\(\int (d+e x)^3 (f+g x)^2 (70 e^2 f^2-60 d e f g+18 d^2 g^2-8 e g (-10 e f+3 d g) x+28 e^2 g^2 x^2)^2 \, dx\) [753]

Optimal result

Integrand size = 65, antiderivative size = 64 \[ \int (d+e x)^3 (f+g x)^2 \left (70 e^2 f^2-60 d e f g+18 d^2 g^2-8 e g (-10 e f+3 d g) x+28 e^2 g^2 x^2\right )^2 \, dx=\frac {(d+e x)^4 \left (35 e^2 f^2-30 d e f g+9 d^2 g^2+4 e g (10 e f-3 d g) x+14 e^2 g^2 x^2\right )^3}{35 e^3} \] Output:

1/35*(e*x+d)^4*(35*e^2*f^2-30*d*e*f*g+9*d^2*g^2+4*e*g*(-3*d*g+10*e*f)*x+14 
*e^2*g^2*x^2)^3/e^3
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(403\) vs. \(2(64)=128\).

Time = 0.18 (sec) , antiderivative size = 403, normalized size of antiderivative = 6.30 \[ \int (d+e x)^3 (f+g x)^2 \left (70 e^2 f^2-60 d e f g+18 d^2 g^2-8 e g (-10 e f+3 d g) x+28 e^2 g^2 x^2\right )^2 \, dx=\frac {1}{35} x \left (3780 d^7 g^4 \left (3 f^2+3 f g x+g^2 x^2\right )+10 d e^6 x^2 \left (14 f^2+7 f g x+2 g^2 x^2\right ) \left (35 f^2+40 f g x+14 g^2 x^2\right )^2+e^7 x^3 \left (35 f^2+40 f g x+14 g^2 x^2\right )^3+945 d^6 e g^3 \left (-80 f^3-78 f^2 g x-24 f g^2 x^2+g^3 x^3\right )+126 d^5 e^2 g^2 \left (1700 f^4+1600 f^3 g x+430 f^2 g^2 x^2-55 f g^3 x^3-2 g^4 x^4\right )+35 d^4 e^3 g \left (-8400 f^5-5700 f^4 g x+2000 f^3 g^2 x^2+3435 f^2 g^3 x^3+1176 f g^4 x^4+190 g^5 x^5\right )+15 d^2 e^5 x \left (17150 f^6+29400 f^5 g x+17325 f^4 g^2 x^2+1120 f^3 g^3 x^3-2604 f^2 g^4 x^4-864 f g^5 x^5-28 g^6 x^6\right )+20 d^3 e^4 \left (8575 f^6-3675 f^5 g x-19425 f^4 g^2 x^2-15925 f^3 g^3 x^3-4746 f^2 g^4 x^4-210 f g^5 x^5+160 g^6 x^6\right )\right ) \] Input:

Integrate[(d + e*x)^3*(f + g*x)^2*(70*e^2*f^2 - 60*d*e*f*g + 18*d^2*g^2 - 
8*e*g*(-10*e*f + 3*d*g)*x + 28*e^2*g^2*x^2)^2,x]
 

Output:

(x*(3780*d^7*g^4*(3*f^2 + 3*f*g*x + g^2*x^2) + 10*d*e^6*x^2*(14*f^2 + 7*f* 
g*x + 2*g^2*x^2)*(35*f^2 + 40*f*g*x + 14*g^2*x^2)^2 + e^7*x^3*(35*f^2 + 40 
*f*g*x + 14*g^2*x^2)^3 + 945*d^6*e*g^3*(-80*f^3 - 78*f^2*g*x - 24*f*g^2*x^ 
2 + g^3*x^3) + 126*d^5*e^2*g^2*(1700*f^4 + 1600*f^3*g*x + 430*f^2*g^2*x^2 
- 55*f*g^3*x^3 - 2*g^4*x^4) + 35*d^4*e^3*g*(-8400*f^5 - 5700*f^4*g*x + 200 
0*f^3*g^2*x^2 + 3435*f^2*g^3*x^3 + 1176*f*g^4*x^4 + 190*g^5*x^5) + 15*d^2* 
e^5*x*(17150*f^6 + 29400*f^5*g*x + 17325*f^4*g^2*x^2 + 1120*f^3*g^3*x^3 - 
2604*f^2*g^4*x^4 - 864*f*g^5*x^5 - 28*g^6*x^6) + 20*d^3*e^4*(8575*f^6 - 36 
75*f^5*g*x - 19425*f^4*g^2*x^2 - 15925*f^3*g^3*x^3 - 4746*f^2*g^4*x^4 - 21 
0*f*g^5*x^5 + 160*g^6*x^6)))/35
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(163\) vs. \(2(64)=128\).

