Integrand size = 35, antiderivative size = 97 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^3 d^3 x}{e^3}+\frac {\left (c d^2-a e^2\right )^3}{2 e^4 (d+e x)^2}-\frac {3 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) \log (d+e x)}{e^4} \] Output:
c^3*d^3*x/e^3+1/2*(-a*e^2+c*d^2)^3/e^4/(e*x+d)^2-3*c*d*(-a*e^2+c*d^2)^2/e^ 4/(e*x+d)-3*c^2*d^2*(-a*e^2+c*d^2)*ln(e*x+d)/e^4
Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {-a^3 e^6-3 a^2 c d e^4 (d+2 e x)+3 a c^2 d^3 e^2 (3 d+4 e x)+c^3 d^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )-6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^2 \log (d+e x)}{2 e^4 (d+e x)^2} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^6,x]
Output:
(-(a^3*e^6) - 3*a^2*c*d*e^4*(d + 2*e*x) + 3*a*c^2*d^3*e^2*(3*d + 4*e*x) + c^3*d^3*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) - 6*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^2*Log[d + e*x])/(2*e^4*(d + e*x)^2)
Time = 0.45 (sec) , antiderivative size = 97, 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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3}{(d+e x)^6} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^2}+\frac {\left (a e^2-c d^2\right )^3}{e^3 (d+e x)^3}+\frac {c^3 d^3}{e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) \log (d+e x)}{e^4}-\frac {3 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)}+\frac {\left (c d^2-a e^2\right )^3}{2 e^4 (d+e x)^2}+\frac {c^3 d^3 x}{e^3}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^6,x]
Output:
(c^3*d^3*x)/e^3 + (c*d^2 - a*e^2)^3/(2*e^4*(d + e*x)^2) - (3*c*d*(c*d^2 - a*e^2)^2)/(e^4*(d + e*x)) - (3*c^2*d^2*(c*d^2 - a*e^2)*Log[d + e*x])/e^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 = 1.36 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.37
method | result | size |
default | \(\frac {c^{3} d^{3} x}{e^{3}}+\frac {3 c^{2} d^{2} \left (a \,e^{2}-c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{4}}-\frac {3 d c \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{e^{4} \left (e x +d \right )}-\frac {e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}}{2 e^{4} \left (e x +d \right )^{2}}\) | \(133\) |
risch | \(\frac {c^{3} d^{3} x}{e^{3}}+\frac {\left (-3 d \,e^{4} a^{2} c +6 d^{3} e^{2} a \,c^{2}-3 d^{5} c^{3}\right ) x -\frac {e^{6} a^{3}+3 d^{2} e^{4} a^{2} c -9 d^{4} e^{2} a \,c^{2}+5 d^{6} c^{3}}{2 e}}{e^{3} \left (e x +d \right )^{2}}+\frac {3 c^{2} d^{2} \ln \left (e x +d \right ) a}{e^{2}}-\frac {3 c^{3} d^{4} \ln \left (e x +d \right )}{e^{4}}\) | \(138\) |
parallelrisch | \(\frac {6 \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{2} e^{4}-6 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}+2 c^{3} d^{3} e^{3} x^{3}+12 \ln \left (e x +d \right ) x a \,c^{2} d^{3} e^{3}-12 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +6 \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{2}-6 \ln \left (e x +d \right ) c^{3} d^{6}-6 x \,a^{2} c d \,e^{5}+12 x a \,c^{2} d^{3} e^{3}-12 c^{3} d^{5} e x -e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +9 d^{4} e^{2} a \,c^{2}-9 d^{6} c^{3}}{2 e^{4} \left (e x +d \right )^{2}}\) | \(210\) |
norman | \(\frac {e^{2} d^{3} c^{3} x^{6}-\frac {d^{3} \left (a^{3} e^{7}+3 a^{2} e^{5} c \,d^{2}-9 a \,e^{3} c^{2} d^{4}+15 d^{6} c^{3} e \right )}{2 e^{5}}-\frac {\left (a^{3} e^{7}+21 a^{2} e^{5} c \,d^{2}-45 a \,e^{3} c^{2} d^{4}+103 d^{6} c^{3} e \right ) x^{3}}{2 e^{2}}-\frac {d \left (3 e^{5} c \,a^{2}-6 e^{3} a \,c^{2} d^{2}+18 d^{4} e \,c^{3}\right ) x^{4}}{e}-\frac {d \left (3 a^{3} e^{7}+27 a^{2} e^{5} c \,d^{2}-63 a \,e^{3} c^{2} d^{4}+123 d^{6} c^{3} e \right ) x^{2}}{2 e^{3}}-\frac {d^{2} \left (3 a^{3} e^{7}+15 a^{2} e^{5} c \,d^{2}-39 a \,e^{3} c^{2} d^{4}+69 d^{6} c^{3} e \right ) x}{2 e^{4}}}{\left (e x +d \right )^{5}}+\frac {3 c^{2} d^{2} \left (a \,e^{2}-c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(293\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3/(e*x+d)^6,x,method=_RETURNVERBOSE)
Output:
c^3*d^3*x/e^3+3/e^4*c^2*d^2*(a*e^2-c*d^2)*ln(e*x+d)-3*d/e^4*c*(a^2*e^4-2*a *c*d^2*e^2+c^2*d^4)/(e*x+d)-1/2*(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c ^3*d^6)/e^4/(e*x+d)^2
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (95) = 190\).
