\(\int (a+b x)^2 (c+d x) (e+f x)^{3/2} (g+h x) \, dx\) [61]

Optimal result

Integrand size = 27, antiderivative size = 247 \[ \int (a+b x)^2 (c+d x) (e+f x)^{3/2} (g+h x) \, dx=-\frac {2 (b e-a f)^2 (d e-c f) (f g-e h) (e+f x)^{5/2}}{5 f^5}+\frac {2 (b e-a f) (b d e (3 f g-4 e h)-b c f (2 f g-3 e h)-a f (d f g-2 d e h+c f h)) (e+f x)^{7/2}}{7 f^5}+\frac {2 \left (a^2 d f^2 h+2 a b f (d f g-3 d e h+c f h)+b^2 (c f (f g-3 e h)-3 d e (f g-2 e h))\right ) (e+f x)^{9/2}}{9 f^5}+\frac {2 b (2 a d f h+b (d f g-4 d e h+c f h)) (e+f x)^{11/2}}{11 f^5}+\frac {2 b^2 d h (e+f x)^{13/2}}{13 f^5} \] Output:

-2/5*(-a*f+b*e)^2*(-c*f+d*e)*(-e*h+f*g)*(f*x+e)^(5/2)/f^5+2/7*(-a*f+b*e)*( 
b*d*e*(-4*e*h+3*f*g)-b*c*f*(-3*e*h+2*f*g)-a*f*(c*f*h-2*d*e*h+d*f*g))*(f*x+ 
e)^(7/2)/f^5+2/9*(a^2*d*f^2*h+2*a*b*f*(c*f*h-3*d*e*h+d*f*g)+b^2*(c*f*(-3*e 
*h+f*g)-3*d*e*(-2*e*h+f*g)))*(f*x+e)^(9/2)/f^5+2/11*b*(2*a*d*f*h+b*(c*f*h- 
4*d*e*h+d*f*g))*(f*x+e)^(11/2)/f^5+2/13*b^2*d*h*(f*x+e)^(13/2)/f^5
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.26 \[ \int (a+b x)^2 (c+d x) (e+f x)^{3/2} (g+h x) \, dx=\frac {2 (e+f x)^{5/2} \left (143 a^2 f^2 \left (9 c f (7 f g-2 e h+5 f h x)+d \left (8 e^2 h+5 f^2 x (9 g+7 h x)-2 e f (9 g+10 h x)\right )\right )+26 a b f \left (11 c f \left (8 e^2 h+5 f^2 x (9 g+7 h x)-2 e f (9 g+10 h x)\right )+d \left (-48 e^3 h+35 f^3 x^2 (11 g+9 h x)+8 e^2 f (11 g+15 h x)-10 e f^2 x (22 g+21 h x)\right )\right )+b^2 \left (3 d \left (128 e^4 h+105 f^4 x^3 (13 g+11 h x)-70 e f^3 x^2 (13 g+12 h x)+40 e^2 f^2 x (13 g+14 h x)-16 e^3 f (13 g+20 h x)\right )+13 c f \left (-48 e^3 h+35 f^3 x^2 (11 g+9 h x)+8 e^2 f (11 g+15 h x)-10 e f^2 x (22 g+21 h x)\right )\right )\right )}{45045 f^5} \] Input:

Integrate[(a + b*x)^2*(c + d*x)*(e + f*x)^(3/2)*(g + h*x),x]
 

Output:

(2*(e + f*x)^(5/2)*(143*a^2*f^2*(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))) + 26*a*b*f*(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))) + b^2*(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^5)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 247, 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.

\(\displaystyle \int (a+b x)^2 (c+d x) (e+f x)^{3/2} (g+h x) \, dx\)

\(\Big \downarrow \) 159

\(\displaystyle \int \left (\frac {(e+f x)^{7/2} \left (a^2 d f^2 h+2 a b f (c f h-3 d e h+d f g)+b^2 (c f (f g-3 e h)-3 d e (f g-2 e h))\right )}{f^4}+\frac {b (e+f x)^{9/2} (2 a d f h+b (c f h-4 d e h+d f g))}{f^4}+\frac {(e+f x)^{5/2} (b e-a f) (-a f (c f h-2 d e h+d f g)-b c f (2 f g-3 e h)+b d e (3 f g-4 e h))}{f^4}+\frac {(e+f x)^{3/2} (a f-b e)^2 (c f-d e) (f g-e h)}{f^4}+\frac {b^2 d h (e+f x)^{11/2}}{f^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 (e+f x)^{9/2} \left (a^2 d f^2 h+2 a b f (c f h-3 d e h+d f g)+b^2 (c f (f g-3 e h)-3 d e (f g-2 e h))\right )}{9 f^5}+\frac {2 b (e+f x)^{11/2} (2 a d f h+b (c f h-4 d e h+d f g))}{11 f^5}+\frac {2 (e+f x)^{7/2} (b e-a f) (-a f (c f h-2 d e h+d f g)-b c f (2 f g-3 e h)+b d e (3 f g-4 e h))}{7 f^5}-\frac {2 (e+f x)^{5/2} (b e-a f)^2 (d e-c f) (f g-e h)}{5 f^5}+\frac {2 b^2 d h (e+f x)^{13/2}}{13 f^5}\)

Input:

Int[(a + b*x)^2*(c + d*x)*(e + f*x)^(3/2)*(g + h*x),x]
 

Output:

(-2*(b*e - a*f)^2*(d*e - c*f)*(f*g - e*h)*(e + f*x)^(5/2))/(5*f^5) + (2*(b 
*e - a*f)*(b*d*e*(3*f*g - 4*e*h) - b*c*f*(2*f*g - 3*e*h) - a*f*(d*f*g - 2* 
d*e*h + c*f*h))*(e + f*x)^(7/2))/(7*f^5) + (2*(a^2*d*f^2*h + 2*a*b*f*(d*f* 
g - 3*d*e*h + c*f*h) + b^2*(c*f*(f*g - 3*e*h) - 3*d*e*(f*g - 2*e*h)))*(e + 
 f*x)^(9/2))/(9*f^5) + (2*b*(2*a*d*f*h + b*(d*f*g - 4*d*e*h + c*f*h))*(e + 
 f*x)^(11/2))/(11*f^5) + (2*b^2*d*h*(e + f*x)^(13/2))/(13*f^5)
 

Defintions of rubi rules used

rule 159

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])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.03

method	result	size
derivativedivides	\(\frac {\frac {2 d h \,b^{2} \left (f x +e \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (2 b \left (a f -b e \right ) d +b^{2} \left (c f -d e \right )\right ) h +d \,b^{2} \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (\left (a f -b e \right )^{2} d +2 b \left (a f -b e \right ) \left (c f -d e \right )\right ) h +\left (2 b \left (a f -b e \right ) d +b^{2} \left (c f -d e \right )\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a f -b e \right )^{2} \left (c f -d e \right ) h +\left (\left (a f -b e \right )^{2} d +2 b \left (a f -b e \right ) \left (c f -d e \right )\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a f -b e \right )^{2} \left (c f -d e \right ) \left (-e h +f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{5}}{f^{5}}\)	\(255\)
default	\(\frac {\frac {2 d h \,b^{2} \left (f x +e \right )^{\frac {13}{2}}}{13}-\frac {2 \left (-\left (2 b \left (a f -b e \right ) d +b^{2} \left (c f -d e \right )\right ) h +d \,b^{2} \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {11}{2}}}{11}-\frac {2 \left (-\left (\left (a f -b e \right )^{2} d +2 b \left (a f -b e \right ) \left (c f -d e \right )\right ) h +\left (2 b \left (a f -b e \right ) d +b^{2} \left (c f -d e \right )\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}-\frac {2 \left (-\left (a f -b e \right )^{2} \left (c f -d e \right ) h +\left (\left (a f -b e \right )^{2} d +2 b \left (a f -b e \right ) \left (c f -d e \right )\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {2 \left (a f -b e \right )^{2} \left (c f -d e \right ) \left (e h -f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{5}}{f^{5}}\)	\(258\)
pseudoelliptic	\(-\frac {4 \left (\left (-\frac {35 \left (\frac {9 d h \,x^{2}}{13}+\frac {9 \left (c h +d g \right ) x}{11}+c g \right ) x^{2} b^{2}}{18}-5 a x \left (\frac {7 d h \,x^{2}}{11}+\frac {7 \left (c h +d g \right ) x}{9}+c g \right ) b -\frac {7 a^{2} \left (\frac {5 d h \,x^{2}}{9}+\frac {5 \left (c h +d g \right ) x}{7}+c g \right )}{2}\right ) f^{4}+\left (\frac {10 x \left (\frac {126 d h \,x^{2}}{143}+\frac {21 \left (c h +d g \right ) x}{22}+c g \right ) b^{2}}{9}+2 a \left (\frac {35 d h \,x^{2}}{33}+\frac {10 \left (c h +d g \right ) x}{9}+c g \right ) b +a^{2} \left (d g +c h +\frac {10}{9} d h x \right )\right ) e \,f^{3}-\frac {4 \left (\left (\frac {210 d h \,x^{2}}{143}+\frac {15 \left (c h +d g \right ) x}{11}+c g \right ) b^{2}+2 \left (\frac {15}{11} d h x +c h +d g \right ) a b +a^{2} d h \right ) e^{2} f^{2}}{9}+\frac {16 \left (\left (\frac {10}{13} d h x +\frac {1}{2} c h +\frac {1}{2} d g \right ) b +a d h \right ) b \,e^{3} f}{33}-\frac {64 b^{2} d \,e^{4} h}{429}\right ) \left (f x +e \right )^{\frac {5}{2}}}{35 f^{5}}\)	\(268\)