Time = 1.13 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.55, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {1195, 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.

\(\displaystyle \int (d+e x)^3 (f+g x)^2 \left (18 d^2 g^2-8 e g x (3 d g-10 e f)-60 d e f g+70 e^2 f^2+28 e^2 g^2 x^2\right )^2 \, dx\)

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (\frac {6048 g^5 (d+e x)^8 (e f-d g)}{e^2}+\frac {20064 g^4 (d+e x)^7 (e f-d g)^2}{e^2}+\frac {36320 g^3 (d+e x)^6 (e f-d g)^3}{e^2}+\frac {37620 g^2 (d+e x)^5 (e f-d g)^4}{e^2}+\frac {21000 g (d+e x)^4 (e f-d g)^5}{e^2}+\frac {4900 (d+e x)^3 (e f-d g)^6}{e^2}+\frac {784 g^6 (d+e x)^9}{e^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {672 g^5 (d+e x)^9 (e f-d g)}{e^3}+\frac {2508 g^4 (d+e x)^8 (e f-d g)^2}{e^3}+\frac {36320 g^3 (d+e x)^7 (e f-d g)^3}{7 e^3}+\frac {6270 g^2 (d+e x)^6 (e f-d g)^4}{e^3}+\frac {4200 g (d+e x)^5 (e f-d g)^5}{e^3}+\frac {1225 (d+e x)^4 (e f-d g)^6}{e^3}+\frac {392 g^6 (d+e x)^{10}}{5 e^3}\)

Input:

Int[(d + e*x)^3*(f + g*x)^2*(70*e^2*f^2 - 60*d*e*f*g + 18*d^2*g^2 - 8*e*g* 
(-10*e*f + 3*d*g)*x + 28*e^2*g^2*x^2)^2,x]
 

Output:

(1225*(e*f - d*g)^6*(d + e*x)^4)/e^3 + (4200*g*(e*f - d*g)^5*(d + e*x)^5)/ 
e^3 + (6270*g^2*(e*f - d*g)^4*(d + e*x)^6)/e^3 + (36320*g^3*(e*f - d*g)^3* 
(d + e*x)^7)/(7*e^3) + (2508*g^4*(e*f - d*g)^2*(d + e*x)^8)/e^3 + (672*g^5 
*(e*f - d*g)*(d + e*x)^9)/e^3 + (392*g^6*(d + e*x)^10)/(5*e^3)
 

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(577\) vs. \(2(62)=124\).

Time = 3.31 (sec) , antiderivative size = 578, normalized size of antiderivative = 9.03

