Integrand size = 24, antiderivative size = 101 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=\frac {\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac {\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac {3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac {1}{640 c^4 d^{12} (b+2 c x)^5} \]
1/1408*(-4*a*c+b^2)^3/c^4/d^12/(2*c*x+b)^11-1/384*(-4*a*c+b^2)^2/c^4/d^12/ (2*c*x+b)^9+3/896*(-4*a*c+b^2)/c^4/d^12/(2*c*x+b)^7-1/640/c^4/d^12/(2*c*x+ b)^5
Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=\frac {105 \left (b^2-4 a c\right )^3-385 \left (b^2-4 a c\right )^2 (b+2 c x)^2+495 \left (b^2-4 a c\right ) (b+2 c x)^4-231 (b+2 c x)^6}{147840 c^4 d^{12} (b+2 c x)^{11}} \]
(105*(b^2 - 4*a*c)^3 - 385*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 495*(b^2 - 4*a* c)*(b + 2*c*x)^4 - 231*(b + 2*c*x)^6)/(147840*c^4*d^12*(b + 2*c*x)^11)
Time = 0.30 (sec) , antiderivative size = 101, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx\) |
| \(\Big \downarrow \) 1107 |
| \(\displaystyle \int \left (\frac {\left (4 a c-b^2\right )^3}{64 c^3 d^{12} (b+2 c x)^{12}}+\frac {3 \left (4 a c-b^2\right )^2}{64 c^3 d^{12} (b+2 c x)^{10}}+\frac {3 \left (4 a c-b^2\right )}{64 c^3 d^{12} (b+2 c x)^8}+\frac {1}{64 c^3 d^{12} (b+2 c x)^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac {\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac {3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac {1}{640 c^4 d^{12} (b+2 c x)^5}\) |
(b^2 - 4*a*c)^3/(1408*c^4*d^12*(b + 2*c*x)^11) - (b^2 - 4*a*c)^2/(384*c^4* d^12*(b + 2*c*x)^9) + (3*(b^2 - 4*a*c))/(896*c^4*d^12*(b + 2*c*x)^7) - 1/( 640*c^4*d^12*(b + 2*c*x)^5)
3.12.53.3.1 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] /; 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 = 2.52 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(\frac {-\frac {1}{640 c^{4} \left (2 c x +b \right )^{5}}-\frac {12 a c -3 b^{2}}{896 c^{4} \left (2 c x +b \right )^{7}}-\frac {64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{1408 c^{4} \left (2 c x +b \right )^{11}}-\frac {48 a^{2} c^{2}-24 a \,b^{2} c +3 b^{4}}{1152 c^{4} \left (2 c x +b \right )^{9}}}{d^{12}}\) | \(121\) |
| risch | \(\frac {-\frac {c^{2} x^{6}}{10}-\frac {3 b c \,x^{5}}{10}+\left (-\frac {3 a c}{14}-\frac {9 b^{2}}{28}\right ) x^{4}-\frac {b \left (3 a c +b^{2}\right ) x^{3}}{7 c}-\frac {\left (7 a^{2} c^{2}+10 a \,b^{2} c +b^{4}\right ) x^{2}}{42 c^{2}}-\frac {b \left (70 a^{2} c^{2}+10 a \,b^{2} c +b^{4}\right ) x}{420 c^{3}}-\frac {420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}}{9240 c^{4}}}{d^{12} \left (2 c x +b \right )^{11}}\) | \(148\) |
| gosper | \(-\frac {924 c^{6} x^{6}+2772 b \,c^{5} x^{5}+1980 a \,c^{5} x^{4}+2970 b^{2} c^{4} x^{4}+3960 a b \,c^{4} x^{3}+1320 x^{3} b^{3} c^{3}+1540 a^{2} c^{4} x^{2}+2200 a \,b^{2} c^{3} x^{2}+220 x^{2} b^{4} c^{2}+1540 a^{2} b \,c^{3} x +220 x a \,b^{3} c^{2}+22 x \,b^{5} c +420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}}{9240 \left (2 c x +b \right )^{11} d^{12} c^{4}}\) | \(166\) |
