Integrand size = 27, antiderivative size = 341 \[ \int (a+b x)^3 (c+d x) (e+f x)^{3/2} (g+h x) \, dx=\frac {2 (b e-a f)^3 (d e-c f) (f g-e h) (e+f x)^{5/2}}{5 f^6}-\frac {2 (b e-a f)^2 (b d e (4 f g-5 e h)-b c f (3 f g-4 e h)-a f (d f g-2 d e h+c f h)) (e+f x)^{7/2}}{7 f^6}-\frac {2 (b e-a f) \left (a^2 d f^2 h+a b f (3 d f g-8 d e h+3 c f h)-b^2 (2 d e (3 f g-5 e h)-3 c f (f g-2 e h))\right ) (e+f x)^{9/2}}{9 f^6}+\frac {2 b \left (3 a^2 d f^2 h+3 a b f (d f g-4 d e h+c f h)-b^2 (2 d e (2 f g-5 e h)-c f (f g-4 e h))\right ) (e+f x)^{11/2}}{11 f^6}+\frac {2 b^2 (3 a d f h+b (d f g-5 d e h+c f h)) (e+f x)^{13/2}}{13 f^6}+\frac {2 b^3 d h (e+f x)^{15/2}}{15 f^6} \] Output:
2/5*(-a*f+b*e)^3*(-c*f+d*e)*(-e*h+f*g)*(f*x+e)^(5/2)/f^6-2/7*(-a*f+b*e)^2* (b*d*e*(-5*e*h+4*f*g)-b*c*f*(-4*e*h+3*f*g)-a*f*(c*f*h-2*d*e*h+d*f*g))*(f*x +e)^(7/2)/f^6-2/9*(-a*f+b*e)*(a^2*d*f^2*h+a*b*f*(3*c*f*h-8*d*e*h+3*d*f*g)- b^2*(2*d*e*(-5*e*h+3*f*g)-3*c*f*(-2*e*h+f*g)))*(f*x+e)^(9/2)/f^6+2/11*b*(3 *a^2*d*f^2*h+3*a*b*f*(c*f*h-4*d*e*h+d*f*g)-b^2*(2*d*e*(-5*e*h+2*f*g)-c*f*( -4*e*h+f*g)))*(f*x+e)^(11/2)/f^6+2/13*b^2*(3*a*d*f*h+b*(c*f*h-5*d*e*h+d*f* g))*(f*x+e)^(13/2)/f^6+2/15*b^3*d*h*(f*x+e)^(15/2)/f^6
Time = 0.47 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.43 \[ \int (a+b x)^3 (c+d x) (e+f x)^{3/2} (g+h x) \, dx=\frac {2 (e+f x)^{5/2} \left (143 a^3 f^3 \left (9 c f (7 f g-2 e h+5 f h x)+d \left (8 e^2 h+5 f^2 x (9 g+7 h x)-2 e f (9 g+10 h x)\right )\right )+b^3 \left (d \left (-256 e^5 h+1680 e^2 f^3 x^2 (g+h x)+128 e^4 f (3 g+5 h x)-160 e^3 f^2 x (6 g+7 h x)-210 e f^4 x^3 (12 g+11 h x)+231 f^5 x^4 (15 g+13 h x)\right )+3 c f \left (128 e^4 h+105 f^4 x^3 (13 g+11 h x)-70 e f^3 x^2 (13 g+12 h x)+40 e^2 f^2 x (13 g+14 h x)-16 e^3 f (13 g+20 h x)\right )\right )+39 a^2 b f^2 \left (11 c f \left (8 e^2 h+5 f^2 x (9 g+7 h x)-2 e f (9 g+10 h x)\right )+d \left (-48 e^3 h+35 f^3 x^2 (11 g+9 h x)+8 e^2 f (11 g+15 h x)-10 e f^2 x (22 g+21 h x)\right )\right )+3 a b^2 f \left (3 d \left (128 e^4 h+105 f^4 x^3 (13 g+11 h x)-70 e f^3 x^2 (13 g+12 h x)+40 e^2 f^2 x (13 g+14 h x)-16 e^3 f (13 g+20 h x)\right )+13 c f \left (-48 e^3 h+35 f^3 x^2 (11 g+9 h x)+8 e^2 f (11 g+15 h x)-10 e f^2 x (22 g+21 h x)\right )\right )\right )}{45045 f^6} \] Input:
Integrate[(a + b*x)^3*(c + d*x)*(e + f*x)^(3/2)*(g + h*x),x]
Output:
(2*(e + f*x)^(5/2)*(143*a^3*f^3*(9*c*f*(7*f*g - 2*e*h + 5*f*h*x) + d*(8*e^ 2*h + 5*f^2*x*(9*g + 7*h*x) - 2*e*f*(9*g + 10*h*x))) + b^3*(d*(-256*e^5*h + 1680*e^2*f^3*x^2*(g + h*x) + 128*e^4*f*(3*g + 5*h*x) - 160*e^3*f^2*x*(6* g + 7*h*x) - 210*e*f^4*x^3*(12*g + 11*h*x) + 231*f^5*x^4*(15*g + 13*h*x)) + 3*c*f*(128*e^4*h + 105*f^4*x^3*(13*g + 11*h*x) - 70*e*f^3*x^2*(13*g + 12 *h*x) + 40*e^2*f^2*x*(13*g + 14*h*x) - 16*e^3*f*(13*g + 20*h*x))) + 39*a^2 *b*f^2*(11*c*f*(8*e^2*h + 5*f^2*x*(9*g + 7*h*x) - 2*e*f*(9*g + 10*h*x)) + d*(-48*e^3*h + 35*f^3*x^2*(11*g + 9*h*x) + 8*e^2*f*(11*g + 15*h*x) - 10*e* f^2*x*(22*g + 21*h*x))) + 3*a*b^2*f*(3*d*(128*e^4*h + 105*f^4*x^3*(13*g + 11*h*x) - 70*e*f^3*x^2*(13*g + 12*h*x) + 40*e^2*f^2*x*(13*g + 14*h*x) - 16 *e^3*f*(13*g + 20*h*x)) + 13*c*f*(-48*e^3*h + 35*f^3*x^2*(11*g + 9*h*x) + 8*e^2*f*(11*g + 15*h*x) - 10*e*f^2*x*(22*g + 21*h*x)))))/(45045*f^6)
Time = 0.62 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {159, 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.
