Integrand size = 29, antiderivative size = 579 \[ \int (a+b x)^3 (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=-\frac {2 (b e-a f)^3 (d e-c f)^2 (f g-e h) (e+f x)^{5/2}}{5 f^7}+\frac {2 (b e-a f)^2 (d e-c f) (b d e (5 f g-6 e h)-b c f (3 f g-4 e h)-a f (2 d f g-3 d e h+c f h)) (e+f x)^{7/2}}{7 f^7}-\frac {2 (b e-a f) \left (a^2 d f^2 (d f g-3 d e h+2 c f h)+a b f \left (3 c^2 f^2 h-d^2 e (8 f g-15 e h)+2 c d f (3 f g-8 e h)\right )-b^2 \left (4 c d e f (3 f g-5 e h)-5 d^2 e^2 (2 f g-3 e h)-3 c^2 f^2 (f g-2 e h)\right )\right ) (e+f x)^{9/2}}{9 f^7}+\frac {2 \left (a^3 d^2 f^3 h+3 a^2 b d f^2 (d f g-4 d e h+2 c f h)+3 a b^2 f \left (c^2 f^2 h-2 d^2 e (2 f g-5 e h)+2 c d f (f g-4 e h)\right )-b^3 \left (4 c d e f (2 f g-5 e h)-c^2 f^2 (f g-4 e h)-10 d^2 e^2 (f g-2 e h)\right )\right ) (e+f x)^{11/2}}{11 f^7}+\frac {2 b \left (3 a^2 d^2 f^2 h+3 a b d f (d f g-5 d e h+2 c f h)+b^2 \left (c^2 f^2 h+2 c d f (f g-5 e h)-5 d^2 e (f g-3 e h)\right )\right ) (e+f x)^{13/2}}{13 f^7}+\frac {2 b^2 d (3 a d f h+b (d f g-6 d e h+2 c f h)) (e+f x)^{15/2}}{15 f^7}+\frac {2 b^3 d^2 h (e+f x)^{17/2}}{17 f^7} \] Output:
-2/5*(-a*f+b*e)^3*(-c*f+d*e)^2*(-e*h+f*g)*(f*x+e)^(5/2)/f^7+2/7*(-a*f+b*e) ^2*(-c*f+d*e)*(b*d*e*(-6*e*h+5*f*g)-b*c*f*(-4*e*h+3*f*g)-a*f*(c*f*h-3*d*e* h+2*d*f*g))*(f*x+e)^(7/2)/f^7-2/9*(-a*f+b*e)*(a^2*d*f^2*(2*c*f*h-3*d*e*h+d *f*g)+a*b*f*(3*c^2*f^2*h-d^2*e*(-15*e*h+8*f*g)+2*c*d*f*(-8*e*h+3*f*g))-b^2 *(4*c*d*e*f*(-5*e*h+3*f*g)-5*d^2*e^2*(-3*e*h+2*f*g)-3*c^2*f^2*(-2*e*h+f*g) ))*(f*x+e)^(9/2)/f^7+2/11*(a^3*d^2*f^3*h+3*a^2*b*d*f^2*(2*c*f*h-4*d*e*h+d* f*g)+3*a*b^2*f*(c^2*f^2*h-2*d^2*e*(-5*e*h+2*f*g)+2*c*d*f*(-4*e*h+f*g))-b^3 *(4*c*d*e*f*(-5*e*h+2*f*g)-c^2*f^2*(-4*e*h+f*g)-10*d^2*e^2*(-2*e*h+f*g)))* (f*x+e)^(11/2)/f^7+2/13*b*(3*a^2*d^2*f^2*h+3*a*b*d*f*(2*c*f*h-5*d*e*h+d*f* g)+b^2*(c^2*f^2*h+2*c*d*f*(-5*e*h+f*g)-5*d^2*e*(-3*e*h+f*g)))*(f*x+e)^(13/ 2)/f^7+2/15*b^2*d*(3*a*d*f*h+b*(2*c*f*h-6*d*e*h+d*f*g))*(f*x+e)^(15/2)/f^7 +2/17*b^3*d^2*h*(f*x+e)^(17/2)/f^7
Time = 0.93 (sec) , antiderivative size = 850, normalized size of antiderivative = 1.47 \[ \int (a+b x)^3 (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\frac {2 (e+f x)^{5/2} \left (221 a^3 f^3 \left (99 c^2 f^2 (7 f g-2 e h+5 f h x)+22 c d f \left (8 e^2 h+5 f^2 x (9 g+7 h x)-2 e f (9 g+10 h x)\right )+d^2 \left (-48 e^3 h+35 f^3 x^2 (11 g+9 h x)+8 e^2 f (11 g+15 h x)-10 e f^2 x (22 g+21 h x)\right )\right )+51 a^2 b f^2 \left (143 c^2 f^2 \left (8 e^2 h+5 f^2 x (9 g+7 h x)-2 e f (9 g+10 h x)\right )+3 d^2 \left (128 e^4 h+105 f^4 x^3 (13 g+11 h x)-70 e f^3 x^2 (13 g+12 h x)+40 e^2 f^2 x (13 g+14 h x)-16 e^3 f (13 g+20 h x)\right )+26 c d f \left (-48 e^3 h+35 f^3 x^2 (11 g+9 h x)+8 e^2 f (11 g+15 h x)-10 e f^2 x (22 g+21 h x)\right )\right )+51 a b^2 f \left (d^2 \left (-256 e^5 h+1680 e^2 f^3 x^2 (g+h x)+128 e^4 f (3 g+5 h x)-160 e^3 f^2 x (6 g+7 h x)-210 e f^4 x^3 (12 g+11 h x)+231 f^5 x^4 (15 g+13 h x)\right )+6 c d f \left (128 e^4 h+105 f^4 x^3 (13 g+11 h x)-70 e f^3 x^2 (13 g+12 h x)+40 e^2 f^2 x (13 g+14 h x)-16 e^3 f (13 g+20 h x)\right )+13 c^2 f^2 \left (-48 e^3 h+35 f^3 x^2 (11 g+9 h x)+8 e^2 f (11 g+15 h x)-10 e f^2 x (22 g+21 h x)\right )\right )+b^3 \left (34 c d f \left (-256 e^5 h+1680 