| norman | \(\frac {\frac {a^{3} x}{b d}+\frac {\left (60 a^{3} c^{2}+10 a^{2} b^{2} c +a \,b^{4}\right ) x^{3}}{b^{3} d}+\frac {\left (20 c \,a^{3}+3 a^{2} b^{2}\right ) x^{2}}{2 b^{2} d}+\frac {\left (960 c^{3} a^{3}+160 a^{2} b^{2} c^{2}+22 a \,b^{4} c +b^{6}\right ) x^{4}}{4 b^{4} d}+\frac {128 c^{6} \left (420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{10}}{105 b^{10} d}+\frac {64 c^{5} \left (420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{9}}{21 b^{9} d}+\frac {32 c^{4} \left (420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{8}}{7 b^{8} d}+\frac {32 c^{3} \left (420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{7}}{7 b^{7} d}+\frac {c \left (6720 c^{3} a^{3}+1120 a^{2} b^{2} c^{2}+160 a \,b^{4} c +13 b^{6}\right ) x^{5}}{10 b^{5} d}+\frac {c^{2} \left (13440 c^{3} a^{3}+2240 a^{2} b^{2} c^{2}+320 a \,b^{4} c +31 b^{6}\right ) x^{6}}{10 b^{6} d}+\frac {256 c^{7} \left (420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{11}}{1155 b^{11} d}}{d^{11} \left (2 c x +b \right )^{11}}\) | \(435\) |
| parallelrisch | \(\frac {430080 x^{11} a^{3} c^{10}+71680 x^{11} a^{2} b^{2} c^{9}+10240 x^{11} a \,b^{4} c^{8}+1024 x^{11} b^{6} c^{7}+2365440 x^{10} a^{3} b \,c^{9}+394240 x^{10} a^{2} b^{3} c^{8}+56320 x^{10} a \,b^{5} c^{7}+5632 x^{10} b^{7} c^{6}+5913600 x^{9} a^{3} b^{2} c^{8}+985600 x^{9} a^{2} b^{4} c^{7}+140800 x^{9} a \,b^{6} c^{6}+14080 x^{9} b^{8} c^{5}+8870400 x^{8} a^{3} b^{3} c^{7}+1478400 x^{8} a^{2} b^{5} c^{6}+211200 x^{8} a \,b^{7} c^{5}+21120 x^{8} b^{9} c^{4}+8870400 x^{7} a^{3} b^{4} c^{6}+1478400 x^{7} a^{2} b^{6} c^{5}+211200 x^{7} a \,b^{8} c^{4}+21120 x^{7} b^{10} c^{3}+6209280 x^{6} a^{3} b^{5} c^{5}+1034880 x^{6} a^{2} b^{7} c^{4}+147840 x^{6} a \,b^{9} c^{3}+14322 x^{6} b^{11} c^{2}+3104640 x^{5} a^{3} b^{6} c^{4}+517440 x^{5} a^{2} b^{8} c^{3}+73920 x^{5} a \,b^{10} c^{2}+6006 x^{5} b^{12} c +1108800 x^{4} a^{3} b^{7} c^{3}+184800 x^{4} a^{2} b^{9} c^{2}+25410 x^{4} a \,b^{11} c +1155 x^{4} b^{13}+277200 x^{3} a^{3} b^{8} c^{2}+46200 x^{3} a^{2} b^{10} c +4620 x^{3} a \,b^{12}+46200 x^{2} a^{3} b^{9} c +6930 x^{2} a^{2} b^{11}+4620 a^{3} b^{10} x}{4620 b^{11} d^{12} \left (2 c x +b \right )^{11}}\) | \(481\) |
1/d^12*(-1/640/c^4/(2*c*x+b)^5-1/896*(12*a*c-3*b^2)/c^4/(2*c*x+b)^7-1/1408 *(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^4/(2*c*x+b)^11-1/1152*(48*a^ 2*c^2-24*a*b^2*c+3*b^4)/c^4/(2*c*x+b)^9)
Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (93) = 186\).
Time = 0.33 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.03 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=-\frac {924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3} + 990 \, {\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 1320 \, {\left (b^{3} c^{3} + 3 \, a b c^{4}\right )} x^{3} + 220 \, {\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} + 22 \, {\left (b^{5} c + 10 \, a b^{3} c^{2} + 70 \, a^{2} b c^{3}\right )} x}{9240 \, {\left (2048 \, c^{15} d^{12} x^{11} + 11264 \, b c^{14} d^{12} x^{10} + 28160 \, b^{2} c^{13} d^{12} x^{9} + 42240 \, b^{3} c^{12} d^{12} x^{8} + 42240 \, b^{4} c^{11} d^{12} x^{7} + 29568 \, b^{5} c^{10} d^{12} x^{6} + 14784 \, b^{6} c^{9} d^{12} x^{5} + 5280 \, b^{7} c^{8} d^{12} x^{4} + 1320 \, b^{8} c^{7} d^{12} x^{3} + 220 \, b^{9} c^{6} d^{12} x^{2} + 22 \, b^{10} c^{5} d^{12} x + b^{11} c^{4} d^{12}\right )}} \]
-1/9240*(924*c^6*x^6 + 2772*b*c^5*x^5 + b^6 + 10*a*b^4*c + 70*a^2*b^2*c^2 + 420*a^3*c^3 + 990*(3*b^2*c^4 + 2*a*c^5)*x^4 + 1320*(b^3*c^3 + 3*a*b*c^4) *x^3 + 220*(b^4*c^2 + 10*a*b^2*c^3 + 7*a^2*c^4)*x^2 + 22*(b^5*c + 10*a*b^3 *c^2 + 70*a^2*b*c^3)*x)/(2048*c^15*d^12*x^11 + 11264*b*c^14*d^12*x^10 + 28 160*b^2*c^13*d^12*x^9 + 42240*b^3*c^12*d^12*x^8 + 42240*b^4*c^11*d^12*x^7 + 29568*b^5*c^10*d^12*x^6 + 14784*b^6*c^9*d^12*x^5 + 5280*b^7*c^8*d^12*x^4 + 1320*b^8*c^7*d^12*x^3 + 220*b^9*c^6*d^12*x^2 + 22*b^10*c^5*d^12*x + b^1 1*c^4*d^12)
Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (100) = 200\).
