Integrand size = 35, antiderivative size = 89 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^4} \, dx=\frac {c d \left (c d^2-a e^2\right )^2 x}{e^3}+\frac {1}{2} \left (a-\frac {c d^2}{e^2}\right ) (a e+c d x)^2+\frac {(a e+c d x)^3}{3 e}-\frac {\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4} \]
c*d*(-a*e^2+c*d^2)^2*x/e^3+1/2*(a-c*d^2/e^2)*(c*d*x+a*e)^2+1/3*(c*d*x+a*e) ^3/e-(-a*e^2+c*d^2)^3*ln(e*x+d)/e^4
Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^4} \, dx=\frac {c d e x \left (18 a^2 e^4+9 a c d e^2 (-2 d+e x)+c^2 d^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 \left (c d^2-a e^2\right )^3 \log (d+e x)}{6 e^4} \]
(c*d*e*x*(18*a^2*e^4 + 9*a*c*d*e^2*(-2*d + e*x) + c^2*d^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) - 6*(c*d^2 - a*e^2)^3*Log[d + e*x])/(6*e^4)
Time = 0.25 (sec) , antiderivative size = 89, 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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3}{(d+e x)^4} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {c d \left (c d^2-a e^2\right ) (a e+c d x)}{e^2}+\frac {\left (a e^2-c d^2\right )^3}{e^3 (d+e x)}+\frac {c d \left (c d^2-a e^2\right )^2}{e^3}+\frac {c d (a e+c d x)^2}{e}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (a-\frac {c d^2}{e^2}\right ) (a e+c d x)^2-\frac {\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4}+\frac {c d x \left (c d^2-a e^2\right )^2}{e^3}+\frac {(a e+c d x)^3}{3 e}\) |
(c*d*(c*d^2 - a*e^2)^2*x)/e^3 + ((a - (c*d^2)/e^2)*(a*e + c*d*x)^2)/2 + (a *e + c*d*x)^3/(3*e) - ((c*d^2 - a*e^2)^3*Log[d + e*x])/e^4
3.19.57.3.1 Defintions of rubi rules used
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 = 2.84 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.39
method | result | size |
default | \(\frac {c d \left (\frac {1}{3} x^{3} c^{2} d^{2} e^{2}+\frac {3}{2} x^{2} a c d \,e^{3}-\frac {1}{2} x^{2} c^{2} d^{3} e +3 a^{2} e^{4} x -3 a c \,d^{2} e^{2} x +c^{2} d^{4} x \right )}{e^{3}}+\frac {\left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(124\) |
risch | \(\frac {c^{3} d^{3} x^{3}}{3 e}+\frac {3 a \,c^{2} d^{2} x^{2}}{2}-\frac {c^{3} d^{4} x^{2}}{2 e^{2}}+3 a^{2} x c d e -\frac {3 c^{2} d^{3} a x}{e}+\frac {c^{3} d^{5} x}{e^{3}}+e^{2} \ln \left (e x +d \right ) a^{3}-3 \ln \left (e x +d \right ) d^{2} a^{2} c +\frac {3 \ln \left (e x +d \right ) d^{4} c^{2} a}{e^{2}}-\frac {\ln \left (e x +d \right ) c^{3} d^{6}}{e^{4}}\) | \(138\) |
parallelrisch | \(\frac {2 x^{3} c^{3} d^{3} e^{3}+9 x^{2} a \,c^{2} d^{2} e^{4}-3 x^{2} c^{3} d^{4} e^{2}+6 \ln \left (e x +d \right ) a^{3} e^{6}-18 \ln \left (e x +d \right ) a^{2} c \,d^{2} e^{4}+18 \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{2}-6 \ln \left (e x +d \right ) c^{3} d^{6}+18 x \,a^{2} c d \,e^{5}-18 x a \,c^{2} d^{3} e^{3}+6 x \,c^{3} d^{5} e}{6 e^{4}}\) | \(148\) |
