Integrand size = 27, antiderivative size = 248 \[ \int (a+b x) (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=-\frac {2 (b e-a f) (d e-c f)^2 (f g-e h) (e+f x)^{5/2}}{5 f^5}+\frac {2 (d e-c f) (b d e (3 f g-4 e h)-b c f (f g-2 e h)-a f (2 d f g-3 d e h+c f h)) (e+f x)^{7/2}}{7 f^5}+\frac {2 \left (a d f (d f g-3 d e h+2 c f h)+b \left (c^2 f^2 h+2 c d f (f g-3 e h)-3 d^2 e (f g-2 e h)\right )\right ) (e+f x)^{9/2}}{9 f^5}+\frac {2 d (a d f h+b (d f g-4 d e h+2 c f h)) (e+f x)^{11/2}}{11 f^5}+\frac {2 b d^2 h (e+f x)^{13/2}}{13 f^5} \] Output:
-2/5*(-a*f+b*e)*(-c*f+d*e)^2*(-e*h+f*g)*(f*x+e)^(5/2)/f^5+2/7*(-c*f+d*e)*( b*d*e*(-4*e*h+3*f*g)-b*c*f*(-2*e*h+f*g)-a*f*(c*f*h-3*d*e*h+2*d*f*g))*(f*x+ e)^(7/2)/f^5+2/9*(a*d*f*(2*c*f*h-3*d*e*h+d*f*g)+b*(c^2*f^2*h+2*c*d*f*(-3*e *h+f*g)-3*d^2*e*(-2*e*h+f*g)))*(f*x+e)^(9/2)/f^5+2/11*d*(a*d*f*h+b*(2*c*f* h-4*d*e*h+d*f*g))*(f*x+e)^(11/2)/f^5+2/13*b*d^2*h*(f*x+e)^(13/2)/f^5
Time = 0.27 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.27 \[ \int (a+b x) (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\frac {2 (e+f x)^{5/2} \left (13 a f \left (99 c^2 f^2 (7 f g-2 e h+5 f h x)+22 c d f \left (8 e^2 h+5 f^2 x (9 g+7 h x)-2 e f (9 g+10 h x)\right )+d^2 \left (-48 e^3 h+35 f^3 x^2 (11 g+9 h x)+8 e^2 f (11 g+15 h x)-10 e f^2 x (22 g+21 h x)\right )\right )+b \left (143 c^2 f^2 \left (8 e^2 h+5 f^2 x (9 g+7 h x)-2 e f (9 g+10 h x)\right )+3 d^2 \left (128 e^4 h+105 f^4 x^3 (13 g+11 h x)-70 e f^3 x^2 (13 g+12 h x)+40 e^2 f^2 x (13 g+14 h x)-16 e^3 f (13 g+20 h x)\right )+26 c d f \left (-48 e^3 h+35 f^3 x^2 (11 g+9 h x)+8 e^2 f (11 g+15 h x)-10 e f^2 x (22 g+21 h x)\right )\right )\right )}{45045 f^5} \] Input:
Integrate[(a + b*x)*(c + d*x)^2*(e + f*x)^(3/2)*(g + h*x),x]
Output:
(2*(e + f*x)^(5/2)*(13*a*f*(99*c^2*f^2*(7*f*g - 2*e*h + 5*f*h*x) + 22*c*d* f*(8*e^2*h + 5*f^2*x*(9*g + 7*h*x) - 2*e*f*(9*g + 10*h*x)) + d^2*(-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))) + b*(143*c^2*f^2*(8*e^2*h + 5*f^2*x*(9*g + 7*h*x) - 2*e*f*(9* g + 10*h*x)) + 3*d^2*(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)) + 26*c*d*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^5)
Time = 0.47 (sec) , antiderivative size = 248, 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.
| \(\displaystyle \int (a+b x) (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx\) |
| \(\Big \downarrow \) 159 |
| \(\displaystyle \int \left (\frac {(e+f x)^{7/2} \left (a d f (2 c f h-3 d e h+d f g)+b \left (c^2 f^2 h+2 c d f (f g-3 e h)-3 d^2 e (f g-2 e h)\right )\right )}{f^4}+\frac {d (e+f x)^{9/2} (a d f h+b (2 c f h-4 d e h+d f g))}{f^4}+\frac {(e+f x)^{5/2} (d e-c f) (-a f (c f h-3 d e h+2 d f g)-b c f (f g-2 e h)+b d e (3 f g-4 e h))}{f^4}+\frac {(e+f x)^{3/2} (a f-b e) (c f-d e)^2 (f g-e h)}{f^4}+\frac {b d^2 h (e+f x)^{11/2}}{f^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (e+f x)^{9/2} \left (a d f (2 c f h-3 d e h+d f g)+b \left (c^2 f^2 h+2 c d f (f g-3 e h)-3 d^2 e (f g-2 e h)\right )\right )}{9 f^5}+\frac {2 d (e+f x)^{11/2} (a d f h+b (2 c f h-4 d e h+d f g))}{11 f^5}+\frac {2 (e+f x)^{7/2} (d e-c f) (-a f (c f h-3 d e h+2 d f g)-b c f (f g-2 e h)+b d e (3 f g-4 e h))}{7 f^5}-\frac {2 (e+f x)^{5/2} (b e-a f) (d e-c f)^2 (f g-e h)}{5 f^5}+\frac {2 b d^2 h (e+f x)^{13/2}}{13 f^5}\) |
Input:
Int[(a + b*x)*(c + d*x)^2*(e + f*x)^(3/2)*(g + h*x),x]
