Integrand size = 20, antiderivative size = 151 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^2}{5 e^5 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{2 e^5 (d+e x)^4}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{3 e^5 (d+e x)^3}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}-\frac {c^2}{e^5 (d+e x)} \] Output:
-1/5*(a*e^2-b*d*e+c*d^2)^2/e^5/(e*x+d)^5+1/2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d ^2)/e^5/(e*x+d)^4-1/3*(6*c^2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))/e^5/(e*x+d)^3 +c*(-b*e+2*c*d)/e^5/(e*x+d)^2-c^2/e^5/(e*x+d)
Time = 0.05 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {6 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+e^2 \left (6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )\right )+c e \left (2 a e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )}{30 e^5 (d+e x)^5} \] Input:
Integrate[(a + b*x + c*x^2)^2/(d + e*x)^6,x]
Output:
-1/30*(6*c^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + e^2*(6*a^2*e^2 + 3*a*b*e*(d + 5*e*x) + b^2*(d^2 + 5*d*e*x + 10*e^2*x^2) ) + c*e*(2*a*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*b*(d^3 + 5*d^2*e*x + 10*d* e^2*x^2 + 10*e^3*x^3)))/(e^5*(d + e*x)^5)
Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^4 (d+e x)^4}+\frac {2 (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)^5}+\frac {\left (a e^2-b d e+c d^2\right )^2}{e^4 (d+e x)^6}-\frac {2 c (2 c d-b e)}{e^4 (d+e x)^3}+\frac {c^2}{e^4 (d+e x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^5 (d+e x)^3}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^5}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}-\frac {c^2}{e^5 (d+e x)}\) |
Input:
Int[(a + b*x + c*x^2)^2/(d + e*x)^6,x]
Output:
-1/5*(c*d^2 - b*d*e + a*e^2)^2/(e^5*(d + e*x)^5) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2))/(2*e^5*(d + e*x)^4) - (6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))/(3*e^5*(d + e*x)^3) + (c*(2*c*d - b*e))/(e^5*(d + e*x)^2) - c^2/( e^5*(d + e*x))
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 = 1.03 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.19
| method | result | size |
| risch | \(\frac {-\frac {c^{2} x^{4}}{e}-\frac {c \left (b e +2 c d \right ) x^{3}}{e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) x^{2}}{3 e^{3}}-\frac {\left (3 a \,e^{3} b +2 a d \,e^{2} c +d \,e^{2} b^{2}+3 d^{2} e b c +6 c^{2} d^{3}\right ) x}{6 e^{4}}-\frac {6 a^{2} e^{4}+3 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}+3 b c \,d^{3} e +6 c^{2} d^{4}}{30 e^{5}}}{\left (e x +d \right )^{5}}\) | \(179\) |
| norman | \(\frac {-\frac {c^{2} x^{4}}{e}-\frac {\left (b c e +2 c^{2} d \right ) x^{3}}{e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) x^{2}}{3 e^{3}}-\frac {\left (3 a \,e^{3} b +2 a d \,e^{2} c +d \,e^{2} b^{2}+3 d^{2} e b c +6 c^{2} d^{3}\right ) x}{6 e^{4}}-\frac {6 a^{2} e^{4}+3 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}+3 b c \,d^{3} e +6 c^{2} d^{4}}{30 e^{5}}}{\left (e x +d \right )^{5}}\) | \(181\) |
| gosper | \(-\frac {30 c^{2} x^{4} e^{4}+30 x^{3} b c \,e^{4}+60 d \,c^{2} x^{3} e^{3}+20 x^{2} a c \,e^{4}+10 x^{2} b^{2} e^{4}+30 x^{2} b c d \,e^{3}+60 x^{2} c^{2} d^{2} e^{2}+15 x a b \,e^{4}+10 x a c d \,e^{3}+5 x \,b^{2} d \,e^{3}+15 x b c \,d^{2} e^{2}+30 x \,c^{2} d^{3} e +6 a^{2} e^{4}+3 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}+3 b c \,d^{3} e +6 c^{2} d^{4}}{30 \left (e x +d \right )^{5} e^{5}}\) | \(193\) |
