Integrand size = 24, antiderivative size = 73 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=-\frac {\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac {b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac {1}{96 c^3 d^8 (b+2 c x)^3} \] Output:
-1/224*(-4*a*c+b^2)^2/c^3/d^8/(2*c*x+b)^7+1/80*(-4*a*c+b^2)/c^3/d^8/(2*c*x +b)^5-1/96/c^3/d^8/(2*c*x+b)^3
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=\frac {-15 \left (b^2-4 a c\right )^2+42 \left (b^2-4 a c\right ) (b+2 c x)^2-35 (b+2 c x)^4}{3360 c^3 d^8 (b+2 c x)^7} \] Input:
Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^8,x]
Output:
(-15*(b^2 - 4*a*c)^2 + 42*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 35*(b + 2*c*x)^4)/ (3360*c^3*d^8*(b + 2*c*x)^7)
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.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx\) |
\(\Big \downarrow \) 1107 |
\(\displaystyle \int \left (\frac {\left (4 a c-b^2\right )^2}{16 c^2 d^8 (b+2 c x)^8}+\frac {4 a c-b^2}{8 c^2 d^8 (b+2 c x)^6}+\frac {1}{16 c^2 d^8 (b+2 c x)^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac {b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac {1}{96 c^3 d^8 (b+2 c x)^3}\) |
Input:
Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^8,x]
Output:
-1/224*(b^2 - 4*a*c)^2/(c^3*d^8*(b + 2*c*x)^7) + (b^2 - 4*a*c)/(80*c^3*d^8 *(b + 2*c*x)^5) - 1/(96*c^3*d^8*(b + 2*c*x)^3)
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 = 0.83 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {-\frac {16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}{224 c^{3} \left (2 c x +b \right )^{7}}-\frac {4 a c -b^{2}}{80 c^{3} \left (2 c x +b \right )^{5}}-\frac {1}{96 c^{3} \left (2 c x +b \right )^{3}}}{d^{8}}\) | \(74\) |
risch | \(\frac {-\frac {c \,x^{4}}{6}-\frac {b \,x^{3}}{3}-\frac {\left (a c +b^{2}\right ) x^{2}}{5 c}-\frac {b \left (6 a c +b^{2}\right ) x}{30 c^{2}}-\frac {30 a^{2} c^{2}+6 c a \,b^{2}+b^{4}}{420 c^{3}}}{d^{8} \left (2 c x +b \right )^{7}}\) | \(80\) |
gosper | \(-\frac {70 c^{4} x^{4}+140 b \,c^{3} x^{3}+84 a \,c^{3} x^{2}+84 b^{2} c^{2} x^{2}+84 a b \,c^{2} x +14 b^{3} c x +30 a^{2} c^{2}+6 c a \,b^{2}+b^{4}}{420 \left (2 c x +b \right )^{7} d^{8} c^{3}}\) | \(88\) |
norman | \(\frac {\frac {a^{2} x}{b d}+\frac {\left (6 a^{2} c +a \,b^{2}\right ) x^{2}}{b^{2} d}+\frac {\left (60 a^{2} c^{2}+12 c a \,b^{2}+b^{4}\right ) x^{3}}{3 d \,b^{3}}+\frac {16 c^{3} \left (30 a^{2} c^{2}+6 c a \,b^{2}+b^{4}\right ) x^{6}}{15 b^{6} d}+\frac {8 c^{2} \left (30 a^{2} c^{2}+6 c a \,b^{2}+b^{4}\right ) x^{5}}{5 b^{5} d}+\frac {c \left (240 a^{2} c^{2}+48 c a \,b^{2}+7 b^{4}\right ) x^{4}}{6 b^{4} d}+\frac {32 c^{4} \left (30 a^{2} c^{2}+6 c a \,b^{2}+b^{4}\right ) x^{7}}{105 b^{7} d}}{d^{7} \left (2 c x +b \right )^{7}}\) | \(209\) |
