5 + 6*a^2*c ^2*d^4*e^7 - 4*a^3*c*d^2*e^9 + a^4*e^11)*x^3 + 3*(c^4*d^9*e^2 - 4*a*c^3*d^ 7*e^4 + 6*a^2*c^2*d^5*e^6 - 4*a^3*c*d^3*e^8 + a^4*d*e^10)*x^2 + 3*(c^4*d^1 0*e - 4*a*c^3*d^8*e^3 + 6*a^2*c^2*d^6*e^5 - 4*a^3*c*d^4*e^7 + a^4*d^2*e^9) *x)
Leaf count of result is larger than twice the leaf count of optimal. 672 vs. \(2 (121) = 242\).
Time = 0.80 (sec) , antiderivative size = 672, normalized size of antiderivative = 4.83 \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=- \frac {c^{3} d^{3} \log {\left (x + \frac {- \frac {a^{5} c^{3} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {5 a^{4} c^{4} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {10 a^{3} c^{5} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {10 a^{2} c^{6} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {5 a c^{7} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + a c^{3} d^{3} e^{2} + \frac {c^{8} d^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + c^{4} d^{5}}{2 c^{4} d^{4} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {c^{3} d^{3} \log {\left (x + \frac {\frac {a^{5} c^{3} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {5 a^{4} c^{4} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {10 a^{3} c^{5} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {10 a^{2} c^{6} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {5 a c^{7} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + a c^{3} d^{3} e^{2} - \frac {c^{8} d^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + c^{4} d^{5}}{2 c^{4} d^{4} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {- 2 a^{2} e^{4} + 7 a c d^{2} e^{2} - 11 c^{2} d^{4} - 6 c^{2} d^{2} e^{2} x^{2} + x \left (3 a c d e^{3} - 15 c^{2} d^{3} e\right )}{6 a^{3} d^{3} e^{6} - 18 a^{2} c d^{5} e^{4} + 18 a c^{2} d^{7} e^{2} - 6 c^{3} d^{9} + x^{3} \cdot \left (6 a^{3} e^{9} - 18 a^{2} c d^{2} e^{7} + 18 a c^{2} d^{4} e^{5} - 6 c^{3} d^{6} e^{3}\right ) + x^{2} \cdot \left (18 a^{3} d e^{8} - 54 a^{2} c d^{3} e^{6} + 54 a c^{2} d^{5} e^{4} - 18 c^{3} d^{7} e^{2}\right ) + x \left (18 a^{3} d^{2} e^{7} - 54 a^{2} c d^{4} e^{5} + 54 a c^{2} d^{6} e^{3} - 18 c^{3} d^{8} e\right )} \] Input:
integrate(1/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
Output:
-c**3*d**3*log(x + (-a**5*c**3*d**3*e**10/(a*e**2 - c*d**2)**4 + 5*a**4*c* *4*d**5*e**8/(a*e**2 - c*d**2)**4 - 10*a**3*c**5*d**7*e**6/(a*e**2 - c*d** 2)**4 + 10*a**2*c**6*d**9*e**4/(a*e**2 - c*d**2)**4 - 5*a*c**7*d**11*e**2/ (a*e**2 - c*d**2)**4 + a*c**3*d**3*e**2 + c**8*d**13/(a*e**2 - c*d**2)**4 + c**4*d**5)/(2*c**4*d**4*e))/(a*e**2 - c*d**2)**4 + c**3*d**3*log(x + (a* *5*c**3*d**3*e**10/(a*e**2 - c*d**2)**4 - 5*a**4*c**4*d**5*e**8/(a*e**2 - c*d**2)**4 + 10*a**3*c**5*d**7*e**6/(a*e**2 - c*d**2)**4 - 10*a**2*c**6*d* *9*e**4/(a*e**2 - c*d**2)**4 + 5*a*c**7*d**11*e**2/(a*e**2 - c*d**2)**4 + a*c**3*d**3*e**2 - c**8*d**13/(a*e**2 - c*d**2)**4 + c**4*d**5)/(2*c**4*d* *4*e))/(a*e**2 - c*d**2)**4 + (-2*a**2*e**4 + 7*a*c*d**2*e**2 - 11*c**2*d* *4 - 6*c**2*d**2*e**2*x**2 + x*(3*a*c*d*e**3 - 15*c**2*d**3*e))/(6*a**3*d* *3*e**6 - 18*a**2*c*d**5*e**4 + 18*a*c**2*d**7*e**2 - 6*c**3*d**9 + x**3*( 6*a**3*e**9 - 18*a**2*c*d**2*e**7 + 18*a*c**2*d**4*e**5 - 6*c**3*d**6*e**3 ) + x**2*(18*a**3*d*e**8 - 54*a**2*c*d**3*e**6 + 54*a*c**2*d**5*e**4 - 18* c**3*d**7*e**2) + x*(18*a**3*d**2*e**7 - 54*a**2*c*d**4*e**5 + 54*a*c**2*d **6*e**3 - 18*c**3*d**8*e))
Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (135) = 270\).
