Integrand size = 19, antiderivative size = 137 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac {d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{5 e^5 (d+e x)^5}+\frac {c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {c^2}{3 e^5 (d+e x)^3} \]
-1/7*d^2*(-b*e+c*d)^2/e^5/(e*x+d)^7+1/3*d*(-b*e+c*d)*(-b*e+2*c*d)/e^5/(e*x +d)^6+1/5*(-b^2*e^2+6*b*c*d*e-6*c^2*d^2)/e^5/(e*x+d)^5+1/2*c*(-b*e+2*c*d)/ e^5/(e*x+d)^4-1/3*c^2/e^5/(e*x+d)^3
Time = 0.05 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 b c e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+2 c^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )}{210 e^5 (d+e x)^7} \]
-1/210*(2*b^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*b*c*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 2*c^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 3 5*d*e^3*x^3 + 35*e^4*x^4))/(e^5*(d + e*x)^7)
Time = 0.30 (sec) , antiderivative size = 137, 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 = {1140, 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.
\(\displaystyle \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^8} \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (\frac {b^2 e^2-6 b c d e+6 c^2 d^2}{e^4 (d+e x)^6}+\frac {d^2 (c d-b e)^2}{e^4 (d+e x)^8}-\frac {2 c (2 c d-b e)}{e^4 (d+e x)^5}+\frac {2 d (c d-b e) (b e-2 c d)}{e^4 (d+e x)^7}+\frac {c^2}{e^4 (d+e x)^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b^2 e^2-6 b c d e+6 c^2 d^2}{5 e^5 (d+e x)^5}-\frac {d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac {c (2 c d-b e)}{2 e^5 (d+e x)^4}+\frac {d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac {c^2}{3 e^5 (d+e x)^3}\) |
-1/7*(d^2*(c*d - b*e)^2)/(e^5*(d + e*x)^7) + (d*(c*d - b*e)*(2*c*d - b*e)) /(3*e^5*(d + e*x)^6) - (6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)/(5*e^5*(d + e*x)^ 5) + (c*(2*c*d - b*e))/(2*e^5*(d + e*x)^4) - c^2/(3*e^5*(d + e*x)^3)
3.3.43.3.1 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*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 2.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {-\frac {c^{2} x^{4}}{3 e}-\frac {c \left (3 b e +2 c d \right ) x^{3}}{6 e^{2}}-\frac {\left (2 b^{2} e^{2}+3 b c d e +2 c^{2} d^{2}\right ) x^{2}}{10 e^{3}}-\frac {d \left (2 b^{2} e^{2}+3 b c d e +2 c^{2} d^{2}\right ) x}{30 e^{4}}-\frac {d^{2} \left (2 b^{2} e^{2}+3 b c d e +2 c^{2} d^{2}\right )}{210 e^{5}}}{\left (e x +d \right )^{7}}\) | \(131\) |
gosper | \(-\frac {70 c^{2} x^{4} e^{4}+105 x^{3} b c \,e^{4}+70 x^{3} c^{2} d \,e^{3}+42 x^{2} b^{2} e^{4}+63 x^{2} b c d \,e^{3}+42 x^{2} c^{2} d^{2} e^{2}+14 x \,b^{2} d \,e^{3}+21 x b c \,d^{2} e^{2}+14 x \,c^{2} d^{3} e +2 b^{2} d^{2} e^{2}+3 d^{3} e b c +2 c^{2} d^{4}}{210 e^{5} \left (e x +d \right )^{7}}\) | \(141\) |
default | \(\frac {d \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right )}{3 e^{5} \left (e x +d \right )^{6}}-\frac {c^{2}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {c \left (b e -2 c d \right )}{2 e^{5} \left (e x +d \right )^{4}}-\frac {d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}{7 e^{5} \left (e x +d \right )^{7}}-\frac {b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{5 e^{5} \left (e x +d \right )^{5}}\) | \(143\) |
parallelrisch | \(\frac {-70 x^{4} c^{2} e^{6}-105 b c \,e^{6} x^{3}-70 c^{2} d \,e^{5} x^{3}-42 b^{2} e^{6} x^{2}-63 b c d \,e^{5} x^{2}-42 c^{2} d^{2} e^{4} x^{2}-14 b^{2} d \,e^{5} x -21 b c \,d^{2} e^{4} x -14 c^{2} d^{3} e^{3} x -2 b^{2} d^{2} e^{4}-3 b c \,d^{3} e^{3}-2 c^{2} d^{4} e^{2}}{210 e^{7} \left (e x +d \right )^{7}}\) | \(148\) |
norman | \(\frac {-\frac {c^{2} x^{4}}{3 e}-\frac {\left (3 e^{3} b c +2 d \,e^{2} c^{2}\right ) x^{3}}{6 e^{4}}-\frac {\left (2 e^{4} b^{2}+3 d \,e^{3} b c +2 d^{2} e^{2} c^{2}\right ) x^{2}}{10 e^{5}}-\frac {d \left (2 e^{4} b^{2}+3 d \,e^{3} b c +2 d^{2} e^{2} c^{2}\right ) x}{30 e^{6}}-\frac {d^{2} \left (2 e^{4} b^{2}+3 d \,e^{3} b c +2 d^{2} e^{2} c^{2}\right )}{210 e^{7}}}{\left (e x +d \right )^{7}}\) | \(153\) |
