Integrand size = 24, antiderivative size = 73 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^9} \, dx=-\frac {\left (b^2-4 a c\right )^2}{256 c^3 d^9 (b+2 c x)^8}+\frac {b^2-4 a c}{96 c^3 d^9 (b+2 c x)^6}-\frac {1}{128 c^3 d^9 (b+2 c x)^4} \] Output:
-1/256*(-4*a*c+b^2)^2/c^3/d^9/(2*c*x+b)^8+1/96*(-4*a*c+b^2)/c^3/d^9/(2*c*x +b)^6-1/128/c^3/d^9/(2*c*x+b)^4
Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^9} \, dx=\frac {-3 \left (b^2-4 a c\right )^2+8 \left (b^2-4 a c\right ) (b+2 c x)^2-6 (b+2 c x)^4}{768 c^3 d^9 (b+2 c x)^8} \] Input:
Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^9,x]
Output:
(-3*(b^2 - 4*a*c)^2 + 8*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 6*(b + 2*c*x)^4)/(76 8*c^3*d^9*(b + 2*c*x)^8)
Time = 0.24 (sec) , antiderivative size = 73, 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 = {1107, 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.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^9} \, dx\) |
\(\Big \downarrow \) 1107 |
\(\displaystyle \int \left (\frac {\left (4 a c-b^2\right )^2}{16 c^2 d^9 (b+2 c x)^9}+\frac {4 a c-b^2}{8 c^2 d^9 (b+2 c x)^7}+\frac {1}{16 c^2 d^9 (b+2 c x)^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (b^2-4 a c\right )^2}{256 c^3 d^9 (b+2 c x)^8}+\frac {b^2-4 a c}{96 c^3 d^9 (b+2 c x)^6}-\frac {1}{128 c^3 d^9 (b+2 c x)^4}\) |
Input:
Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^9,x]
Output:
-1/256*(b^2 - 4*a*c)^2/(c^3*d^9*(b + 2*c*x)^8) + (b^2 - 4*a*c)/(96*c^3*d^9 *(b + 2*c*x)^6) - 1/(128*c^3*d^9*(b + 2*c*x)^4)
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] /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b*e, 0] && IGtQ[p, 0] && !(EqQ[ m, 3] && NeQ[p, 1])
Time = 0.84 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {-\frac {1}{128 c^{3} \left (2 c x +b \right )^{4}}-\frac {16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}{256 c^{3} \left (2 c x +b \right )^{8}}-\frac {4 a c -b^{2}}{96 c^{3} \left (2 c x +b \right )^{6}}}{d^{9}}\) | \(74\) |
risch | \(\frac {-\frac {c \,x^{4}}{8}-\frac {b \,x^{3}}{4}-\frac {\left (8 a c +7 b^{2}\right ) x^{2}}{48 c}-\frac {b \left (8 a c +b^{2}\right ) x}{48 c^{2}}-\frac {48 a^{2} c^{2}+8 c a \,b^{2}+b^{4}}{768 c^{3}}}{d^{9} \left (2 c x +b \right )^{8}}\) | \(83\) |