| \(\displaystyle \int (a+b x)^3 (c+d x) (e+f x)^{3/2} (g+h x) \, dx\) |
| \(\Big \downarrow \) 159 |
| \(\displaystyle \int \left (\frac {b (e+f x)^{9/2} \left (3 a^2 d f^2 h+3 a b f (c f h-4 d e h+d f g)-\left (b^2 (2 d e (2 f g-5 e h)-c f (f g-4 e h))\right )\right )}{f^5}+\frac {(e+f x)^{7/2} (b e-a f) \left (-a^2 d f^2 h-a b f (3 c f h-8 d e h+3 d f g)+b^2 (2 d e (3 f g-5 e h)-3 c f (f g-2 e h))\right )}{f^5}+\frac {b^2 (e+f x)^{11/2} (3 a d f h+b (c f h-5 d e h+d f g))}{f^5}+\frac {(e+f x)^{5/2} (b e-a f)^2 (a f (c f h-2 d e h+d f g)+b c f (3 f g-4 e h)-b d e (4 f g-5 e h))}{f^5}+\frac {(e+f x)^{3/2} (a f-b e)^3 (c f-d e) (f g-e h)}{f^5}+\frac {b^3 d h (e+f x)^{13/2}}{f^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b (e+f x)^{11/2} \left (3 a^2 d f^2 h+3 a b f (c f h-4 d e h+d f g)-\left (b^2 (2 d e (2 f g-5 e h)-c f (f g-4 e h))\right )\right )}{11 f^6}-\frac {2 (e+f x)^{9/2} (b e-a f) \left (a^2 d f^2 h+a b f (3 c f h-8 d e h+3 d f g)-\left (b^2 (2 d e (3 f g-5 e h)-3 c f (f g-2 e h))\right )\right )}{9 f^6}+\frac {2 b^2 (e+f x)^{13/2} (3 a d f h+b (c f h-5 d e h+d f g))}{13 f^6}-\frac {2 (e+f x)^{7/2} (b e-a f)^2 (-a f (c f h-2 d e h+d f g)-b c f (3 f g-4 e h)+b d e (4 f g-5 e h))}{7 f^6}+\frac {2 (e+f x)^{5/2} (b e-a f)^3 (d e-c f) (f g-e h)}{5 f^6}+\frac {2 b^3 d h (e+f x)^{15/2}}{15 f^6}\) |
Input:
Int[(a + b*x)^3*(c + d*x)*(e + f*x)^(3/2)*(g + h*x),x]
Output:
(2*(b*e - a*f)^3*(d*e - c*f)*(f*g - e*h)*(e + f*x)^(5/2))/(5*f^6) - (2*(b* e - a*f)^2*(b*d*e*(4*f*g - 5*e*h) - b*c*f*(3*f*g - 4*e*h) - a*f*(d*f*g - 2 *d*e*h + c*f*h))*(e + f*x)^(7/2))/(7*f^6) - (2*(b*e - a*f)*(a^2*d*f^2*h + a*b*f*(3*d*f*g - 8*d*e*h + 3*c*f*h) - b^2*(2*d*e*(3*f*g - 5*e*h) - 3*c*f*( f*g - 2*e*h)))*(e + f*x)^(9/2))/(9*f^6) + (2*b*(3*a^2*d*f^2*h + 3*a*b*f*(d *f*g - 4*d*e*h + c*f*h) - b^2*(2*d*e*(2*f*g - 5*e*h) - c*f*(f*g - 4*e*h))) *(e + f*x)^(11/2))/(11*f^6) + (2*b^2*(3*a*d*f*h + b*(d*f*g - 5*d*e*h + c*f *h))*(e + f*x)^(13/2))/(13*f^6) + (2*b^3*d*h*(e + f*x)^(15/2))/(15*f^6)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n *(e + f*x)*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && (IGtQ [m, 0] || IntegersQ[m, n])
Time = 0.94 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.04
| method | result | size |
| derivativedivides | \(\frac {\frac {2 d h \,b^{3} \left (f x +e \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (3 \left (a f -b e \right ) b^{2} d +b^{3} \left (c f -d e \right )\right ) h +b^{3} d \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (3 \left (a f -b e \right )^{2} b d +3 \left (a f -b e \right ) b^{2} \left (c f -d e \right )\right ) h +\left (3 \left (a f -b e \right ) b^{2} d +b^{3} \left (c f -d e \right )\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (\left (a f -b e \right )^{3} d +3 \left (a f -b e \right )^{2} b \left (c f -d e \right )\right ) h +\left (3 \left (a f -b e \right )^{2} b d +3 \left (a f -b e \right ) b^{2} \left (c f -d e \right )\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a f -b e \right )^{3} \left (c f -d e \right ) h +\left (\left (a f -b e \right )^{3} d +3 \left (a f -b e \right )^{2} b \left (c f -d e \right )\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a f -b e \right )^{3} \left (c f -d e \right ) \left (-e h +f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{5}}{f^{6}}\) | \(356\) |
