Integrand size = 35, antiderivative size = 139 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}+\frac {e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac {e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \] Output:
-1/3/(-a*e^2+c*d^2)/(c*d*x+a*e)^3+1/2*e/(-a*e^2+c*d^2)^2/(c*d*x+a*e)^2-e^2 /(-a*e^2+c*d^2)^3/(c*d*x+a*e)-e^3*ln(c*d*x+a*e)/(-a*e^2+c*d^2)^4+e^3*ln(e* x+d)/(-a*e^2+c*d^2)^4
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {\frac {\left (c d^2-a e^2\right ) \left (11 a^2 e^4+a c d e^2 (-7 d+15 e x)+c^2 d^2 \left (2 d^2-3 d e x+6 e^2 x^2\right )\right )}{(a e+c d x)^3}+6 e^3 \log (a e+c d x)-6 e^3 \log (d+e x)}{6 \left (c d^2-a e^2\right )^4} \] Input:
Integrate[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
Output:
-1/6*(((c*d^2 - a*e^2)*(11*a^2*e^4 + a*c*d*e^2*(-7*d + 15*e*x) + c^2*d^2*( 2*d^2 - 3*d*e*x + 6*e^2*x^2)))/(a*e + c*d*x)^3 + 6*e^3*Log[a*e + c*d*x] - 6*e^3*Log[d + e*x])/(c*d^2 - a*e^2)^4
Time = 0.51 (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.
| \(\displaystyle \int \frac {(d+e x)^3}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {c d e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac {c d e}{\left (c d^2-a e^2\right )^2 (a e+c d x)^3}+\frac {c d}{\left (c d^2-a e^2\right ) (a e+c d x)^4}+\frac {e^4}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac {c d e^3}{\left (c d^2-a e^2\right )^4 (a e+c d x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}+\frac {e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac {1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}-\frac {e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4}\) |
Input:
Int[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
Output:
-1/3*1/((c*d^2 - a*e^2)*(a*e + c*d*x)^3) + e/(2*(c*d^2 - a*e^2)^2*(a*e + c *d*x)^2) - e^2/((c*d^2 - a*e^2)^3*(a*e + c*d*x)) - (e^3*Log[a*e + c*d*x])/ (c*d^2 - a*e^2)^4 + (e^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 = 1.63 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(\frac {1}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (c d x +a e \right )^{3}}+\frac {e}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (c d x +a e \right )^{2}}+\frac {e^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c d x +a e \right )}-\frac {e^{3} \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{4}}+\frac {e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{4}}\) | \(135\) |
| risch | \(\frac {\frac {e^{2} c^{2} d^{2} x^{2}}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}}+\frac {c d \left (5 a \,e^{2}-c \,d^{2}\right ) e x}{2 e^{6} a^{3}-6 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} a \,c^{2}-2 d^{6} c^{3}}+\frac {11 a^{2} e^{4}-7 a c \,d^{2} e^{2}+2 c^{2} d^{4}}{6 e^{6} a^{3}-18 d^{2} e^{4} a^{2} c +18 d^{4} e^{2} a \,c^{2}-6 d^{6} c^{3}}}{\left (c d x +a e \right )^{3}}+\frac {e^{3} \ln \left (-e x -d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {e^{3} \ln \left (c d x +a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\) | \(338\) |
| parallelrisch | \(\frac {12 x^{2} a^{2} c^{4} d^{5} e^{4}-3 x^{2} a \,c^{5} d^{7} e^{2}+6 \ln \left (e x +d \right ) a^{5} c d \,e^{8}-6 \ln \left (c d x +a e \right ) a^{5} c d \,e^{8}-5 x^{3} a^{2} c^{4} d^{4} e^{5}+6 x^{3} a \,c^{5} d^{6} e^{3}-9 x^{2} a^{3} c^{3} d^{3} e^{6}-x^{3} c^{6} d^{8} e +6 a^{5} c d \,e^{8}-12 a^{4} c^{2} d^{3} e^{6}+8 a^{3} c^{3} d^{5} e^{4}-2 a^{2} c^{4} d^{7} e^{2}+6 \ln \left (e x +d \right ) x^{3} a^{2} c^{4} d^{4} e^{5}-6 \ln \left (c d x +a e \right ) x^{3} a^{2} c^{4} d^{4} e^{5}+18 \ln \left (e x +d \right ) x^{2} a^{3} c^{3} d^{3} e^{6}-18 \ln \left (c d x +a e \right ) x^{2} a^{3} c^{3} d^{3} e^{6}+18 \ln \left (e x +d \right ) x \,a^{4} c^{2} d^{2} e^{7}-18 \ln \left (c d x +a e \right ) x \,a^{4} c^{2} d^{2} e^{7}}{6 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d x +a e \right )^{3} a^{2} e^{2} c d}\) | \(404\) |