e^2 f^3 x^2 (g+h x)+128 e^4 f (3 g+5 h x)-160 e^3 f^2 x (6 g+7 h x)-210 e f^4 x^3 (12 g+11 h x)+231 f^5 x^4 (15 g+13 h x)\right )+51 c^2 f^2 \left (128 e^4 h+105 f^4 x^3 (13 g+11 h x)-70 e f^3 x^2 (13 g+12 h x)+40 e^2 f^2 x (13 g+14 h x)-16 e^3 f (13 g+20 h x)\right )+d^2 \left (3072 e^6 h+3003 f^6 x^5 (17 g+15 h x)-1120 e^3 f^3 x^2 (17 g+18 h x)+640 e^4 f^2 x (17 g+21 h x)-256 e^5 f (17 g+30 h x)+840 e^2 f^4 x^3 (34 g+33 h x)-462 e f^5 x^4 (85 g+78 h x)\right )\right )\right )}{765765 f^7} \] Input:
Integrate[(a + b*x)^3*(c + d*x)^2*(e + f*x)^(3/2)*(g + h*x),x]
Output:
(2*(e + f*x)^(5/2)*(221*a^3*f^3*(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))) + 51*a^2*b*f^2*(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))) + 51*a*b^2*f*(d^2*(-2 56*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)) + 6*c*d*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) ) + 13*c^2*f^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^3*(34*c*d*f*(-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)) + 51*c^2* f^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)) + d^2*(3072*e ^6*h + 3003*f^6*x^5*(17*g + 15*h*x) - 1120*e^3*f^3*x^2*(17*g + 18*h*x) + 6 40*e^4*f^2*x*(17*g + 21*h*x) - 256*e^5*f*(17*g + 30*h*x) + 840*e^2*f^4*...
Time = 1.00 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {165, 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.
| \(\displaystyle \int (a+b x)^3 (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx\) |
| \(\Big \downarrow \) 165 |
| \(\displaystyle \int \left (\frac {(e+f x)^{7/2} (b e-a f) \left (-a^2 d f^2 (2 c f h-3 d e h+d f g)-a b f \left (3 c^2 f^2 h+2 c d f (3 f g-8 e h)+d^2 (-e) (8 f g-15 e h)\right )+b^2 \left (-3 c^2 f^2 (f g-2 e h)+4 c d e f (3 f g-5 e h)-5 d^2 e^2 (2 f g-3 e h)\right )\right )}{f^6}+\frac {b (e+f x)^{11/2} \left (3 a^2 d^2 f^2 h+3 a b d f (2 c f h-5 d e h+d f g)+b^2 \left (c^2 f^2 h+2 c d f (f g-5 e h)-5 d^2 e (f g-3 e h)\right )\right )}{f^6}+\frac {(e+f x)^{9/2} \left (a^3 d^2 f^3 h+3 a^2 b d f^2 (2 c f h-4 d e h+d f g)+3 a b^2 f \left (c^2 f^2 h+2 c d f (f g-4 e h)-2 d^2 e (2 f g-5 e h)\right )-\left (b^3 \left (-c^2 f^2 (f g-4 e h)+4 c d e f (2 f g-5 e h)-10 d^2 e^2 (f g-2 e h)\right )\right )\right )}{f^6}+\frac {b^2 d (e+f x)^{13/2} (3 a d f h+b (2 c f h-6 d e h+d f g))}{f^6}+\frac {(e+f x)^{5/2} (b e-a f)^2 (d e-c f) (-a f (c f h-3 d e h+2 d f g)-b c f (3 f g-4 e h)+b d e (5 f g-6 e h))}{f^6}+\frac {(e+f x)^{3/2} (a f-b e)^3 (c f-d e)^2 (f g-e h)}{f^6}+\frac {b^3 d^2 h (e+f x)^{15/2}}{f^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (e+f x)^{9/2} (b e-a f) \left (a^2 d f^2 (2 c f h-3 d e h+d f g)+a b f \left (3 c^2 f^2 h+2 c d f (3 f g-8 e h)+d^2 (-e) (8 f g-15 e h)\right )-\left (b^2 \left (-3 c^2 f^2 (f g-2 e h)+4 c d e f (3 f g-5 e h)-5 d^2 e^2 (2 f g-3 e h)\right )\right )\right )}{9 f^7}+\frac {2 b (e+f x)^{13/2} \left (3 a^2 d^2 f^2 h+3 a b d f (2 c f h-5 d e h+d f g)+b^2 \left (c^2 f^2 h+2 c d f (f g-5 e h)-5 d^2 e (f g-3 e h)\right )\right )}{13 f^7}+\frac {2 (e+f x)^{11/2} \left (a^3 d^2 f^3 h+3 a^2 b d f^2 (2 c f h-4 d e h+d f g)+3 a b^2 f \left (c^2 f^2 h+2 c d f (f g-4 e h)-2 d^2 e (2 f g-5 e h)\right )-\left (b^3 \left (-c^2 f^2 (f g-4 e h)+4 c d e f (2 f g-5 e h)-10 d^2 e^2 (f g-2 e h)\right )\right )\right )}{11 f^7}+\frac {2 b^2 d (e+f x)^{15/2} (3 a d f h+b (2 c f h-6 d e h+d f g))}{15 f^7}+\frac {2 (e+f x)^{7/2} (b e-a f)^2 (d e-c f) (-a f (c f h-3 d e h+2 d f g)-b c f (3 f g-4 e h)+b d e (5 f g-6 e h))}{7 f^7}-\frac {2 (e+f x)^{5/2} (b e-a f)^3 (d e-c f)^2 (f g-e h)}{5 f^7}+\frac {2 b^3 d^2 h (e+f x)^{17/2}}{17 f^7}\) |