| default | \(\frac {\frac {2 d h \,b^{3} \left (f x +e \right )^{\frac {15}{2}}}{15}-\frac {2 \left (-\left (3 \left (a f -b e \right ) b^{2} d +b^{3} \left (c f -d e \right )\right ) h +b^{3} d \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {13}{2}}}{13}-\frac {2 \left (-\left (3 \left (a f -b e \right )^{2} b d +3 \left (a f -b e \right ) b^{2} \left (c f -d e \right )\right ) h +\left (3 \left (a f -b e \right ) b^{2} d +b^{3} \left (c f -d e \right )\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {11}{2}}}{11}-\frac {2 \left (-\left (\left (a f -b e \right )^{3} d +3 \left (a f -b e \right )^{2} b \left (c f -d e \right )\right ) h +\left (3 \left (a f -b e \right )^{2} b d +3 \left (a f -b e \right ) b^{2} \left (c f -d e \right )\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}-\frac {2 \left (-\left (a f -b e \right )^{3} \left (c f -d e \right ) h +\left (\left (a f -b e \right )^{3} d +3 \left (a f -b e \right )^{2} b \left (c f -d e \right )\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {2 \left (a f -b e \right )^{3} \left (c f -d e \right ) \left (e h -f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{5}}{f^{6}}\) | \(360\) |
| pseudoelliptic | \(-\frac {4 \left (f x +e \right )^{\frac {5}{2}} \left (\frac {\left (7 \left (-\frac {d h \,x^{5}}{3}+\frac {5 \left (-c h -d g \right ) x^{4}}{13}-\frac {5 c g \,x^{3}}{11}\right ) b^{3}-\frac {35 a \,x^{2} \left (\frac {9 d h \,x^{2}}{13}+\frac {9 \left (c h +d g \right ) x}{11}+c g \right ) b^{2}}{3}-15 a^{2} x \left (\frac {7 d h \,x^{2}}{11}+\frac {7 \left (c h +d g \right ) x}{9}+c g \right ) b -7 a^{3} \left (\frac {5 d h \,x^{2}}{9}+\frac {5 \left (c h +d g \right ) x}{7}+c g \right )\right ) f^{5}}{2}+\left (\frac {35 x^{2} \left (\frac {11 d h \,x^{2}}{13}+\frac {12 \left (c h +d g \right ) x}{13}+c g \right ) b^{3}}{33}+\frac {10 a x \left (\frac {126 d h \,x^{2}}{143}+\frac {21 \left (c h +d g \right ) x}{22}+c g \right ) b^{2}}{3}+3 a^{2} \left (\frac {35 d h \,x^{2}}{33}+\frac {10 \left (c h +d g \right ) x}{9}+c g \right ) b +a^{3} \left (d g +c h +\frac {10}{9} d h x \right )\right ) e \,f^{4}-\frac {4 e^{2} \left (\frac {15 x \left (\frac {14 d h \,x^{2}}{13}+\frac {14 \left (c h +d g \right ) x}{13}+c g \right ) b^{3}}{11}+3 a \left (\frac {210 d h \,x^{2}}{143}+\frac {15 \left (c h +d g \right ) x}{11}+c g \right ) b^{2}+3 \left (\frac {15}{11} d h x +c h +d g \right ) a^{2} b +d h \,a^{3}\right ) f^{3}}{9}+\frac {8 \left (\frac {\left (\frac {70 d h \,x^{2}}{39}+\frac {20 \left (c h +d g \right ) x}{13}+c g \right ) b^{2}}{3}+a \left (\frac {20}{13} d h x +c h +d g \right ) b +a^{2} d h \right ) b \,e^{3} f^{2}}{11}-\frac {64 \left (\frac {\left (\frac {5}{3} d h x +c h +d g \right ) b}{3}+a d h \right ) b^{2} e^{4} f}{143}+\frac {128 b^{3} d \,e^{5} h}{1287}\right )}{35 f^{6}}\) | \(427\) |
| gosper | \(-\frac {2 \left (f x +e \right )^{\frac {5}{2}} \left (-3003 d h \,b^{3} x^{5} f^{5}-10395 a \,b^{2} d \,f^{5} h \,x^{4}-3465 b^{3} c \,f^{5} h \,x^{4}+2310 b^{3} d e \,f^{4} h \,x^{4}-3465 b^{3} d \,f^{5} g \,x^{4}-12285 a^{2} b d \,f^{5} h \,x^{3}-12285 a \,b^{2} c \,f^{5} h \,x^{3}+7560 a \,b^{2} d e \,f^{4} h \,x^{3}-12285 a \,b^{2} d \,f^{5} g \,x^{3}+2520 b^{3} c e \,f^{4} h \,x^{3}-4095 b^{3} c \,f^{5} g \,x^{3}-1680 b^{3} d \,e^{2} f^{3} h \,x^{3}+2520 b^{3} d e \,f^{4} g \,x^{3}-5005 a^{3} d \,f^{5} h \,x^{2}-15015 a^{2} b c \,f^{5} h \,x^{2}+8190 a^{2} b d e \,f^{4} h \,x^{2}-15015 a^{2} b d \,f^{5} g \,x^{2}+8190 a \,b^{2} c e \,f^{4} h \,x^{2}-15015 a \,b^{2} c \,f^{5} g \,x^{2}-5040 a \,b^{2} d \,e^{2} f^{3} h \,x^{2}+8190 a \,b^{2} d e \,f^{4} g \,x^{2}-1680 b^{3} c \,e^{2} f^{3} h \,x^{2}+2730 b^{3} c e \,f^{4} g \,x^{2}+1120 b^{3} d \,e^{3} f^{2} h \,x^{2}-1680 b^{3} d \,e^{2} f^{3} g \,x^{2}-6435 a^{3} c \,f^{5} h x +2860 a^{3} d e \,f^{4} h x -6435 