gosper	\(-\frac {2 \left (f x +e \right )^{\frac {5}{2}} \left (-3465 d h \,b^{2} x^{4} f^{4}-8190 a b d \,f^{4} h \,x^{3}-4095 b^{2} c \,f^{4} h \,x^{3}+2520 b^{2} d e \,f^{3} h \,x^{3}-4095 b^{2} d \,f^{4} g \,x^{3}-5005 a^{2} d \,f^{4} h \,x^{2}-10010 a b c \,f^{4} h \,x^{2}+5460 a b d e \,f^{3} h \,x^{2}-10010 a b d \,f^{4} g \,x^{2}+2730 b^{2} c e \,f^{3} h \,x^{2}-5005 b^{2} c \,f^{4} g \,x^{2}-1680 b^{2} d \,e^{2} f^{2} h \,x^{2}+2730 b^{2} d e \,f^{3} g \,x^{2}-6435 a^{2} c \,f^{4} h x +2860 a^{2} d e \,f^{3} h x -6435 a^{2} d \,f^{4} g x +5720 a b c e \,f^{3} h x -12870 a b c \,f^{4} g x -3120 a b d \,e^{2} f^{2} h x +5720 a b d e \,f^{3} g x -1560 b^{2} c \,e^{2} f^{2} h x +2860 b^{2} c e \,f^{3} g x +960 b^{2} d \,e^{3} f h x -1560 b^{2} d \,e^{2} f^{2} g x +2574 a^{2} c e \,f^{3} h -9009 c g \,a^{2} f^{4}-1144 a^{2} d \,e^{2} f^{2} h +2574 a^{2} d e \,f^{3} g -2288 a b c \,e^{2} f^{2} h +5148 a b c e \,f^{3} g +1248 a b d \,e^{3} f h -2288 a b d \,e^{2} f^{2} g +624 b^{2} c \,e^{3} f h -1144 b^{2} c \,e^{2} f^{2} g -384 b^{2} d \,e^{4} h +624 b^{2} d \,e^{3} f g \right )}{45045 f^{5}}\)	\(451\)
orering	\(-\frac {2 \left (f x +e \right )^{\frac {5}{2}} \left (-3465 d h \,b^{2} x^{4} f^{4}-8190 a b d \,f^{4} h \,x^{3}-4095 b^{2} c \,f^{4} h \,x^{3}+2520 b^{2} d e \,f^{3} h \,x^{3}-4095 b^{2} d \,f^{4} g \,x^{3}-5005 a^{2} d \,f^{4} h \,x^{2}-10010 a b c \,f^{4} h \,x^{2}+5460 a b d e \,f^{3} h \,x^{2}-10010 a b d \,f^{4} g \,x^{2}+2730 b^{2} c e \,f^{3} h \,x^{2}-5005 b^{2} c \,f^{4} g \,x^{2}-1680 b^{2} d \,e^{2} f^{2} h \,x^{2}+2730 b^{2} d e \,f^{3} g \,x^{2}-6435 a^{2} c \,f^{4} h x +2860 a^{2} d e \,f^{3} h x -6435 a^{2} d \,f^{4} g x +5720 a b c e \,f^{3} h x -12870 a b c \,f^{4} g x -3120 a b d \,e^{2} f^{2} h x +5720 a b d e \,f^{3} g x -1560 b^{2} c \,e^{2} f^{2} h x +2860 b^{2} c e \,f^{3} g x +960 b^{2} d \,e^{3} f h x -1560 b^{2} d \,e^{2} f^{2} g x +2574 a^{2} c e \,f^{3} h -9009 c g \,a^{2} f^{4}-1144 a^{2} d \,e^{2} f^{2} h +2574 a^{2} d e \,f^{3} g -2288 a b c \,e^{2} f^{2} h +5148 a b c e \,f^{3} g +1248 a b d \,e^{3} f h -2288 a b d \,e^{2} f^{2} g +624 b^{2} c \,e^{3} f h -1144 b^{2} c \,e^{2} f^{2} g -384 b^{2} d \,e^{4} h +624 b^{2} d \,e^{3} f g \right )}{45045 f^{5}}\)	\(451\)
trager	\(-\frac {2 \left (-3465 b^{2} d \,f^{6} h \,x^{6}-8190 a b d \,f^{6} h \,x^{5}-4095 b^{2} c \,f^{6} h \,x^{5}-4410 b^{2} d e \,f^{5} h \,x^{5}-4095 b^{2} d \,f^{6} g \,x^{5}-5005 a^{2} d \,f^{6} h \,x^{4}-10010 a b c \,f^{6} h \,x^{4}-10920 a b d e \,f^{5} h \,x^{4}-10010 a b d \,f^{6} g \,x^{4}-5460 b^{2} c e \,f^{5} h \,x^{4}-5005 b^{2} c \,f^{6} g \,x^{4}-105 b^{2} d \,e^{2} f^{4} h \,x^{4}-5460 b^{2} d e \,f^{5} g \,x^{4}-6435 a^{2} c \,f^{6} h \,x^{3}-7150 a^{2} d e \,f^{5} h \,x^{3}-6435 a^{2} d \,f^{6} g \,x^{3}-14300 a b c e \,f^{5} h \,x^{3}-12870 a b c \,f^{6} g \,x^{3}-390 a b d \,e^{2} f^{4} h \,x^{3}-14300 a b d e \,f^{5} g \,x^{3}-195 b^{2} c \,e^{2} f^{4} h \,x^{3}-7150 b^{2} c e \,f^{5} g \,x^{3}+120 