Time = 0.08 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.15 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {2 \, c^{3} d^{3} e^{3} x^{3} + 4 \, c^{3} d^{4} e^{2} x^{2} - 5 \, c^{3} d^{6} + 9 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 2 \, {\left (2 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 6 \, {\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} + {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{3} d^{5} e - a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^6,x, algorithm="fric as")
Output:
1/2*(2*c^3*d^3*e^3*x^3 + 4*c^3*d^4*e^2*x^2 - 5*c^3*d^6 + 9*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - a^3*e^6 - 2*(2*c^3*d^5*e - 6*a*c^2*d^3*e^3 + 3*a^2*c*d* e^5)*x - 6*(c^3*d^6 - a*c^2*d^4*e^2 + (c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 2*(c^3*d^5*e - a*c^2*d^3*e^3)*x)*log(e*x + d))/(e^6*x^2 + 2*d*e^5*x + d^2* e^4)
Time = 0.52 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^{3} d^{3} x}{e^{3}} + \frac {3 c^{2} d^{2} \left (a e^{2} - c d^{2}\right ) \log {\left (d + e x \right )}}{e^{4}} + \frac {- a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 9 a c^{2} d^{4} e^{2} - 5 c^{3} d^{6} + x \left (- 6 a^{2} c d e^{5} + 12 a c^{2} d^{3} e^{3} - 6 c^{3} d^{5} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**6,x)
Output:
c**3*d**3*x/e**3 + 3*c**2*d**2*(a*e**2 - c*d**2)*log(d + e*x)/e**4 + (-a** 3*e**6 - 3*a**2*c*d**2*e**4 + 9*a*c**2*d**4*e**2 - 5*c**3*d**6 + x*(-6*a** 2*c*d*e**5 + 12*a*c**2*d**3*e**3 - 6*c**3*d**5*e))/(2*d**2*e**4 + 4*d*e**5 *x + 2*e**6*x**2)
Time = 0.03 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^{3} d^{3} x}{e^{3}} - \frac {5 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} - \frac {3 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^6,x, algorithm="maxi ma")
Output:
c^3*d^3*x/e^3 - 1/2*(5*c^3*d^6 - 9*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e ^6 + 6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^6*x^2 + 2*d*e^5*x + d^2*e^4) - 3*(c^3*d^4 - a*c^2*d^2*e^2)*log(e*x + d)/e^4
Time = 0.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^{3} d^{3} x}{e^{3}} - \frac {3 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{4}} - \frac {5 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{4}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^6,x, algorithm="giac ")
Output:
c^3*d^3*x/e^3 - 3*(c^3*d^4 - a*c^2*d^2*e^2)*log(abs(e*x + d))/e^4 - 1/2*(5 *c^3*d^6 - 9*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6 + 6*(c^3*d^5*e - 2* a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/((e*x + d)^2*e^4)
Time = 5.40 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^3\,d^3\,x}{e^3}-\frac {\ln \left (d+e\,x\right )\,\left (3\,c^3\,d^4-3\,a\,c^2\,d^2\,e^2\right )}{e^4}-\frac {\frac {a^3\,e^6+3\,a^2\,c\,d^2\,e^4-9\,a\,c^2\,d^4\,e^2+5\,c^3\,d^6}{2\,e}+x\,\left (3\,a^2\,c\,d\,e^4-6\,a\,c^2\,d^3\,e^2+3\,c^3\,d^5\right )}{d^2\,e^3+2\,d\,e^4\,x+e^5\,x^2} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3/(d + e*x)^6,x)
Output:
(c^3*d^3*x)/e^3 - (log(d + e*x)*(3*c^3*d^4 - 3*a*c^2*d^2*e^2))/e^4 - ((a^3 *e^6 + 5*c^3*d^6 - 9*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)/(2*e) + x*(3*c^3*d^5 - 6*a*c^2*d^3*e^2 + 3*a^2*c*d*e^4))/(d^2*e^3 + e^5*x^2 + 2*d*e^4*x)
Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.22 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {6 \,\mathrm {log}\left (e x +d \right ) a \,c^{2} d^{4} e^{2}+12 \,\mathrm {log}\left (e x +d \right ) a \,c^{2} d^{3} e^{3} x +6 \,\mathrm {log}\left (e x +d \right ) a \,c^{2} d^{2} e^{4} x^{2}-6 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{6}-12 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{5} e x -6 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{4} e^{2} x^{2}-a^{3} e^{6}+3 a^{2} c \,e^{6} x^{2}+3 a \,c^{2} d^{4} e^{2}-6 a \,c^{2} d^{2} e^{4} x^{2}-3 c^{3} d^{6}+6 c^{3} d^{4} e^{2} x^{2}+2 c^{3} d^{3} e^{3} x^{3}}{2 e^{4} \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^6,x)
Output:
(6*log(d + e*x)*a*c**2*d**4*e**2 + 12*log(d + e*x)*a*c**2*d**3*e**3*x + 6* log(d + e*x)*a*c**2*d**2*e**4*x**2 - 6*log(d + e*x)*c**3*d**6 - 12*log(d + e*x)*c**3*d**5*e*x - 6*log(d + e*x)*c**3*d**4*e**2*x**2 - a**3*e**6 + 3*a **2*c*e**6*x**2 + 3*a*c**2*d**4*e**2 - 6*a*c**2*d**2*e**4*x**2 - 3*c**3*d* *6 + 6*c**3*d**4*e**2*x**2 + 2*c**3*d**3*e**3*x**3)/(2*e**4*(d**2 + 2*d*e* x + e**2*x**2))