Input:
Int[(a + b*x)^3*(c + d*x)^2*(e + f*x)^(3/2)*(g + h*x),x]
Output:
(-2*(b*e - a*f)^3*(d*e - c*f)^2*(f*g - e*h)*(e + f*x)^(5/2))/(5*f^7) + (2* (b*e - a*f)^2*(d*e - c*f)*(b*d*e*(5*f*g - 6*e*h) - b*c*f*(3*f*g - 4*e*h) - a*f*(2*d*f*g - 3*d*e*h + c*f*h))*(e + f*x)^(7/2))/(7*f^7) - (2*(b*e - a*f )*(a^2*d*f^2*(d*f*g - 3*d*e*h + 2*c*f*h) + a*b*f*(3*c^2*f^2*h - d^2*e*(8*f *g - 15*e*h) + 2*c*d*f*(3*f*g - 8*e*h)) - b^2*(4*c*d*e*f*(3*f*g - 5*e*h) - 5*d^2*e^2*(2*f*g - 3*e*h) - 3*c^2*f^2*(f*g - 2*e*h)))*(e + f*x)^(9/2))/(9 *f^7) + (2*(a^3*d^2*f^3*h + 3*a^2*b*d*f^2*(d*f*g - 4*d*e*h + 2*c*f*h) + 3* a*b^2*f*(c^2*f^2*h - 2*d^2*e*(2*f*g - 5*e*h) + 2*c*d*f*(f*g - 4*e*h)) - b^ 3*(4*c*d*e*f*(2*f*g - 5*e*h) - c^2*f^2*(f*g - 4*e*h) - 10*d^2*e^2*(f*g - 2 *e*h)))*(e + f*x)^(11/2))/(11*f^7) + (2*b*(3*a^2*d^2*f^2*h + 3*a*b*d*f*(d* f*g - 5*d*e*h + 2*c*f*h) + b^2*(c^2*f^2*h + 2*c*d*f*(f*g - 5*e*h) - 5*d^2* e*(f*g - 3*e*h)))*(e + f*x)^(13/2))/(13*f^7) + (2*b^2*d*(3*a*d*f*h + b*(d* f*g - 6*d*e*h + 2*c*f*h))*(e + f*x)^(15/2))/(15*f^7) + (2*b^3*d^2*h*(e + f *x)^(17/2))/(17*f^7)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d* x)^n*(e + f*x)^p*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && (IntegersQ[m, n, p] || (IGtQ[n, 0] && IGtQ[p, 0]))
Time = 1.09 (sec) , antiderivative size = 569, normalized size of antiderivative = 0.98
| method | result | size |
| derivativedivides | \(\frac {\frac {2 h \,d^{2} b^{3} \left (f x +e \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (3 \left (a f -b e \right ) b^{2} d^{2}+2 b^{3} d \left (c f -d e \right )\right ) h +b^{3} d^{2} \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (3 \left (a f -b e \right )^{2} b \,d^{2}+6 \left (a f -b e \right ) b^{2} d \left (c f -d e \right )+b^{3} \left (c f -d e \right )^{2}\right ) h +\left (3 \left (a f -b e \right ) b^{2} d^{2}+2 b^{3} d \left (c f -d e \right )\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (\left (a f -b e \right )^{3} d^{2}+6 \left (a f -b e \right )^{2} b d \left (c f -d e \right )+3 \left (a f -b e \right ) b^{2} \left (c f -d e \right )^{2}\right ) h +\left (3 \left (a f -b e \right )^{2} b \,d^{2}+6 \left (a f -b e \right ) b^{2} d \left (c f -d e \right )+b^{3} \left (c f -d e \right )^{2}\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (2 \left (a f -b e \right )^{3} d \left (c f -d e \right )+3 \left (a f -b e \right )^{2} b \left (c f -d e \right )^{2}\right ) h +\left (\left (a f -b e \right )^{3} d^{2}+6 \left (a f -b e \right )^{2} b d \left (c f -d e \right )+3 \left (a f -b e \right ) b^{2} \left (c f -d e \right )^{2}\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a f -b e \right )^{3} \left (c f -d e \right )^{2} h +\left (2 \left (a f -b e \right )^{3} d \left (c f -d e \right )+3 \left (a f -b e \right )^{2} b \left (c f -d e \right )^{2}\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a f -b e \right )^{3} \left (c f -d e \right )^{2} \left (-e h +f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{5}}{f^{7}}\) | \(569\) |
