Integrand size = 35, antiderivative size = 77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=-\frac {\left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^5}+\frac {c d \left (c d^2-a e^2\right )}{2 e^3 (d+e x)^4}-\frac {c^2 d^2}{3 e^3 (d+e x)^3} \] Output:
-1/5*(-a*e^2+c*d^2)^2/e^3/(e*x+d)^5+1/2*c*d*(-a*e^2+c*d^2)/e^3/(e*x+d)^4-1 /3*c^2*d^2/e^3/(e*x+d)^3
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=-\frac {6 a^2 e^4+3 a c d e^2 (d+5 e x)+c^2 d^2 \left (d^2+5 d e x+10 e^2 x^2\right )}{30 e^3 (d+e x)^5} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^8,x]
Output:
-1/30*(6*a^2*e^4 + 3*a*c*d*e^2*(d + 5*e*x) + c^2*d^2*(d^2 + 5*d*e*x + 10*e ^2*x^2))/(e^3*(d + e*x)^5)
Time = 0.39 (sec) , antiderivative size = 77, 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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2}{(d+e x)^8} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^5}+\frac {\left (a e^2-c d^2\right )^2}{e^2 (d+e x)^6}+\frac {c^2 d^2}{e^2 (d+e x)^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c d \left (c d^2-a e^2\right )}{2 e^3 (d+e x)^4}-\frac {\left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^5}-\frac {c^2 d^2}{3 e^3 (d+e x)^3}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^8,x]
Output:
-1/5*(c*d^2 - a*e^2)^2/(e^3*(d + e*x)^5) + (c*d*(c*d^2 - a*e^2))/(2*e^3*(d + e*x)^4) - (c^2*d^2)/(3*e^3*(d + e*x)^3)
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 = 1.31 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(-\frac {10 x^{2} c^{2} d^{2} e^{2}+15 x a c d \,e^{3}+5 x \,c^{2} d^{3} e +6 a^{2} e^{4}+3 a c \,d^{2} e^{2}+c^{2} d^{4}}{30 e^{3} \left (e x +d \right )^{5}}\) | \(72\) |
risch | \(\frac {-\frac {c^{2} d^{2} x^{2}}{3 e}-\frac {d c \left (3 a \,e^{2}+c \,d^{2}\right ) x}{6 e^{2}}-\frac {6 a^{2} e^{4}+3 a c \,d^{2} e^{2}+c^{2} d^{4}}{30 e^{3}}}{\left (e x +d \right )^{5}}\) | \(75\) |
parallelrisch | \(\frac {-10 d^{2} c^{2} x^{2} e^{4}-15 a c d \,e^{5} x -5 c^{2} d^{3} e^{3} x -6 a^{2} e^{6}-3 a c \,d^{2} e^{4}-c^{2} d^{4} e^{2}}{30 e^{5} \left (e x +d \right )^{5}}\) | \(78\) |
default | \(-\frac {c^{2} d^{2}}{3 e^{3} \left (e x +d \right )^{3}}-\frac {d c \left (a \,e^{2}-c \,d^{2}\right )}{2 e^{3} \left (e x +d \right )^{4}}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{5 e^{3} \left (e x +d \right )^{5}}\) | \(83\) |
orering | \(-\frac {\left (10 x^{2} c^{2} d^{2} e^{2}+15 x a c d \,e^{3}+5 x \,c^{2} d^{3} e +6 a^{2} e^{4}+3 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{2}}{30 e^{3} \left (c d x +a e \right )^{2} \left (e x +d \right )^{7}}\) | \(109\) |
norman | \(\frac {-\frac {d^{2} \left (6 e^{8} a^{2}+3 a \,d^{2} c \,e^{6}+c^{2} d^{4} e^{4}\right )}{30 e^{7}}-\frac {\left (2 e^{8} a^{2}+11 a \,d^{2} c \,e^{6}+7 c^{2} d^{4} e^{4}\right ) x^{2}}{10 e^{5}}-\frac {d \left (3 a c \,e^{6}+5 c^{2} d^{2} e^{4}\right ) x^{3}}{6 e^{4}}-\frac {d \left (12 e^{8} a^{2}+21 a \,d^{2} c \,e^{6}+7 c^{2} d^{4} e^{4}\right ) x}{30 e^{6}}-\frac {d^{2} e \,c^{2} x^{4}}{3}}{\left (e x +d \right )^{7}}\) | \(162\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2/(e*x+d)^8,x,method=_RETURNVERBOSE)
Output:
-1/30/e^3*(10*c^2*d^2*e^2*x^2+15*a*c*d*e^3*x+5*c^2*d^3*e*x+6*a^2*e^4+3*a*c *d^2*e^2+c^2*d^4)/(e*x+d)^5
Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=-\frac {10 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 3 \, a c d^{2} e^{2} + 6 \, a^{2} e^{4} + 5 \, {\left (c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x}{30 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^8,x, algorithm="fric as")
Output:
-1/30*(10*c^2*d^2*e^2*x^2 + c^2*d^4 + 3*a*c*d^2*e^2 + 6*a^2*e^4 + 5*(c^2*d ^3*e + 3*a*c*d*e^3)*x)/(e^8*x^5 + 5*d*e^7*x^4 + 10*d^2*e^6*x^3 + 10*d^3*e^ 5*x^2 + 5*d^4*e^4*x + d^5*e^3)
Time = 0.80 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=\frac {- 6 a^{2} e^{4} - 3 a c d^{2} e^{2} - c^{2} d^{4} - 10 c^{2} d^{2} e^{2} x^{2} + x \left (- 15 a c d e^{3} - 5 c^{2} d^{3} e\right )}{30 d^{5} e^{3} + 150 d^{4} e^{4} x + 300 d^{3} e^{5} x^{2} + 300 d^{2} e^{6} x^{3} + 150 d e^{7} x^{4} + 30 e^{8} x^{5}} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d)**8,x)
Output:
(-6*a**2*e**4 - 3*a*c*d**2*e**2 - c**2*d**4 - 10*c**2*d**2*e**2*x**2 + x*( -15*a*c*d*e**3 - 5*c**2*d**3*e))/(30*d**5*e**3 + 150*d**4*e**4*x + 300*d** 3*e**5*x**2 + 300*d**2*e**6*x**3 + 150*d*e**7*x**4 + 30*e**8*x**5)
Time = 0.05 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=-\frac {10 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 3 \, a c d^{2} e^{2} + 6 \, a^{2} e^{4} + 5 \, {\left (c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x}{30 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^8,x, algorithm="maxi ma")
Output:
-1/30*(10*c^2*d^2*e^2*x^2 + c^2*d^4 + 3*a*c*d^2*e^2 + 6*a^2*e^4 + 5*(c^2*d ^3*e + 3*a*c*d*e^3)*x)/(e^8*x^5 + 5*d*e^7*x^4 + 10*d^2*e^6*x^3 + 10*d^3*e^ 5*x^2 + 5*d^4*e^4*x + d^5*e^3)
Time = 0.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=-\frac {10 \, c^{2} d^{2} e^{2} x^{2} + 5 \, c^{2} d^{3} e x + 15 \, a c d e^{3} x + c^{2} d^{4} + 3 \, a c d^{2} e^{2} + 6 \, a^{2} e^{4}}{30 \, {\left (e x + d\right )}^{5} e^{3}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^8,x, algorithm="giac ")
Output:
-1/30*(10*c^2*d^2*e^2*x^2 + 5*c^2*d^3*e*x + 15*a*c*d*e^3*x + c^2*d^4 + 3*a *c*d^2*e^2 + 6*a^2*e^4)/((e*x + d)^5*e^3)
Time = 5.56 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=-\frac {\frac {6\,a^2\,e^4+3\,a\,c\,d^2\,e^2+c^2\,d^4}{30\,e^3}+\frac {c^2\,d^2\,x^2}{3\,e}+\frac {c\,d\,x\,\left (c\,d^2+3\,a\,e^2\right )}{6\,e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2/(d + e*x)^8,x)
Output:
-((6*a^2*e^4 + c^2*d^4 + 3*a*c*d^2*e^2)/(30*e^3) + (c^2*d^2*x^2)/(3*e) + ( c*d*x*(3*a*e^2 + c*d^2))/(6*e^2))/(d^5 + e^5*x^5 + 5*d*e^4*x^4 + 10*d^3*e^ 2*x^2 + 10*d^2*e^3*x^3 + 5*d^4*e*x)
Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=\frac {-10 c^{2} d^{2} e^{2} x^{2}-15 a c d \,e^{3} x -5 c^{2} d^{3} e x -6 a^{2} e^{4}-3 a c \,d^{2} e^{2}-c^{2} d^{4}}{30 e^{3} \left (e^{5} x^{5}+5 d \,e^{4} x^{4}+10 d^{2} e^{3} x^{3}+10 d^{3} e^{2} x^{2}+5 d^{4} e x +d^{5}\right )} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^8,x)
Output:
( - 6*a**2*e**4 - 3*a*c*d**2*e**2 - 15*a*c*d*e**3*x - c**2*d**4 - 5*c**2*d **3*e*x - 10*c**2*d**2*e**2*x**2)/(30*e**3*(d**5 + 5*d**4*e*x + 10*d**3*e* *2*x**2 + 10*d**2*e**3*x**3 + 5*d*e**4*x**4 + e**5*x**5))