Integrand size = 24, antiderivative size = 122 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^5} \, dx=\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{4 e^4 (d+e x)^4}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{3 e^4 (d+e x)^3}+\frac {3 c (2 c d-b e)}{2 e^4 (d+e x)^2}-\frac {2 c^2}{e^4 (d+e x)} \]
1/4*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)/e^4/(e*x+d)^4+1/3*(-6*c^2*d^2-b^2*e^2 +2*c*e*(-a*e+3*b*d))/e^4/(e*x+d)^3+3/2*c*(-b*e+2*c*d)/e^4/(e*x+d)^2-2*c^2/ e^4/(e*x+d)
Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.82 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^5} \, dx=-\frac {6 c^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+b e^2 (3 a e+b (d+4 e x))+c e \left (2 a e (d+4 e x)+3 b \left (d^2+4 d e x+6 e^2 x^2\right )\right )}{12 e^4 (d+e x)^4} \]
-1/12*(6*c^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) + b*e^2*(3*a*e + b*(d + 4*e*x)) + c*e*(2*a*e*(d + 4*e*x) + 3*b*(d^2 + 4*d*e*x + 6*e^2*x^2)) )/(e^4*(d + e*x)^4)
Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
\(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^5} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^3 (d+e x)^4}+\frac {(b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^3 (d+e x)^5}-\frac {3 c (2 c d-b e)}{e^3 (d+e x)^3}+\frac {2 c^2}{e^3 (d+e x)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^4 (d+e x)^3}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac {3 c (2 c d-b e)}{2 e^4 (d+e x)^2}-\frac {2 c^2}{e^4 (d+e x)}\) |
((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2))/(4*e^4*(d + e*x)^4) - (6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))/(3*e^4*(d + e*x)^3) + (3*c*(2*c*d - b*e))/ (2*e^4*(d + e*x)^2) - (2*c^2)/(e^4*(d + e*x))
3.16.3.3.1 Defintions of rubi rules used
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 = 0.54 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {-\frac {2 c^{2} x^{3}}{e}-\frac {3 \left (b e +2 c d \right ) c \,x^{2}}{2 e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) x}{3 e^{3}}-\frac {3 a b \,e^{3}+2 a c d \,e^{2}+b^{2} d \,e^{2}+3 b c \,d^{2} e +6 c^{2} d^{3}}{12 e^{4}}}{\left (e x +d \right )^{4}}\) | \(118\) |
norman | \(\frac {-\frac {2 c^{2} x^{3}}{e}-\frac {3 \left (b c e +2 c^{2} d \right ) x^{2}}{2 e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) x}{3 e^{3}}-\frac {3 a b \,e^{3}+2 a c d \,e^{2}+b^{2} d \,e^{2}+3 b c \,d^{2} e +6 c^{2} d^{3}}{12 e^{4}}}{\left (e x +d \right )^{4}}\) | \(120\) |
gosper | \(-\frac {24 c^{2} x^{3} e^{3}+18 x^{2} b c \,e^{3}+36 x^{2} c^{2} d \,e^{2}+8 x a c \,e^{3}+4 x \,b^{2} e^{3}+12 x b c d \,e^{2}+24 x \,c^{2} d^{2} e +3 a b \,e^{3}+2 a c d \,e^{2}+b^{2} d \,e^{2}+3 b c \,d^{2} e +6 c^{2} d^{3}}{12 \left (e x +d \right )^{4} e^{4}}\) | \(122\) |
parallelrisch | \(\frac {-24 c^{2} x^{3} e^{3}-18 x^{2} b c \,e^{3}-36 x^{2} c^{2} d \,e^{2}-8 x a c \,e^{3}-4 x \,b^{2} e^{3}-12 x b c d \,e^{2}-24 x \,c^{2} d^{2} e -3 a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}-3 b c \,d^{2} e -6 c^{2} d^{3}}{12 e^{4} \left (e x +d \right )^{4}}\) | \(123\) |
default | \(-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{3 e^{4} \left (e x +d \right )^{3}}-\frac {2 c^{2}}{e^{4} \left (e x +d \right )}-\frac {a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}+3 b c \,d^{2} e -2 c^{2} d^{3}}{4 e^{4} \left (e x +d \right )^{4}}-\frac {3 c \left (b e -2 c d \right )}{2 e^{4} \left (e x +d \right )^{2}}\) | \(131\) |
(-2*c^2*x^3/e-3/2*(b*e+2*c*d)*c/e^2*x^2-1/3*(2*a*c*e^2+b^2*e^2+3*b*c*d*e+6 *c^2*d^2)/e^3*x-1/12*(3*a*b*e^3+2*a*c*d*e^2+b^2*d*e^2+3*b*c*d^2*e+6*c^2*d^ 3)/e^4)/(e*x+d)^4