gosper | \(-\frac {96 c^{4} x^{4}+192 b \,c^{3} x^{3}+128 a \,c^{3} x^{2}+112 b^{2} c^{2} x^{2}+128 a b \,c^{2} x +16 b^{3} c x +48 a^{2} c^{2}+8 c a \,b^{2}+b^{4}}{768 \left (2 c x +b \right )^{8} d^{9} c^{3}}\) | \(88\) |
norman | \(\frac {\frac {a^{2} x}{b d}+\frac {\left (7 a^{2} c +a \,b^{2}\right ) x^{2}}{b^{2} d}+\frac {\left (84 a^{2} c^{2}+14 c a \,b^{2}+b^{4}\right ) x^{3}}{3 d \,b^{3}}+\frac {4 c^{4} \left (48 a^{2} c^{2}+8 c a \,b^{2}+b^{4}\right ) x^{7}}{3 b^{7} d}+\frac {7 c^{3} \left (48 a^{2} c^{2}+8 c a \,b^{2}+b^{4}\right ) x^{6}}{3 b^{6} d}+\frac {7 c^{2} \left (48 a^{2} c^{2}+8 c a \,b^{2}+b^{4}\right ) x^{5}}{3 b^{5} d}+\frac {c \left (210 a^{2} c^{2}+35 c a \,b^{2}+4 b^{4}\right ) x^{4}}{3 b^{4} d}+\frac {c^{5} \left (48 a^{2} c^{2}+8 c a \,b^{2}+b^{4}\right ) x^{8}}{3 b^{8} d}}{d^{8} \left (2 c x +b \right )^{8}}\) | \(242\) |
parallelrisch | \(\frac {48 x^{8} a^{2} c^{7}+8 x^{8} a \,b^{2} c^{6}+x^{8} b^{4} c^{5}+192 x^{7} a^{2} b \,c^{6}+32 x^{7} a \,b^{3} c^{5}+4 x^{7} b^{5} c^{4}+336 x^{6} a^{2} b^{2} c^{5}+56 x^{6} a \,b^{4} c^{4}+7 x^{6} b^{6} c^{3}+336 x^{5} a^{2} b^{3} c^{4}+56 x^{5} a \,b^{5} c^{3}+7 x^{5} b^{7} c^{2}+210 x^{4} a^{2} b^{4} c^{3}+35 x^{4} a \,b^{6} c^{2}+4 x^{4} b^{8} c +84 x^{3} a^{2} b^{5} c^{2}+14 x^{3} a \,b^{7} c +x^{3} b^{9}+21 x^{2} a^{2} b^{6} c +3 x^{2} a \,b^{8}+3 x \,a^{2} b^{7}}{3 b^{8} d^{9} \left (2 c x +b \right )^{8}}\) | \(256\) |
orering | \(\frac {x \left (48 a^{2} c^{7} x^{7}+8 a \,b^{2} c^{6} x^{7}+b^{4} c^{5} x^{7}+192 a^{2} b \,c^{6} x^{6}+32 a \,b^{3} c^{5} x^{6}+4 b^{5} c^{4} x^{6}+336 a^{2} b^{2} c^{5} x^{5}+56 a \,b^{4} c^{4} x^{5}+7 b^{6} c^{3} x^{5}+336 a^{2} b^{3} c^{4} x^{4}+56 a \,b^{5} c^{3} x^{4}+7 b^{7} c^{2} x^{4}+210 a^{2} b^{4} c^{3} x^{3}+35 a \,b^{6} c^{2} x^{3}+4 b^{8} c \,x^{3}+84 a^{2} b^{5} c^{2} x^{2}+14 a \,b^{7} c \,x^{2}+b^{9} x^{2}+21 a^{2} b^{6} c x +3 a \,b^{8} x +3 a^{2} b^{7}\right ) \left (2 c x +b \right )}{3 b^{8} \left (2 c d x +b d \right )^{9}}\) | \(258\) |
Input:
int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^9,x,method=_RETURNVERBOSE)
Output:
1/d^9*(-1/128/c^3/(2*c*x+b)^4-1/256*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+ b)^8-1/96*(4*a*c-b^2)/c^3/(2*c*x+b)^6)
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (67) = 134\).