| orering | \(-\frac {30 c^{2} x^{4} e^{4}+30 x^{3} b c \,e^{4}+60 d \,c^{2} x^{3} e^{3}+20 x^{2} a c \,e^{4}+10 x^{2} b^{2} e^{4}+30 x^{2} b c d \,e^{3}+60 x^{2} c^{2} d^{2} e^{2}+15 x a b \,e^{4}+10 x a c d \,e^{3}+5 x \,b^{2} d \,e^{3}+15 x b c \,d^{2} e^{2}+30 x \,c^{2} d^{3} e +6 a^{2} e^{4}+3 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}+3 b c \,d^{3} e +6 c^{2} d^{4}}{30 \left (e x +d \right )^{5} e^{5}}\) | \(193\) |
| parallelrisch | \(\frac {-30 c^{2} x^{4} e^{4}-30 x^{3} b c \,e^{4}-60 d \,c^{2} x^{3} e^{3}-20 x^{2} a c \,e^{4}-10 x^{2} b^{2} e^{4}-30 x^{2} b c d \,e^{3}-60 x^{2} c^{2} d^{2} e^{2}-15 x a b \,e^{4}-10 x a c d \,e^{3}-5 x \,b^{2} d \,e^{3}-15 x b c \,d^{2} e^{2}-30 x \,c^{2} d^{3} e -6 a^{2} e^{4}-3 d \,e^{3} a b -2 a c \,d^{2} e^{2}-d^{2} e^{2} b^{2}-3 b c \,d^{3} e -6 c^{2} d^{4}}{30 e^{5} \left (e x +d \right )^{5}}\) | \(194\) |
| default | \(-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {2 a \,e^{3} b -4 a d \,e^{2} c -2 d \,e^{2} b^{2}+6 d^{2} e b c -4 c^{2} d^{3}}{4 e^{5} \left (e x +d \right )^{4}}-\frac {a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {c^{2}}{e^{5} \left (e x +d \right )}-\frac {c \left (b e -2 c d \right )}{e^{5} \left (e x +d \right )^{2}}\) | \(195\) |
Input:
int((c*x^2+b*x+a)^2/(e*x+d)^6,x,method=_RETURNVERBOSE)
Output:
(-c^2*x^4/e-c*(b*e+2*c*d)/e^2*x^3-1/3*(2*a*c*e^2+b^2*e^2+3*b*c*d*e+6*c^2*d ^2)/e^3*x^2-1/6*(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*x-1/30*(6*a^2*e^4+3*a*b*d*e^3+2*a*c*d^2*e^2+b^2*d^2*e^2+3*b*c*d^3*e+6*c^ 2*d^4)/e^5)/(e*x+d)^5
Time = 0.08 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + 3 \, a b d e^{3} + 6 \, a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 30 \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \, {\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 5 \, {\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 3 \, a b e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{30 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \] Input:
integrate((c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="fricas")
Output:
-1/30*(30*c^2*e^4*x^4 + 6*c^2*d^4 + 3*b*c*d^3*e + 3*a*b*d*e^3 + 6*a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2 + 30*(2*c^2*d*e^3 + b*c*e^4)*x^3 + 10*(6*c^2*d^2*e ^2 + 3*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 5*(6*c^2*d^3*e + 3*b*c*d^2*e^2 + 3*a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^ 8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6*x + d^5*e^5)
Time = 17.87 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=\frac {- 6 a^{2} e^{4} - 3 a b d e^{3} - 2 a c d^{2} e^{2} - b^{2} d^{2} e^{2} - 3 b c d^{3} e - 6 c^{2} d^{4} - 30 c^{2} e^{4} x^{4} + x^{3} \left (- 30 b c e^{4} - 60 c^{2} d e^{3}\right ) + x^{2} \left (- 20 a c e^{4} - 10 b^{2} e^{4} - 30 b c d e^{3} - 60 c^{2} d^{2} e^{2}\right ) + x \left (- 15 a b e^{4} - 10 a c d e^{3} - 5 b^{2} d e^{3} - 15 b c d^{2} e^{2} - 30 c^{2} d^{3} e\right )}{30 d^{5} e^{5} + 150 d^{4} e^{6} x + 300 d^{3} e^{7} x^{2} + 300 d^{2} e^{8} x^{3} + 150 d e^{9} x^{4} + 30 e^{10} x^{5}} \] Input:
integrate((c*x**2+b*x+a)**2/(e*x+d)**6,x)
Output:
(-6*a**2*e**4 - 3*a*b*d*e**3 - 2*a*c*d**2*e**2 - b**2*d**2*e**2 - 3*b*c*d* *3*e - 6*c**2*d**4 - 30*c**2*e**4*x**4 + x**3*(-30*b*c*e**4 - 60*c**2*d*e* *3) + x**2*(-20*a*c*e**4 - 10*b**2*e**4 - 30*b*c*d*e**3 - 60*c**2*d**2*e** 2) + x*(-15*a*b*e**4 - 10*a*c*d*e**3 - 5*b**2*d*e**3 - 15*b*c*d**2*e**2 - 30*c**2*d**3*e))/(30*d**5*e**5 + 150*d**4*e**6*x + 300*d**3*e**7*x**2 + 30 0*d**2*e**8*x**3 + 150*d*e**9*x**4 + 30*e**10*x**5)