a^{3} d \,f^{5} g x +8580 a^{2} b c e \,f^{4} h x -19305 a^{2} b c \,f^{5} g x -4680 a^{2} b d \,e^{2} f^{3} h x +8580 a^{2} b d e \,f^{4} g x -4680 a \,b^{2} c \,e^{2} f^{3} h x +8580 a \,b^{2} c e \,f^{4} g x +2880 a \,b^{2} d \,e^{3} f^{2} h x -4680 a \,b^{2} d \,e^{2} f^{3} g x +960 b^{3} c \,e^{3} f^{2} h x -1560 b^{3} c \,e^{2} f^{3} g x -640 b^{3} d \,e^{4} f h x +960 b^{3} d \,e^{3} f^{2} g x +2574 a^{3} c e \,f^{4} h -9009 c g \,a^{3} f^{5}-1144 a^{3} d \,e^{2} f^{3} h +2574 a^{3} d e \,f^{4} g -3432 a^{2} b c \,e^{2} f^{3} h +7722 a^{2} b c e \,f^{4} g +1872 a^{2} b d \,e^{3} f^{2} h -3432 a^{2} b d \,e^{2} f^{3} g +1872 a \,b^{2} c \,e^{3} f^{2} h -3432 a \,b^{2} c \,e^{2} f^{3} g -1152 a \,b^{2} d \,e^{4} f h +1872 a \,b^{2} d \,e^{3} f^{2} g -384 b^{3} c \,e^{4} f h +624 b^{3} c \,e^{3} f^{2} g +256 b^{3} d \,e^{5} h -384 b^{3} d \,e^{4} f g \right )}{45045 f^{6}}\) | \(771\) |
| orering | \(-\frac {2 \left (f x +e \right )^{\frac {5}{2}} \left (-3003 d h \,b^{3} x^{5} f^{5}-10395 a \,b^{2} d \,f^{5} h \,x^{4}-3465 b^{3} c \,f^{5} h \,x^{4}+2310 b^{3} d e \,f^{4} h \,x^{4}-3465 b^{3} d \,f^{5} g \,x^{4}-12285 a^{2} b d \,f^{5} h \,x^{3}-12285 a \,b^{2} c \,f^{5} h \,x^{3}+7560 a \,b^{2} d e \,f^{4} h \,x^{3}-12285 a \,b^{2} d \,f^{5} g \,x^{3}+2520 b^{3} c e \,f^{4} h \,x^{3}-4095 b^{3} c \,f^{5} g \,x^{3}-1680 b^{3} d \,e^{2} f^{3} h \,x^{3}+2520 b^{3} d e \,f^{4} g \,x^{3}-5005 a^{3} d \,f^{5} h \,x^{2}-15015 a^{2} b c \,f^{5} h \,x^{2}+8190 a^{2} b d e \,f^{4} h \,x^{2}-15015 a^{2} b d \,f^{5} g \,x^{2}+8190 a \,b^{2} c e \,f^{4} h \,x^{2}-15015 a \,b^{2} c \,f^{5} g \,x^{2}-5040 a \,b^{2} d \,e^{2} f^{3} h \,x^{2}+8190 a \,b^{2} d e \,f^{4} g \,x^{2}-1680 b^{3} c \,e^{2} f^{3} h \,x^{2}+2730 b^{3} c e \,f^{4} g \,x^{2}+1120 b^{3} d \,e^{3} f^{2} h \,x^{2}-1680 b^{3} d \,e^{2} f^{3} g \,x^{2}-6435 a^{3} c \,f^{5} h x +2860 a^{3} d e \,f^{4} h x -6435 a^{3} d \,f^{5} g x +8580 a^{2} b c e \,f^{4} h x -19305 a^{2} b c \,f^{5} g x -4680 a^{2} b d \,e^{2} f^{3} h x +8580 a^{2} b d e \,f^{4} g x -4680 a \,b^{2} c \,e^{2} f^{3} h x +8580 a \,b^{2} c e \,f^{4} g x +2880 a \,b^{2} d \,e^{3} f^{2} h x -4680 a \,b^{2} d \,e^{2} f^{3} g x +960 b^{3} c \,e^{3} f^{2} h x -1560 b^{3} c \,e^{2} f^{3} g x -640 b^{3} d \,e^{4} f h x +960 b^{3} d \,e^{3} f^{2} g x +2574 a^{3} c e \,f^{4} h -9009 c g \,a^{3} f^{5}-1144 a^{3} d \,e^{2} f^{3} h +2574 a^{3} d e \,f^{4} g -3432 a^{2} b c \,e^{2} f^{3} h +7722 a^{2} b c e \,f^{4} g +1872 a^{2} b d \,e^{3} f^{2} h -3432 a^{2} b d \,e^{2} f^{3} g +1872 a \,b^{2} c \,e^{3} f^{2} h -3432 a \,b^{2} c \,e^{2} f^{3} g -1152 a \,b^{2} d \,e^{4} f h +1872 a \,b^{2} d \,e^{3} f^{2} g -384 b^{3} c \,e^{4} f h +624 b^{3} c \,e^{3} f^{2} g +256 b^{3} d \,e^{5} h -384 b^{3} d \,e^{4} f g \right )}{45045 f^{6}}\) | \(771\) |
| trager | \(\text {Expression too large to display}\) | \(1295\) |
| risch | \(\text {Expression too large to display}\) | \(1295\) |
Input:
int((b*x+a)^3*(d*x+c)*(f*x+e)^(3/2)*(h*x+g),x,method=_RETURNVERBOSE)
Output:
2/f^6*(1/15*d*h*b^3*(f*x+e)^(15/2)+1/13*((3*(a*f-b*e)*b^2*d+b^3*(c*f-d*e)) *h+b^3*d*(-e*h+f*g))*(f*x+e)^(13/2)+1/11*((3*(a*f-b*e)^2*b*d+3*(a*f-b*e)*b ^2*(c*f-d*e))*h+(3*(a*f-b*e)*b^2*d+b^3*(c*f-d*e))*(-e*h+f*g))*(f*x+e)^(11/ 2)+1/9*(((a*f-b*e)^3*d+3*(a*f-b*e)^2*b*(c*f-d*e))*h+(3*(a*f-b*e)^2*b*d+3*( a*f-b*e)*b^2*(c*f-d*e))*(-e*h+f*g))*(f*x+e)^(9/2)+1/7*((a*f-b*e)^3*(c*f-d* e)*h+((a*f-b*e)^3*d+3*(a*f-b*e)^2*b*(c*f-d*e))*(-e*h+f*g))*(f*x+e)^(7/2)+1 /5*(a*f-b*e)^3*(c*f-d*e)*(-e*h+f*g)*(f*x+e)^(5/2))
Leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (317) = 634\).