Time = 10.40 (sec) , antiderivative size = 328, normalized size of antiderivative = 3.25 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=\frac {- 420 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} - 10 a b^{4} c - b^{6} - 2772 b c^{5} x^{5} - 924 c^{6} x^{6} + x^{4} \left (- 1980 a c^{5} - 2970 b^{2} c^{4}\right ) + x^{3} \left (- 3960 a b c^{4} - 1320 b^{3} c^{3}\right ) + x^{2} \left (- 1540 a^{2} c^{4} - 2200 a b^{2} c^{3} - 220 b^{4} c^{2}\right ) + x \left (- 1540 a^{2} b c^{3} - 220 a b^{3} c^{2} - 22 b^{5} c\right )}{9240 b^{11} c^{4} d^{12} + 203280 b^{10} c^{5} d^{12} x + 2032800 b^{9} c^{6} d^{12} x^{2} + 12196800 b^{8} c^{7} d^{12} x^{3} + 48787200 b^{7} c^{8} d^{12} x^{4} + 136604160 b^{6} c^{9} d^{12} x^{5} + 273208320 b^{5} c^{10} d^{12} x^{6} + 390297600 b^{4} c^{11} d^{12} x^{7} + 390297600 b^{3} c^{12} d^{12} x^{8} + 260198400 b^{2} c^{13} d^{12} x^{9} + 104079360 b c^{14} d^{12} x^{10} + 18923520 c^{15} d^{12} x^{11}} \]
(-420*a**3*c**3 - 70*a**2*b**2*c**2 - 10*a*b**4*c - b**6 - 2772*b*c**5*x** 5 - 924*c**6*x**6 + x**4*(-1980*a*c**5 - 2970*b**2*c**4) + x**3*(-3960*a*b *c**4 - 1320*b**3*c**3) + x**2*(-1540*a**2*c**4 - 2200*a*b**2*c**3 - 220*b **4*c**2) + x*(-1540*a**2*b*c**3 - 220*a*b**3*c**2 - 22*b**5*c))/(9240*b** 11*c**4*d**12 + 203280*b**10*c**5*d**12*x + 2032800*b**9*c**6*d**12*x**2 + 12196800*b**8*c**7*d**12*x**3 + 48787200*b**7*c**8*d**12*x**4 + 136604160 *b**6*c**9*d**12*x**5 + 273208320*b**5*c**10*d**12*x**6 + 390297600*b**4*c **11*d**12*x**7 + 390297600*b**3*c**12*d**12*x**8 + 260198400*b**2*c**13*d **12*x**9 + 104079360*b*c**14*d**12*x**10 + 18923520*c**15*d**12*x**11)
Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (93) = 186\).