Time = 0.05 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.83 \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {c^{3} d^{3} \log \left (c d x + a e\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} - \frac {c^{3} d^{3} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {6 \, c^{2} d^{2} e^{2} x^{2} + 11 \, c^{2} d^{4} - 7 \, a c d^{2} e^{2} + 2 \, a^{2} e^{4} + 3 \, {\left (5 \, c^{2} d^{3} e - a c d e^{3}\right )} x}{6 \, {\left (c^{3} d^{9} - 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} - a^{3} d^{3} e^{6} + {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} - 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e - 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )}} \] Input:
integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="maxi ma")
Output:
c^3*d^3*log(c*d*x + a*e)/(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8) - c^3*d^3*log(e*x + d)/(c^4*d^8 - 4*a*c^3*d^6*e ^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8) + 1/6*(6*c^2*d^2*e^2*x ^2 + 11*c^2*d^4 - 7*a*c*d^2*e^2 + 2*a^2*e^4 + 3*(5*c^2*d^3*e - a*c*d*e^3)* x)/(c^3*d^9 - 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4 - a^3*d^3*e^6 + (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)*x^3 + 3*(c^3*d^7*e^2 - 3 *a*c^2*d^5*e^4 + 3*a^2*c*d^3*e^6 - a^3*d*e^8)*x^2 + 3*(c^3*d^8*e - 3*a*c^2 *d^6*e^3 + 3*a^2*c*d^4*e^5 - a^3*d^2*e^7)*x)
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (135) = 270\).
Time = 0.12 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.98 \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {c^{4} d^{4} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{9} - 4 \, a c^{4} d^{7} e^{2} + 6 \, a^{2} c^{3} d^{5} e^{4} - 4 \, a^{3} c^{2} d^{3} e^{6} + a^{4} c d e^{8}} - \frac {c^{3} d^{3} e \log \left ({\left | e x + d \right |}\right )}{c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} + \frac {11 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (5 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{6 \, {\left (c d^{2} - a e^{2}\right )}^{4} {\left (e x + d\right )}^{3}} \] Input:
integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac ")
Output:
c^4*d^4*log(abs(c*d*x + a*e))/(c^5*d^9 - 4*a*c^4*d^7*e^2 + 6*a^2*c^3*d^5*e ^4 - 4*a^3*c^2*d^3*e^6 + a^4*c*d*e^8) - c^3*d^3*e*log(abs(e*x + d))/(c^4*d ^8*e - 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 - 4*a^3*c*d^2*e^7 + a^4*e^9) + 1/6*(11*c^3*d^6 - 18*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 2*a^3*e^6 + 6*(c^3* d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 3*(5*c^3*d^5*e - 6*a*c^2*d^3*e^3 + a^2*c*d* e^5)*x)/((c*d^2 - a*e^2)^4*(e*x + d)^3)
Time = 5.28 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.58 \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {2\,c^3\,d^3\,\mathrm {atanh}\left (\frac {a^4\,e^8-2\,a^3\,c\,d^2\,e^6+2\,a\,c^3\,d^6\,e^2-c^4\,d^8}{{\left (a\,e^2-c\,d^2\right )}^4}+\frac {2\,c\,d\,e\,x\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^4}\right )}{{\left (a\,e^2-c\,d^2\right )}^4}-\frac {\frac {2\,a^2\,e^4-7\,a\,c\,d^2\,e^2+11\,c^2\,d^4}{6\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}-\frac {c\,d\,x\,\left (a\,e^3-5\,c\,d^2\,e\right )}{2\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}+\frac {c^2\,d^2\,e^2\,x^2}{a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \] Input:
