Integrand size = 19, antiderivative size = 103 \[ \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx=\frac {d \left (c d^2-b d e+a e^2\right )}{5 e^4 (d+e x)^5}-\frac {3 c d^2-e (2 b d-a e)}{4 e^4 (d+e x)^4}+\frac {3 c d-b e}{3 e^4 (d+e x)^3}-\frac {c}{2 e^4 (d+e x)^2} \] Output:
1/5*d*(a*e^2-b*d*e+c*d^2)/e^4/(e*x+d)^5-1/4*(3*c*d^2-e*(-a*e+2*b*d))/e^4/( e*x+d)^4+1/3*(-b*e+3*c*d)/e^4/(e*x+d)^3-1/2*c/e^4/(e*x+d)^2
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.77 \[ \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx=-\frac {3 c \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+e \left (3 a e (d+5 e x)+2 b \left (d^2+5 d e x+10 e^2 x^2\right )\right )}{60 e^4 (d+e x)^5} \] Input:
Integrate[(x*(a + b*x + c*x^2))/(d + e*x)^6,x]
Output:
-1/60*(3*c*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + e*(3*a*e*(d + 5 *e*x) + 2*b*(d^2 + 5*d*e*x + 10*e^2*x^2)))/(e^4*(d + e*x)^5)
Time = 0.43 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1195, 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.
\(\displaystyle \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {3 c d^2-e (2 b d-a e)}{e^3 (d+e x)^5}-\frac {d \left (a e^2-b d e+c d^2\right )}{e^3 (d+e x)^6}+\frac {b e-3 c d}{e^3 (d+e x)^4}+\frac {c}{e^3 (d+e x)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 c d^2-e (2 b d-a e)}{4 e^4 (d+e x)^4}+\frac {d \left (a e^2-b d e+c d^2\right )}{5 e^4 (d+e x)^5}+\frac {3 c d-b e}{3 e^4 (d+e x)^3}-\frac {c}{2 e^4 (d+e x)^2}\) |
Input:
Int[(x*(a + b*x + c*x^2))/(d + e*x)^6,x]
Output:
(d*(c*d^2 - b*d*e + a*e^2))/(5*e^4*(d + e*x)^5) - (3*c*d^2 - e*(2*b*d - a* e))/(4*e^4*(d + e*x)^4) + (3*c*d - b*e)/(3*e^4*(d + e*x)^3) - c/(2*e^4*(d + e*x)^2)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.78 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\frac {-\frac {c \,x^{3}}{2 e}-\frac {\left (2 b e +3 c d \right ) x^{2}}{6 e^{2}}-\frac {\left (3 a \,e^{2}+2 b d e +3 c \,d^{2}\right ) x}{12 e^{3}}-\frac {d \left (3 a \,e^{2}+2 b d e +3 c \,d^{2}\right )}{60 e^{4}}}{\left (e x +d \right )^{5}}\) | \(84\) |
gosper | \(-\frac {30 c \,e^{3} x^{3}+20 x^{2} b \,e^{3}+30 c d \,e^{2} x^{2}+15 a \,e^{3} x +10 d x b \,e^{2}+15 c \,d^{2} e x +3 a d \,e^{2}+2 b \,d^{2} e +3 c \,d^{3}}{60 e^{4} \left (e x +d \right )^{5}}\) | \(85\) |
orering | \(-\frac {30 c \,e^{3} x^{3}+20 x^{2} b \,e^{3}+30 c d \,e^{2} x^{2}+15 a \,e^{3} x +10 d x b \,e^{2}+15 c \,d^{2} e x +3 a d \,e^{2}+2 b \,d^{2} e +3 c \,d^{3}}{60 e^{4} \left (e x +d \right )^{5}}\) | \(85\) |
parallelrisch | \(\frac {-30 c \,x^{3} e^{4}-20 b \,e^{4} x^{2}-30 c d \,e^{3} x^{2}-15 a \,e^{4} x -10 b d \,e^{3} x -15 c \,d^{2} e^{2} x -3 a d \,e^{3}-2 d^{2} e^{2} b -3 c \,d^{3} e}{60 e^{5} \left (e x +d \right )^{5}}\) | \(90\) |
default | \(-\frac {b e -3 c d}{3 e^{4} \left (e x +d \right )^{3}}-\frac {a \,e^{2}-2 b d e +3 c \,d^{2}}{4 e^{4} \left (e x +d \right )^{4}}+\frac {d \left (a \,e^{2}-b d e +c \,d^{2}\right )}{5 e^{4} \left (e x +d \right )^{5}}-\frac {c}{2 e^{4} \left (e x +d \right )^{2}}\) | \(93\) |
norman | \(\frac {-\frac {c \,x^{3}}{2 e}-\frac {\left (2 b \,e^{2}+3 d e c \right ) x^{2}}{6 e^{3}}-\frac {\left (3 a \,e^{3}+2 b d \,e^{2}+3 c \,d^{2} e \right ) x}{12 e^{4}}-\frac {d \left (3 a \,e^{3}+2 b d \,e^{2}+3 c \,d^{2} e \right )}{60 e^{5}}}{\left (e x +d \right )^{5}}\) | \(93\) |
Input:
int(x*(c*x^2+b*x+a)/(e*x+d)^6,x,method=_RETURNVERBOSE)
Output:
(-1/2*c*x^3/e-1/6/e^2*(2*b*e+3*c*d)*x^2-1/12/e^3*(3*a*e^2+2*b*d*e+3*c*d^2) *x-1/60*d/e^4*(3*a*e^2+2*b*d*e+3*c*d^2))/(e*x+d)^5
Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.28 \[ \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx=-\frac {30 \, c e^{3} x^{3} + 3 \, c d^{3} + 2 \, b d^{2} e + 3 \, a d e^{2} + 10 \, {\left (3 \, c d e^{2} + 2 \, b e^{3}\right )} x^{2} + 5 \, {\left (3 \, c d^{2} e + 2 \, b d e^{2} + 3 \, a e^{3}\right )} x}{60 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \] Input:
integrate(x*(c*x^2+b*x+a)/(e*x+d)^6,x, algorithm="fricas")
Output:
-1/60*(30*c*e^3*x^3 + 3*c*d^3 + 2*b*d^2*e + 3*a*d*e^2 + 10*(3*c*d*e^2 + 2* b*e^3)*x^2 + 5*(3*c*d^2*e + 2*b*d*e^2 + 3*a*e^3)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)
Time = 1.89 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.37 \[ \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx=\frac {- 3 a d e^{2} - 2 b d^{2} e - 3 c d^{3} - 30 c e^{3} x^{3} + x^{2} \left (- 20 b e^{3} - 30 c d e^{2}\right ) + x \left (- 15 a e^{3} - 10 b d e^{2} - 15 c d^{2} e\right )}{60 d^{5} e^{4} + 300 d^{4} e^{5} x + 600 d^{3} e^{6} x^{2} + 600 d^{2} e^{7} x^{3} + 300 d e^{8} x^{4} + 60 e^{9} x^{5}} \] Input:
integrate(x*(c*x**2+b*x+a)/(e*x+d)**6,x)
Output:
(-3*a*d*e**2 - 2*b*d**2*e - 3*c*d**3 - 30*c*e**3*x**3 + x**2*(-20*b*e**3 - 30*c*d*e**2) + x*(-15*a*e**3 - 10*b*d*e**2 - 15*c*d**2*e))/(60*d**5*e**4 + 300*d**4*e**5*x + 600*d**3*e**6*x**2 + 600*d**2*e**7*x**3 + 300*d*e**8*x **4 + 60*e**9*x**5)
Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.28 \[ \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx=-\frac {30 \, c e^{3} x^{3} + 3 \, c d^{3} + 2 \, b d^{2} e + 3 \, a d e^{2} + 10 \, {\left (3 \, c d e^{2} + 2 \, b e^{3}\right )} x^{2} + 5 \, {\left (3 \, c d^{2} e + 2 \, b d e^{2} + 3 \, a e^{3}\right )} x}{60 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \] Input:
integrate(x*(c*x^2+b*x+a)/(e*x+d)^6,x, algorithm="maxima")
Output:
-1/60*(30*c*e^3*x^3 + 3*c*d^3 + 2*b*d^2*e + 3*a*d*e^2 + 10*(3*c*d*e^2 + 2* b*e^3)*x^2 + 5*(3*c*d^2*e + 2*b*d*e^2 + 3*a*e^3)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)
Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.82 \[ \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx=-\frac {30 \, c e^{3} x^{3} + 30 \, c d e^{2} x^{2} + 20 \, b e^{3} x^{2} + 15 \, c d^{2} e x + 10 \, b d e^{2} x + 15 \, a e^{3} x + 3 \, c d^{3} + 2 \, b d^{2} e + 3 \, a d e^{2}}{60 \, {\left (e x + d\right )}^{5} e^{4}} \] Input:
integrate(x*(c*x^2+b*x+a)/(e*x+d)^6,x, algorithm="giac")
Output:
-1/60*(30*c*e^3*x^3 + 30*c*d*e^2*x^2 + 20*b*e^3*x^2 + 15*c*d^2*e*x + 10*b* d*e^2*x + 15*a*e^3*x + 3*c*d^3 + 2*b*d^2*e + 3*a*d*e^2)/((e*x + d)^5*e^4)
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.24 \[ \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx=-\frac {\frac {c\,x^3}{2\,e}+\frac {d\,\left (3\,c\,d^2+2\,b\,d\,e+3\,a\,e^2\right )}{60\,e^4}+\frac {x\,\left (3\,c\,d^2+2\,b\,d\,e+3\,a\,e^2\right )}{12\,e^3}+\frac {x^2\,\left (2\,b\,e+3\,c\,d\right )}{6\,e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \] Input:
int((x*(a + b*x + c*x^2))/(d + e*x)^6,x)
Output:
-((c*x^3)/(2*e) + (d*(3*a*e^2 + 3*c*d^2 + 2*b*d*e))/(60*e^4) + (x*(3*a*e^2 + 3*c*d^2 + 2*b*d*e))/(12*e^3) + (x^2*(2*b*e + 3*c*d))/(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 = 0.23 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.24 \[ \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx=\frac {-30 c \,e^{3} x^{3}-20 b \,e^{3} x^{2}-30 c d \,e^{2} x^{2}-15 a \,e^{3} x -10 b d \,e^{2} x -15 c \,d^{2} e x -3 a d \,e^{2}-2 b \,d^{2} e -3 c \,d^{3}}{60 e^{4} \left (e^{5} x^{5}+5 d \,e^{4} x^{4}+10 d^{2} e^{3} x^{3}+10 d^{3} e^{2} x^{2}+5 d^{4} e x +d^{5}\right )} \] Input:
int(x*(c*x^2+b*x+a)/(e*x+d)^6,x)
Output:
( - 3*a*d*e**2 - 15*a*e**3*x - 2*b*d**2*e - 10*b*d*e**2*x - 20*b*e**3*x**2 - 3*c*d**3 - 15*c*d**2*e*x - 30*c*d*e**2*x**2 - 30*c*e**3*x**3)/(60*e**4* (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))