b^{2} d \,e^{3} f^{3} h \,x^{3}-195 b^{2} d \,e^{2} f^{4} g \,x^{3}-10296 a^{2} c e \,f^{5} h \,x^{2}-9009 a^{2} c \,f^{6} g \,x^{2}-429 a^{2} d \,e^{2} f^{4} h \,x^{2}-10296 a^{2} d e \,f^{5} g \,x^{2}-858 a b c \,e^{2} f^{4} h \,x^{2}-20592 a b c e \,f^{5} g \,x^{2}+468 a b d \,e^{3} f^{3} h \,x^{2}-858 a b d \,e^{2} f^{4} g \,x^{2}+234 b^{2} c \,e^{3} f^{3} h \,x^{2}-429 b^{2} c \,e^{2} f^{4} g \,x^{2}-144 b^{2} d \,e^{4} f^{2} h \,x^{2}+234 b^{2} d \,e^{3} f^{3} g \,x^{2}-1287 a^{2} c \,e^{2} f^{4} h x -18018 a^{2} c e \,f^{5} g x +572 a^{2} d \,e^{3} f^{3} h x -1287 a^{2} d \,e^{2} f^{4} g x +1144 a b c \,e^{3} f^{3} h x -2574 a b c \,e^{2} f^{4} g x -624 a b d \,e^{4} f^{2} h x +1144 a b d \,e^{3} f^{3} g x -312 b^{2} c \,e^{4} f^{2} h x +572 b^{2} c \,e^{3} f^{3} g x +192 b^{2} d \,e^{5} f h x -312 b^{2} d \,e^{4} f^{2} g x +2574 a^{2} c \,e^{3} f^{3} h -9009 a^{2} c \,e^{2} f^{4} g -1144 a^{2} d \,e^{4} f^{2} h +2574 a^{2} d \,e^{3} f^{3} g -2288 a b c \,e^{4} f^{2} h +5148 a b c \,e^{3} f^{3} g +1248 a b d \,e^{5} f h -2288 a b d \,e^{4} f^{2} g +624 b^{2} c \,e^{5} f h -1144 b^{2} c \,e^{4} f^{2} g -384 b^{2} d \,e^{6} h +624 b^{2} d \,e^{5} f g \right ) \sqrt {f x +e}}{45045 f^{5}}\)	\(823\)
risch	\(-\frac {2 \left (-3465 b^{2} d \,f^{6} h \,x^{6}-8190 a b d \,f^{6} h \,x^{5}-4095 b^{2} c \,f^{6} h \,x^{5}-4410 b^{2} d e \,f^{5} h \,x^{5}-4095 b^{2} d \,f^{6} g \,x^{5}-5005 a^{2} d \,f^{6} h \,x^{4}-10010 a b c \,f^{6} h \,x^{4}-10920 a b d e \,f^{5} h \,x^{4}-10010 a b d \,f^{6} g \,x^{4}-5460 b^{2} c e \,f^{5} h \,x^{4}-5005 b^{2} c \,f^{6} g \,x^{4}-105 b^{2} d \,e^{2} f^{4} h \,x^{4}-5460 b^{2} d e \,f^{5} g \,x^{4}-6435 a^{2} c \,f^{6} h \,x^{3}-7150 a^{2} d e \,f^{5} h \,x^{3}-6435 a^{2} d \,f^{6} g \,x^{3}-14300 a b c e \,f^{5} h \,x^{3}-12870 a b c \,f^{6} g \,x^{3}-390 a b d \,e^{2} f^{4} h \,x^{3}-14300 a b d e \,f^{5} g \,x^{3}-195 b^{2} c \,e^{2} f^{4} h \,x^{3}-7150 b^{2} c e \,f^{5} g \,x^{3}+120 b^{2} d \,e^{3} f^{3} h \,x^{3}-195 b^{2} d \,e^{2} f^{4} g \,x^{3}-10296 a^{2} c e \,f^{5} h \,x^{2}-9009 a^{2} c \,f^{6} g \,x^{2}-429 a^{2} d \,e^{2} f^{4} h \,x^{2}-10296 a^{2} d e \,f^{5} g \,x^{2}-858 a b c \,e^{2} f^{4} h \,x^{2}-20592 a b c e \,f^{5} g \,x^{2}+468 a b d \,e^{3} f^{3} h \,x^{2}-858 a b d \,e^{2} f^{4} g \,x^{2}+234 b^{2} c \,e^{3} f^{3} h \,x^{2}-429 b^{2} c \,e^{2} f^{4} g \,x^{2}-144 b^{2} d \,e^{4} f^{2} h \,x^{2}+234 b^{2} d \,e^{3} f^{3} g \,x^{2}-1287 a^{2} c \,e^{2} f^{4} h x -18018 a^{2} c e \,f^{5} g x +572 a^{2} d \,e^{3} f^{3} h x -1287 a^{2} d \,e^{2} f^{4} g x +1144 a b c \,e^{3} f^{3} h x -2574 a b c \,e^{2} f^{4} g x -624 a b d \,e^{4} f^{2} h x +1144 a b d \,e^{3} f^{3} g x -312 b^{2} c \,e^{4} f^{2} h x +572 b^{2} c \,e^{3} f^{3} g x +192 b^{2} d \,e^{5} f h x -312 b^{2} d \,e^{4} f^{2} g x +2574 a^{2} c \,e^{3} f^{3} h -9009 a^{2} c \,e^{2} f^{4} g -1144 a^{2} d \,e^{4} f^{2} h +2574 a^{2} d \,e^{3} f^{3} g -2288 a b c \,e^{4} f^{2} h +5148 a b c \,e^{3} f^{3} g +1248 a b d \,e^{5} f h -2288 a b d \,e^{4} f^{2} g +624 b^{2} c \,e^{5} f h -1144 b^{2} c \,e^{4} f^{2} g -384 b^{2} d \,e^{6} h +624 b^{2} d \,e^{5} f g \right ) \sqrt {f x +e}}{45045 f^{5}}\)	\(823\)