Time = 0.12 (sec) , antiderivative size = 998, normalized size of antiderivative = 2.93 \[ \int (a+b x)^3 (c+d x) (e+f x)^{3/2} (g+h x) \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^3*(d*x+c)*(f*x+e)^(3/2)*(h*x+g),x, algorithm="fricas")
Output:
2/45045*(3003*b^3*d*f^7*h*x^7 + 231*(15*b^3*d*f^7*g + (16*b^3*d*e*f^6 + 15 *(b^3*c + 3*a*b^2*d)*f^7)*h)*x^6 + 63*(5*(14*b^3*d*e*f^6 + 13*(b^3*c + 3*a *b^2*d)*f^7)*g + (b^3*d*e^2*f^5 + 70*(b^3*c + 3*a*b^2*d)*e*f^6 + 195*(a*b^ 2*c + a^2*b*d)*f^7)*h)*x^5 + 35*(3*(b^3*d*e^2*f^5 + 52*(b^3*c + 3*a*b^2*d) *e*f^6 + 143*(a*b^2*c + a^2*b*d)*f^7)*g - (2*b^3*d*e^3*f^4 - 3*(b^3*c + 3* a*b^2*d)*e^2*f^5 - 468*(a*b^2*c + a^2*b*d)*e*f^6 - 143*(3*a^2*b*c + a^3*d) *f^7)*h)*x^4 - 5*(3*(8*b^3*d*e^3*f^4 - 13*(b^3*c + 3*a*b^2*d)*e^2*f^5 - 14 30*(a*b^2*c + a^2*b*d)*e*f^6 - 429*(3*a^2*b*c + a^3*d)*f^7)*g - (16*b^3*d* e^4*f^3 + 1287*a^3*c*f^7 - 24*(b^3*c + 3*a*b^2*d)*e^3*f^4 + 117*(a*b^2*c + a^2*b*d)*e^2*f^5 + 1430*(3*a^2*b*c + a^3*d)*e*f^6)*h)*x^3 + 3*(3*(16*b^3* d*e^4*f^3 + 1001*a^3*c*f^7 - 26*(b^3*c + 3*a*b^2*d)*e^3*f^4 + 143*(a*b^2*c + a^2*b*d)*e^2*f^5 + 1144*(3*a^2*b*c + a^3*d)*e*f^6)*g - (32*b^3*d*e^5*f^ 2 - 3432*a^3*c*e*f^6 - 48*(b^3*c + 3*a*b^2*d)*e^4*f^3 + 234*(a*b^2*c + a^2 *b*d)*e^3*f^4 - 143*(3*a^2*b*c + a^3*d)*e^2*f^5)*h)*x^2 + 3*(128*b^3*d*e^6 *f + 3003*a^3*c*e^2*f^5 - 208*(b^3*c + 3*a*b^2*d)*e^5*f^2 + 1144*(a*b^2*c + a^2*b*d)*e^4*f^3 - 858*(3*a^2*b*c + a^3*d)*e^3*f^4)*g - 2*(128*b^3*d*e^7 + 1287*a^3*c*e^3*f^4 - 192*(b^3*c + 3*a*b^2*d)*e^6*f + 936*(a*b^2*c + a^2 *b*d)*e^5*f^2 - 572*(3*a^2*b*c + a^3*d)*e^4*f^3)*h - (3*(64*b^3*d*e^5*f^2 - 6006*a^3*c*e*f^6 - 104*(b^3*c + 3*a*b^2*d)*e^4*f^3 + 572*(a*b^2*c + a^2* b*d)*e^3*f^4 - 429*(3*a^2*b*c + a^3*d)*e^2*f^5)*g - (128*b^3*d*e^6*f + ...
Leaf count of result is larger than twice the leaf count of optimal. 1013 vs. \(2 (355) = 710\).
