Integrand size = 24, antiderivative size = 94 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^6} \, dx=\frac {x}{64 c^3 d^6}+\frac {\left (b^2-4 a c\right )^3}{640 c^4 d^6 (b+2 c x)^5}-\frac {\left (b^2-4 a c\right )^2}{128 c^4 d^6 (b+2 c x)^3}+\frac {3 \left (b^2-4 a c\right )}{128 c^4 d^6 (b+2 c x)} \] Output:
1/64*x/c^3/d^6+1/640*(-4*a*c+b^2)^3/c^4/d^6/(2*c*x+b)^5-1/128*(-4*a*c+b^2) ^2/c^4/d^6/(2*c*x+b)^3+3/128*(-4*a*c+b^2)/c^4/d^6/(2*c*x+b)
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^6} \, dx=\frac {10 c x+\frac {\left (b^2-4 a c\right )^3}{(b+2 c x)^5}-\frac {5 \left (b^2-4 a c\right )^2}{(b+2 c x)^3}+\frac {15 \left (b^2-4 a c\right )}{b+2 c x}}{640 c^4 d^6} \] Input:
Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^6,x]
Output:
(10*c*x + (b^2 - 4*a*c)^3/(b + 2*c*x)^5 - (5*(b^2 - 4*a*c)^2)/(b + 2*c*x)^ 3 + (15*(b^2 - 4*a*c))/(b + 2*c*x))/(640*c^4*d^6)
Time = 0.28 (sec) , antiderivative size = 94, 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.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^6} \, dx\) |
\(\Big \downarrow \) 1107 |
\(\displaystyle \int \left (\frac {\left (4 a c-b^2\right )^3}{64 c^3 d^6 (b+2 c x)^6}+\frac {3 \left (4 a c-b^2\right )^2}{64 c^3 d^6 (b+2 c x)^4}+\frac {3 \left (4 a c-b^2\right )}{64 c^3 d^6 (b+2 c x)^2}+\frac {1}{64 c^3 d^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (b^2-4 a c\right )^3}{640 c^4 d^6 (b+2 c x)^5}-\frac {\left (b^2-4 a c\right )^2}{128 c^4 d^6 (b+2 c x)^3}+\frac {3 \left (b^2-4 a c\right )}{128 c^4 d^6 (b+2 c x)}+\frac {x}{64 c^3 d^6}\) |
Input:
Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^6,x]
Output:
x/(64*c^3*d^6) + (b^2 - 4*a*c)^3/(640*c^4*d^6*(b + 2*c*x)^5) - (b^2 - 4*a* c)^2/(128*c^4*d^6*(b + 2*c*x)^3) + (3*(b^2 - 4*a*c))/(128*c^4*d^6*(b + 2*c *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] /; 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 = 114, normalized size of antiderivative = 1.21
method | result | size |
default | \(\frac {\frac {x}{64 c^{3}}-\frac {64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{640 c^{4} \left (2 c x +b \right )^{5}}-\frac {48 a^{2} c^{2}-24 c a \,b^{2}+3 b^{4}}{384 c^{4} \left (2 c x +b \right )^{3}}-\frac {12 a c -3 b^{2}}{128 c^{4} \left (2 c x +b \right )}}{d^{6}}\) | \(114\) |