Output:
(-2*(b*e - a*f)*(d*e - c*f)^2*(f*g - e*h)*(e + f*x)^(5/2))/(5*f^5) + (2*(d *e - c*f)*(b*d*e*(3*f*g - 4*e*h) - b*c*f*(f*g - 2*e*h) - a*f*(2*d*f*g - 3* d*e*h + c*f*h))*(e + f*x)^(7/2))/(7*f^5) + (2*(a*d*f*(d*f*g - 3*d*e*h + 2* c*f*h) + b*(c^2*f^2*h + 2*c*d*f*(f*g - 3*e*h) - 3*d^2*e*(f*g - 2*e*h)))*(e + f*x)^(9/2))/(9*f^5) + (2*d*(a*d*f*h + b*(d*f*g - 4*d*e*h + 2*c*f*h))*(e + f*x)^(11/2))/(11*f^5) + (2*b*d^2*h*(e + f*x)^(13/2))/(13*f^5)
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 = 0.88 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {\frac {2 h b \,d^{2} \left (f x +e \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (\left (a f -b e \right ) d^{2}+2 b d \left (c f -d e \right )\right ) h +b \,d^{2} \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (2 \left (a f -b e \right ) d \left (c f -d e \right )+b \left (c f -d e \right )^{2}\right ) h +\left (\left (a f -b e \right ) d^{2}+2 b d \left (c f -d e \right )\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a f -b e \right ) \left (c f -d e \right )^{2} h +\left (2 \left (a f -b e \right ) d \left (c f -d e \right )+b \left (c f -d e \right )^{2}\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a f -b e \right ) \left (c f -d e \right )^{2} \left (-e h +f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{5}}{f^{5}}\) | \(255\) |
| default | \(\frac {\frac {2 h b \,d^{2} \left (f x +e \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-\left (\left (-a f +b e \right ) d^{2}-2 b d \left (c f -d e \right )\right ) h -b \,d^{2} \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-\left (2 \left (-a f +b e \right ) d \left (c f -d e \right )-b \left (c f -d e \right )^{2}\right ) h +\left (\left (-a f +b e \right ) d^{2}-2 b d \left (c f -d e \right )\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-\left (-a f +b e \right ) \left (c f -d e \right )^{2} h +\left (2 \left (-a f +b e \right ) d \left (c f -d e \right )-b \left (c f -d e \right )^{2}\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-a f +b e \right ) \left (c f -d e \right )^{2} \left (e h -f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{5}}{f^{5}}\) | \(261\) |
| pseudoelliptic | \(-\frac {4 \left (f x +e \right )^{\frac {5}{2}} \left (\left (-\frac {35 \left (\frac {9 b h \,x^{2}}{13}+\frac {9 \left (a h +b g \right ) x}{11}+g a \right ) x^{2} d^{2}}{18}-5 \left (\frac {7 b h \,x^{2}}{11}+\frac {7 \left (a h +b g \right ) x}{9}+g a \right ) x c d -\frac {7 c^{2} \left (\frac {5 b h \,x^{2}}{9}+\frac {5 \left (a h +b g \right ) x}{7}+g a \right )}{2}\right ) f^{4}+e \left (\frac {10 x \left (\frac {126 b h \,x^{2}}{143}+\frac {21 \left (a h +b g \right ) x}{22}+g a \right ) d^{2}}{9}+2 c \left (\frac {35 b h \,x^{2}}{33}+\frac {10 \left (a h +b g \right ) x}{9}+g a \right ) d +c^{2} \left (a h +b g +\frac {10}{9} b h x \right )\right ) f^{3}-\frac {8 \left (\left (\frac {105 b h \,x^{2}}{143}+\frac {15 \left (a h +b g \right ) x}{22}+\frac {g a}{2}\right ) d^{2}+c \left (\frac {15}{11} b h x +a h +b g \right ) d +\frac {h b \,c^{2}}{2}\right ) e^{2} f^{2}}{9}+\frac {8 d \left (\left (\frac {20}{13} b h x +a h +b g \right ) d +2 b c h \right ) e^{3} f}{33}-\frac {64 b \,d^{2} e^{4} h}{429}\right )}{35 f^{5}}\) | \(268\) |