method	result	size
norman	\(\left (\frac {640}{7} d^{3} e^{4} g^{6}-\frac {2592}{7} d^{2} e^{5} f \,g^{5}+\frac {31488}{7} d \,e^{6} f^{2} g^{4}+\frac {36320}{7} e^{7} f^{3} g^{3}\right ) x^{7}+\left (-\frac {36}{5} d^{5} e^{2} g^{6}+1176 d^{4} e^{3} f \,g^{5}-2712 d^{3} f^{2} e^{4} g^{4}+480 d^{2} e^{5} f^{3} g^{3}+16620 d \,e^{6} f^{4} g^{2}+4200 e^{7} f^{5} g \right ) x^{5}+\left (112 d \,e^{6} g^{6}+672 e^{7} f \,g^{5}\right ) x^{9}+\left (-12 d^{2} e^{5} g^{6}+1032 d \,e^{6} f \,g^{5}+2508 e^{7} f^{2} g^{4}\right ) x^{8}+\left (190 d^{4} e^{3} g^{6}-120 d^{3} e^{4} f \,g^{5}-1116 d^{2} e^{5} f^{2} g^{4}+11240 d \,e^{6} f^{3} g^{3}+6270 e^{7} f^{4} g^{2}\right ) x^{6}+\left (324 d^{7} f^{2} g^{4}-2160 d^{6} e \,f^{3} g^{3}+6120 d^{5} e^{2} f^{4} g^{2}-8400 d^{4} e^{3} f^{5} g +4900 d^{3} e^{4} f^{6}\right ) x +\left (324 d^{7} f \,g^{5}-2106 d^{6} e \,f^{2} g^{4}+5760 d^{5} e^{2} f^{3} g^{3}-5700 d^{4} e^{3} f^{4} g^{2}-2100 d^{3} e^{4} f^{5} g +7350 d^{2} e^{5} f^{6}\right ) x^{2}+\left (27 d^{6} e \,g^{6}-198 d^{5} e^{2} f \,g^{5}+3435 d^{4} e^{3} f^{2} g^{4}-9100 d^{3} e^{4} f^{3} g^{3}+7425 d^{2} e^{5} f^{4} g^{2}+13650 d \,e^{6} f^{5} g +1225 e^{7} f^{6}\right ) x^{4}+\left (108 d^{7} g^{6}-648 d^{6} e f \,g^{5}+1548 d^{5} e^{2} f^{2} g^{4}+2000 d^{4} e^{3} f^{3} g^{3}-11100 d^{3} e^{4} f^{4} g^{2}+12600 d^{2} e^{5} f^{5} g +4900 d \,e^{6} f^{6}\right ) x^{3}+\frac {392 e^{7} g^{6} x^{10}}{5}\)	\(578\)
gosper	\(\frac {x \left (2744 e^{7} g^{6} x^{9}+3920 d \,e^{6} g^{6} x^{8}+23520 e^{7} f \,g^{5} x^{8}-420 d^{2} e^{5} g^{6} x^{7}+36120 d \,e^{6} f \,g^{5} x^{7}+87780 e^{7} f^{2} g^{4} x^{7}+3200 x^{6} d^{3} e^{4} g^{6}-12960 x^{6} d^{2} e^{5} f \,g^{5}+157440 x^{6} d \,e^{6} f^{2} g^{4}+181600 x^{6} e^{7} f^{3} g^{3}+6650 x^{5} d^{4} e^{3} g^{6}-4200 x^{5} d^{3} e^{4} f \,g^{5}-39060 x^{5} d^{2} e^{5} f^{2} g^{4}+393400 x^{5} d \,e^{6} f^{3} g^{3}+219450 x^{5} e^{7} f^{4} g^{2}-252 x^{4} d^{5} e^{2} g^{6}+41160 x^{4} d^{4} e^{3} f \,g^{5}-94920 x^{4} d^{3} f^{2} e^{4} g^{4}+16800 x^{4} d^{2} e^{5} f^{3} g^{3}+581700 x^{4} d \,e^{6} f^{4} g^{2}+147000 x^{4} e^{7} f^{5} g +945 x^{3} d^{6} e \,g^{6}-6930 x^{3} d^{5} e^{2} f \,g^{5}+120225 x^{3} d^{4} e^{3} f^{2} g^{4}-318500 x^{3} d^{3} e^{4} f^{3} g^{3}+259875 x^{3} d^{2} e^{5} f^{4} g^{2}+477750 x^{3} d \,e^{6} f^{5} g +42875 x^{3} e^{7} f^{6}+3780 d^{7} g^{6} x^{2}-22680 d^{6} e f \,g^{5} x^{2}+54180 d^{5} e^{2} f^{2} g^{4} x^{2}+70000 d^{4} e^{3} f^{3} g^{3} x^{2}-388500 d^{3} e^{4} f^{4} g^{2} x^{2}+441000 d^{2} e^{5} f^{5} g \,x^{2}+171500 d \,e^{6} f^{6} x^{2}+11340 x \,d^{7} f \,g^{5}-73710 x \,d^{6} e \,f^{2} g^{4}+201600 x \,d^{5} e^{2} f^{3} g^{3}-199500 x \,d^{4} e^{3} f^{4} g^{2}-73500 x \,d^{3} e^{4} f^{5} g +257250 x \,d^{2} e^{5} f^{6}+11340 d^{7} f^{2} g^{4}-75600 d^{6} e \,f^{3} g^{3}+214200 d^{5} e^{2} f^{4} g^{2}-294000 d^{4} e^{3} f^{5} g +171500 d^{3} e^{4} f^{6}\right )}{35}\)	\(646\)