Input:

int((b*x+a)^2*(d*x+c)*(f*x+e)^(3/2)*(h*x+g),x,method=_RETURNVERBOSE)
 

Output:

2/f^5*(1/13*d*h*b^2*(f*x+e)^(13/2)+1/11*((2*b*(a*f-b*e)*d+b^2*(c*f-d*e))*h 
+d*b^2*(-e*h+f*g))*(f*x+e)^(11/2)+1/9*(((a*f-b*e)^2*d+2*b*(a*f-b*e)*(c*f-d 
*e))*h+(2*b*(a*f-b*e)*d+b^2*(c*f-d*e))*(-e*h+f*g))*(f*x+e)^(9/2)+1/7*((a*f 
-b*e)^2*(c*f-d*e)*h+((a*f-b*e)^2*d+2*b*(a*f-b*e)*(c*f-d*e))*(-e*h+f*g))*(f 
*x+e)^(7/2)+1/5*(a*f-b*e)^2*(c*f-d*e)*(-e*h+f*g)*(f*x+e)^(5/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (229) = 458\).

Time = 0.09 (sec) , antiderivative size = 658, normalized size of antiderivative = 2.66 \[ \int (a+b x)^2 (c+d x) (e+f x)^{3/2} (g+h x) \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^2*(d*x+c)*(f*x+e)^(3/2)*(h*x+g),x, algorithm="fricas")
 

Output:

2/45045*(3465*b^2*d*f^6*h*x^6 + 315*(13*b^2*d*f^6*g + (14*b^2*d*e*f^5 + 13 
*(b^2*c + 2*a*b*d)*f^6)*h)*x^5 + 35*(13*(12*b^2*d*e*f^5 + 11*(b^2*c + 2*a* 
b*d)*f^6)*g + (3*b^2*d*e^2*f^4 + 156*(b^2*c + 2*a*b*d)*e*f^5 + 143*(2*a*b* 
c + a^2*d)*f^6)*h)*x^4 + 5*(13*(3*b^2*d*e^2*f^4 + 110*(b^2*c + 2*a*b*d)*e* 
f^5 + 99*(2*a*b*c + a^2*d)*f^6)*g - (24*b^2*d*e^3*f^3 - 1287*a^2*c*f^6 - 3 
9*(b^2*c + 2*a*b*d)*e^2*f^4 - 1430*(2*a*b*c + a^2*d)*e*f^5)*h)*x^3 - 3*(13 
*(6*b^2*d*e^3*f^3 - 231*a^2*c*f^6 - 11*(b^2*c + 2*a*b*d)*e^2*f^4 - 264*(2* 
a*b*c + a^2*d)*e*f^5)*g - (48*b^2*d*e^4*f^2 + 3432*a^2*c*e*f^5 - 78*(b^2*c 
 + 2*a*b*d)*e^3*f^3 + 143*(2*a*b*c + a^2*d)*e^2*f^4)*h)*x^2 - 13*(48*b^2*d 
*e^5*f - 693*a^2*c*e^2*f^4 - 88*(b^2*c + 2*a*b*d)*e^4*f^2 + 198*(2*a*b*c + 
 a^2*d)*e^3*f^3)*g + 2*(192*b^2*d*e^6 - 1287*a^2*c*e^3*f^3 - 312*(b^2*c + 
2*a*b*d)*e^5*f + 572*(2*a*b*c + a^2*d)*e^4*f^2)*h + (13*(24*b^2*d*e^4*f^2 
+ 1386*a^2*c*e*f^5 - 44*(b^2*c + 2*a*b*d)*e^3*f^3 + 99*(2*a*b*c + a^2*d)*e 
^2*f^4)*g - (192*b^2*d*e^5*f - 1287*a^2*c*e^2*f^4 - 312*(b^2*c + 2*a*b*d)* 
e^4*f^2 + 572*(2*a*b*c + a^2*d)*e^3*f^3)*h)*x)*sqrt(f*x + e)/f^5
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (262) = 524\).

Time = 1.77 (sec) , antiderivative size = 620, normalized size of antiderivative = 2.51 \[ \int (a+b x)^2 (c+d x) (e+f x)^{3/2} (g+h x) \, dx=\begin {cases} \frac {2 \left (\frac {b^{2} d h \left (e + f x\right )^{\frac {13}{2}}}{13 f^{4}} + \frac {\left (e + f x\right )^{\frac {11}{2}} \cdot \left (2 a b d f h + b^{2} c f h - 4 b^{2} d e h + b^{2} d f g\right )}{11 f^{4}} + \frac {\left (e + f x\right )^{\frac {9}{2}} \left (a^{2} d f^{2} h + 2 a b c f^{2} h - 6 a b d e f h + 2 a b d f^{2} g - 3 b^{2} c e f h + b^{2} c f^{2} g + 6 b^{2} d e^{2} h - 3 b^{2} d e f g\right )}{9 f^{4}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \left (a^{2} c f^{3} h - 2 a^{2} d e f^{2} h + a^{2} d f^{3} g - 4 a b c e f^{2} h + 2 a b c f^{3} g + 6 a b d e^{2} f h - 4 a b d e f^{2} g + 3 b^{2} c e^{2} f h - 2 b^{2} c e f^{2} g - 4 b^{2} d e^{3} h + 3 b^{2} d e^{2} f g\right )}{7 f^{4}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (- a^{2} c e f^{3} h + a^{2} c f^{4} g + a^{2} d e^{2} f^{2} h - a^{2} d e f^{3} g + 2 a b c e^{2} f^{2} h - 2 a b c e f^{3} g - 2 a b d e^{3} f h + 2 a b d e^{2} f^{2} g - b^{2} c e^{3} f h + b^{2} c e^{2} f^{2} g + b^{2} d e^{4} h - b^{2} d e^{3} f g\right )}{5 f^{4}}\right )}{f} & \text {for}\: f \neq 0 \\e^{\frac {3}{2}} \left (a^{2} c g x + \frac {b^{2} d h x^{5}}{5} + \frac {x^{4} \cdot \left (2 a b d h + b^{2} c h + b^{2} d g\right )}{4} + \frac {x^{3} \left (a^{2} d h + 2 a b c h + 2 a b d g + b^{2} c g\right )}{3} + \frac {x^{2} \left (a^{2} c h + a^{2} d g + 2 a b c g\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((b*x+a)**2*(d*x+c)*(f*x+e)**(3/2)*(h*x+g),x)
 