Time = 0.05 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + 3 \, a b d e^{3} + 6 \, a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 30 \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \, {\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 5 \, {\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 3 \, a b e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{30 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \] Input:
integrate((c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="maxima")
Output:
-1/30*(30*c^2*e^4*x^4 + 6*c^2*d^4 + 3*b*c*d^3*e + 3*a*b*d*e^3 + 6*a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2 + 30*(2*c^2*d*e^3 + b*c*e^4)*x^3 + 10*(6*c^2*d^2*e ^2 + 3*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 5*(6*c^2*d^3*e + 3*b*c*d^2*e^2 + 3*a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^ 8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6*x + d^5*e^5)
Time = 0.34 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {30 \, c^{2} e^{4} x^{4} + 60 \, c^{2} d e^{3} x^{3} + 30 \, b c e^{4} x^{3} + 60 \, c^{2} d^{2} e^{2} x^{2} + 30 \, b c d e^{3} x^{2} + 10 \, b^{2} e^{4} x^{2} + 20 \, a c e^{4} x^{2} + 30 \, c^{2} d^{3} e x + 15 \, b c d^{2} e^{2} x + 5 \, b^{2} d e^{3} x + 10 \, a c d e^{3} x + 15 \, a b e^{4} x + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + 3 \, a b d e^{3} + 6 \, a^{2} e^{4}}{30 \, {\left (e x + d\right )}^{5} e^{5}} \] Input:
integrate((c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="giac")
Output:
-1/30*(30*c^2*e^4*x^4 + 60*c^2*d*e^3*x^3 + 30*b*c*e^4*x^3 + 60*c^2*d^2*e^2 *x^2 + 30*b*c*d*e^3*x^2 + 10*b^2*e^4*x^2 + 20*a*c*e^4*x^2 + 30*c^2*d^3*e*x + 15*b*c*d^2*e^2*x + 5*b^2*d*e^3*x + 10*a*c*d*e^3*x + 15*a*b*e^4*x + 6*c^ 2*d^4 + 3*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 + 3*a*b*d*e^3 + 6*a^2*e^ 4)/((e*x + d)^5*e^5)
Time = 0.05 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {\frac {6\,a^2\,e^4+3\,a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2+3\,b\,c\,d^3\,e+6\,c^2\,d^4}{30\,e^5}+\frac {x\,\left (b^2\,d\,e^2+3\,b\,c\,d^2\,e+3\,a\,b\,e^3+6\,c^2\,d^3+2\,a\,c\,d\,e^2\right )}{6\,e^4}+\frac {c^2\,x^4}{e}+\frac {x^2\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{3\,e^3}+\frac {c\,x^3\,\left (b\,e+2\,c\,d\right )}{e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \] Input:
int((a + b*x + c*x^2)^2/(d + e*x)^6,x)
Output:
-((6*a^2*e^4 + 6*c^2*d^4 + b^2*d^2*e^2 + 3*a*b*d*e^3 + 3*b*c*d^3*e + 2*a*c *d^2*e^2)/(30*e^5) + (x*(6*c^2*d^3 + b^2*d*e^2 + 3*a*b*e^3 + 2*a*c*d*e^2 + 3*b*c*d^2*e))/(6*e^4) + (c^2*x^4)/e + (x^2*(b^2*e^2 + 6*c^2*d^2 + 2*a*c*e ^2 + 3*b*c*d*e))/(3*e^3) + (c*x^3*(b*e + 2*c*d))/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 = 0.22 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=\frac {6 c^{2} e^{4} x^{5}-30 b c d \,e^{3} x^{3}-20 a c d \,e^{3} x^{2}-10 b^{2} d \,e^{3} x^{2}-30 b c \,d^{2} e^{2} x^{2}-15 a b d \,e^{3} x -10 a c \,d^{2} e^{2} x -5 b^{2} d^{2} e^{2} x -15 b c \,d^{3} e x -6 a^{2} d \,e^{3}-3 a b \,d^{2} e^{2}-2 a c \,d^{3} e -b^{2} d^{3} e -3 b c \,d^{4}}{30 d \,e^{4} \left (e^{5} x^{5}+5 d \,e^{4} x^{4}+10 d^{2} e^{3} x^{3}+10 d^{3} e^{2} x^{2}+5 d^{4} e x +d^{5}\right )} \] Input:
int((c*x^2+b*x+a)^2/(e*x+d)^6,x)
Output:
( - 6*a**2*d*e**3 - 3*a*b*d**2*e**2 - 15*a*b*d*e**3*x - 2*a*c*d**3*e - 10* a*c*d**2*e**2*x - 20*a*c*d*e**3*x**2 - b**2*d**3*e - 5*b**2*d**2*e**2*x - 10*b**2*d*e**3*x**2 - 3*b*c*d**4 - 15*b*c*d**3*e*x - 30*b*c*d**2*e**2*x**2 - 30*b*c*d*e**3*x**3 + 6*c**2*e**4*x**5)/(30*d*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))