risch	\(\frac {640}{7} x^{7} d^{3} e^{4} g^{6}+\frac {36320}{7} x^{7} e^{7} f^{3} g^{3}+13650 d \,e^{6} f^{5} g \,x^{4}-648 d^{6} e f \,g^{5} x^{3}+1548 d^{5} e^{2} f^{2} g^{4} x^{3}+2000 d^{4} e^{3} f^{3} g^{3} x^{3}-11100 d^{3} e^{4} f^{4} g^{2} x^{3}+12600 d^{2} e^{5} f^{5} g \,x^{3}-2106 d^{6} e \,f^{2} g^{4} x^{2}+5760 d^{5} e^{2} f^{3} g^{3} x^{2}-5700 d^{4} e^{3} f^{4} g^{2} x^{2}-2100 d^{3} e^{4} f^{5} g \,x^{2}-2160 d^{6} e \,f^{3} g^{3} x +6120 d^{5} e^{2} f^{4} g^{2} x -8400 d^{4} e^{3} f^{5} g x -\frac {36}{5} x^{5} d^{5} e^{2} g^{6}+4200 x^{5} e^{7} f^{5} g +112 d \,e^{6} g^{6} x^{9}+672 e^{7} f \,g^{5} x^{9}-12 d^{2} e^{5} g^{6} x^{8}+2508 e^{7} f^{2} g^{4} x^{8}+190 d^{4} e^{3} g^{6} x^{6}+6270 e^{7} f^{4} g^{2} x^{6}+27 d^{6} e \,g^{6} x^{4}+4900 d \,e^{6} f^{6} x^{3}+324 d^{7} f \,g^{5} x^{2}+7350 d^{2} e^{5} f^{6} x^{2}+324 d^{7} f^{2} g^{4} x +4900 d^{3} e^{4} f^{6} x +\frac {392}{5} e^{7} g^{6} x^{10}+1032 d \,e^{6} f \,g^{5} x^{8}-\frac {2592}{7} x^{7} d^{2} e^{5} f \,g^{5}+\frac {31488}{7} x^{7} d \,e^{6} f^{2} g^{4}+1176 x^{5} d^{4} e^{3} f \,g^{5}-2712 x^{5} d^{3} f^{2} e^{4} g^{4}+480 x^{5} d^{2} e^{5} f^{3} g^{3}+16620 x^{5} d \,e^{6} f^{4} g^{2}-120 d^{3} e^{4} f \,g^{5} x^{6}-1116 d^{2} e^{5} f^{2} g^{4} x^{6}+11240 d \,e^{6} f^{3} g^{3} x^{6}-198 d^{5} e^{2} f \,g^{5} x^{4}+3435 d^{4} e^{3} f^{2} g^{4} x^{4}-9100 d^{3} e^{4} f^{3} g^{3} x^{4}+7425 d^{2} e^{5} f^{4} g^{2} x^{4}+1225 e^{7} f^{6} x^{4}+108 d^{7} g^{6} x^{3}\)	\(660\)
parallelrisch	\(\frac {640}{7} x^{7} d^{3} e^{4} g^{6}+\frac {36320}{7} x^{7} e^{7} f^{3} g^{3}+13650 d \,e^{6} f^{5} g \,x^{4}-648 d^{6} e f \,g^{5} x^{3}+1548 d^{5} e^{2} f^{2} g^{4} x^{3}+2000 d^{4} e^{3} f^{3} g^{3} x^{3}-11100 d^{3} e^{4} f^{4} g^{2} x^{3}+12600 d^{2} e^{5} f^{5} g \,x^{3}-2106 d^{6} e \,f^{2} g^{4} x^{2}+5760 d^{5} e^{2} f^{3} g^{3} x^{2}-5700 d^{4} e^{3} f^{4} g^{2} x^{2}-2100 d^{3} e^{4} f^{5} g \,x^{2}-2160 d^{6} e \,f^{3} g^{3} x +6120 d^{5} e^{2} f^{4} g^{2} x -8400 d^{4} e^{3} f^{5} g x -\frac {36}{5} x^{5} d^{5} e^{2} g^{6}+4200 x^{5} e^{7} f^{5} g +112 d \,e^{6} g^{6} x^{9}+672 e^{7} f \,g^{5} x^{9}-12 d^{2} e^{5} g^{6} x^{8}+2508 e^{7} f^{2} g^{4} x^{8}+190 d^{4} e^{3} g^{6} x^{6}+6270 