| gosper | \(-\frac {2 \left (f x +e \right )^{\frac {5}{2}} \left (-3465 h b \,d^{2} x^{4} f^{4}-4095 a \,d^{2} f^{4} h \,x^{3}-8190 b c d \,f^{4} h \,x^{3}+2520 b \,d^{2} e \,f^{3} h \,x^{3}-4095 b \,d^{2} f^{4} g \,x^{3}-10010 a c d \,f^{4} h \,x^{2}+2730 a \,d^{2} e \,f^{3} h \,x^{2}-5005 a \,d^{2} f^{4} g \,x^{2}-5005 b \,c^{2} f^{4} h \,x^{2}+5460 b c d e \,f^{3} h \,x^{2}-10010 b c d \,f^{4} g \,x^{2}-1680 b \,d^{2} e^{2} f^{2} h \,x^{2}+2730 b \,d^{2} e \,f^{3} g \,x^{2}-6435 a \,c^{2} f^{4} h x +5720 a c d e \,f^{3} h x -12870 a c d \,f^{4} g x -1560 a \,d^{2} e^{2} f^{2} h x +2860 a \,d^{2} e \,f^{3} g x +2860 b \,c^{2} e \,f^{3} h x -6435 b \,c^{2} f^{4} g x -3120 b c d \,e^{2} f^{2} h x +5720 b c d e \,f^{3} g x +960 b \,d^{2} e^{3} f h x -1560 b \,d^{2} e^{2} f^{2} g x +2574 a \,c^{2} e \,f^{3} h -9009 g a \,c^{2} f^{4}-2288 a c d \,e^{2} f^{2} h +5148 a c d e \,f^{3} g +624 a \,d^{2} e^{3} f h -1144 a \,d^{2} e^{2} f^{2} g -1144 b \,c^{2} e^{2} f^{2} h +2574 b \,c^{2} e \,f^{3} g +1248 b c d \,e^{3} f h -2288 b c d \,e^{2} f^{2} g -384 b \,d^{2} e^{4} h +624 b \,d^{2} e^{3} f g \right )}{45045 f^{5}}\) | \(451\) |
| orering | \(-\frac {2 \left (f x +e \right )^{\frac {5}{2}} \left (-3465 h b \,d^{2} x^{4} f^{4}-4095 a \,d^{2} f^{4} h \,x^{3}-8190 b c d \,f^{4} h \,x^{3}+2520 b \,d^{2} e \,f^{3} h \,x^{3}-4095 b \,d^{2} f^{4} g \,x^{3}-10010 a c d \,f^{4} h \,x^{2}+2730 a \,d^{2} e \,f^{3} h \,x^{2}-5005 a \,d^{2} f^{4} g \,x^{2}-5005 b \,c^{2} f^{4} h \,x^{2}+5460 b c d e \,f^{3} h \,x^{2}-10010 b c d \,f^{4} g \,x^{2}-1680 b \,d^{2} e^{2} f^{2} h \,x^{2}+2730 b \,d^{2} e \,f^{3} g \,x^{2}-6435 a \,c^{2} f^{4} h x +5720 a c d e \,f^{3} h x -12870 a c d \,f^{4} g x -1560 a \,d^{2} e^{2} f^{2} h x +2860 a \,d^{2} e \,f^{3} g x +2860 b \,c^{2} e \,f^{3} h x -6435 b \,c^{2} f^{4} g x -3120 b c d \,e^{2} f^{2} h x +5720 b c d e \,f^{3} g x +960 b \,d^{2} e^{3} f h x -1560 b \,d^{2} e^{2} f^{2} g x +2574 a \,c^{2} e \,f^{3} h -9009 g a \,c^{2} f^{4}-2288 a c d \,e^{2} f^{2} h +5148 a c d e \,f^{3} g +624 a \,d^{2} e^{3} f h -1144 a \,d^{2} e^{2} f^{2} g -1144 b \,c^{2} e^{2} f^{2} h +2574 b \,c^{2} e \,f^{3} g +1248 b c d \,e^{3} f h -2288 b c d \,e^{2} f^{2} g -384 b \,d^{2} e^{4} h +624 b \,d^{2} e^{3} f g \right )}{45045 f^{5}}\) | \(451\) |
| trager | \(-\frac {2 \left (-3465 b \,d^{2} f^{6} h \,x^{6}-4095 a \,d^{2} f^{6} h \,x^{5}-8190 b c d \,f^{6} h \,x^{5}-4410 b \,d^{2} e \,f^{5} h \,x^{5}-4095 b \,d^{2} f^{6} g \,x^{5}-10010 a c d \,f^{6} h \,x^{4}-5460 a \,d^{2} e \,f^{5} h \,x^{4}-5005 a \,d^{2} f^{6} g \,x^{4}-5005 b \,c^{2} f^{6} h \,x^{4}-10920 b c d e \,f^{5} h \,x^{4}-10010 b c d \,f^{6} g \,x^{4}-105 b \,d^{2} e^{2} f^{4} h \,x^{4}-5460 b \,d^{2} e \,f^{5} g \,x^{4}-6435 a \,c^{2} f^{6} h \,x^{3}-14300 a c d e \,f^{5} h \,x^{3}-12870 a c d \,f^{6} g \,x^{3}-195 a \,d^{2} e^{2} f^{4} h \,x^{3}-7150 a \,d^{2} e \,f^{5} g \,x^{3}-7150 b \,c^{2} e \,f^{5} h \,x^{3}-6435 b \,c^{2} f^{6} g \,x^{3}-390 b c d \,e^{2} f^{4} h \,x^{3}-14300 b c d e \,f^{5} g \,x^{3}+120 b \,d^{2} e^{3} f^{3} h \,x^{3}-195 b \,d^{2} e^{2} f^{4} g \,x^{3}-10296 a \,c^{2} e \,f^{5} h \,x^{2}-9009 a \,c^{2} f^{6} g \,x^{2}-858 a c d \,e^{2} f^{4} h \,x^{2}-20592 a c d e \,f^{5} g \,x^{2}+234 a \,d^{2} e^{3} f^{3} h \,x^{2}-429 a \,d^{2} e^{2} f^{4} g \,x^{2}-429 b \,c^{2} e^{2} f^{4} h \,x^{2}-10296 b \,c^{2} e \,f^{5} g \,x^{2}+468 b c d \,e^{3} f^{3} h \,x^{2}-858 b c d \,e^{2} f^{4} g \,x^{2}-144 b \,d^{2} e^{4} f^{2} h \,x^{2}+234 b \,d^{2} e^{3} f^{3} g \,x^{2}-1287 a \,c^{2} e^{2} f^{4} h x -18018 a \,c^{2} e \,f^{5} g x +1144 a c d \,e^{3} f^{3} h x -2574 a c d \,e^{2} f^{4} g x -312 a \,d^{2} e^{4} f^{2} h x +572 a \,d^{2} e^{3} f^{3} g x +572 b \,c^{2} e^{3} f^{3} h x -1287 b \,c^{2} e^{2} f^{4} g x -624 b c d \,e^{4} f^{2} h x +1144 b c d \,e^{3} f^{3} g x +192 b \,d^{2} e^{5} f h x -312 b \,d^{2} e^{4} f^{2} g x +2574 a \,c^{2} e^{3} f^{3} h -9009 a \,c^{2} e^{2} f^{4} g -2288 a c d \,e^{4} f^{2} h +5148 a c d \,e^{3} f^{3} g +624 a \,d^{2} e^{5} f h -1144 a \,d^{2} e^{4} f^{2} g -1144 b \,c^{2} e^{4} f^{2} h +2574 b \,c^{2} e^{3} f^{3} g +1248 b c d \,e^{5} f h -2288 b c d \,e^{4} f^{2} g -384 b \,d^{2} e^{6} h +624 b \,d^{2} e^{5} f g \right ) \sqrt {f x +e}}{45045 f^{5}}\) | \(823\) |
| risch | \(-\frac {2 \left (-3465 b \,d^{2} f^{6} h \,x^{6}-4095 a \,d^{2} f^{6} h \,x^{5}-8190 b c d \,f^{6} h \,x^{5}-4410 b \,d^{2} e \,f^{5} h \,x^{5}-4095 b \,d^{2} f^{6} g \,x^{5}-10010 a c d \,f^{6} h \,x^{4}-5460 a \,d^{2} e \,f^{5} h \,x^{4}-5005 a \,d^{2} f^{6} g \,x^{4}-5005 b \,c^{2} f^{6} h \,x^{4}-10920 b c d e \,f^{5} h \,x^{4}-10010 b c d \,f^{6} g \,x^{4}-105 b \,d^{2} e^{2} f^{4} h \,x^{4}-5460 b \,d^{2} e \,f^{5} g \,x^{4}-6435 a \,c^{2} f^{6} h \,x^{3}-14300 a c d e \,f^{5} h \,x^{3}-12870 a c d \,f^{6} g \,x^{3}-195 a \,d^{2} e^{2} f^{4} h \,x^{3}-7150 a \,d^{2} e \,f^{5} g \,x^{3}-7150 b \,c^{2} e \,f^{5} h \,x^{3}-6435 b \,c^{2} f^{6} g \,x^{3}-390 b c d \,e^{2} f^{4} h \,x^{3}-14300 b c d e \,f^{5} g \,x^{3}+120 b \,d^{2} e^{3} f^{3} h \,x^{3}-195 b \,d^{2} e^{2} f^{4} g \,x^{3}-10296 a \,c^{2} e \,f^{5} h \,x^{2}-9009 a \,c^{2} f^{6} g \,x^{2}-858 a c d \,e^{2} f^{4} h \,x^{2}-20592 a c d e \,f^{5} g \,x^{2}+234 a \,d^{2} e^{3} f^{3} h \,x^{2}-429 a \,d^{2} e^{2} f^{4} g \,x^{2}-429 b \,c^{2} e^{2} f^{4} h \,x^{2}-10296 b \,c^{2} e \,f^{5} g \,x^{2}+468 b c d \,e^{3} f^{3} h \,x^{2}-858 b c d \,e^{2} f^{4} g \,x^{2}-144 b \,d^{2} e^{4} f^{2} h \,x^{2}+234 b \,d^{2} e^{3} f^{3} g \,x^{2}-1287 a \,c^{2} e^{2} f^{4} h x -18018 a \,c^{2} e \,f^{5} g x +1144 a c d \,e^{3} f^{3} h x -2574 a c d \,e^{2} f^{4} g x -312 a \,d^{2} e^{4} f^{2} h x +572 a \,d^{2} e^{3} f^{3} g x +572 b \,c^{2} e^{3} f^{3} h x -1287 b \,c^{2} e^{2} f^{4} g x -624 b c d \,e^{4} f^{2} h x +1144 b c d \,e^{3} f^{3} g x +192 b \,d^{2} e^{5} f h x -312 b \,d^{2} e^{4} f^{2} g x +2574 a \,c^{2} e^{3} f^{3} h -9009 a \,c^{2} e^{2} f^{4} g -2288 a c d \,e^{4} f^{2} h +5148 a c d \,e^{3} f^{3} g +624 a \,d^{2} e^{5} f h -1144 a \,d^{2} e^{4} f^{2} g -1144 b \,c^{2} e^{4} f^{2} h +2574 b \,c^{2} e^{3} f^{3} g +1248 b c d \,e^{5} f h -2288 b c d \,e^{4} f^{2} g -384 b \,d^{2} e^{6} h +624 b \,d^{2} e^{5} f g \right ) \sqrt {f x +e}}{45045 f^{5}}\) | \(823\) |
Input:
int((b*x+a)*(d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x,method=_RETURNVERBOSE)
Output:
2/f^5*(1/13*h*b*d^2*(f*x+e)^(13/2)+1/11*(((a*f-b*e)*d^2+2*b*d*(c*f-d*e))*h +b*d^2*(-e*h+f*g))*(f*x+e)^(11/2)+1/9*((2*(a*f-b*e)*d*(c*f-d*e)+b*(c*f-d*e )^2)*h+((a*f-b*e)*d^2+2*b*d*(c*f-d*e))*(-e*h+f*g))*(f*x+e)^(9/2)+1/7*((a*f -b*e)*(c*f-d*e)^2*h+(2*(a*f-b*e)*d*(c*f-d*e)+b*(c*f-d*e)^2)*(-e*h+f*g))*(f *x+e)^(7/2)+1/5*(a*f-b*e)*(c*f-d*e)^2*(-e*h+f*g)*(f*x+e)^(5/2))
Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (228) = 456\).