Output:

Piecewise((2*(b**2*d*h*(e + f*x)**(13/2)/(13*f**4) + (e + f*x)**(11/2)*(2* 
a*b*d*f*h + b**2*c*f*h - 4*b**2*d*e*h + b**2*d*f*g)/(11*f**4) + (e + f*x)* 
*(9/2)*(a**2*d*f**2*h + 2*a*b*c*f**2*h - 6*a*b*d*e*f*h + 2*a*b*d*f**2*g - 
3*b**2*c*e*f*h + b**2*c*f**2*g + 6*b**2*d*e**2*h - 3*b**2*d*e*f*g)/(9*f**4 
) + (e + f*x)**(7/2)*(a**2*c*f**3*h - 2*a**2*d*e*f**2*h + a**2*d*f**3*g - 
4*a*b*c*e*f**2*h + 2*a*b*c*f**3*g + 6*a*b*d*e**2*f*h - 4*a*b*d*e*f**2*g + 
3*b**2*c*e**2*f*h - 2*b**2*c*e*f**2*g - 4*b**2*d*e**3*h + 3*b**2*d*e**2*f* 
g)/(7*f**4) + (e + f*x)**(5/2)*(-a**2*c*e*f**3*h + a**2*c*f**4*g + a**2*d* 
e**2*f**2*h - a**2*d*e*f**3*g + 2*a*b*c*e**2*f**2*h - 2*a*b*c*e*f**3*g - 2 
*a*b*d*e**3*f*h + 2*a*b*d*e**2*f**2*g - b**2*c*e**3*f*h + b**2*c*e**2*f**2 
*g + b**2*d*e**4*h - b**2*d*e**3*f*g)/(5*f**4))/f, Ne(f, 0)), (e**(3/2)*(a 
**2*c*g*x + b**2*d*h*x**5/5 + x**4*(2*a*b*d*h + b**2*c*h + b**2*d*g)/4 + x 
**3*(a**2*d*h + 2*a*b*c*h + 2*a*b*d*g + b**2*c*g)/3 + x**2*(a**2*c*h + a** 
2*d*g + 2*a*b*c*g)/2), True))
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.52 \[ \int (a+b x)^2 (c+d x) (e+f x)^{3/2} (g+h x) \, dx=\frac {2 \, {\left (3465 \, {\left (f x + e\right )}^{\frac {13}{2}} b^{2} d h + 4095 \, {\left (b^{2} d f g - {\left (4 \, b^{2} d e - {\left (b^{2} c + 2 \, a b d\right )} f\right )} h\right )} {\left (f x + e\right )}^{\frac {11}{2}} - 5005 \, {\left ({\left (3 \, b^{2} d e f - {\left (b^{2} c + 2 \, a b d\right )} f^{2}\right )} g - {\left (6 \, b^{2} d e^{2} - 3 \, {\left (b^{2} c + 2 \, a b d\right )} e f + {\left (2 \, a b c + a^{2} d\right )} f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {9}{2}} + 6435 \, {\left ({\left (3 \, b^{2} d e^{2} f - 2 \, {\left (b^{2} c + 2 \, a b d\right )} e f^{2} + {\left (2 \, a b c + a^{2} d\right )} f^{3}\right )} g - {\left (4 \, b^{2} d e^{3} - a^{2} c f^{3} - 3 \, {\left (b^{2} c + 2 \, a b d\right )} e^{2} f + 2 \, {\left (2 \, a b c + a^{2} d\right )} e f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {7}{2}} - 9009 \, {\left ({\left (b^{2} d e^{3} f - a^{2} c f^{4} - {\left (b^{2} c + 2 \, a b d\right )} e^{2} f^{2} + {\left (2 \, a b c + a^{2} d\right )} e f^{3}\right )} g - {\left (b^{2} d e^{4} - a^{2} c e f^{3} - {\left (b^{2} c + 2 \, a b d\right )} e^{3} f + {\left (2 \, a b c + a^{2} d\right )} e^{2} f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {5}{2}}\right )}}{45045 \, f^{5}} \] Input:

integrate((b*x+a)^2*(d*x+c)*(f*x+e)^(3/2)*(h*x+g),x, algorithm="maxima")
 

Output:

2/45045*(3465*(f*x + e)^(13/2)*b^2*d*h + 4095*(b^2*d*f*g - (4*b^2*d*e - (b 
^2*c + 2*a*b*d)*f)*h)*(f*x + e)^(11/2) - 5005*((3*b^2*d*e*f - (b^2*c + 2*a 
*b*d)*f^2)*g - (6*b^2*d*e^2 - 3*(b^2*c + 2*a*b*d)*e*f + (2*a*b*c + a^2*d)* 
f^2)*h)*(f*x + e)^(9/2) + 6435*((3*b^2*d*e^2*f - 2*(b^2*c + 2*a*b*d)*e*f^2 
 + (2*a*b*c + a^2*d)*f^3)*g - (4*b^2*d*e^3 - a^2*c*f^3 - 3*(b^2*c + 2*a*b* 
d)*e^2*f + 2*(2*a*b*c + a^2*d)*e*f^2)*h)*(f*x + e)^(7/2) - 9009*((b^2*d*e^ 
3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*g - 
 (b^2*d*e^4 - a^2*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c + a^2*d)*e^ 
2*f^2)*h)*(f*x + e)^(5/2))/f^5
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1968 vs. \(2 (229) = 458\).