parallelrisch | \(\frac {1920 x^{7} a^{2} c^{6}+384 x^{7} a \,b^{2} c^{5}+64 x^{7} b^{4} c^{4}+6720 x^{6} a^{2} b \,c^{5}+1344 x^{6} a \,b^{3} c^{4}+224 b^{5} c^{3} x^{6}+10080 x^{5} a^{2} b^{2} c^{4}+2016 a \,b^{4} c^{3} x^{5}+336 b^{6} c^{2} x^{5}+8400 x^{4} a^{2} b^{3} c^{3}+1680 x^{4} a \,b^{5} c^{2}+245 x^{4} b^{7} c +4200 x^{3} a^{2} b^{4} c^{2}+840 x^{3} a \,b^{6} c +70 x^{3} b^{8}+1260 x^{2} a^{2} b^{5} c +210 x^{2} a \,b^{7}+210 a^{2} b^{6} x}{210 b^{7} d^{8} \left (2 c x +b \right )^{7}}\) | \(221\) |
orering | \(\frac {x \left (1920 a^{2} c^{6} x^{6}+384 a \,b^{2} c^{5} x^{6}+64 b^{4} c^{4} x^{6}+6720 a^{2} b \,c^{5} x^{5}+1344 a \,b^{3} c^{4} x^{5}+224 b^{5} c^{3} x^{5}+10080 a^{2} b^{2} c^{4} x^{4}+2016 a \,b^{4} c^{3} x^{4}+336 b^{6} c^{2} x^{4}+8400 a^{2} b^{3} c^{3} x^{3}+1680 a \,b^{5} c^{2} x^{3}+245 b^{7} c \,x^{3}+4200 a^{2} b^{4} c^{2} x^{2}+840 a \,b^{6} c \,x^{2}+70 b^{8} x^{2}+1260 a^{2} b^{5} c x +210 a \,b^{7} x +210 a^{2} b^{6}\right ) \left (2 c x +b \right )}{210 b^{7} \left (2 c d x +b d \right )^{8}}\) | \(223\) |
Input:
int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^8,x,method=_RETURNVERBOSE)
Output:
1/d^8*(-1/224*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b)^7-1/80*(4*a*c-b^2)/ c^3/(2*c*x+b)^5-1/96/c^3/(2*c*x+b)^3)
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (67) = 134\).
Time = 0.09 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.41 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=-\frac {70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \, {\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \, {\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \, {\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \] Input:
integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^8,x, algorithm="fricas")
Output:
-1/420*(70*c^4*x^4 + 140*b*c^3*x^3 + b^4 + 6*a*b^2*c + 30*a^2*c^2 + 84*(b^ 2*c^2 + a*c^3)*x^2 + 14*(b^3*c + 6*a*b*c^2)*x)/(128*c^10*d^8*x^7 + 448*b*c ^9*d^8*x^6 + 672*b^2*c^8*d^8*x^5 + 560*b^3*c^7*d^8*x^4 + 280*b^4*c^6*d^8*x ^3 + 84*b^5*c^5*d^8*x^2 + 14*b^6*c^4*d^8*x + b^7*c^3*d^8)
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (70) = 140\).
Time = 0.95 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.59 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=\frac {- 30 a^{2} c^{2} - 6 a b^{2} c - b^{4} - 140 b c^{3} x^{3} - 70 c^{4} x^{4} + x^{2} \left (- 84 a c^{3} - 84 b^{2} c^{2}\right ) + x \left (- 84 a b c^{2} - 14 b^{3} c\right )}{420 b^{7} c^{3} d^{8} + 5880 b^{6} c^{4} d^{8} x + 35280 b^{5} c^{5} d^{8} x^{2} + 117600 b^{4} c^{6} d^{8} x^{3} + 235200 b^{3} c^{7} d^{8} x^{4} + 282240 b^{2} c^{8} d^{8} x^{5} + 188160 b c^{9} d^{8} x^{6} + 53760 c^{10} d^{8} x^{7}} \] Input:
integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**8,x)
Output:
(-30*a**2*c**2 - 6*a*b**2*c - b**4 - 140*b*c**3*x**3 - 70*c**4*x**4 + x**2 *(-84*a*c**3 - 84*b**2*c**2) + x*(-84*a*b*c**2 - 14*b**3*c))/(420*b**7*c** 3*d**8 + 5880*b**6*c**4*d**8*x + 35280*b**5*c**5*d**8*x**2 + 117600*b**4*c **6*d**8*x**3 + 235200*b**3*c**7*d**8*x**4 + 282240*b**2*c**8*d**8*x**5 + 188160*b*c**9*d**8*x**6 + 53760*c**10*d**8*x**7)
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (67) = 134\).