Time = 0.22 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.03 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=-\frac {924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3} + 990 \, {\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 1320 \, {\left (b^{3} c^{3} + 3 \, a b c^{4}\right )} x^{3} + 220 \, {\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} + 22 \, {\left (b^{5} c + 10 \, a b^{3} c^{2} + 70 \, a^{2} b c^{3}\right )} x}{9240 \, {\left (2048 \, c^{15} d^{12} x^{11} + 11264 \, b c^{14} d^{12} x^{10} + 28160 \, b^{2} c^{13} d^{12} x^{9} + 42240 \, b^{3} c^{12} d^{12} x^{8} + 42240 \, b^{4} c^{11} d^{12} x^{7} + 29568 \, b^{5} c^{10} d^{12} x^{6} + 14784 \, b^{6} c^{9} d^{12} x^{5} + 5280 \, b^{7} c^{8} d^{12} x^{4} + 1320 \, b^{8} c^{7} d^{12} x^{3} + 220 \, b^{9} c^{6} d^{12} x^{2} + 22 \, b^{10} c^{5} d^{12} x + b^{11} c^{4} d^{12}\right )}} \]
-1/9240*(924*c^6*x^6 + 2772*b*c^5*x^5 + b^6 + 10*a*b^4*c + 70*a^2*b^2*c^2 + 420*a^3*c^3 + 990*(3*b^2*c^4 + 2*a*c^5)*x^4 + 1320*(b^3*c^3 + 3*a*b*c^4) *x^3 + 220*(b^4*c^2 + 10*a*b^2*c^3 + 7*a^2*c^4)*x^2 + 22*(b^5*c + 10*a*b^3 *c^2 + 70*a^2*b*c^3)*x)/(2048*c^15*d^12*x^11 + 11264*b*c^14*d^12*x^10 + 28 160*b^2*c^13*d^12*x^9 + 42240*b^3*c^12*d^12*x^8 + 42240*b^4*c^11*d^12*x^7 + 29568*b^5*c^10*d^12*x^6 + 14784*b^6*c^9*d^12*x^5 + 5280*b^7*c^8*d^12*x^4 + 1320*b^8*c^7*d^12*x^3 + 220*b^9*c^6*d^12*x^2 + 22*b^10*c^5*d^12*x + b^1 1*c^4*d^12)
Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=-\frac {924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + 2970 \, b^{2} c^{4} x^{4} + 1980 \, a c^{5} x^{4} + 1320 \, b^{3} c^{3} x^{3} + 3960 \, a b c^{4} x^{3} + 220 \, b^{4} c^{2} x^{2} + 2200 \, a b^{2} c^{3} x^{2} + 1540 \, a^{2} c^{4} x^{2} + 22 \, b^{5} c x + 220 \, a b^{3} c^{2} x + 1540 \, a^{2} b c^{3} x + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3}}{9240 \, {\left (2 \, c x + b\right )}^{11} c^{4} d^{12}} \]
-1/9240*(924*c^6*x^6 + 2772*b*c^5*x^5 + 2970*b^2*c^4*x^4 + 1980*a*c^5*x^4 + 1320*b^3*c^3*x^3 + 3960*a*b*c^4*x^3 + 220*b^4*c^2*x^2 + 2200*a*b^2*c^3*x ^2 + 1540*a^2*c^4*x^2 + 22*b^5*c*x + 220*a*b^3*c^2*x + 1540*a^2*b*c^3*x + b^6 + 10*a*b^4*c + 70*a^2*b^2*c^2 + 420*a^3*c^3)/((2*c*x + b)^11*c^4*d^12)
Time = 0.39 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.89 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=-\frac {\frac {420\,a^3\,c^3+70\,a^2\,b^2\,c^2+10\,a\,b^4\,c+b^6}{9240\,c^4}+x^4\,\left (\frac {9\,b^2}{28}+\frac {3\,a\,c}{14}\right )+\frac {c^2\,x^6}{10}+\frac {x^3\,\left (b^3+3\,a\,c\,b\right )}{7\,c}+\frac {x^2\,\left (7\,a^2\,c^2+10\,a\,b^2\,c+b^4\right )}{42\,c^2}+\frac {3\,b\,c\,x^5}{10}+\frac {b\,x\,\left (70\,a^2\,c^2+10\,a\,b^2\,c+b^4\right )}{420\,c^3}}{b^{11}\,d^{12}+22\,b^{10}\,c\,d^{12}\,x+220\,b^9\,c^2\,d^{12}\,x^2+1320\,b^8\,c^3\,d^{12}\,x^3+5280\,b^7\,c^4\,d^{12}\,x^4+14784\,b^6\,c^5\,d^{12}\,x^5+29568\,b^5\,c^6\,d^{12}\,x^6+42240\,b^4\,c^7\,d^{12}\,x^7+42240\,b^3\,c^8\,d^{12}\,x^8+28160\,b^2\,c^9\,d^{12}\,x^9+11264\,b\,c^{10}\,d^{12}\,x^{10}+2048\,c^{11}\,d^{12}\,x^{11}} \]
-((b^6 + 420*a^3*c^3 + 70*a^2*b^2*c^2 + 10*a*b^4*c)/(9240*c^4) + x^4*((3*a *c)/14 + (9*b^2)/28) + (c^2*x^6)/10 + (x^3*(b^3 + 3*a*b*c))/(7*c) + (x^2*( b^4 + 7*a^2*c^2 + 10*a*b^2*c))/(42*c^2) + (3*b*c*x^5)/10 + (b*x*(b^4 + 70* a^2*c^2 + 10*a*b^2*c))/(420*c^3))/(b^11*d^12 + 2048*c^11*d^12*x^11 + 11264 *b*c^10*d^12*x^10 + 220*b^9*c^2*d^12*x^2 + 1320*b^8*c^3*d^12*x^3 + 5280*b^ 7*c^4*d^12*x^4 + 14784*b^6*c^5*d^12*x^5 + 29568*b^5*c^6*d^12*x^6 + 42240*b ^4*c^7*d^12*x^7 + 42240*b^3*c^8*d^12*x^8 + 28160*b^2*c^9*d^12*x^9 + 22*b^1 0*c*d^12*x)