Time = 0.15 (sec) , antiderivative size = 1968, normalized size of antiderivative = 7.97 \[ \int (a+b x)^2 (c+d x) (e+f x)^{3/2} (g+h x) \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^2*(d*x+c)*(f*x+e)^(3/2)*(h*x+g),x, algorithm="giac")
 

Output:

2/45045*(45045*sqrt(f*x + e)*a^2*c*e^2*g + 30030*((f*x + e)^(3/2) - 3*sqrt 
(f*x + e)*e)*a^2*c*e*g + 30030*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a*b*c 
*e^2*g/f + 15015*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a^2*d*e^2*g/f + 150 
15*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a^2*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^2*c*g + 3003*(3*(f 
*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*b^2*c*e^2*g/f 
^2 + 6006*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2 
)*a*b*d*e^2*g/f^2 + 12012*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*s 
qrt(f*x + e)*e^2)*a*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^2*d*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*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^2*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^2* 
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^2*d*e^2*g/f^3 + 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)*b^ 
2*c*e*g/f^2 + 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)*a*b*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*b*c*g/f + 1287*(5*(f*x + e)^(7/2) - 21*(f*x + e)^(5/2)*e + 35*(f*x + ...
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.05 \[ \int (a+b x)^2 (c+d x) (e+f x)^{3/2} (g+h x) \, dx=\frac {{\left (e+f\,x\right )}^{11/2}\,\left (2\,b^2\,c\,f\,h-8\,b^2\,d\,e\,h+2\,b^2\,d\,f\,g+4\,a\,b\,d\,f\,h\right )}{11\,f^5}+\frac {{\left (e+f\,x\right )}^{9/2}\,\left (2\,b^2\,c\,f^2\,g+2\,a^2\,d\,f^2\,h+12\,b^2\,d\,e^2\,h+4\,a\,b\,c\,f^2\,h+4\,a\,b\,d\,f^2\,g-6\,b^2\,c\,e\,f\,h-6\,b^2\,d\,e\,f\,g-12\,a\,b\,d\,e\,f\,h\right )}{9\,f^5}+\frac {2\,{\left (e+f\,x\right )}^{7/2}\,\left (a\,f-b\,e\right )\,\left (a\,c\,f^2\,h+a\,d\,f^2\,g+2\,b\,c\,f^2\,g+4\,b\,d\,e^2\,h-2\,a\,d\,e\,f\,h-3\,b\,c\,e\,f\,h-3\,b\,d\,e\,f\,g\right )}{7\,f^5}+\frac {2\,b^2\,d\,h\,{\left (e+f\,x\right )}^{13/2}}{13\,f^5}-\frac {2\,{\left (e+f\,x\right )}^{5/2}\,{\left (a\,f-b\,e\right )}^2\,\left (c\,f-d\,e\right )\,\left (e\,h-f\,g\right )}{5\,f^5} \] Input:

int((e + f*x)^(3/2)*(g + h*x)*(a + b*x)^2*(c + d*x),x)
 

Output:

((e + f*x)^(11/2)*(2*b^2*c*f*h - 8*b^2*d*e*h + 2*b^2*d*f*g + 4*a*b*d*f*h)) 
/(11*f^5) + ((e + f*x)^(9/2)*(2*b^2*c*f^2*g + 2*a^2*d*f^2*h + 12*b^2*d*e^2 
*h + 4*a*b*c*f^2*h + 4*a*b*d*f^2*g - 6*b^2*c*e*f*h - 6*b^2*d*e*f*g - 12*a* 
b*d*e*f*h))/(9*f^5) + (2*(e + f*x)^(7/2)*(a*f - b*e)*(a*c*f^2*h + a*d*f^2* 
g + 2*b*c*f^2*g + 4*b*d*e^2*h - 2*a*d*e*f*h - 3*b*c*e*f*h - 3*b*d*e*f*g))/ 
(7*f^5) + (2*b^2*d*h*(e + f*x)^(13/2))/(13*f^5) - (2*(e + f*x)^(5/2)*(a*f 
- b*e)^2*(c*f - d*e)*(e*h - f*g))/(5*f^5)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 821, normalized size of antiderivative = 3.32 \[ \int (a+b x)^2 (c+d x) (e+f x)^{3/2} (g+h x) \, dx=\frac {2 \sqrt {f x +e}\, \left (3465 b^{2} d \,f^{6} h \,x^{6}+8190 a b d \,f^{6} h \,x^{5}+4095 b^{2} c \,f^{6} h \,x^{5}+4410 b^{2} d e \,f^{5} h \,x^{5}+4095 b^{2} d \,f^{6} g \,x^{5}+5005 a^{2} d \,f^{6} h \,x^{4}+10010 a b c \,f^{6} h \,x^{4}+10920 a b d e \,f^{5} h \,x^{4}+10010 a b d \,f^{6} g \,x^{4}+5460 b^{2} c e \,f^{5} h \,x^{4}+5005 b^{2} c \,f^{6} g \,x^{4}+105 b^{2} d \,e^{2} f^{4} h \,x^{4}+5460 b^{2} d e \,f^{5} g \,x^{4}+6435 a^{2} c \,f^{6} h \,x^{3}+7150 a^{2} d e \,f^{5} h \,x^{3}+6435 a^{2} d \,f^{6} g \,x^{3}+14300 a b c e \,f^{5} h \,x^{3}+12870 a b c \,f^{6} g \,x^{3}+390 a b d \,e^{2} f^{4} h \,x^{3}+14300 a b d e \,f^{5} g \,x^{3}+195 b^{2} c \,e^{2} f^{4} h \,x^{3}+7150 b^{2} c e \,f^{5} g \,x^{3}-120 b^{2} d \,e^{3} f^{3} h \,x^{3}+195 b^{2} d \,e^{2} f^{4} g \,x^{3}+10296 a^{2} c e \,f^{5} h \,x^{2}+9009 a^{2} c \,f^{6} g \,x^{2}+429 a^{2} d \,e^{2} f^{4} h \,x^{2}+10296 a^{2} d e \,f^{5} g \,x^{2}+858 a b c \,e^{2} f^{4} h \,x^{2}+20592 a b c e \,f^{5} g \,x^{2}-468 a b d \,e^{3} f^{3} h \,x^{2}+858 a b d \,e^{2} f^{4} g \,x^{2}-234 b^{2} c \,e^{3} f^{3} h \,x^{2}+429 b^{2} c \,e^{2} f^{4} g \,x^{2}+144 b^{2} d \,e^{4} f^{2} h \,x^{2}-234 b^{2} d \,e^{3} f^{3} g \,x^{2}+1287 a^{2} c \,e^{2} f^{4} h x +18018 a^{2} c e \,f^{5} g x -572 a^{2} d \,e^{3} f^{3} h x +1287 a^{2} d \,e^{2} f^{4} g x -1144 a b c \,e^{3} f^{3} h x +2574 a b c \,e^{2} f^{4} g x +624 a b d \,e^{4} f^{2} h x -1144 a b d \,e^{3} f^{3} g x +312 b^{2} c \,e^{4} f^{2} h x -572 b^{2} c \,e^{3} f^{3} g x -192 b^{2} d \,e^{5} f h x +312 b^{2} d \,e^{4} f^{2} g x -2574 a^{2} c \,e^{3} f^{3} h +9009 a^{2} c \,e^{2} f^{4} g +1144 a^{2} d \,e^{4} f^{2} h -2574 a^{2} d \,e^{3} f^{3} g +2288 a b c \,e^{4} f^{2} h -5148 a b c \,e^{3} f^{3} g -1248 a b d \,e^{5} f h +2288 a b d \,e^{4} f^{2} g -624 b^{2} c \,e^{5} f h +1144 b^{2} c \,e^{4} f^{2} g +384 b^{2} d \,e^{6} h -624 b^{2} d \,e^{5} f g \right )}{45045 f^{5}} \] Input:

int((b*x+a)^2*(d*x+c)*(f*x+e)^(3/2)*(h*x+g),x)
 

Output:

(2*sqrt(e + f*x)*( - 2574*a**2*c*e**3*f**3*h + 9009*a**2*c*e**2*f**4*g + 1 
287*a**2*c*e**2*f**4*h*x + 18018*a**2*c*e*f**5*g*x + 10296*a**2*c*e*f**5*h 
*x**2 + 9009*a**2*c*f**6*g*x**2 + 6435*a**2*c*f**6*h*x**3 + 1144*a**2*d*e* 
*4*f**2*h - 2574*a**2*d*e**3*f**3*g - 572*a**2*d*e**3*f**3*h*x + 1287*a**2 
*d*e**2*f**4*g*x + 429*a**2*d*e**2*f**4*h*x**2 + 10296*a**2*d*e*f**5*g*x** 
2 + 7150*a**2*d*e*f**5*h*x**3 + 6435*a**2*d*f**6*g*x**3 + 5005*a**2*d*f**6 
*h*x**4 + 2288*a*b*c*e**4*f**2*h - 5148*a*b*c*e**3*f**3*g - 1144*a*b*c*e** 
3*f**3*h*x + 2574*a*b*c*e**2*f**4*g*x + 858*a*b*c*e**2*f**4*h*x**2 + 20592 
*a*b*c*e*f**5*g*x**2 + 14300*a*b*c*e*f**5*h*x**3 + 12870*a*b*c*f**6*g*x**3 
 + 10010*a*b*c*f**6*h*x**4 - 1248*a*b*d*e**5*f*h + 2288*a*b*d*e**4*f**2*g 
+ 624*a*b*d*e**4*f**2*h*x - 1144*a*b*d*e**3*f**3*g*x - 468*a*b*d*e**3*f**3 
*h*x**2 + 858*a*b*d*e**2*f**4*g*x**2 + 390*a*b*d*e**2*f**4*h*x**3 + 14300* 
a*b*d*e*f**5*g*x**3 + 10920*a*b*d*e*f**5*h*x**4 + 10010*a*b*d*f**6*g*x**4 
+ 8190*a*b*d*f**6*h*x**5 - 624*b**2*c*e**5*f*h + 1144*b**2*c*e**4*f**2*g + 
 312*b**2*c*e**4*f**2*h*x - 572*b**2*c*e**3*f**3*g*x - 234*b**2*c*e**3*f** 
3*h*x**2 + 429*b**2*c*e**2*f**4*g*x**2 + 195*b**2*c*e**2*f**4*h*x**3 + 715 
0*b**2*c*e*f**5*g*x**3 + 5460*b**2*c*e*f**5*h*x**4 + 5005*b**2*c*f**6*g*x* 
*4 + 4095*b**2*c*f**6*h*x**5 + 384*b**2*d*e**6*h - 624*b**2*d*e**5*f*g - 1 
92*b**2*d*e**5*f*h*x + 312*b**2*d*e**4*f**2*g*x + 144*b**2*d*e**4*f**2*h*x 
**2 - 234*b**2*d*e**3*f**3*g*x**2 - 120*b**2*d*e**3*f**3*h*x**3 + 195*b...