Time = 2.08 (sec) , antiderivative size = 1013, normalized size of antiderivative = 2.97 \[ \int (a+b x)^3 (c+d x) (e+f x)^{3/2} (g+h x) \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)**3*(d*x+c)*(f*x+e)**(3/2)*(h*x+g),x)
Output:
Piecewise((2*(b**3*d*h*(e + f*x)**(15/2)/(15*f**5) + (e + f*x)**(13/2)*(3* a*b**2*d*f*h + b**3*c*f*h - 5*b**3*d*e*h + b**3*d*f*g)/(13*f**5) + (e + f* x)**(11/2)*(3*a**2*b*d*f**2*h + 3*a*b**2*c*f**2*h - 12*a*b**2*d*e*f*h + 3* a*b**2*d*f**2*g - 4*b**3*c*e*f*h + b**3*c*f**2*g + 10*b**3*d*e**2*h - 4*b* *3*d*e*f*g)/(11*f**5) + (e + f*x)**(9/2)*(a**3*d*f**3*h + 3*a**2*b*c*f**3* h - 9*a**2*b*d*e*f**2*h + 3*a**2*b*d*f**3*g - 9*a*b**2*c*e*f**2*h + 3*a*b* *2*c*f**3*g + 18*a*b**2*d*e**2*f*h - 9*a*b**2*d*e*f**2*g + 6*b**3*c*e**2*f *h - 3*b**3*c*e*f**2*g - 10*b**3*d*e**3*h + 6*b**3*d*e**2*f*g)/(9*f**5) + (e + f*x)**(7/2)*(a**3*c*f**4*h - 2*a**3*d*e*f**3*h + a**3*d*f**4*g - 6*a* *2*b*c*e*f**3*h + 3*a**2*b*c*f**4*g + 9*a**2*b*d*e**2*f**2*h - 6*a**2*b*d* e*f**3*g + 9*a*b**2*c*e**2*f**2*h - 6*a*b**2*c*e*f**3*g - 12*a*b**2*d*e**3 *f*h + 9*a*b**2*d*e**2*f**2*g - 4*b**3*c*e**3*f*h + 3*b**3*c*e**2*f**2*g + 5*b**3*d*e**4*h - 4*b**3*d*e**3*f*g)/(7*f**5) + (e + f*x)**(5/2)*(-a**3*c *e*f**4*h + a**3*c*f**5*g + a**3*d*e**2*f**3*h - a**3*d*e*f**4*g + 3*a**2* b*c*e**2*f**3*h - 3*a**2*b*c*e*f**4*g - 3*a**2*b*d*e**3*f**2*h + 3*a**2*b* d*e**2*f**3*g - 3*a*b**2*c*e**3*f**2*h + 3*a*b**2*c*e**2*f**3*g + 3*a*b**2 *d*e**4*f*h - 3*a*b**2*d*e**3*f**2*g + b**3*c*e**4*f*h - b**3*c*e**3*f**2* g - b**3*d*e**5*h + b**3*d*e**4*f*g)/(5*f**5))/f, Ne(f, 0)), (e**(3/2)*(a* *3*c*g*x + b**3*d*h*x**6/6 + x**5*(3*a*b**2*d*h + b**3*c*h + b**3*d*g)/5 + x**4*(3*a**2*b*d*h + 3*a*b**2*c*h + 3*a*b**2*d*g + b**3*c*g)/4 + x**3*...
Time = 0.04 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.80 \[ \int (a+b x)^3 (c+d x) (e+f x)^{3/2} (g+h x) \, dx=\frac {2 \, {\left (3003 \, {\left (f x + e\right )}^{\frac {15}{2}} b^{3} d h + 3465 \, {\left (b^{3} d f g - {\left (5 \, b^{3} d e - {\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} h\right )} {\left (f x + e\right )}^{\frac {13}{2}} - 4095 \, {\left ({\left (4 \, b^{3} d e f - {\left (b^{3} c + 3 \, a b^{2} d\right )} f^{2}\right )} g - {\left (10 \, b^{3} d e^{2} - 4 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e f + 3 \, {\left (a b^{2} c + a^{2} b d\right )} f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {11}{2}} + 5005 \, {\left (3 \, {\left (2 \, b^{3} d e^{2} f - {\left (b^{3} c + 3 \, a b^{2} d\right )} e f^{2} + {\left (a b^{2} c + a^{2} b d\right )} f^{3}\right )} g - {\left (10 \, b^{3} d e^{3} - 6 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{2} f + 9 \, {\left (a b^{2} c + a^{2} b d\right )} e f^{2} - {\left (3 \, a^{2} b c + a^{3} d\right )} f^{3}\right )} h\right )} {\left (f x + e\right )}^{\frac {9}{2}} - 6435 \, {\left ({\left (4 \, b^{3} d e^{3} f - 3 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{2} f^{2} + 6 \, {\left (a b^{2} c + a^{2} b d\right )} e f^{3} - {\left (3 \, a^{2} b c + a^{3} d\right )} f^{4}\right )} g - {\left (5 \, b^{3} d e^{4} + a^{3} c f^{4} - 4 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{3} f + 9 \, {\left (a b^{2} c + a^{2} b d\right )} e^{2} f^{2} - 2 \, {\left (3 \, a^{2} b c + a^{3} d\right )} e f^{3}\right )} h\right )} {\left (f x + e\right )}^{\frac {7}{2}} + 9009 \, {\left ({\left (b^{3} d e^{4} f + a^{3} c