Time = 0.08 (sec) , antiderivative size = 658, normalized size of antiderivative = 2.65 \[ \int (a+b x) (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)*(d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x, algorithm="fricas")
Output:
2/45045*(3465*b*d^2*f^6*h*x^6 + 315*(13*b*d^2*f^6*g + (14*b*d^2*e*f^5 + 13 *(2*b*c*d + a*d^2)*f^6)*h)*x^5 + 35*(13*(12*b*d^2*e*f^5 + 11*(2*b*c*d + a* d^2)*f^6)*g + (3*b*d^2*e^2*f^4 + 156*(2*b*c*d + a*d^2)*e*f^5 + 143*(b*c^2 + 2*a*c*d)*f^6)*h)*x^4 + 5*(13*(3*b*d^2*e^2*f^4 + 110*(2*b*c*d + a*d^2)*e* f^5 + 99*(b*c^2 + 2*a*c*d)*f^6)*g - (24*b*d^2*e^3*f^3 - 1287*a*c^2*f^6 - 3 9*(2*b*c*d + a*d^2)*e^2*f^4 - 1430*(b*c^2 + 2*a*c*d)*e*f^5)*h)*x^3 - 3*(13 *(6*b*d^2*e^3*f^3 - 231*a*c^2*f^6 - 11*(2*b*c*d + a*d^2)*e^2*f^4 - 264*(b* c^2 + 2*a*c*d)*e*f^5)*g - (48*b*d^2*e^4*f^2 + 3432*a*c^2*e*f^5 - 78*(2*b*c *d + a*d^2)*e^3*f^3 + 143*(b*c^2 + 2*a*c*d)*e^2*f^4)*h)*x^2 - 13*(48*b*d^2 *e^5*f - 693*a*c^2*e^2*f^4 - 88*(2*b*c*d + a*d^2)*e^4*f^2 + 198*(b*c^2 + 2 *a*c*d)*e^3*f^3)*g + 2*(192*b*d^2*e^6 - 1287*a*c^2*e^3*f^3 - 312*(2*b*c*d + a*d^2)*e^5*f + 572*(b*c^2 + 2*a*c*d)*e^4*f^2)*h + (13*(24*b*d^2*e^4*f^2 + 1386*a*c^2*e*f^5 - 44*(2*b*c*d + a*d^2)*e^3*f^3 + 99*(b*c^2 + 2*a*c*d)*e ^2*f^4)*g - (192*b*d^2*e^5*f - 1287*a*c^2*e^2*f^4 - 312*(2*b*c*d + a*d^2)* e^4*f^2 + 572*(b*c^2 + 2*a*c*d)*e^3*f^3)*h)*x)*sqrt(f*x + e)/f^5
Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (265) = 530\).
Time = 1.70 (sec) , antiderivative size = 620, normalized size of antiderivative = 2.50 \[ \int (a+b x) (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\begin {cases} \frac {2 \left (\frac {b d^{2} h \left (e + f x\right )^{\frac {13}{2}}}{13 f^{4}} + \frac {\left (e + f x\right )^{\frac {11}{2}} \left (a d^{2} f h + 2 b c d f h - 4 b d^{2} e h + b d^{2} f g\right )}{11 f^{4}} + \frac {\left (e + f x\right )^{\frac {9}{2}} \cdot \left (2 a c d f^{2} h - 3 a d^{2} e f h + a d^{2} f^{2} g + b c^{2} f^{2} h - 6 b c d e f h + 2 b c d f^{2} g + 6 b d^{2} e^{2} h - 3 b d^{2} e f g\right )}{9 f^{4}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \left (a c^{2} f^{3} h - 4 a c d e f^{2} h + 2 a c d f^{3} g + 3 a d^{2} e^{2} f h - 2 a d^{2} e f^{2} g - 2 b c^{2} e f^{2} h + b c^{2} f^{3} g + 6 b c d e^{2} f h - 4 b c d e f^{2} g - 4 b d^{2} e^{3} h + 3 b d^{2} e^{2} f g\right )}{7 f^{4}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (- a c^{2} e f^{3} h + a c^{2} f^{4} g + 2 a c d e^{2} f^{2} h - 2 a c d e f^{3} g - a d^{2} e^{3} f h + a d^{2} e^{2} f^{2} g + b c^{2} e^{2} f^{2} h - b c^{2} e f^{3} g - 2 b c d e^{3} f h + 2 b c d e^{2} f^{2} g + b d^{2} e^{4} h - b d^{2} e^{3} f g\right )}{5 f^{4}}\right )}{f} & \text {for}\: f \neq 0 \\e^{\frac {3}{2}} \left (a c^{2} g x + \frac {b d^{2} h x^{5}}{5} + \frac {x^{4} \left (a d^{2} h + 2 b c d h + b d^{2} g\right )}{4} + \frac {x^{3} \cdot \left (2 a c d h + a d^{2} g + b c^{2} h + 2 b c d g\right )}{3} + \frac {x^{2} \left (a c^{2} h + 2 a c d g + b c^{2} g\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)*(d*x+c)**2*(f*x+e)**(3/2)*(h*x+g),x)
Output:
Piecewise((2*(b*d**2*h*(e + f*x)**(13/2)/(13*f**4) + (e + f*x)**(11/2)*(a* d**2*f*h + 2*b*c*d*f*h - 4*b*d**2*e*h + b*d**2*f*g)/(11*f**4) + (e + f*x)* *(9/2)*(2*a*c*d*f**2*h - 3*a*d**2*e*f*h + a*d**2*f**2*g + b*c**2*f**2*h - 6*b*c*d*e*f*h + 2*b*c*d*f**2*g + 6*b*d**2*e**2*h - 3*b*d**2*e*f*g)/(9*f**4 ) + (e + f*x)**(7/2)*(a*c**2*f**3*h - 4*a*c*d*e*f**2*h + 2*a*c*d*f**3*g + 3*a*d**2*e**2*f*h - 2*a*d**2*e*f**2*g - 2*b*c**2*e*f**2*h + b*c**2*f**3*g + 6*b*c*d*e**2*f*h - 4*b*c*d*e*f**2*g - 4*b*d**2*e**3*h + 3*b*d**2*e**2*f* g)/(7*f**4) + (e + f*x)**(5/2)*(-a*c**2*e*f**3*h + a*c**2*f**4*g + 2*a*c*d *e**2*f**2*h - 2*a*c*d*e*f**3*g - a*d**2*e**3*f*h + a*d**2*e**2*f**2*g + b *c**2*e**2*f**2*h - b*c**2*e*f**3*g - 2*b*c*d*e**3*f*h + 2*b*c*d*e**2*f**2 *g + b*d**2*e**4*h - b*d**2*e**3*f*g)/(5*f**4))/f, Ne(f, 0)), (e**(3/2)*(a *c**2*g*x + b*d**2*h*x**5/5 + x**4*(a*d**2*h + 2*b*c*d*h + b*d**2*g)/4 + x **3*(2*a*c*d*h + a*d**2*g + b*c**2*h + 2*b*c*d*g)/3 + x**2*(a*c**2*h + 2*a *c*d*g + b*c**2*g)/2), True))
Time = 0.05 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.52 \[ \int (a+b x) (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\frac {2 \, {\left (3465 \, {\left (f x + e\right )}^{\frac {13}{2}} b d^{2} h + 4095 \, {\left (b d^{2} f g - {\left (4 \, b d^{2} e - {\left (2 \, b c d + a d^{2}\right )} f\right )} h\right )} {\left (f x + e\right )}^{\frac {11}{2}} - 5005 \, {\left ({\left (3 \, b d^{2} e f - {\left (2 \, b c d + a d^{2}\right )} f^{2}\right )} g - {\left (6 \, b d^{2} e^{2} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e f + {\left (b c^{2} + 2 \, a c d\right )} f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {9}{2}} + 6435 \, {\left ({\left (3 \, b d^{2} e^{2} f - 2 \, {\left (2 \, b c d + a d^{2}\right )} e f^{2} + {\left (b c^{2} + 2 \, a c d\right )} f^{3}\right )} g - {\left (4 \, b d^{2} e^{3} - a c^{2} f^{3} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f + 2 \, {\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {7}{2}} - 9009 \, {\left ({\left (b d^{2} e^{3} f - a c^{2} f^{4} - {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{2} + {\left (b c^{2} + 2 \, a c d\right )} e f^{3}\right )} g - {\left (b d^{2} e^{4} - a c^{2} e f^{3} - {\left (2 \, b c d + a d^{2}\right )} e^{3} f + {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {5}{2}}\right )}}{45045 \, f^{5}} \] Input:
integrate((b*x+a)*(d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x, algorithm="maxima")
Output:
2/45045*(3465*(f*x + e)^(13/2)*b*d^2*h + 4095*(b*d^2*f*g - (4*b*d^2*e - (2 *b*c*d + a*d^2)*f)*h)*(f*x + e)^(11/2) - 5005*((3*b*d^2*e*f - (2*b*c*d + a *d^2)*f^2)*g - (6*b*d^2*e^2 - 3*(2*b*c*d + a*d^2)*e*f + (b*c^2 + 2*a*c*d)* f^2)*h)*(f*x + e)^(9/2) + 6435*((3*b*d^2*e^2*f - 2*(2*b*c*d + a*d^2)*e*f^2 + (b*c^2 + 2*a*c*d)*f^3)*g - (4*b*d^2*e^3 - a*c^2*f^3 - 3*(2*b*c*d + a*d^ 2)*e^2*f + 2*(b*c^2 + 2*a*c*d)*e*f^2)*h)*(f*x + e)^(7/2) - 9009*((b*d^2*e^ 3*f - a*c^2*f^4 - (2*b*c*d + a*d^2)*e^2*f^2 + (b*c^2 + 2*a*c*d)*e*f^3)*g - (b*d^2*e^4 - a*c^2*e*f^3 - (2*b*c*d + a*d^2)*e^3*f + (b*c^2 + 2*a*c*d)*e^ 2*f^2)*h)*(f*x + e)^(5/2))/f^5
Leaf count of result is larger than twice the leaf count of optimal. 1968 vs. \(2 (228) = 456\).