e^{7} f^{4} g^{2} x^{6}+27 d^{6} e \,g^{6} x^{4}+4900 d \,e^{6} f^{6} x^{3}+324 d^{7} f \,g^{5} x^{2}+7350 d^{2} e^{5} f^{6} x^{2}+324 d^{7} f^{2} g^{4} x +4900 d^{3} e^{4} f^{6} x +\frac {392}{5} e^{7} g^{6} x^{10}+1032 d \,e^{6} f \,g^{5} x^{8}-\frac {2592}{7} x^{7} d^{2} e^{5} f \,g^{5}+\frac {31488}{7} x^{7} d \,e^{6} f^{2} g^{4}+1176 x^{5} d^{4} e^{3} f \,g^{5}-2712 x^{5} d^{3} f^{2} e^{4} g^{4}+480 x^{5} d^{2} e^{5} f^{3} g^{3}+16620 x^{5} d \,e^{6} f^{4} g^{2}-120 d^{3} e^{4} f \,g^{5} x^{6}-1116 d^{2} e^{5} f^{2} g^{4} x^{6}+11240 d \,e^{6} f^{3} g^{3} x^{6}-198 d^{5} e^{2} f \,g^{5} x^{4}+3435 d^{4} e^{3} f^{2} g^{4} x^{4}-9100 d^{3} e^{4} f^{3} g^{3} x^{4}+7425 d^{2} e^{5} f^{4} g^{2} x^{4}+1225 e^{7} f^{6} x^{4}+108 d^{7} g^{6} x^{3}\)	\(660\)
orering	\(\frac {x \left (2744 e^{7} g^{6} x^{9}+3920 d \,e^{6} g^{6} x^{8}+23520 e^{7} f \,g^{5} x^{8}-420 d^{2} e^{5} g^{6} x^{7}+36120 d \,e^{6} f \,g^{5} x^{7}+87780 e^{7} f^{2} g^{4} x^{7}+3200 x^{6} d^{3} e^{4} g^{6}-12960 x^{6} d^{2} e^{5} f \,g^{5}+157440 x^{6} d \,e^{6} f^{2} g^{4}+181600 x^{6} e^{7} f^{3} g^{3}+6650 x^{5} d^{4} e^{3} g^{6}-4200 x^{5} d^{3} e^{4} f \,g^{5}-39060 x^{5} d^{2} e^{5} f^{2} g^{4}+393400 x^{5} d \,e^{6} f^{3} g^{3}+219450 x^{5} e^{7} f^{4} g^{2}-252 x^{4} d^{5} e^{2} g^{6}+41160 x^{4} d^{4} e^{3} f \,g^{5}-94920 x^{4} d^{3} f^{2} e^{4} g^{4}+16800 x^{4} d^{2} e^{5} f^{3} g^{3}+581700 x^{4} d \,e^{6} f^{4} g^{2}+147000 x^{4} e^{7} f^{5} g +945 x^{3} d^{6} e \,g^{6}-6930 x^{3} d^{5} e^{2} f \,g^{5}+120225 x^{3} d^{4} e^{3} f^{2} g^{4}-318500 x^{3} d^{3} e^{4} f^{3} g^{3}+259875 x^{3} d^{2} e^{5} f^{4} g^{2}+477750 x^{3} d \,e^{6} f^{5} g +42875 x^{3} e^{7} f^{6}+3780 d^{7} g^{6} x^{2}-22680 d^{6} e f \,g^{5} x^{2}+54180 d^{5} e^{2} f^{2} g^{4} x^{2}+70000 d^{4} e^{3} f^{3} g^{3} x^{2}-388500 d^{3} e^{4} f^{4} g^{2} x^{2}+441000 d^{2} e^{5} f^{5} g \,x^{2}+171500 d \,e^{6} f^{6} x^{2}+11340 x \,d^{7} f \,g^{5}-73710 x \,d^{6} e \,f^{2} g^{4}+201600 x \,d^{5} e^{2} f^{3} g^{3}-199500 x \,d^{4} e^{3} f^{4} g^{2}-73500 x \,d^{3} e^{4} f^{5} g +257250 x \,d^{2} e^{5} f^{6}+11340 d^{7} f^{2} g^{4}-75600 d^{6} e \,f^{3} g^{3}+214200 d^{5} e^{2} f^{4} g^{2}-294000 d^{4} e^{3} f^{5} g +171500 d^{3} e^{4} f^{6}\right ) \left (70 e^{2} f^{2}-60 d e f g +18 d^{2} g^{2}-8 e g \left (3 d g -10 e f \right ) x +28 e^{2} g^{2} x^{2}\right )^{2}}{140 \left (14 e^{2} g^{2} x^{2}-12 d e \,g^{2} x +40 e^{2} f g x +9 d^{2} g^{2}-30 d e f g +35 e^{2} f^{2}\right )^{2}}\)	\(748\)
default	\(\text {Expression too large to display}\)	\(1354\)