Time = 0.08 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.63 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^9} \, dx=-\frac {96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + b^{4} + 8 \, a b^{2} c + 48 \, a^{2} c^{2} + 16 \, {\left (7 \, b^{2} c^{2} + 8 \, a c^{3}\right )} x^{2} + 16 \, {\left (b^{3} c + 8 \, a b c^{2}\right )} x}{768 \, {\left (256 \, c^{11} d^{9} x^{8} + 1024 \, b c^{10} d^{9} x^{7} + 1792 \, b^{2} c^{9} d^{9} x^{6} + 1792 \, b^{3} c^{8} d^{9} x^{5} + 1120 \, b^{4} c^{7} d^{9} x^{4} + 448 \, b^{5} c^{6} d^{9} x^{3} + 112 \, b^{6} c^{5} d^{9} x^{2} + 16 \, b^{7} c^{4} d^{9} x + b^{8} c^{3} d^{9}\right )}} \] Input:
integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^9,x, algorithm="fricas")
Output:
-1/768*(96*c^4*x^4 + 192*b*c^3*x^3 + b^4 + 8*a*b^2*c + 48*a^2*c^2 + 16*(7* b^2*c^2 + 8*a*c^3)*x^2 + 16*(b^3*c + 8*a*b*c^2)*x)/(256*c^11*d^9*x^8 + 102 4*b*c^10*d^9*x^7 + 1792*b^2*c^9*d^9*x^6 + 1792*b^3*c^8*d^9*x^5 + 1120*b^4* c^7*d^9*x^4 + 448*b^5*c^6*d^9*x^3 + 112*b^6*c^5*d^9*x^2 + 16*b^7*c^4*d^9*x + b^8*c^3*d^9)
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (70) = 140\).
Time = 1.67 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.79 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^9} \, dx=\frac {- 48 a^{2} c^{2} - 8 a b^{2} c - b^{4} - 192 b c^{3} x^{3} - 96 c^{4} x^{4} + x^{2} \left (- 128 a c^{3} - 112 b^{2} c^{2}\right ) + x \left (- 128 a b c^{2} - 16 b^{3} c\right )}{768 b^{8} c^{3} d^{9} + 12288 b^{7} c^{4} d^{9} x + 86016 b^{6} c^{5} d^{9} x^{2} + 344064 b^{5} c^{6} d^{9} x^{3} + 860160 b^{4} c^{7} d^{9} x^{4} + 1376256 b^{3} c^{8} d^{9} x^{5} + 1376256 b^{2} c^{9} d^{9} x^{6} + 786432 b c^{10} d^{9} x^{7} + 196608 c^{11} d^{9} x^{8}} \] Input:
integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**9,x)
Output:
(-48*a**2*c**2 - 8*a*b**2*c - b**4 - 192*b*c**3*x**3 - 96*c**4*x**4 + x**2 *(-128*a*c**3 - 112*b**2*c**2) + x*(-128*a*b*c**2 - 16*b**3*c))/(768*b**8* c**3*d**9 + 12288*b**7*c**4*d**9*x + 86016*b**6*c**5*d**9*x**2 + 344064*b* *5*c**6*d**9*x**3 + 860160*b**4*c**7*d**9*x**4 + 1376256*b**3*c**8*d**9*x* *5 + 1376256*b**2*c**9*d**9*x**6 + 786432*b*c**10*d**9*x**7 + 196608*c**11 *d**9*x**8)
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (67) = 134\).
Time = 0.05 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.63 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^9} \, dx=-\frac {96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + b^{4} + 8 \, a b^{2} c + 48 \, a^{2} c^{2} + 16 \, {\left (7 \, b^{2} c^{2} + 8 \, a c^{3}\right )} x^{2} + 16 \, {\left (b^{3} c + 8 \, a b c^{2}\right )} x}{768 \, {\left (256 \, c^{11} d^{9} x^{8} + 1024 \, b c^{10} d^{9} x^{7} + 1792 \, b^{2} c^{9} d^{9} x^{6} + 1792 \, b^{3} c^{8} d^{9} x^{5} + 1120 \, b^{4} c^{7} d^{9} x^{4} + 448 \, b^{5} c^{6} d^{9} x^{3} + 112 \, b^{6} c^{5} d^{9} x^{2} + 16 \, b^{7} c^{4} d^{9} x + b^{8} c^{3} d^{9}\right )}} \] Input:
integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^9,x, algorithm="maxima")
Output:
-1/768*(96*c^4*x^4 + 192*b*c^3*x^3 + b^4 + 8*a*b^2*c + 48*a^2*c^2 + 16*(7* b^2*c^2 + 8*a*c^3)*x^2 + 16*(b^3*c + 8*a*b*c^2)*x)/(256*c^11*d^9*x^8 + 102 4*b*c^10*d^9*x^7 + 1792*b^2*c^9*d^9*x^6 + 1792*b^3*c^8*d^9*x^5 + 1120*b^4* c^7*d^9*x^4 + 448*b^5*c^6*d^9*x^3 + 112*b^6*c^5*d^9*x^2 + 16*b^7*c^4*d^9*x + b^8*c^3*d^9)
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^9} \, dx=-\frac {96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + 112 \, b^{2} c^{2} x^{2} + 128 \, a c^{3} x^{2} + 16 \, b^{3} c x + 128 \, a b c^{2} x + b^{4} + 8 \, a b^{2} c + 48 \, a^{2} c^{2}}{768 \, {\left (2 \, c x + b\right )}^{8} c^{3} d^{9}} \] Input:
integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^9,x, algorithm="giac")
Output:
-1/768*(96*c^4*x^4 + 192*b*c^3*x^3 + 112*b^2*c^2*x^2 + 128*a*c^3*x^2 + 16* b^3*c*x + 128*a*b*c^2*x + b^4 + 8*a*b^2*c + 48*a^2*c^2)/((2*c*x + b)^8*c^3 *d^9)
Time = 5.27 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.53 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^9} \, dx=-\frac {\frac {48\,a^2\,c^2+8\,a\,b^2\,c+b^4}{768\,c^3}+\frac {b\,x^3}{4}+\frac {c\,x^4}{8}+\frac {x^2\,\left (7\,b^2+8\,a\,c\right )}{48\,c}+\frac {b\,x\,\left (b^2+8\,a\,c\right )}{48\,c^2}}{b^8\,d^9+16\,b^7\,c\,d^9\,x+112\,b^6\,c^2\,d^9\,x^2+448\,b^5\,c^3\,d^9\,x^3+1120\,b^4\,c^4\,d^9\,x^4+1792\,b^3\,c^5\,d^9\,x^5+1792\,b^2\,c^6\,d^9\,x^6+1024\,b\,c^7\,d^9\,x^7+256\,c^8\,d^9\,x^8} \] Input:
int((a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^9,x)
Output:
-((b^4 + 48*a^2*c^2 + 8*a*b^2*c)/(768*c^3) + (b*x^3)/4 + (c*x^4)/8 + (x^2* (8*a*c + 7*b^2))/(48*c) + (b*x*(8*a*c + b^2))/(48*c^2))/(b^8*d^9 + 256*c^8 *d^9*x^8 + 1024*b*c^7*d^9*x^7 + 112*b^6*c^2*d^9*x^2 + 448*b^5*c^3*d^9*x^3 + 1120*b^4*c^4*d^9*x^4 + 1792*b^3*c^5*d^9*x^5 + 1792*b^2*c^6*d^9*x^6 + 16* b^7*c*d^9*x)
Time = 0.22 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.27 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^9} \, dx=\frac {-96 c^{4} x^{4}-192 b \,c^{3} x^{3}-128 a \,c^{3} x^{2}-112 b^{2} c^{2} x^{2}-128 a b \,c^{2} x -16 b^{3} c x -48 a^{2} c^{2}-8 a \,b^{2} c -b^{4}}{768 c^{3} d^{9} \left (256 c^{8} x^{8}+1024 b \,c^{7} x^{7}+1792 b^{2} c^{6} x^{6}+1792 b^{3} c^{5} x^{5}+1120 b^{4} c^{4} x^{4}+448 b^{5} c^{3} x^{3}+112 b^{6} c^{2} x^{2}+16 b^{7} c x +b^{8}\right )} \] Input:
int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^9,x)
Output:
( - 48*a**2*c**2 - 8*a*b**2*c - 128*a*b*c**2*x - 128*a*c**3*x**2 - b**4 - 16*b**3*c*x - 112*b**2*c**2*x**2 - 192*b*c**3*x**3 - 96*c**4*x**4)/(768*c* *3*d**9*(b**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3*x**3 + 11 20*b**4*c**4*x**4 + 1792*b**3*c**5*x**5 + 1792*b**2*c**6*x**6 + 1024*b*c** 7*x**7 + 256*c**8*x**8))