Time = 0.14 (sec) , antiderivative size = 1968, normalized size of antiderivative = 7.94 \[ \int (a+b x) (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*(d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x, algorithm="giac")
Output:
2/45045*(45045*sqrt(f*x + e)*a*c^2*e^2*g + 30030*((f*x + e)^(3/2) - 3*sqrt (f*x + e)*e)*a*c^2*e*g + 15015*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*b*c^2 *e^2*g/f + 30030*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a*c*d*e^2*g/f + 150 15*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a*c^2*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*c^2*g + 6006*(3*(f *x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*b*c*d*e^2*g/f ^2 + 3003*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2 )*a*d^2*e^2*g/f^2 + 6006*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sq rt(f*x + e)*e^2)*b*c^2*e*g/f + 12012*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/ 2)*e + 15*sqrt(f*x + e)*e^2)*a*c*d*e*g/f + 3003*(3*(f*x + e)^(5/2) - 10*(f *x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*b*c^2*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*c*d*e^2*h/f^2 + 6 006*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a*c^ 2*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*d^2*e^2*g/f^3 + 5148*(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* c*d*e*g/f^2 + 2574*(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*d^2*e*g/f^2 + 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*c^2*g/f + 2574*(5*(f*x + e)^(7/2) - 21*(f*x + e)^(5/2)*e + 35*(f*x + ...
Time = 0.07 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.04 \[ \int (a+b x) (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\frac {{\left (e+f\,x\right )}^{11/2}\,\left (2\,a\,d^2\,f\,h-8\,b\,d^2\,e\,h+2\,b\,d^2\,f\,g+4\,b\,c\,d\,f\,h\right )}{11\,f^5}+\frac {{\left (e+f\,x\right )}^{9/2}\,\left (2\,a\,d^2\,f^2\,g+2\,b\,c^2\,f^2\,h+12\,b\,d^2\,e^2\,h+4\,a\,c\,d\,f^2\,h+4\,b\,c\,d\,f^2\,g-6\,a\,d^2\,e\,f\,h-6\,b\,d^2\,e\,f\,g-12\,b\,c\,d\,e\,f\,h\right )}{9\,f^5}+\frac {2\,{\left (e+f\,x\right )}^{7/2}\,\left (c\,f-d\,e\right )\,\left (a\,c\,f^2\,h+2\,a\,d\,f^2\,g+b\,c\,f^2\,g+4\,b\,d\,e^2\,h-3\,a\,d\,e\,f\,h-2\,b\,c\,e\,f\,h-3\,b\,d\,e\,f\,g\right )}{7\,f^5}+\frac {2\,b\,d^2\,h\,{\left (e+f\,x\right )}^{13/2}}{13\,f^5}-\frac {2\,{\left (e+f\,x\right )}^{5/2}\,\left (a\,f-b\,e\right )\,{\left (c\,f-d\,e\right )}^2\,\left (e\,h-f\,g\right )}{5\,f^5} \] Input:
int((e + f*x)^(3/2)*(g + h*x)*(a + b*x)*(c + d*x)^2,x)
Output:
((e + f*x)^(11/2)*(2*a*d^2*f*h - 8*b*d^2*e*h + 2*b*d^2*f*g + 4*b*c*d*f*h)) /(11*f^5) + ((e + f*x)^(9/2)*(2*a*d^2*f^2*g + 2*b*c^2*f^2*h + 12*b*d^2*e^2 *h + 4*a*c*d*f^2*h + 4*b*c*d*f^2*g - 6*a*d^2*e*f*h - 6*b*d^2*e*f*g - 12*b* c*d*e*f*h))/(9*f^5) + (2*(e + f*x)^(7/2)*(c*f - d*e)*(a*c*f^2*h + 2*a*d*f^ 2*g + b*c*f^2*g + 4*b*d*e^2*h - 3*a*d*e*f*h - 2*b*c*e*f*h - 3*b*d*e*f*g))/ (7*f^5) + (2*b*d^2*h*(e + f*x)^(13/2))/(13*f^5) - (2*(e + f*x)^(5/2)*(a*f - b*e)*(c*f - d*e)^2*(e*h - f*g))/(5*f^5)