| default | \(\frac {\frac {2 h \,d^{2} b^{3} \left (f x +e \right )^{\frac {17}{2}}}{17}-\frac {2 \left (-\left (3 \left (a f -b e \right ) b^{2} d^{2}+2 b^{3} d \left (c f -d e \right )\right ) h +b^{3} d^{2} \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {15}{2}}}{15}-\frac {2 \left (-\left (3 \left (a f -b e \right )^{2} b \,d^{2}+6 \left (a f -b e \right ) b^{2} d \left (c f -d e \right )+b^{3} \left (c f -d e \right )^{2}\right ) h +\left (3 \left (a f -b e \right ) b^{2} d^{2}+2 b^{3} d \left (c f -d e \right )\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {13}{2}}}{13}-\frac {2 \left (-\left (\left (a f -b e \right )^{3} d^{2}+6 \left (a f -b e \right )^{2} b d \left (c f -d e \right )+3 \left (a f -b e \right ) b^{2} \left (c f -d e \right )^{2}\right ) h +\left (3 \left (a f -b e \right )^{2} b \,d^{2}+6 \left (a f -b e \right ) b^{2} d \left (c f -d e \right )+b^{3} \left (c f -d e \right )^{2}\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {11}{2}}}{11}-\frac {2 \left (-\left (2 \left (a f -b e \right )^{3} d \left (c f -d e \right )+3 \left (a f -b e \right )^{2} b \left (c f -d e \right )^{2}\right ) h +\left (\left (a f -b e \right )^{3} d^{2}+6 \left (a f -b e \right )^{2} b d \left (c f -d e \right )+3 \left (a f -b e \right ) b^{2} \left (c f -d e \right )^{2}\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}-\frac {2 \left (-\left (a f -b e \right )^{3} \left (c f -d e \right )^{2} h +\left (2 \left (a f -b e \right )^{3} d \left (c f -d e \right )+3 \left (a f -b e \right )^{2} b \left (c f -d e \right )^{2}\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {2 \left (a f -b e \right )^{3} \left (c f -d e \right )^{2} \left (e h -f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{5}}{f^{7}}\) | \(574\) |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(736\) |
| gosper | \(\text {Expression too large to display}\) | \(1511\) |
| orering | \(\text {Expression too large to display}\) | \(1511\) |
| trager | \(\text {Expression too large to display}\) | \(2379\) |
| risch | \(\text {Expression too large to display}\) | \(2379\) |
Input:
int((b*x+a)^3*(d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x,method=_RETURNVERBOSE)
Output:
2/f^7*(1/17*h*d^2*b^3*(f*x+e)^(17/2)+1/15*((3*(a*f-b*e)*b^2*d^2+2*b^3*d*(c *f-d*e))*h+b^3*d^2*(-e*h+f*g))*(f*x+e)^(15/2)+1/13*((3*(a*f-b*e)^2*b*d^2+6 *(a*f-b*e)*b^2*d*(c*f-d*e)+b^3*(c*f-d*e)^2)*h+(3*(a*f-b*e)*b^2*d^2+2*b^3*d *(c*f-d*e))*(-e*h+f*g))*(f*x+e)^(13/2)+1/11*(((a*f-b*e)^3*d^2+6*(a*f-b*e)^ 2*b*d*(c*f-d*e)+3*(a*f-b*e)*b^2*(c*f-d*e)^2)*h+(3*(a*f-b*e)^2*b*d^2+6*(a*f -b*e)*b^2*d*(c*f-d*e)+b^3*(c*f-d*e)^2)*(-e*h+f*g))*(f*x+e)^(11/2)+1/9*((2* (a*f-b*e)^3*d*(c*f-d*e)+3*(a*f-b*e)^2*b*(c*f-d*e)^2)*h+((a*f-b*e)^3*d^2+6* (a*f-b*e)^2*b*d*(c*f-d*e)+3*(a*f-b*e)*b^2*(c*f-d*e)^2)*(-e*h+f*g))*(f*x+e) ^(9/2)+1/7*((a*f-b*e)^3*(c*f-d*e)^2*h+(2*(a*f-b*e)^3*d*(c*f-d*e)+3*(a*f-b* e)^2*b*(c*f-d*e)^2)*(-e*h+f*g))*(f*x+e)^(7/2)+1/5*(a*f-b*e)^3*(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. 1773 vs. \(2 (548) = 1096\).