f^{5} - {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{3} f^{2} + 3 \, {\left (a b^{2} c + a^{2} b d\right )} e^{2} f^{3} - {\left (3 \, a^{2} b c + a^{3} d\right )} e f^{4}\right )} g - {\left (b^{3} d e^{5} + a^{3} c e f^{4} - {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{4} f + 3 \, {\left (a b^{2} c + a^{2} b d\right )} e^{3} f^{2} - {\left (3 \, a^{2} b c + a^{3} d\right )} e^{2} f^{3}\right )} h\right )} {\left (f x + e\right )}^{\frac {5}{2}}\right )}}{45045 \, f^{6}} \] Input:
integrate((b*x+a)^3*(d*x+c)*(f*x+e)^(3/2)*(h*x+g),x, algorithm="maxima")
Output:
2/45045*(3003*(f*x + e)^(15/2)*b^3*d*h + 3465*(b^3*d*f*g - (5*b^3*d*e - (b ^3*c + 3*a*b^2*d)*f)*h)*(f*x + e)^(13/2) - 4095*((4*b^3*d*e*f - (b^3*c + 3 *a*b^2*d)*f^2)*g - (10*b^3*d*e^2 - 4*(b^3*c + 3*a*b^2*d)*e*f + 3*(a*b^2*c + a^2*b*d)*f^2)*h)*(f*x + e)^(11/2) + 5005*(3*(2*b^3*d*e^2*f - (b^3*c + 3* a*b^2*d)*e*f^2 + (a*b^2*c + a^2*b*d)*f^3)*g - (10*b^3*d*e^3 - 6*(b^3*c + 3 *a*b^2*d)*e^2*f + 9*(a*b^2*c + a^2*b*d)*e*f^2 - (3*a^2*b*c + a^3*d)*f^3)*h )*(f*x + e)^(9/2) - 6435*((4*b^3*d*e^3*f - 3*(b^3*c + 3*a*b^2*d)*e^2*f^2 + 6*(a*b^2*c + a^2*b*d)*e*f^3 - (3*a^2*b*c + a^3*d)*f^4)*g - (5*b^3*d*e^4 + a^3*c*f^4 - 4*(b^3*c + 3*a*b^2*d)*e^3*f + 9*(a*b^2*c + a^2*b*d)*e^2*f^2 - 2*(3*a^2*b*c + a^3*d)*e*f^3)*h)*(f*x + e)^(7/2) + 9009*((b^3*d*e^4*f + a^ 3*c*f^5 - (b^3*c + 3*a*b^2*d)*e^3*f^2 + 3*(a*b^2*c + a^2*b*d)*e^2*f^3 - (3 *a^2*b*c + a^3*d)*e*f^4)*g - (b^3*d*e^5 + a^3*c*e*f^4 - (b^3*c + 3*a*b^2*d )*e^4*f + 3*(a*b^2*c + a^2*b*d)*e^3*f^2 - (3*a^2*b*c + a^3*d)*e^2*f^3)*h)* (f*x + e)^(5/2))/f^6
Leaf count of result is larger than twice the leaf count of optimal. 2956 vs. \(2 (317) = 634\).
Time = 0.16 (sec) , antiderivative size = 2956, normalized size of antiderivative = 8.67 \[ \int (a+b x)^3 (c+d x) (e+f x)^{3/2} (g+h x) \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^3*(d*x+c)*(f*x+e)^(3/2)*(h*x+g),x, algorithm="giac")
Output:
2/45045*(45045*sqrt(f*x + e)*a^3*c*e^2*g + 30030*((f*x + e)^(3/2) - 3*sqrt (f*x + e)*e)*a^3*c*e*g + 45045*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a^2*b *c*e^2*g/f + 15015*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a^3*d*e^2*g/f + 1 5015*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a^3*c*e^2*h/f + 3003*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a^3*c*g + 9009*(3* (f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a*b^2*c*e^2 *g/f^2 + 9009*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e) *e^2)*a^2*b*d*e^2*g/f^2 + 18018*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a^2*b*c*e*g/f + 6006*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a^3*d*e*g/f + 9009*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a^2*b*c*e^2*h/f^2 + 3003*( 3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a^3*d*e^2 *h/f^2 + 6006*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e) *e^2)*a^3*c*e*h/f + 1287*(5*(f*x + e)^(7/2) - 21*(f*x + e)^(5/2)*e + 35*(f *x + e)^(3/2)*e^2 - 35*sqrt(f*x + e)*e^3)*b^3*c*e^2*g/f^3 + 3861*(5*(f*x + e)^(7/2) - 21*(f*x + e)^(5/2)*e + 35*(f*x + e)^(3/2)*e^2 - 35*sqrt(f*x + e)*e^3)*a*b^2*d*e^2*g/f^3 + 7722*(5*(f*x + e)^(7/2) - 21*(f*x + e)^(5/2)*e + 35*(f*x + e)^(3/2)*e^2 - 35*sqrt(f*x + e)*e^3)*a*b^2*c*e*g/f^2 + 7722*( 5*(f*x + e)^(7/2) - 21*(f*x + e)^(5/2)*e + 35*(f*x + e)^(3/2)*e^2 - 35*sqr t(f*x + e)*e^3)*a^2*b*d*e*g/f^2 + 3861*(5*(f*x + e)^(7/2) - 21*(f*x + e...