norman | \(\frac {\frac {a^{3} x}{b d}+\frac {\left (8 c^{2} a^{3}+4 a^{2} b^{2} c +a \,b^{4}\right ) x^{3}}{b^{3} d}+\frac {c^{2} x^{6}}{2 d}+\frac {\left (8 c \,a^{3}+3 a^{2} b^{2}\right ) x^{2}}{2 b^{2} d}+\frac {\left (32 a^{3} c^{3}+16 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{4}}{4 b^{4} d}+\frac {c \left (32 a^{3} c^{3}+16 a^{2} b^{2} c^{2}+16 a \,b^{4} c +7 b^{6}\right ) x^{5}}{10 b^{5} d}}{d^{5} \left (2 c x +b \right )^{5}}\) | \(180\) |
risch | \(\frac {x}{64 c^{3} d^{6}}+\frac {\left (-96 a \,d^{6} c^{4}+24 b^{2} d^{6} c^{3}\right ) x^{4}-48 b \,d^{6} c^{2} \left (4 a c -b^{2}\right ) x^{3}+\left (-32 a^{2} d^{6} c^{3}-128 a \,b^{2} c^{2} d^{6}+34 b^{4} c \,d^{6}\right ) x^{2}-2 d^{6} b \left (16 a^{2} c^{2}+16 c a \,b^{2}-5 b^{4}\right ) x -\frac {d^{6} \left (64 a^{3} c^{3}+32 a^{2} b^{2} c^{2}+32 a \,b^{4} c -11 b^{6}\right )}{10 c}}{64 d^{12} c^{3} \left (2 c x +b \right )^{5}}\) | \(180\) |
gosper | \(\frac {x \left (10 b^{5} c^{2} x^{5}+64 a^{3} c^{4} x^{4}+32 a^{2} b^{2} c^{3} x^{4}+32 x^{4} a \,b^{4} c^{2}+14 b^{6} c \,x^{4}+160 a^{3} b \,c^{3} x^{3}+80 x^{3} a^{2} b^{3} c^{2}+50 a \,b^{5} c \,x^{3}+5 x^{3} b^{7}+160 a^{3} b^{2} c^{2} x^{2}+80 a^{2} b^{4} c \,x^{2}+20 a \,b^{6} x^{2}+80 a^{3} b^{3} c x +30 a^{2} b^{5} x +20 a^{3} b^{4}\right )}{20 \left (2 c x +b \right )^{5} d^{6} b^{5}}\) | \(182\) |
parallelrisch | \(\frac {10 c^{2} x^{6} b^{5}+64 x^{5} a^{3} c^{4}+32 x^{5} a^{2} b^{2} c^{3}+32 x^{5} a \,b^{4} c^{2}+14 b^{6} c \,x^{5}+160 x^{4} a^{3} b \,c^{3}+80 x^{4} a^{2} b^{3} c^{2}+50 a \,b^{5} c \,x^{4}+5 x^{4} b^{7}+160 x^{3} a^{3} b^{2} c^{2}+80 a^{2} b^{4} c \,x^{3}+20 x^{3} a \,b^{6}+80 a^{3} b^{3} c \,x^{2}+30 x^{2} a^{2} b^{5}+20 a^{3} b^{4} x}{20 b^{5} d^{6} \left (2 c x +b \right )^{5}}\) | \(186\) |
orering | \(\frac {x \left (10 b^{5} c^{2} x^{5}+64 a^{3} c^{4} x^{4}+32 a^{2} b^{2} c^{3} x^{4}+32 x^{4} a \,b^{4} c^{2}+14 b^{6} c \,x^{4}+160 a^{3} b \,c^{3} x^{3}+80 x^{3} a^{2} b^{3} c^{2}+50 a \,b^{5} c \,x^{3}+5 x^{3} b^{7}+160 a^{3} b^{2} c^{2} x^{2}+80 a^{2} b^{4} c \,x^{2}+20 a \,b^{6} x^{2}+80 a^{3} b^{3} c x +30 a^{2} b^{5} x +20 a^{3} b^{4}\right ) \left (2 c x +b \right )}{20 b^{5} \left (2 c d x +b d \right )^{6}}\) | \(188\) |
Input:
int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^6,x,method=_RETURNVERBOSE)
Output:
1/d^6*(1/64*x/c^3-1/640/c^4*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(2* c*x+b)^5-1/384*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2*c*x+b)^3-1/128*(12*a*c -3*b^2)/c^4/(2*c*x+b))
Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (86) = 172\).