Time = 0.17 (sec) , antiderivative size = 821, normalized size of antiderivative = 3.31 \[ \int (a+b x) (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\frac {2 \sqrt {f x +e}\, \left (3465 b \,d^{2} f^{6} h \,x^{6}+4095 a \,d^{2} f^{6} h \,x^{5}+8190 b c d \,f^{6} h \,x^{5}+4410 b \,d^{2} e \,f^{5} h \,x^{5}+4095 b \,d^{2} f^{6} g \,x^{5}+10010 a c d \,f^{6} h \,x^{4}+5460 a \,d^{2} e \,f^{5} h \,x^{4}+5005 a \,d^{2} f^{6} g \,x^{4}+5005 b \,c^{2} f^{6} h \,x^{4}+10920 b c d e \,f^{5} h \,x^{4}+10010 b c d \,f^{6} g \,x^{4}+105 b \,d^{2} e^{2} f^{4} h \,x^{4}+5460 b \,d^{2} e \,f^{5} g \,x^{4}+6435 a \,c^{2} f^{6} h \,x^{3}+14300 a c d e \,f^{5} h \,x^{3}+12870 a c d \,f^{6} g \,x^{3}+195 a \,d^{2} e^{2} f^{4} h \,x^{3}+7150 a \,d^{2} e \,f^{5} g \,x^{3}+7150 b \,c^{2} e \,f^{5} h \,x^{3}+6435 b \,c^{2} f^{6} g \,x^{3}+390 b c d \,e^{2} f^{4} h \,x^{3}+14300 b c d e \,f^{5} g \,x^{3}-120 b \,d^{2} e^{3} f^{3} h \,x^{3}+195 b \,d^{2} e^{2} f^{4} g \,x^{3}+10296 a \,c^{2} e \,f^{5} h \,x^{2}+9009 a \,c^{2} f^{6} g \,x^{2}+858 a c d \,e^{2} f^{4} h \,x^{2}+20592 a c d e \,f^{5} g \,x^{2}-234 a \,d^{2} e^{3} f^{3} h \,x^{2}+429 a \,d^{2} e^{2} f^{4} g \,x^{2}+429 b \,c^{2} e^{2} f^{4} h \,x^{2}+10296 b \,c^{2} e \,f^{5} g \,x^{2}-468 b c d \,e^{3} f^{3} h \,x^{2}+858 b c d \,e^{2} f^{4} g \,x^{2}+144 b \,d^{2} e^{4} f^{2} h \,x^{2}-234 b \,d^{2} e^{3} f^{3} g \,x^{2}+1287 a \,c^{2} e^{2} f^{4} h x +18018 a \,c^{2} e \,f^{5} g x -1144 a c d \,e^{3} f^{3} h x +2574 a c d \,e^{2} f^{4} g x +312 a \,d^{2} e^{4} f^{2} h x -572 a \,d^{2} e^{3} f^{3} g x -572 b \,c^{2} e^{3} f^{3} h x +1287 b \,c^{2} e^{2} f^{4} g x +624 b c d \,e^{4} f^{2} h x -1144 b c d \,e^{3} f^{3} g x -192 b \,d^{2} e^{5} f h x +312 b \,d^{2} e^{4} f^{2} g x -2574 a \,c^{2} e^{3} f^{3} h +9009 a \,c^{2} e^{2} f^{4} g +2288 a c d \,e^{4} f^{2} h -5148 a c d \,e^{3} f^{3} g -624 a \,d^{2} e^{5} f h +1144 a \,d^{2} e^{4} f^{2} g +1144 b \,c^{2} e^{4} f^{2} h -2574 b \,c^{2} e^{3} f^{3} g -1248 b c d \,e^{5} f h +2288 b c d \,e^{4} f^{2} g +384 b \,d^{2} e^{6} h -624 b \,d^{2} e^{5} f g \right )}{45045 f^{5}} \] Input:
int((b*x+a)*(d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x)
Output:
(2*sqrt(e + f*x)*( - 2574*a*c**2*e**3*f**3*h + 9009*a*c**2*e**2*f**4*g + 1 287*a*c**2*e**2*f**4*h*x + 18018*a*c**2*e*f**5*g*x + 10296*a*c**2*e*f**5*h *x**2 + 9009*a*c**2*f**6*g*x**2 + 6435*a*c**2*f**6*h*x**3 + 2288*a*c*d*e** 4*f**2*h - 5148*a*c*d*e**3*f**3*g - 1144*a*c*d*e**3*f**3*h*x + 2574*a*c*d* e**2*f**4*g*x + 858*a*c*d*e**2*f**4*h*x**2 + 20592*a*c*d*e*f**5*g*x**2 + 1 4300*a*c*d*e*f**5*h*x**3 + 12870*a*c*d*f**6*g*x**3 + 10010*a*c*d*f**6*h*x* *4 - 624*a*d**2*e**5*f*h + 1144*a*d**2*e**4*f**2*g + 312*a*d**2*e**4*f**2* h*x - 572*a*d**2*e**3*f**3*g*x - 234*a*d**2*e**3*f**3*h*x**2 + 429*a*d**2* e**2*f**4*g*x**2 + 195*a*d**2*e**2*f**4*h*x**3 + 7150*a*d**2*e*f**5*g*x**3 + 5460*a*d**2*e*f**5*h*x**4 + 5005*a*d**2*f**6*g*x**4 + 4095*a*d**2*f**6* h*x**5 + 1144*b*c**2*e**4*f**2*h - 2574*b*c**2*e**3*f**3*g - 572*b*c**2*e* *3*f**3*h*x + 1287*b*c**2*e**2*f**4*g*x + 429*b*c**2*e**2*f**4*h*x**2 + 10 296*b*c**2*e*f**5*g*x**2 + 7150*b*c**2*e*f**5*h*x**3 + 6435*b*c**2*f**6*g* x**3 + 5005*b*c**2*f**6*h*x**4 - 1248*b*c*d*e**5*f*h + 2288*b*c*d*e**4*f** 2*g + 624*b*c*d*e**4*f**2*h*x - 1144*b*c*d*e**3*f**3*g*x - 468*b*c*d*e**3* f**3*h*x**2 + 858*b*c*d*e**2*f**4*g*x**2 + 390*b*c*d*e**2*f**4*h*x**3 + 14 300*b*c*d*e*f**5*g*x**3 + 10920*b*c*d*e*f**5*h*x**4 + 10010*b*c*d*f**6*g*x **4 + 8190*b*c*d*f**6*h*x**5 + 384*b*d**2*e**6*h - 624*b*d**2*e**5*f*g - 1 92*b*d**2*e**5*f*h*x + 312*b*d**2*e**4*f**2*g*x + 144*b*d**2*e**4*f**2*h*x **2 - 234*b*d**2*e**3*f**3*g*x**2 - 120*b*d**2*e**3*f**3*h*x**3 + 195*b...