Time = 0.04 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.41 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=-\frac {70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \, {\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \, {\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \, {\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \] Input:
integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^8,x, algorithm="maxima")
Output:
-1/420*(70*c^4*x^4 + 140*b*c^3*x^3 + b^4 + 6*a*b^2*c + 30*a^2*c^2 + 84*(b^ 2*c^2 + a*c^3)*x^2 + 14*(b^3*c + 6*a*b*c^2)*x)/(128*c^10*d^8*x^7 + 448*b*c ^9*d^8*x^6 + 672*b^2*c^8*d^8*x^5 + 560*b^3*c^7*d^8*x^4 + 280*b^4*c^6*d^8*x ^3 + 84*b^5*c^5*d^8*x^2 + 14*b^6*c^4*d^8*x + b^7*c^3*d^8)
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=-\frac {70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + 84 \, b^{2} c^{2} x^{2} + 84 \, a c^{3} x^{2} + 14 \, b^{3} c x + 84 \, a b c^{2} x + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2}}{420 \, {\left (2 \, c x + b\right )}^{7} c^{3} d^{8}} \] Input:
integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^8,x, algorithm="giac")
Output:
-1/420*(70*c^4*x^4 + 140*b*c^3*x^3 + 84*b^2*c^2*x^2 + 84*a*c^3*x^2 + 14*b^ 3*c*x + 84*a*b*c^2*x + b^4 + 6*a*b^2*c + 30*a^2*c^2)/((2*c*x + b)^7*c^3*d^ 8)
Time = 5.52 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.30 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=-\frac {\frac {30\,a^2\,c^2+6\,a\,b^2\,c+b^4}{420\,c^3}+\frac {b\,x^3}{3}+\frac {c\,x^4}{6}+\frac {x^2\,\left (b^2+a\,c\right )}{5\,c}+\frac {b\,x\,\left (b^2+6\,a\,c\right )}{30\,c^2}}{b^7\,d^8+14\,b^6\,c\,d^8\,x+84\,b^5\,c^2\,d^8\,x^2+280\,b^4\,c^3\,d^8\,x^3+560\,b^3\,c^4\,d^8\,x^4+672\,b^2\,c^5\,d^8\,x^5+448\,b\,c^6\,d^8\,x^6+128\,c^7\,d^8\,x^7} \] Input:
int((a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^8,x)
Output:
-((b^4 + 30*a^2*c^2 + 6*a*b^2*c)/(420*c^3) + (b*x^3)/3 + (c*x^4)/6 + (x^2* (a*c + b^2))/(5*c) + (b*x*(6*a*c + b^2))/(30*c^2))/(b^7*d^8 + 128*c^7*d^8* x^7 + 448*b*c^6*d^8*x^6 + 84*b^5*c^2*d^8*x^2 + 280*b^4*c^3*d^8*x^3 + 560*b ^3*c^4*d^8*x^4 + 672*b^2*c^5*d^8*x^5 + 14*b^6*c*d^8*x)
Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.12 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=\frac {-70 c^{4} x^{4}-140 b \,c^{3} x^{3}-84 a \,c^{3} x^{2}-84 b^{2} c^{2} x^{2}-84 a b \,c^{2} x -14 b^{3} c x -30 a^{2} c^{2}-6 a \,b^{2} c -b^{4}}{420 c^{3} d^{8} \left (128 c^{7} x^{7}+448 b \,c^{6} x^{6}+672 b^{2} c^{5} x^{5}+560 b^{3} c^{4} x^{4}+280 b^{4} c^{3} x^{3}+84 b^{5} c^{2} x^{2}+14 b^{6} c x +b^{7}\right )} \] Input:
int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^8,x)
Output:
( - 30*a**2*c**2 - 6*a*b**2*c - 84*a*b*c**2*x - 84*a*c**3*x**2 - b**4 - 14 *b**3*c*x - 84*b**2*c**2*x**2 - 140*b*c**3*x**3 - 70*c**4*x**4)/(420*c**3* d**8*(b**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560*b* *3*c**4*x**4 + 672*b**2*c**5*x**5 + 448*b*c**6*x**6 + 128*c**7*x**7))