Time = 0.10 (sec) , antiderivative size = 1773, normalized size of antiderivative = 3.06 \[ \int (a+b x)^3 (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^3*(d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x, algorithm="fricas")
Output:
2/765765*(45045*b^3*d^2*f^8*h*x^8 + 3003*(17*b^3*d^2*f^8*g + (18*b^3*d^2*e *f^7 + 17*(2*b^3*c*d + 3*a*b^2*d^2)*f^8)*h)*x^7 + 231*(17*(16*b^3*d^2*e*f^ 7 + 15*(2*b^3*c*d + 3*a*b^2*d^2)*f^8)*g + (3*b^3*d^2*e^2*f^6 + 272*(2*b^3* c*d + 3*a*b^2*d^2)*e*f^7 + 255*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*f^8)* h)*x^6 + 63*(17*(b^3*d^2*e^2*f^6 + 70*(2*b^3*c*d + 3*a*b^2*d^2)*e*f^7 + 65 *(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*f^8)*g - (12*b^3*d^2*e^3*f^5 - 17*( 2*b^3*c*d + 3*a*b^2*d^2)*e^2*f^6 - 1190*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d ^2)*e*f^7 - 1105*(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*f^8)*h)*x^5 - 35*(1 7*(2*b^3*d^2*e^3*f^5 - 3*(2*b^3*c*d + 3*a*b^2*d^2)*e^2*f^6 - 156*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e*f^7 - 143*(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3* d^2)*f^8)*g - (24*b^3*d^2*e^4*f^4 - 34*(2*b^3*c*d + 3*a*b^2*d^2)*e^3*f^5 + 51*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e^2*f^6 + 2652*(3*a*b^2*c^2 + 6* a^2*b*c*d + a^3*d^2)*e*f^7 + 2431*(3*a^2*b*c^2 + 2*a^3*c*d)*f^8)*h)*x^4 + 5*(17*(16*b^3*d^2*e^4*f^4 - 24*(2*b^3*c*d + 3*a*b^2*d^2)*e^3*f^5 + 39*(b^3 *c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e^2*f^6 + 1430*(3*a*b^2*c^2 + 6*a^2*b*c* d + a^3*d^2)*e*f^7 + 1287*(3*a^2*b*c^2 + 2*a^3*c*d)*f^8)*g - (192*b^3*d^2* e^5*f^3 - 21879*a^3*c^2*f^8 - 272*(2*b^3*c*d + 3*a*b^2*d^2)*e^4*f^4 + 408* (b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e^3*f^5 - 663*(3*a*b^2*c^2 + 6*a^2*b *c*d + a^3*d^2)*e^2*f^6 - 24310*(3*a^2*b*c^2 + 2*a^3*c*d)*e*f^7)*h)*x^3 - 3*(17*(32*b^3*d^2*e^5*f^3 - 3003*a^3*c^2*f^8 - 48*(2*b^3*c*d + 3*a*b^2*...
Leaf count of result is larger than twice the leaf count of optimal. 1909 vs. \(2 (620) = 1240\).
Time = 2.60 (sec) , antiderivative size = 1909, normalized size of antiderivative = 3.30 \[ \int (a+b x)^3 (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)**3*(d*x+c)**2*(f*x+e)**(3/2)*(h*x+g),x)
Output:
Piecewise((2*(b**3*d**2*h*(e + f*x)**(17/2)/(17*f**6) + (e + f*x)**(15/2)* (3*a*b**2*d**2*f*h + 2*b**3*c*d*f*h - 6*b**3*d**2*e*h + b**3*d**2*f*g)/(15 *f**6) + (e + f*x)**(13/2)*(3*a**2*b*d**2*f**2*h + 6*a*b**2*c*d*f**2*h - 1 5*a*b**2*d**2*e*f*h + 3*a*b**2*d**2*f**2*g + b**3*c**2*f**2*h - 10*b**3*c* d*e*f*h + 2*b**3*c*d*f**2*g + 15*b**3*d**2*e**2*h - 5*b**3*d**2*e*f*g)/(13 *f**6) + (e + f*x)**(11/2)*(a**3*d**2*f**3*h + 6*a**2*b*c*d*f**3*h - 12*a* *2*b*d**2*e*f**2*h + 3*a**2*b*d**2*f**3*g + 3*a*b**2*c**2*f**3*h - 24*a*b* *2*c*d*e*f**2*h + 6*a*b**2*c*d*f**3*g + 30*a*b**2*d**2*e**2*f*h - 12*a*b** 2*d**2*e*f**2*g - 4*b**3*c**2*e*f**2*h + b**3*c**2*f**3*g + 20*b**3*c*d*e* *2*f*h - 8*b**3*c*d*e*f**2*g - 20*b**3*d**2*e**3*h + 10*b**3*d**2*e**2*f*g )/(11*f**6) + (e + f*x)**(9/2)*(2*a**3*c*d*f**4*h - 3*a**3*d**2*e*f**3*h + a**3*d**2*f**4*g + 3*a**2*b*c**2*f**4*h - 18*a**2*b*c*d*e*f**3*h + 6*a**2 *b*c*d*f**4*g + 18*a**2*b*d**2*e**2*f**2*h - 9*a**2*b*d**2*e*f**3*g - 9*a* b**2*c**2*e*f**3*h + 3*a*b**2*c**2*f**4*g + 36*a*b**2*c*d*e**2*f**2*h - 18 *a*b**2*c*d*e*f**3*g - 30*a*b**2*d**2*e**3*f*h + 18*a*b**2*d**2*e**2*f**2* g + 6*b**3*c**2*e**2*f**2*h - 3*b**3*c**2*e*f**3*g - 20*b**3*c*d*e**3*f*h + 12*b**3*c*d*e**2*f**2*g + 15*b**3*d**2*e**4*h - 10*b**3*d**2*e**3*f*g)/( 9*f**6) + (e + f*x)**(7/2)*(a**3*c**2*f**5*h - 4*a**3*c*d*e*f**4*h + 2*a** 3*c*d*f**5*g + 3*a**3*d**2*e**2*f**3*h - 2*a**3*d**2*e*f**4*g - 6*a**2*b*c **2*e*f**4*h + 3*a**2*b*c**2*f**5*g + 18*a**2*b*c*d*e**2*f**3*h - 12*a*...