Time = 0.12 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.07 \[ \int (a+b x)^3 (c+d x) (e+f x)^{3/2} (g+h x) \, dx=\frac {{\left (e+f\,x\right )}^{11/2}\,\left (2\,b^3\,c\,f^2\,g+20\,b^3\,d\,e^2\,h+6\,a\,b^2\,c\,f^2\,h+6\,a\,b^2\,d\,f^2\,g+6\,a^2\,b\,d\,f^2\,h-8\,b^3\,c\,e\,f\,h-8\,b^3\,d\,e\,f\,g-24\,a\,b^2\,d\,e\,f\,h\right )}{11\,f^6}+\frac {{\left (e+f\,x\right )}^{13/2}\,\left (2\,b^3\,c\,f\,h-10\,b^3\,d\,e\,h+2\,b^3\,d\,f\,g+6\,a\,b^2\,d\,f\,h\right )}{13\,f^6}+\frac {2\,{\left (e+f\,x\right )}^{9/2}\,\left (a\,f-b\,e\right )\,\left (3\,b^2\,c\,f^2\,g+a^2\,d\,f^2\,h+10\,b^2\,d\,e^2\,h+3\,a\,b\,c\,f^2\,h+3\,a\,b\,d\,f^2\,g-6\,b^2\,c\,e\,f\,h-6\,b^2\,d\,e\,f\,g-8\,a\,b\,d\,e\,f\,h\right )}{9\,f^6}+\frac {2\,{\left (e+f\,x\right )}^{7/2}\,{\left (a\,f-b\,e\right )}^2\,\left (a\,c\,f^2\,h+a\,d\,f^2\,g+3\,b\,c\,f^2\,g+5\,b\,d\,e^2\,h-2\,a\,d\,e\,f\,h-4\,b\,c\,e\,f\,h-4\,b\,d\,e\,f\,g\right )}{7\,f^6}+\frac {2\,b^3\,d\,h\,{\left (e+f\,x\right )}^{15/2}}{15\,f^6}-\frac {2\,{\left (e+f\,x\right )}^{5/2}\,{\left (a\,f-b\,e\right )}^3\,\left (c\,f-d\,e\right )\,\left (e\,h-f\,g\right )}{5\,f^6} \] Input:
int((e + f*x)^(3/2)*(g + h*x)*(a + b*x)^3*(c + d*x),x)
Output:
((e + f*x)^(11/2)*(2*b^3*c*f^2*g + 20*b^3*d*e^2*h + 6*a*b^2*c*f^2*h + 6*a* b^2*d*f^2*g + 6*a^2*b*d*f^2*h - 8*b^3*c*e*f*h - 8*b^3*d*e*f*g - 24*a*b^2*d *e*f*h))/(11*f^6) + ((e + f*x)^(13/2)*(2*b^3*c*f*h - 10*b^3*d*e*h + 2*b^3* d*f*g + 6*a*b^2*d*f*h))/(13*f^6) + (2*(e + f*x)^(9/2)*(a*f - b*e)*(3*b^2*c *f^2*g + a^2*d*f^2*h + 10*b^2*d*e^2*h + 3*a*b*c*f^2*h + 3*a*b*d*f^2*g - 6* b^2*c*e*f*h - 6*b^2*d*e*f*g - 8*a*b*d*e*f*h))/(9*f^6) + (2*(e + f*x)^(7/2) *(a*f - b*e)^2*(a*c*f^2*h + a*d*f^2*g + 3*b*c*f^2*g + 5*b*d*e^2*h - 2*a*d* e*f*h - 4*b*c*e*f*h - 4*b*d*e*f*g))/(7*f^6) + (2*b^3*d*h*(e + f*x)^(15/2)) /(15*f^6) - (2*(e + f*x)^(5/2)*(a*f - b*e)^3*(c*f - d*e)*(e*h - f*g))/(5*f ^6)
Time = 0.16 (sec) , antiderivative size = 1293, normalized size of antiderivative = 3.79 \[ \int (a+b x)^3 (c+d x) (e+f x)^{3/2} (g+h x) \, dx =\text {Too large to display} \] Input:
int((b*x+a)^3*(d*x+c)*(f*x+e)^(3/2)*(h*x+g),x)
Output:
(2*sqrt(e + f*x)*( - 2574*a**3*c*e**3*f**4*h + 9009*a**3*c*e**2*f**5*g + 1 287*a**3*c*e**2*f**5*h*x + 18018*a**3*c*e*f**6*g*x + 10296*a**3*c*e*f**6*h *x**2 + 9009*a**3*c*f**7*g*x**2 + 6435*a**3*c*f**7*h*x**3 + 1144*a**3*d*e* *4*f**3*h - 2574*a**3*d*e**3*f**4*g - 572*a**3*d*e**3*f**4*h*x + 1287*a**3 *d*e**2*f**5*g*x + 429*a**3*d*e**2*f**5*h*x**2 + 10296*a**3*d*e*f**6*g*x** 2 + 7150*a**3*d*e*f**6*h*x**3 + 6435*a**3*d*f**7*g*x**3 + 5005*a**3*d*f**7 *h*x**4 + 3432*a**2*b*c*e**4*f**3*h - 7722*a**2*b*c*e**3*f**4*g - 1716*a** 2*b*c*e**3*f**4*h*x + 3861*a**2*b*c*e**2*f**5*g*x + 1287*a**2*b*c*e**2*f** 5*h*x**2 + 30888*a**2*b*c*e*f**6*g*x**2 + 21450*a**2*b*c*e*f**6*h*x**3 + 1 9305*a**2*b*c*f**7*g*x**3 + 15015*a**2*b*c*f**7*h*x**4 - 1872*a**2*b*d*e** 5*f**2*h + 3432*a**2*b*d*e**4*f**3*g + 936*a**2*b*d*e**4*f**3*h*x - 1716*a **2*b*d*e**3*f**4*g*x - 702*a**2*b*d*e**3*f**4*h*x**2 + 1287*a**2*b*d*e**2 *f**5*g*x**2 + 585*a**2*b*d*e**2*f**5*h*x**3 + 21450*a**2*b*d*e*f**6*g*x** 3 + 16380*a**2*b*d*e*f**6*h*x**4 + 15015*a**2*b*d*f**7*g*x**4 + 12285*a**2 *b*d*f**7*h*x**5 - 1872*a*b**2*c*e**5*f**2*h + 3432*a*b**2*c*e**4*f**3*g + 936*a*b**2*c*e**4*f**3*h*x - 1716*a*b**2*c*e**3*f**4*g*x - 702*a*b**2*c*e **3*f**4*h*x**2 + 1287*a*b**2*c*e**2*f**5*g*x**2 + 585*a*b**2*c*e**2*f**5* h*x**3 + 21450*a*b**2*c*e*f**6*g*x**3 + 16380*a*b**2*c*e*f**6*h*x**4 + 150 15*a*b**2*c*f**7*g*x**4 + 12285*a*b**2*c*f**7*h*x**5 + 1152*a*b**2*d*e**6* f*h - 1872*a*b**2*d*e**5*f**2*g - 576*a*b**2*d*e**5*f**2*h*x + 936*a*b*...