Time = 0.34 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.23 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^5} \, dx=-\frac {24 \, c^{2} e^{3} x^{3} + 6 \, c^{2} d^{3} + 3 \, b c d^{2} e + 3 \, a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2} + 18 \, {\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} x^{2} + 4 \, {\left (6 \, c^{2} d^{2} e + 3 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
-1/12*(24*c^2*e^3*x^3 + 6*c^2*d^3 + 3*b*c*d^2*e + 3*a*b*e^3 + (b^2 + 2*a*c )*d*e^2 + 18*(2*c^2*d*e^2 + b*c*e^3)*x^2 + 4*(6*c^2*d^2*e + 3*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x)/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^4*e^4)
Time = 2.84 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.39 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^5} \, dx=\frac {- 3 a b e^{3} - 2 a c d e^{2} - b^{2} d e^{2} - 3 b c d^{2} e - 6 c^{2} d^{3} - 24 c^{2} e^{3} x^{3} + x^{2} \left (- 18 b c e^{3} - 36 c^{2} d e^{2}\right ) + x \left (- 8 a c e^{3} - 4 b^{2} e^{3} - 12 b c d e^{2} - 24 c^{2} d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \]
(-3*a*b*e**3 - 2*a*c*d*e**2 - b**2*d*e**2 - 3*b*c*d**2*e - 6*c**2*d**3 - 2 4*c**2*e**3*x**3 + x**2*(-18*b*c*e**3 - 36*c**2*d*e**2) + x*(-8*a*c*e**3 - 4*b**2*e**3 - 12*b*c*d*e**2 - 24*c**2*d**2*e))/(12*d**4*e**4 + 48*d**3*e* *5*x + 72*d**2*e**6*x**2 + 48*d*e**7*x**3 + 12*e**8*x**4)
Time = 0.19 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.23 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^5} \, dx=-\frac {24 \, c^{2} e^{3} x^{3} + 6 \, c^{2} d^{3} + 3 \, b c d^{2} e + 3 \, a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2} + 18 \, {\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} x^{2} + 4 \, {\left (6 \, c^{2} d^{2} e + 3 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
-1/12*(24*c^2*e^3*x^3 + 6*c^2*d^3 + 3*b*c*d^2*e + 3*a*b*e^3 + (b^2 + 2*a*c )*d*e^2 + 18*(2*c^2*d*e^2 + b*c*e^3)*x^2 + 4*(6*c^2*d^2*e + 3*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x)/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^4*e^4)
Time = 0.26 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.44 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^5} \, dx=-\frac {\frac {18 \, b c}{{\left (e x + d\right )}^{2}} - \frac {24 \, b c d}{{\left (e x + d\right )}^{3}} + \frac {9 \, b c d^{2}}{{\left (e x + d\right )}^{4}} + \frac {24 \, c^{2}}{{\left (e x + d\right )} e} - \frac {36 \, c^{2} d}{{\left (e x + d\right )}^{2} e} + \frac {24 \, c^{2} d^{2}}{{\left (e x + d\right )}^{3} e} - \frac {6 \, c^{2} d^{3}}{{\left (e x + d\right )}^{4} e} + \frac {4 \, b^{2} e}{{\left (e x + d\right )}^{3}} + \frac {8 \, a c e}{{\left (e x + d\right )}^{3}} - \frac {3 \, b^{2} d e}{{\left (e x + d\right )}^{4}} - \frac {6 \, a c d e}{{\left (e x + d\right )}^{4}} + \frac {3 \, a b e^{2}}{{\left (e x + d\right )}^{4}}}{12 \, e^{3}} \]
-1/12*(18*b*c/(e*x + d)^2 - 24*b*c*d/(e*x + d)^3 + 9*b*c*d^2/(e*x + d)^4 + 24*c^2/((e*x + d)*e) - 36*c^2*d/((e*x + d)^2*e) + 24*c^2*d^2/((e*x + d)^3 *e) - 6*c^2*d^3/((e*x + d)^4*e) + 4*b^2*e/(e*x + d)^3 + 8*a*c*e/(e*x + d)^ 3 - 3*b^2*d*e/(e*x + d)^4 - 6*a*c*d*e/(e*x + d)^4 + 3*a*b*e^2/(e*x + d)^4) /e^3
Time = 10.78 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.24 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^5} \, dx=-\frac {\frac {b^2\,d\,e^2+3\,b\,c\,d^2\,e+3\,a\,b\,e^3+6\,c^2\,d^3+2\,a\,c\,d\,e^2}{12\,e^4}+\frac {x\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{3\,e^3}+\frac {2\,c^2\,x^3}{e}+\frac {3\,c\,x^2\,\left (b\,e+2\,c\,d\right )}{2\,e^2}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]