Leaf count of result is larger than twice the leaf count of optimal. 1152 vs. \(2 (548) = 1096\).
Time = 0.04 (sec) , antiderivative size = 1152, normalized size of antiderivative = 1.99 \[ \int (a+b x)^3 (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^3*(d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x, algorithm="maxima")
Output:
2/765765*(45045*(f*x + e)^(17/2)*b^3*d^2*h + 51051*(b^3*d^2*f*g - (6*b^3*d ^2*e - (2*b^3*c*d + 3*a*b^2*d^2)*f)*h)*(f*x + e)^(15/2) - 58905*((5*b^3*d^ 2*e*f - (2*b^3*c*d + 3*a*b^2*d^2)*f^2)*g - (15*b^3*d^2*e^2 - 5*(2*b^3*c*d + 3*a*b^2*d^2)*e*f + (b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*f^2)*h)*(f*x + e)^(13/2) + 69615*((10*b^3*d^2*e^2*f - 4*(2*b^3*c*d + 3*a*b^2*d^2)*e*f^2 + (b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*f^3)*g - (20*b^3*d^2*e^3 - 10*(2*b^ 3*c*d + 3*a*b^2*d^2)*e^2*f + 4*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e*f^2 - (3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*f^3)*h)*(f*x + e)^(11/2) - 85085* ((10*b^3*d^2*e^3*f - 6*(2*b^3*c*d + 3*a*b^2*d^2)*e^2*f^2 + 3*(b^3*c^2 + 6* a*b^2*c*d + 3*a^2*b*d^2)*e*f^3 - (3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*f^4 )*g - (15*b^3*d^2*e^4 - 10*(2*b^3*c*d + 3*a*b^2*d^2)*e^3*f + 6*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e^2*f^2 - 3*(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^ 2)*e*f^3 + (3*a^2*b*c^2 + 2*a^3*c*d)*f^4)*h)*(f*x + e)^(9/2) + 109395*((5* b^3*d^2*e^4*f - 4*(2*b^3*c*d + 3*a*b^2*d^2)*e^3*f^2 + 3*(b^3*c^2 + 6*a*b^2 *c*d + 3*a^2*b*d^2)*e^2*f^3 - 2*(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*e*f^ 4 + (3*a^2*b*c^2 + 2*a^3*c*d)*f^5)*g - (6*b^3*d^2*e^5 - a^3*c^2*f^5 - 5*(2 *b^3*c*d + 3*a*b^2*d^2)*e^4*f + 4*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e^ 3*f^2 - 3*(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*e^2*f^3 + 2*(3*a^2*b*c^2 + 2*a^3*c*d)*e*f^4)*h)*(f*x + e)^(7/2) - 153153*((b^3*d^2*e^5*f - a^3*c^2*f ^6 - (2*b^3*c*d + 3*a*b^2*d^2)*e^4*f^2 + (b^3*c^2 + 6*a*b^2*c*d + 3*a^2...
Leaf count of result is larger than twice the leaf count of optimal. 4992 vs. \(2 (548) = 1096\).
Time = 0.19 (sec) , antiderivative size = 4992, normalized size of antiderivative = 8.62 \[ \int (a+b x)^3 (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^3*(d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x, algorithm="giac")
Output:
2/765765*(765765*sqrt(f*x + e)*a^3*c^2*e^2*g + 510510*((f*x + e)^(3/2) - 3 *sqrt(f*x + e)*e)*a^3*c^2*e*g + 765765*((f*x + e)^(3/2) - 3*sqrt(f*x + e)* e)*a^2*b*c^2*e^2*g/f + 510510*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a^3*c* d*e^2*g/f + 255255*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a^3*c^2*e^2*h/f + 51051*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a ^3*c^2*g + 153153*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a*b^2*c^2*e^2*g/f^2 + 306306*(3*(f*x + e)^(5/2) - 10*(f*x + e)^( 3/2)*e + 15*sqrt(f*x + e)*e^2)*a^2*b*c*d*e^2*g/f^2 + 51051*(3*(f*x + e)^(5 /2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a^3*d^2*e^2*g/f^2 + 306 306*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a^2* b*c^2*e*g/f + 204204*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f *x + e)*e^2)*a^3*c*d*e*g/f + 153153*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2 )*e + 15*sqrt(f*x + e)*e^2)*a^2*b*c^2*e^2*h/f^2 + 102102*(3*(f*x + e)^(5/2 ) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a^3*c*d*e^2*h/f^2 + 10210 2*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a^3*c^ 2*e*h/f + 21879*(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^2*e^2*g/f^3 + 131274*(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*d*e^2*g/f^3 + 65637*(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^2*b*d^2*e^2*g/f^3 + 1...