(-1/3*c^2*x^4/e-1/6/e^2*c*(3*b*e+2*c*d)*x^3-1/10/e^3*(2*b^2*e^2+3*b*c*d*e+ 2*c^2*d^2)*x^2-1/30*d/e^4*(2*b^2*e^2+3*b*c*d*e+2*c^2*d^2)*x-1/210*d^2/e^5* (2*b^2*e^2+3*b*c*d*e+2*c^2*d^2))/(e*x+d)^7
Time = 0.26 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.51 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2} + 35 \, {\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \, {\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{2} + 7 \, {\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x}{210 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]
-1/210*(70*c^2*e^4*x^4 + 2*c^2*d^4 + 3*b*c*d^3*e + 2*b^2*d^2*e^2 + 35*(2*c ^2*d*e^3 + 3*b*c*e^4)*x^3 + 21*(2*c^2*d^2*e^2 + 3*b*c*d*e^3 + 2*b^2*e^4)*x ^2 + 7*(2*c^2*d^3*e + 3*b*c*d^2*e^2 + 2*b^2*d*e^3)*x)/(e^12*x^7 + 7*d*e^11 *x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)
Time = 103.25 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.61 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=\frac {- 2 b^{2} d^{2} e^{2} - 3 b c d^{3} e - 2 c^{2} d^{4} - 70 c^{2} e^{4} x^{4} + x^{3} \left (- 105 b c e^{4} - 70 c^{2} d e^{3}\right ) + x^{2} \left (- 42 b^{2} e^{4} - 63 b c d e^{3} - 42 c^{2} d^{2} e^{2}\right ) + x \left (- 14 b^{2} d e^{3} - 21 b c d^{2} e^{2} - 14 c^{2} d^{3} e\right )}{210 d^{7} e^{5} + 1470 d^{6} e^{6} x + 4410 d^{5} e^{7} x^{2} + 7350 d^{4} e^{8} x^{3} + 7350 d^{3} e^{9} x^{4} + 4410 d^{2} e^{10} x^{5} + 1470 d e^{11} x^{6} + 210 e^{12} x^{7}} \]
(-2*b**2*d**2*e**2 - 3*b*c*d**3*e - 2*c**2*d**4 - 70*c**2*e**4*x**4 + x**3 *(-105*b*c*e**4 - 70*c**2*d*e**3) + x**2*(-42*b**2*e**4 - 63*b*c*d*e**3 - 42*c**2*d**2*e**2) + x*(-14*b**2*d*e**3 - 21*b*c*d**2*e**2 - 14*c**2*d**3* e))/(210*d**7*e**5 + 1470*d**6*e**6*x + 4410*d**5*e**7*x**2 + 7350*d**4*e* *8*x**3 + 7350*d**3*e**9*x**4 + 4410*d**2*e**10*x**5 + 1470*d*e**11*x**6 + 210*e**12*x**7)
Time = 0.21 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.51 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2} + 35 \, {\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \, {\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{2} + 7 \, {\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x}{210 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]
-1/210*(70*c^2*e^4*x^4 + 2*c^2*d^4 + 3*b*c*d^3*e + 2*b^2*d^2*e^2 + 35*(2*c ^2*d*e^3 + 3*b*c*e^4)*x^3 + 21*(2*c^2*d^2*e^2 + 3*b*c*d*e^3 + 2*b^2*e^4)*x ^2 + 7*(2*c^2*d^3*e + 3*b*c*d^2*e^2 + 2*b^2*d*e^3)*x)/(e^12*x^7 + 7*d*e^11 *x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)
Time = 0.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.02 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {70 \, c^{2} e^{4} x^{4} + 70 \, c^{2} d e^{3} x^{3} + 105 \, b c e^{4} x^{3} + 42 \, c^{2} d^{2} e^{2} x^{2} + 63 \, b c d e^{3} x^{2} + 42 \, b^{2} e^{4} x^{2} + 14 \, c^{2} d^{3} e x + 21 \, b c d^{2} e^{2} x + 14 \, b^{2} d e^{3} x + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2}}{210 \, {\left (e x + d\right )}^{7} e^{5}} \]
-1/210*(70*c^2*e^4*x^4 + 70*c^2*d*e^3*x^3 + 105*b*c*e^4*x^3 + 42*c^2*d^2*e ^2*x^2 + 63*b*c*d*e^3*x^2 + 42*b^2*e^4*x^2 + 14*c^2*d^3*e*x + 21*b*c*d^2*e ^2*x + 14*b^2*d*e^3*x + 2*c^2*d^4 + 3*b*c*d^3*e + 2*b^2*d^2*e^2)/((e*x + d )^7*e^5)
Time = 9.57 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.44 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {\frac {x^2\,\left (2\,b^2\,e^2+3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{10\,e^3}+\frac {c^2\,x^4}{3\,e}+\frac {d^2\,\left (2\,b^2\,e^2+3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{210\,e^5}+\frac {c\,x^3\,\left (3\,b\,e+2\,c\,d\right )}{6\,e^2}+\frac {d\,x\,\left (2\,b^2\,e^2+3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{30\,e^4}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]
-((x^2*(2*b^2*e^2 + 2*c^2*d^2 + 3*b*c*d*e))/(10*e^3) + (c^2*x^4)/(3*e) + ( d^2*(2*b^2*e^2 + 2*c^2*d^2 + 3*b*c*d*e))/(210*e^5) + (c*x^3*(3*b*e + 2*c*d ))/(6*e^2) + (d*x*(2*b^2*e^2 + 2*c^2*d^2 + 3*b*c*d*e))/(30*e^4))/(d^7 + e^ 7*x^7 + 7*d*e^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d^3*e^4*x^4 + 2 1*d^2*e^5*x^5 + 7*d^6*e*x)