| norman | \(\frac {\frac {d^{2} c^{2} e^{5} x^{5}}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}}+\frac {11 a^{2} c^{3} d^{3} e^{4}-7 a \,c^{4} d^{5} e^{2}+2 c^{5} d^{7}}{6 c^{3} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}+\frac {\left (5 a \,c^{3} d^{3} e^{8}+5 e^{6} d^{5} c^{4}\right ) x^{4}}{2 d^{2} c^{2} e^{2} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}+\frac {\left (11 a^{2} c^{3} d^{3} e^{6}-2 a \,c^{4} d^{5} e^{4}+e^{2} d^{7} c^{5}\right ) x}{2 e d \,c^{3} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}+\frac {\left (11 a^{2} c^{3} d^{3} e^{8}+8 a \,c^{4} d^{5} e^{6}+e^{4} d^{7} c^{5}\right ) x^{2}}{2 e^{2} d^{2} c^{3} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}+\frac {\left (11 a^{2} c^{3} d^{3} e^{10}+38 a \,c^{4} d^{5} e^{8}+11 d^{7} e^{6} c^{5}\right ) x^{3}}{6 c^{3} d^{3} e^{3} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}}{\left (e x +d \right )^{3} \left (c d x +a e \right )^{3}}+\frac {e^{3} \ln \left (e x +d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {e^{3} \ln \left (c d x +a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\) | \(650\) |
Input:
int((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^4,x,method=_RETURNVERBOSE)
Output:
1/3/(a*e^2-c*d^2)/(c*d*x+a*e)^3+1/2*e/(a*e^2-c*d^2)^2/(c*d*x+a*e)^2+e^2/(a *e^2-c*d^2)^3/(c*d*x+a*e)-e^3/(a*e^2-c*d^2)^4*ln(c*d*x+a*e)+e^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. 484 vs. \(2 (135) = 270\).
Time = 0.10 (sec) , antiderivative size = 484, normalized size of antiderivative = 3.48 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {2 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \, {\left (c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x + 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (c d x + a e\right ) - 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (e x + d\right )}{6 \, {\left (a^{3} c^{4} d^{8} e^{3} - 4 \, a^{4} c^{3} d^{6} e^{5} + 6 \, a^{5} c^{2} d^{4} e^{7} - 4 \, a^{6} c d^{2} e^{9} + a^{7} e^{11} + {\left (c^{7} d^{11} - 4 \, a c^{6} d^{9} e^{2} + 6 \, a^{2} c^{5} d^{7} e^{4} - 4 \, a^{3} c^{4} d^{5} e^{6} + a^{4} c^{3} d^{3} e^{8}\right )} x^{3} + 3 \, {\left (a c^{6} d^{10} e - 4 \, a^{2} c^{5} d^{8} e^{3} + 6 \, a^{3} c^{4} d^{6} e^{5} - 4 \, a^{4} c^{3} d^{4} e^{7} + a^{5} c^{2} d^{2} e^{9}\right )} x^{2} + 3 \, {\left (a^{2} c^{5} d^{9} e^{2} - 4 \, a^{3} c^{4} d^{7} e^{4} + 6 \, a^{4} c^{3} d^{5} e^{6} - 4 \, a^{5} c^{2} d^{3} e^{8} + a^{6} c d e^{10}\right )} x\right )}} \] Input:
integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fric as")
Output:
-1/6*(2*c^3*d^6 - 9*a*c^2*d^4*e^2 + 18*a^2*c*d^2*e^4 - 11*a^3*e^6 + 6*(c^3 *d^4*e^2 - a*c^2*d^2*e^4)*x^2 - 3*(c^3*d^5*e - 6*a*c^2*d^3*e^3 + 5*a^2*c*d *e^5)*x + 6*(c^3*d^3*e^3*x^3 + 3*a*c^2*d^2*e^4*x^2 + 3*a^2*c*d*e^5*x + a^3 *e^6)*log(c*d*x + a*e) - 6*(c^3*d^3*e^3*x^3 + 3*a*c^2*d^2*e^4*x^2 + 3*a^2* c*d*e^5*x + a^3*e^6)*log(e*x + d))/(a^3*c^4*d^8*e^3 - 4*a^4*c^3*d^6*e^5 + 6*a^5*c^2*d^4*e^7 - 4*a^6*c*d^2*e^9 + a^7*e^11 + (c^7*d^11 - 4*a*c^6*d^9*e ^2 + 6*a^2*c^5*d^7*e^4 - 4*a^3*c^4*d^5*e^6 + a^4*c^3*d^3*e^8)*x^3 + 3*(a*c ^6*d^10*e - 4*a^2*c^5*d^8*e^3 + 6*a^3*c^4*d^6*e^5 - 4*a^4*c^3*d^4*e^7 + a^ 5*c^2*d^2*e^9)*x^2 + 3*(a^2*c^5*d^9*e^2 - 4*a^3*c^4*d^7*e^4 + 6*a^4*c^3*d^ 5*e^6 - 4*a^5*c^2*d^3*e^8 + a^6*c*d*e^10)*x)
Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (119) = 238\).