Time = 2.43 (sec) , antiderivative size = 692, normalized size of antiderivative = 1.20 \[ \int (a+b x)^3 (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx=\frac {{\left (e+f\,x\right )}^{13/2}\,\left (6\,h\,a^2\,b\,d^2\,f^2+12\,h\,a\,b^2\,c\,d\,f^2-30\,h\,a\,b^2\,d^2\,e\,f+6\,g\,a\,b^2\,d^2\,f^2+2\,h\,b^3\,c^2\,f^2-20\,h\,b^3\,c\,d\,e\,f+4\,g\,b^3\,c\,d\,f^2+30\,h\,b^3\,d^2\,e^2-10\,g\,b^3\,d^2\,e\,f\right )}{13\,f^7}+\frac {{\left (e+f\,x\right )}^{11/2}\,\left (2\,h\,a^3\,d^2\,f^3+12\,h\,a^2\,b\,c\,d\,f^3-24\,h\,a^2\,b\,d^2\,e\,f^2+6\,g\,a^2\,b\,d^2\,f^3+6\,h\,a\,b^2\,c^2\,f^3-48\,h\,a\,b^2\,c\,d\,e\,f^2+12\,g\,a\,b^2\,c\,d\,f^3+60\,h\,a\,b^2\,d^2\,e^2\,f-24\,g\,a\,b^2\,d^2\,e\,f^2-8\,h\,b^3\,c^2\,e\,f^2+2\,g\,b^3\,c^2\,f^3+40\,h\,b^3\,c\,d\,e^2\,f-16\,g\,b^3\,c\,d\,e\,f^2-40\,h\,b^3\,d^2\,e^3+20\,g\,b^3\,d^2\,e^2\,f\right )}{11\,f^7}+\frac {2\,{\left (e+f\,x\right )}^{9/2}\,\left (a\,f-b\,e\right )\,\left (2\,h\,a^2\,c\,d\,f^3-3\,h\,a^2\,d^2\,e\,f^2+g\,a^2\,d^2\,f^3+3\,h\,a\,b\,c^2\,f^3-16\,h\,a\,b\,c\,d\,e\,f^2+6\,g\,a\,b\,c\,d\,f^3+15\,h\,a\,b\,d^2\,e^2\,f-8\,g\,a\,b\,d^2\,e\,f^2-6\,h\,b^2\,c^2\,e\,f^2+3\,g\,b^2\,c^2\,f^3+20\,h\,b^2\,c\,d\,e^2\,f-12\,g\,b^2\,c\,d\,e\,f^2-15\,h\,b^2\,d^2\,e^3+10\,g\,b^2\,d^2\,e^2\,f\right )}{9\,f^7}-\frac {2\,{\left (e+f\,x\right )}^{5/2}\,{\left (a\,f-b\,e\right )}^3\,{\left (c\,f-d\,e\right )}^2\,\left (e\,h-f\,g\right )}{5\,f^7}+\frac {2\,b^3\,d^2\,h\,{\left (e+f\,x\right )}^{17/2}}{17\,f^7}+\frac {2\,b^2\,d\,{\left (e+f\,x\right )}^{15/2}\,\left (3\,a\,d\,f\,h+2\,b\,c\,f\,h-6\,b\,d\,e\,h+b\,d\,f\,g\right )}{15\,f^7}+\frac {2\,{\left (e+f\,x\right )}^{7/2}\,{\left (a\,f-b\,e\right )}^2\,\left (c\,f-d\,e\right )\,\left (a\,c\,f^2\,h+2\,a\,d\,f^2\,g+3\,b\,c\,f^2\,g+6\,b\,d\,e^2\,h-3\,a\,d\,e\,f\,h-4\,b\,c\,e\,f\,h-5\,b\,d\,e\,f\,g\right )}{7\,f^7} \] Input:
int((e + f*x)^(3/2)*(g + h*x)*(a + b*x)^3*(c + d*x)^2,x)
Output:
((e + f*x)^(13/2)*(2*b^3*c^2*f^2*h + 30*b^3*d^2*e^2*h + 4*b^3*c*d*f^2*g - 10*b^3*d^2*e*f*g + 6*a*b^2*d^2*f^2*g + 6*a^2*b*d^2*f^2*h - 20*b^3*c*d*e*f* h + 12*a*b^2*c*d*f^2*h - 30*a*b^2*d^2*e*f*h))/(13*f^7) + ((e + f*x)^(11/2) *(2*b^3*c^2*f^3*g + 2*a^3*d^2*f^3*h - 40*b^3*d^2*e^3*h + 6*a*b^2*c^2*f^3*h + 6*a^2*b*d^2*f^3*g - 8*b^3*c^2*e*f^2*h + 20*b^3*d^2*e^2*f*g + 12*a*b^2*c *d*f^3*g + 12*a^2*b*c*d*f^3*h - 16*b^3*c*d*e*f^2*g + 40*b^3*c*d*e^2*f*h - 24*a*b^2*d^2*e*f^2*g + 60*a*b^2*d^2*e^2*f*h - 24*a^2*b*d^2*e*f^2*h - 