int(1/((d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)),x)
Output:
(2*c^3*d^3*atanh((a^4*e^8 - c^4*d^8 + 2*a*c^3*d^6*e^2 - 2*a^3*c*d^2*e^6)/( a*e^2 - c*d^2)^4 + (2*c*d*e*x*(a^3*e^6 - c^3*d^6 + 3*a*c^2*d^4*e^2 - 3*a^2 *c*d^2*e^4))/(a*e^2 - c*d^2)^4))/(a*e^2 - c*d^2)^4 - ((2*a^2*e^4 + 11*c^2* d^4 - 7*a*c*d^2*e^2)/(6*(a^3*e^6 - c^3*d^6 + 3*a*c^2*d^4*e^2 - 3*a^2*c*d^2 *e^4)) - (c*d*x*(a*e^3 - 5*c*d^2*e))/(2*(a^3*e^6 - c^3*d^6 + 3*a*c^2*d^4*e ^2 - 3*a^2*c*d^2*e^4)) + (c^2*d^2*e^2*x^2)/(a^3*e^6 - c^3*d^6 + 3*a*c^2*d^ 4*e^2 - 3*a^2*c*d^2*e^4))/(d^3 + e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x)
Time = 0.23 (sec) , antiderivative size = 523, normalized size of antiderivative = 3.76 \[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {6 \,\mathrm {log}\left (c d x +a e \right ) c^{3} d^{6}+18 \,\mathrm {log}\left (c d x +a e \right ) c^{3} d^{5} e x +18 \,\mathrm {log}\left (c d x +a e \right ) c^{3} d^{4} e^{2} x^{2}+6 \,\mathrm {log}\left (c d x +a e \right ) c^{3} d^{3} e^{3} x^{3}-6 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{6}-18 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{5} e x -18 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{4} e^{2} x^{2}-6 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{3} e^{3} x^{3}-2 a^{3} e^{6}+9 a^{2} c \,d^{2} e^{4}+3 a^{2} c d \,e^{5} x -16 a \,c^{2} d^{4} e^{2}-12 a \,c^{2} d^{3} e^{3} x +2 a \,c^{2} d \,e^{5} x^{3}+9 c^{3} d^{6}+9 c^{3} d^{5} e x -2 c^{3} d^{3} e^{3} x^{3}}{6 a^{4} e^{11} x^{3}-24 a^{3} c \,d^{2} e^{9} x^{3}+36 a^{2} c^{2} d^{4} e^{7} x^{3}-24 a \,c^{3} d^{6} e^{5} x^{3}+6 c^{4} d^{8} e^{3} x^{3}+18 a^{4} d \,e^{10} x^{2}-72 a^{3} c \,d^{3} e^{8} x^{2}+108 a^{2} c^{2} d^{5} e^{6} x^{2}-72 a \,c^{3} d^{7} e^{4} x^{2}+18 c^{4} d^{9} e^{2} x^{2}+18 a^{4} d^{2} e^{9} x -72 a^{3} c \,d^{4} e^{7} x +108 a^{2} c^{2} d^{6} e^{5} x -72 a \,c^{3} d^{8} e^{3} x +18 c^{4} d^{10} e x +6 a^{4} d^{3} e^{8}-24 a^{3} c \,d^{5} e^{6}+36 a^{2} c^{2} d^{7} e^{4}-24 a \,c^{3} d^{9} e^{2}+6 c^{4} d^{11}} \] Input:
int(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x)
Output:
(6*log(a*e + c*d*x)*c**3*d**6 + 18*log(a*e + c*d*x)*c**3*d**5*e*x + 18*log (a*e + c*d*x)*c**3*d**4*e**2*x**2 + 6*log(a*e + c*d*x)*c**3*d**3*e**3*x**3 - 6*log(d + e*x)*c**3*d**6 - 18*log(d + e*x)*c**3*d**5*e*x - 18*log(d + e *x)*c**3*d**4*e**2*x**2 - 6*log(d + e*x)*c**3*d**3*e**3*x**3 - 2*a**3*e**6 + 9*a**2*c*d**2*e**4 + 3*a**2*c*d*e**5*x - 16*a*c**2*d**4*e**2 - 12*a*c** 2*d**3*e**3*x + 2*a*c**2*d*e**5*x**3 + 9*c**3*d**6 + 9*c**3*d**5*e*x - 2*c **3*d**3*e**3*x**3)/(6*(a**4*d**3*e**8 + 3*a**4*d**2*e**9*x + 3*a**4*d*e** 10*x**2 + a**4*e**11*x**3 - 4*a**3*c*d**5*e**6 - 12*a**3*c*d**4*e**7*x - 1 2*a**3*c*d**3*e**8*x**2 - 4*a**3*c*d**2*e**9*x**3 + 6*a**2*c**2*d**7*e**4 + 18*a**2*c**2*d**6*e**5*x + 18*a**2*c**2*d**5*e**6*x**2 + 6*a**2*c**2*d** 4*e**7*x**3 - 4*a*c**3*d**9*e**2 - 12*a*c**3*d**8*e**3*x - 12*a*c**3*d**7* e**4*x**2 - 4*a*c**3*d**6*e**5*x**3 + c**4*d**11 + 3*c**4*d**10*e*x + 3*c* *4*d**9*e**2*x**2 + c**4*d**8*e**3*x**3))