Time = 0.88 (sec) , antiderivative size = 668, normalized size of antiderivative = 4.81 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^{3} \log {\left (x + \frac {- \frac {a^{5} e^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {5 a^{4} c d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {10 a^{3} c^{2} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {10 a^{2} c^{3} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {5 a c^{4} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + a e^{5} + \frac {c^{5} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + c d^{2} e^{3}}{2 c d e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {e^{3} \log {\left (x + \frac {\frac {a^{5} e^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {5 a^{4} c d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {10 a^{3} c^{2} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {10 a^{2} c^{3} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {5 a c^{4} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + a e^{5} - \frac {c^{5} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + c d^{2} e^{3}}{2 c d e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {11 a^{2} e^{4} - 7 a c d^{2} e^{2} + 2 c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (15 a c d e^{3} - 3 c^{2} d^{3} e\right )}{6 a^{6} e^{9} - 18 a^{5} c d^{2} e^{7} + 18 a^{4} c^{2} d^{4} e^{5} - 6 a^{3} c^{3} d^{6} e^{3} + x^{3} \cdot \left (6 a^{3} c^{3} d^{3} e^{6} - 18 a^{2} c^{4} d^{5} e^{4} + 18 a c^{5} d^{7} e^{2} - 6 c^{6} d^{9}\right ) + x^{2} \cdot \left (18 a^{4} c^{2} d^{2} e^{7} - 54 a^{3} c^{3} d^{4} e^{5} + 54 a^{2} c^{4} d^{6} e^{3} - 18 a c^{5} d^{8} e\right ) + x \left (18 a^{5} c d e^{8} - 54 a^{4} c^{2} d^{3} e^{6} + 54 a^{3} c^{3} d^{5} e^{4} - 18 a^{2} c^{4} d^{7} e^{2}\right )} \] Input:
integrate((e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)
Output:
e**3*log(x + (-a**5*e**13/(a*e**2 - c*d**2)**4 + 5*a**4*c*d**2*e**11/(a*e* *2 - c*d**2)**4 - 10*a**3*c**2*d**4*e**9/(a*e**2 - c*d**2)**4 + 10*a**2*c* *3*d**6*e**7/(a*e**2 - c*d**2)**4 - 5*a*c**4*d**8*e**5/(a*e**2 - c*d**2)** 4 + a*e**5 + c**5*d**10*e**3/(a*e**2 - c*d**2)**4 + c*d**2*e**3)/(2*c*d*e* *4))/(a*e**2 - c*d**2)**4 - e**3*log(x + (a**5*e**13/(a*e**2 - c*d**2)**4 - 5*a**4*c*d**2*e**11/(a*e**2 - c*d**2)**4 + 10*a**3*c**2*d**4*e**9/(a*e** 2 - c*d**2)**4 - 10*a**2*c**3*d**6*e**7/(a*e**2 - c*d**2)**4 + 5*a*c**4*d* *8*e**5/(a*e**2 - c*d**2)**4 + a*e**5 - c**5*d**10*e**3/(a*e**2 - c*d**2)* *4 + c*d**2*e**3)/(2*c*d*e**4))/(a*e**2 - c*d**2)**4 + (11*a**2*e**4 - 7*a *c*d**2*e**2 + 2*c**2*d**4 + 6*c**2*d**2*e**2*x**2 + x*(15*a*c*d*e**3 - 3* c**2*d**3*e))/(6*a**6*e**9 - 18*a**5*c*d**2*e**7 + 18*a**4*c**2*d**4*e**5 - 6*a**3*c**3*d**6*e**3 + x**3*(6*a**3*c**3*d**3*e**6 - 18*a**2*c**4*d**5* e**4 + 18*a*c**5*d**7*e**2 - 6*c**6*d**9) + x**2*(18*a**4*c**2*d**2*e**7 - 54*a**3*c**3*d**4*e**5 + 54*a**2*c**4*d**6*e**3 - 18*a*c**5*d**8*e) + x*( 18*a**5*c*d*e**8 - 54*a**4*c**2*d**3*e**6 + 54*a**3*c**3*d**5*e**4 - 18*a* *2*c**4*d**7*e**2))
Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (135) = 270\).