48*a* b^2*c*d*e*f^2*h))/(11*f^7) + (2*(e + f*x)^(9/2)*(a*f - b*e)*(a^2*d^2*f^3*g + 3*b^2*c^2*f^3*g - 15*b^2*d^2*e^3*h + 3*a*b*c^2*f^3*h + 2*a^2*c*d*f^3*h - 3*a^2*d^2*e*f^2*h - 6*b^2*c^2*e*f^2*h + 10*b^2*d^2*e^2*f*g + 6*a*b*c*d*f ^3*g - 8*a*b*d^2*e*f^2*g + 15*a*b*d^2*e^2*f*h - 12*b^2*c*d*e*f^2*g + 20*b^ 2*c*d*e^2*f*h - 16*a*b*c*d*e*f^2*h))/(9*f^7) - (2*(e + f*x)^(5/2)*(a*f - b *e)^3*(c*f - d*e)^2*(e*h - f*g))/(5*f^7) + (2*b^3*d^2*h*(e + f*x)^(17/2))/ (17*f^7) + (2*b^2*d*(e + f*x)^(15/2)*(3*a*d*f*h + 2*b*c*f*h - 6*b*d*e*h + b*d*f*g))/(15*f^7) + (2*(e + f*x)^(7/2)*(a*f - b*e)^2*(c*f - d*e)*(a*c*f^2 *h + 2*a*d*f^2*g + 3*b*c*f^2*g + 6*b*d*e^2*h - 3*a*d*e*f*h - 4*b*c*e*f*h - 5*b*d*e*f*g))/(7*f^7)
Time = 0.20 (sec) , antiderivative size = 2377, normalized size of antiderivative = 4.11 \[ \int (a+b x)^3 (c+d x)^2 (e+f x)^{3/2} (g+h x) \, dx =\text {Too large to display} \] Input:
int((b*x+a)^3*(d*x+c)^2*(f*x+e)^(3/2)*(h*x+g),x)
Output:
(2*sqrt(e + f*x)*( - 43758*a**3*c**2*e**3*f**5*h + 153153*a**3*c**2*e**2*f **6*g + 21879*a**3*c**2*e**2*f**6*h*x + 306306*a**3*c**2*e*f**7*g*x + 1750 32*a**3*c**2*e*f**7*h*x**2 + 153153*a**3*c**2*f**8*g*x**2 + 109395*a**3*c* *2*f**8*h*x**3 + 38896*a**3*c*d*e**4*f**4*h - 87516*a**3*c*d*e**3*f**5*g - 19448*a**3*c*d*e**3*f**5*h*x + 43758*a**3*c*d*e**2*f**6*g*x + 14586*a**3* c*d*e**2*f**6*h*x**2 + 350064*a**3*c*d*e*f**7*g*x**2 + 243100*a**3*c*d*e*f **7*h*x**3 + 218790*a**3*c*d*f**8*g*x**3 + 170170*a**3*c*d*f**8*h*x**4 - 1 0608*a**3*d**2*e**5*f**3*h + 19448*a**3*d**2*e**4*f**4*g + 5304*a**3*d**2* e**4*f**4*h*x - 9724*a**3*d**2*e**3*f**5*g*x - 3978*a**3*d**2*e**3*f**5*h* x**2 + 7293*a**3*d**2*e**2*f**6*g*x**2 + 3315*a**3*d**2*e**2*f**6*h*x**3 + 121550*a**3*d**2*e*f**7*g*x**3 + 92820*a**3*d**2*e*f**7*h*x**4 + 85085*a* *3*d**2*f**8*g*x**4 + 69615*a**3*d**2*f**8*h*x**5 + 58344*a**2*b*c**2*e**4 *f**4*h - 131274*a**2*b*c**2*e**3*f**5*g - 29172*a**2*b*c**2*e**3*f**5*h*x + 65637*a**2*b*c**2*e**2*f**6*g*x + 21879*a**2*b*c**2*e**2*f**6*h*x**2 + 525096*a**2*b*c**2*e*f**7*g*x**2 + 364650*a**2*b*c**2*e*f**7*h*x**3 + 3281 85*a**2*b*c**2*f**8*g*x**3 + 255255*a**2*b*c**2*f**8*h*x**4 - 63648*a**2*b *c*d*e**5*f**3*h + 116688*a**2*b*c*d*e**4*f**4*g + 31824*a**2*b*c*d*e**4*f **4*h*x - 58344*a**2*b*c*d*e**3*f**5*g*x - 23868*a**2*b*c*d*e**3*f**5*h*x* *2 + 43758*a**2*b*c*d*e**2*f**6*g*x**2 + 19890*a**2*b*c*d*e**2*f**6*h*x**3 + 729300*a**2*b*c*d*e*f**7*g*x**3 + 556920*a**2*b*c*d*e*f**7*h*x**4 + ...