norman | \(\frac {\left (\frac {3}{2} e^{3} a \,c^{2} d^{2}+\frac {1}{2} d^{4} e \,c^{3}\right ) x^{5}+\left (3 d \,e^{4} a^{2} c +\frac {3}{2} d^{3} e^{2} c^{2} a +\frac {1}{2} d^{5} c^{3}\right ) x^{4}-\frac {d^{3} \left (54 d^{2} e^{4} a^{2} c -27 d^{4} e^{2} c^{2} a +11 c^{3} d^{6}\right )}{6 e^{4}}-\frac {3 d \left (6 d^{2} e^{4} a^{2} c -2 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right ) x^{2}}{e^{2}}-\frac {3 d^{2} \left (16 d^{2} e^{4} a^{2} c -7 d^{4} e^{2} c^{2} a +3 c^{3} d^{6}\right ) x}{2 e^{3}}+\frac {e^{2} c^{3} d^{3} x^{6}}{3}}{\left (e x +d \right )^{3}}+\frac {\left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(260\) |
c*d/e^3*(1/3*x^3*c^2*d^2*e^2+3/2*x^2*a*c*d*e^3-1/2*x^2*c^2*d^3*e+3*a^2*e^4 *x-3*a*c*d^2*e^2*x+c^2*d^4*x)+(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*ln(e*x+d)
Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^4} \, dx=\frac {2 \, c^{3} d^{3} e^{3} x^{3} - 3 \, {\left (c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \, {\left (c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 6 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \]
1/6*(2*c^3*d^3*e^3*x^3 - 3*(c^3*d^4*e^2 - 3*a*c^2*d^2*e^4)*x^2 + 6*(c^3*d^ 5*e - 3*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x - 6*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*log(e*x + d))/e^4
Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^4} \, dx=\frac {c^{3} d^{3} x^{3}}{3 e} + x^{2} \cdot \left (\frac {3 a c^{2} d^{2}}{2} - \frac {c^{3} d^{4}}{2 e^{2}}\right ) + x \left (3 a^{2} c d e - \frac {3 a c^{2} d^{3}}{e} + \frac {c^{3} d^{5}}{e^{3}}\right ) + \frac {\left (a e^{2} - c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{4}} \]
c**3*d**3*x**3/(3*e) + x**2*(3*a*c**2*d**2/2 - c**3*d**4/(2*e**2)) + x*(3* a**2*c*d*e - 3*a*c**2*d**3/e + c**3*d**5/e**3) + (a*e**2 - c*d**2)**3*log( d + e*x)/e**4
Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^4} \, dx=\frac {2 \, c^{3} d^{3} e^{2} x^{3} - 3 \, {\left (c^{3} d^{4} e - 3 \, a c^{2} d^{2} e^{3}\right )} x^{2} + 6 \, {\left (c^{3} d^{5} - 3 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x}{6 \, e^{3}} - \frac {{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (e x + d\right )}{e^{4}} \]
1/6*(2*c^3*d^3*e^2*x^3 - 3*(c^3*d^4*e - 3*a*c^2*d^2*e^3)*x^2 + 6*(c^3*d^5 - 3*a*c^2*d^3*e^2 + 3*a^2*c*d*e^4)*x)/e^3 - (c^3*d^6 - 3*a*c^2*d^4*e^2 + 3 *a^2*c*d^2*e^4 - a^3*e^6)*log(e*x + d)/e^4
Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^4} \, dx=\frac {2 \, c^{3} d^{3} e^{2} x^{3} - 3 \, c^{3} d^{4} e x^{2} + 9 \, a c^{2} d^{2} e^{3} x^{2} + 6 \, c^{3} d^{5} x - 18 \, a c^{2} d^{3} e^{2} x + 18 \, a^{2} c d e^{4} x}{6 \, e^{3}} - \frac {{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{4}} \]
1/6*(2*c^3*d^3*e^2*x^3 - 3*c^3*d^4*e*x^2 + 9*a*c^2*d^2*e^3*x^2 + 6*c^3*d^5 *x - 18*a*c^2*d^3*e^2*x + 18*a^2*c*d*e^4*x)/e^3 - (c^3*d^6 - 3*a*c^2*d^4*e ^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*log(abs(e*x + d))/e^4
Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^4} \, dx=x^2\,\left (\frac {3\,a\,c^2\,d^2}{2}-\frac {c^3\,d^4}{2\,e^2}\right )-x\,\left (\frac {d\,\left (3\,a\,c^2\,d^2-\frac {c^3\,d^4}{e^2}\right )}{e}-3\,a^2\,c\,d\,e\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{e^4}+\frac {c^3\,d^3\,x^3}{3\,e} \]