Time = 0.04 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.96 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {e^{3} \log \left (c d x + a e\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {e^{3} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} - \frac {6 \, c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{4} - 7 \, a c d^{2} e^{2} + 11 \, a^{2} e^{4} - 3 \, {\left (c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x}{6 \, {\left (a^{3} c^{3} d^{6} e^{3} - 3 \, a^{4} c^{2} d^{4} e^{5} + 3 \, a^{5} c d^{2} e^{7} - a^{6} e^{9} + {\left (c^{6} d^{9} - 3 \, a c^{5} d^{7} e^{2} + 3 \, a^{2} c^{4} d^{5} e^{4} - a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (a c^{5} d^{8} e - 3 \, a^{2} c^{4} d^{6} e^{3} + 3 \, a^{3} c^{3} d^{4} e^{5} - a^{4} c^{2} d^{2} e^{7}\right )} x^{2} + 3 \, {\left (a^{2} c^{4} d^{7} e^{2} - 3 \, a^{3} c^{3} d^{5} e^{4} + 3 \, a^{4} c^{2} d^{3} e^{6} - a^{5} c d e^{8}\right )} x\right )}} \] Input:
integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxi ma")
Output:
-e^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) + e^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 + 2* c^2*d^4 - 7*a*c*d^2*e^2 + 11*a^2*e^4 - 3*(c^2*d^3*e - 5*a*c*d*e^3)*x)/(a^3 *c^3*d^6*e^3 - 3*a^4*c^2*d^4*e^5 + 3*a^5*c*d^2*e^7 - a^6*e^9 + (c^6*d^9 - 3*a*c^5*d^7*e^2 + 3*a^2*c^4*d^5*e^4 - a^3*c^3*d^3*e^6)*x^3 + 3*(a*c^5*d^8* e - 3*a^2*c^4*d^6*e^3 + 3*a^3*c^3*d^4*e^5 - a^4*c^2*d^2*e^7)*x^2 + 3*(a^2* c^4*d^7*e^2 - 3*a^3*c^3*d^5*e^4 + 3*a^4*c^2*d^3*e^6 - a^5*c*d*e^8)*x)
Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (135) = 270\).
Time = 0.12 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.96 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {c d e^{3} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{9} - 4 \, a c^{4} d^{7} e^{2} + 6 \, a^{2} c^{3} d^{5} e^{4} - 4 \, a^{3} c^{2} d^{3} e^{6} + a^{4} c d e^{8}} + \frac {e^{4} \log \left ({\left | e x + d \right |}\right )}{c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} - \frac {2 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \, {\left (c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x}{6 \, {\left (c d^{2} - a e^{2}\right )}^{4} {\left (c d x + a e\right )}^{3}} \] Input:
integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac ")
Output:
-c*d*e^3*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) + e^4*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*( 2*c^3*d^6 - 9*a*c^2*d^4*e^2 + 18*a^2*c*d^2*e^4 - 11*a^3*e^6 + 6*(c^3*d^4*e ^2 - a*c^2*d^2*e^4)*x^2 - 3*(c^3*d^5*e - 6*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)* x)/((c*d^2 - a*e^2)^4*(c*d*x + a*e)^3)
Time = 0.13 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.68 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {\frac {11\,a^2\,e^4-7\,a\,c\,d^2\,e^2+2\,c^2\,d^4}{6\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}-\frac {e\,x\,\left (c^2\,d^3-5\,a\,c\,d\,e^2\right )}{2\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}+\frac {c^2\,d^2\,e^2\,x^2}{a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6}}{a^3\,e^3+3\,a^2\,c\,d\,e^2\,x+3\,a\,c^2\,d^2\,e\,x^2+c^3\,d^3\,x^3}-\frac {2\,e^3\,\mathrm {atanh}\left (\frac {a^4\,e^8-2\,a^3\,c\,d^2\,e^6+2\,a\,c^3\,d^6\,e^2-c^4\,d^8}{{\left (a\,e^2-c\,d^2\right )}^4}+\frac {2\,c\,d\,e\,x\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^4}\right )}{{\left (a\,e^2-c\,d^2\right )}^4} \] Input:
int((d + e*x)^3/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)
Output:
((11*a^2*e^4 + 2*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)) - (e*x*(c^2*d^3 - 5*a*c*d*e^2))/(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))/(a^3*e^3 + c^3*d^3*x^3 + 3*a^2*c*d*e^2*x + 3*a*c^2*d^2*e*x^2) - (2*e^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
Time = 0.22 (sec) , antiderivative size = 565, normalized size of antiderivative = 4.06 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {-6 \,\mathrm {log}\left (c d x +a e \right ) a^{4} e^{6}-18 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c d \,e^{5} x -18 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{2} d^{2} e^{4} x^{2}-6 \,\mathrm {log}\left (c d x +a e \right ) a \,c^{3} d^{3} e^{3} x^{3}+6 \,\mathrm {log}\left (e x +d \right ) a^{4} e^{6}+18 \,\mathrm {log}\left (e x +d \right ) a^{3} c d \,e^{5} x +18 \,\mathrm {log}\left (e x +d \right ) a^{2} c^{2} d^{2} e^{4} x^{2}+6 \,\mathrm {log}\left (e x +d \right ) a \,c^{3} d^{3} e^{3} x^{3}+9 a^{4} e^{6}-16 a^{3} c \,d^{2} e^{4}+9 a^{3} c d \,e^{5} x +9 a^{2} c^{2} d^{4} e^{2}-12 a^{2} c^{2} d^{3} e^{3} x -2 a \,c^{3} d^{6}+3 a \,c^{3} d^{5} e x -2 a \,c^{3} d^{3} e^{3} x^{3}+2 c^{4} d^{5} e \,x^{3}}{6 a \left (a^{4} c^{3} d^{3} e^{8} x^{3}-4 a^{3} c^{4} d^{5} e^{6} x^{3}+6 a^{2} c^{5} d^{7} e^{4} x^{3}-4 a \,c^{6} d^{9} e^{2} x^{3}+c^{7} d^{11} x^{3}+3 a^{5} c^{2} d^{2} e^{9} x^{2}-12 a^{4} c^{3} d^{4} e^{7} x^{2}+18 a^{3} c^{4} d^{6} e^{5} x^{2}-12 a^{2} c^{5} d^{8} e^{3} x^{2}+3 a \,c^{6} d^{10} e \,x^{2}+3 a^{6} c d \,e^{10} x -12 a^{5} c^{2} d^{3} e^{8} x +18 a^{4} c^{3} d^{5} e^{6} x -12 a^{3} c^{4} d^{7} e^{4} x +3 a^{2} c^{5} d^{9} e^{2} x +a^{7} e^{11}-4 a^{6} c \,d^{2} e^{9}+6 a^{5} c^{2} d^{4} e^{7}-4 a^{4} c^{3} d^{6} e^{5}+a^{3} c^{4} d^{8} e^{3}\right )} \] Input:
int((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)
Output:
( - 6*log(a*e + c*d*x)*a**4*e**6 - 18*log(a*e + c*d*x)*a**3*c*d*e**5*x - 1 8*log(a*e + c*d*x)*a**2*c**2*d**2*e**4*x**2 - 6*log(a*e + c*d*x)*a*c**3*d* *3*e**3*x**3 + 6*log(d + e*x)*a**4*e**6 + 18*log(d + e*x)*a**3*c*d*e**5*x + 18*log(d + e*x)*a**2*c**2*d**2*e**4*x**2 + 6*log(d + e*x)*a*c**3*d**3*e* *3*x**3 + 9*a**4*e**6 - 16*a**3*c*d**2*e**4 + 9*a**3*c*d*e**5*x + 9*a**2*c **2*d**4*e**2 - 12*a**2*c**2*d**3*e**3*x - 2*a*c**3*d**6 + 3*a*c**3*d**5*e *x - 2*a*c**3*d**3*e**3*x**3 + 2*c**4*d**5*e*x**3)/(6*a*(a**7*e**11 - 4*a* *6*c*d**2*e**9 + 3*a**6*c*d*e**10*x + 6*a**5*c**2*d**4*e**7 - 12*a**5*c**2 *d**3*e**8*x + 3*a**5*c**2*d**2*e**9*x**2 - 4*a**4*c**3*d**6*e**5 + 18*a** 4*c**3*d**5*e**6*x - 12*a**4*c**3*d**4*e**7*x**2 + a**4*c**3*d**3*e**8*x** 3 + a**3*c**4*d**8*e**3 - 12*a**3*c**4*d**7*e**4*x + 18*a**3*c**4*d**6*e** 5*x**2 - 4*a**3*c**4*d**5*e**6*x**3 + 3*a**2*c**5*d**9*e**2*x - 12*a**2*c* *5*d**8*e**3*x**2 + 6*a**2*c**5*d**7*e**4*x**3 + 3*a*c**6*d**10*e*x**2 - 4 *a*c**6*d**9*e**2*x**3 + c**7*d**11*x**3))