Time = 0.08 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.41 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^6} \, dx=\frac {320 \, c^{6} x^{6} + 800 \, b c^{5} x^{5} + 11 \, b^{6} - 32 \, a b^{4} c - 32 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 80 \, {\left (13 \, b^{2} c^{4} - 12 \, a c^{5}\right )} x^{4} + 80 \, {\left (11 \, b^{3} c^{3} - 24 \, a b c^{4}\right )} x^{3} + 40 \, {\left (11 \, b^{4} c^{2} - 32 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} x^{2} + 10 \, {\left (11 \, b^{5} c - 32 \, a b^{3} c^{2} - 32 \, a^{2} b c^{3}\right )} x}{640 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}} \] Input:
integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^6,x, algorithm="fricas")
Output:
1/640*(320*c^6*x^6 + 800*b*c^5*x^5 + 11*b^6 - 32*a*b^4*c - 32*a^2*b^2*c^2 - 64*a^3*c^3 + 80*(13*b^2*c^4 - 12*a*c^5)*x^4 + 80*(11*b^3*c^3 - 24*a*b*c^ 4)*x^3 + 40*(11*b^4*c^2 - 32*a*b^2*c^3 - 8*a^2*c^4)*x^2 + 10*(11*b^5*c - 3 2*a*b^3*c^2 - 32*a^2*b*c^3)*x)/(32*c^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2 *c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 + 10*b^4*c^5*d^6*x + b^5*c^4*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (90) = 180\).
Time = 1.44 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.37 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^6} \, dx=\frac {- 64 a^{3} c^{3} - 32 a^{2} b^{2} c^{2} - 32 a b^{4} c + 11 b^{6} + x^{4} \left (- 960 a c^{5} + 240 b^{2} c^{4}\right ) + x^{3} \left (- 1920 a b c^{4} + 480 b^{3} c^{3}\right ) + x^{2} \left (- 320 a^{2} c^{4} - 1280 a b^{2} c^{3} + 340 b^{4} c^{2}\right ) + x \left (- 320 a^{2} b c^{3} - 320 a b^{3} c^{2} + 100 b^{5} c\right )}{640 b^{5} c^{4} d^{6} + 6400 b^{4} c^{5} d^{6} x + 25600 b^{3} c^{6} d^{6} x^{2} + 51200 b^{2} c^{7} d^{6} x^{3} + 51200 b c^{8} d^{6} x^{4} + 20480 c^{9} d^{6} x^{5}} + \frac {x}{64 c^{3} d^{6}} \] Input:
integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**6,x)
Output:
(-64*a**3*c**3 - 32*a**2*b**2*c**2 - 32*a*b**4*c + 11*b**6 + x**4*(-960*a* c**5 + 240*b**2*c**4) + x**3*(-1920*a*b*c**4 + 480*b**3*c**3) + x**2*(-320 *a**2*c**4 - 1280*a*b**2*c**3 + 340*b**4*c**2) + x*(-320*a**2*b*c**3 - 320 *a*b**3*c**2 + 100*b**5*c))/(640*b**5*c**4*d**6 + 6400*b**4*c**5*d**6*x + 25600*b**3*c**6*d**6*x**2 + 51200*b**2*c**7*d**6*x**3 + 51200*b*c**8*d**6* x**4 + 20480*c**9*d**6*x**5) + x/(64*c**3*d**6)
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (86) = 172\).
Time = 0.05 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.32 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^6} \, dx=\frac {11 \, b^{6} - 32 \, a b^{4} c - 32 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 240 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 480 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 20 \, {\left (17 \, b^{4} c^{2} - 64 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 20 \, {\left (5 \, b^{5} c - 16 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x}{640 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}} + \frac {x}{64 \, c^{3} d^{6}} \] Input:
integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^6,x, algorithm="maxima")
Output:
1/640*(11*b^6 - 32*a*b^4*c - 32*a^2*b^2*c^2 - 64*a^3*c^3 + 240*(b^2*c^4 - 4*a*c^5)*x^4 + 480*(b^3*c^3 - 4*a*b*c^4)*x^3 + 20*(17*b^4*c^2 - 64*a*b^2*c ^3 - 16*a^2*c^4)*x^2 + 20*(5*b^5*c - 16*a*b^3*c^2 - 16*a^2*b*c^3)*x)/(32*c ^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 + 10*b^4*c^5*d^6*x + b^5*c^4*d^6) + 1/64*x/(c^3*d^6)
Time = 0.16 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.70 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^6} \, dx=\frac {x}{64 \, c^{3} d^{6}} + \frac {240 \, b^{2} c^{4} x^{4} - 960 \, a c^{5} x^{4} + 480 \, b^{3} c^{3} x^{3} - 1920 \, a b c^{4} x^{3} + 340 \, b^{4} c^{2} x^{2} - 1280 \, a b^{2} c^{3} x^{2} - 320 \, a^{2} c^{4} x^{2} + 100 \, b^{5} c x - 320 \, a b^{3} c^{2} x - 320 \, a^{2} b c^{3} x + 11 \, b^{6} - 32 \, a b^{4} c - 32 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{640 \, {\left (2 \, c x + b\right )}^{5} c^{4} d^{6}} \] Input:
integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^6,x, algorithm="giac")
Output:
1/64*x/(c^3*d^6) + 1/640*(240*b^2*c^4*x^4 - 960*a*c^5*x^4 + 480*b^3*c^3*x^ 3 - 1920*a*b*c^4*x^3 + 340*b^4*c^2*x^2 - 1280*a*b^2*c^3*x^2 - 320*a^2*c^4* x^2 + 100*b^5*c*x - 320*a*b^3*c^2*x - 320*a^2*b*c^3*x + 11*b^6 - 32*a*b^4* c - 32*a^2*b^2*c^2 - 64*a^3*c^3)/((2*c*x + b)^5*c^4*d^6)
Time = 5.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^6} \, dx=-\frac {b^2\,\left (\frac {a^2\,c^2}{20}+2\,a\,c^3\,x^2-\frac {c^4\,x^4}{4}\right )+b\,\left (\frac {a^2\,c^3\,x}{2}+3\,a\,c^4\,x^3-\frac {7\,c^5\,x^5}{10}\right )+\frac {a^3\,c^3}{10}-\frac {c^6\,x^6}{2}+\frac {3\,a\,c^5\,x^4}{2}+\frac {a^2\,c^4\,x^2}{2}+\frac {a\,b^4\,c}{20}+\frac {a\,b^3\,c^2\,x}{2}}{c^4\,d^6\,{\left (b+2\,c\,x\right )}^5} \] Input:
int((a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^6,x)
Output:
-(b^2*((a^2*c^2)/20 - (c^4*x^4)/4 + 2*a*c^3*x^2) + b*((a^2*c^3*x)/2 - (7*c ^5*x^5)/10 + 3*a*c^4*x^3) + (a^3*c^3)/10 - (c^6*x^6)/2 + (3*a*c^5*x^4)/2 + (a^2*c^4*x^2)/2 + (a*b^4*c)/20 + (a*b^3*c^2*x)/2)/(c^4*d^6*(b + 2*c*x)^5)
Time = 0.18 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.26 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^6} \, dx=\frac {160 b \,c^{6} x^{6}+192 a \,c^{6} x^{5}+192 b^{2} c^{5} x^{5}-480 a \,b^{2} c^{4} x^{3}-80 b^{4} c^{3} x^{3}-160 a^{2} b \,c^{4} x^{2}-400 a \,b^{3} c^{3} x^{2}-40 b^{5} c^{2} x^{2}-160 a^{2} b^{2} c^{3} x -100 a \,b^{4} c^{2} x -10 b^{6} c x -32 a^{3} b \,c^{3}-16 a^{2} b^{3} c^{2}-10 a \,b^{5} c -b^{7}}{320 b \,c^{4} d^{6} \left (32 c^{5} x^{5}+80 b \,c^{4} x^{4}+80 b^{2} c^{3} x^{3}+40 b^{3} c^{2} x^{2}+10 b^{4} c x +b^{5}\right )} \] Input:
int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^6,x)
Output:
( - 32*a**3*b*c**3 - 16*a**2*b**3*c**2 - 160*a**2*b**2*c**3*x - 160*a**2*b *c**4*x**2 - 10*a*b**5*c - 100*a*b**4*c**2*x - 400*a*b**3*c**3*x**2 - 480* a*b**2*c**4*x**3 + 192*a*c**6*x**5 - b**7 - 10*b**6*c*x - 40*b**5*c**2*x** 2 - 80*b**4*c**3*x**3 + 192*b**2*c**5*x**5 + 160*b*c**6*x**6)/(320*b*c**4* d**6*(b**5 + 10*b**4*c*x + 40*b**3*c**2*x**2 + 80*b**2*c**3*x**3 + 80*b*c* *4*x**4 + 32*c**5*x**5))