Input:

int((e*x+d)^3*(g*x+f)^2*(70*e^2*f^2-60*d*e*f*g+18*d^2*g^2-8*e*g*(3*d*g-10* 
e*f)*x+28*e^2*g^2*x^2)^2,x,method=_RETURNVERBOSE)
 

Output:

(640/7*d^3*e^4*g^6-2592/7*d^2*e^5*f*g^5+31488/7*d*e^6*f^2*g^4+36320/7*e^7* 
f^3*g^3)*x^7+(-36/5*d^5*e^2*g^6+1176*d^4*e^3*f*g^5-2712*d^3*f^2*e^4*g^4+48 
0*d^2*e^5*f^3*g^3+16620*d*e^6*f^4*g^2+4200*e^7*f^5*g)*x^5+(112*d*e^6*g^6+6 
72*e^7*f*g^5)*x^9+(-12*d^2*e^5*g^6+1032*d*e^6*f*g^5+2508*e^7*f^2*g^4)*x^8+ 
(190*d^4*e^3*g^6-120*d^3*e^4*f*g^5-1116*d^2*e^5*f^2*g^4+11240*d*e^6*f^3*g^ 
3+6270*e^7*f^4*g^2)*x^6+(324*d^7*f^2*g^4-2160*d^6*e*f^3*g^3+6120*d^5*e^2*f 
^4*g^2-8400*d^4*e^3*f^5*g+4900*d^3*e^4*f^6)*x+(324*d^7*f*g^5-2106*d^6*e*f^ 
2*g^4+5760*d^5*e^2*f^3*g^3-5700*d^4*e^3*f^4*g^2-2100*d^3*e^4*f^5*g+7350*d^ 
2*e^5*f^6)*x^2+(27*d^6*e*g^6-198*d^5*e^2*f*g^5+3435*d^4*e^3*f^2*g^4-9100*d 
^3*e^4*f^3*g^3+7425*d^2*e^5*f^4*g^2+13650*d*e^6*f^5*g+1225*e^7*f^6)*x^4+(1 
08*d^7*g^6-648*d^6*e*f*g^5+1548*d^5*e^2*f^2*g^4+2000*d^4*e^3*f^3*g^3-11100 
*d^3*e^4*f^4*g^2+12600*d^2*e^5*f^5*g+4900*d*e^6*f^6)*x^3+392/5*e^7*g^6*x^1 
0
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (62) = 124\).

Time = 0.08 (sec) , antiderivative size = 584, normalized size of antiderivative = 9.12 \[ \int (d+e x)^3 (f+g x)^2 \left (70 e^2 f^2-60 d e f g+18 d^2 g^2-8 e g (-10 e f+3 d g) x+28 e^2 g^2 x^2\right )^2 \, dx=\frac {392}{5} \, e^{7} g^{6} x^{10} + 112 \, {\left (6 \, e^{7} f g^{5} + d e^{6} g^{6}\right )} x^{9} + 12 \, {\left (209 \, e^{7} f^{2} g^{4} + 86 \, d e^{6} f g^{5} - d^{2} e^{5} g^{6}\right )} x^{8} + \frac {32}{7} \, {\left (1135 \, e^{7} f^{3} g^{3} + 984 \, d e^{6} f^{2} g^{4} - 81 \, d^{2} e^{5} f g^{5} + 20 \, d^{3} e^{4} g^{6}\right )} x^{7} + 2 \, {\left (3135 \, e^{7} f^{4} g^{2} + 5620 \, d e^{6} f^{3} g^{3} - 558 \, d^{2} e^{5} f^{2} g^{4} - 60 \, d^{3} e^{4} f g^{5} + 95 \, d^{4} e^{3} g^{6}\right )} x^{6} + \frac {12}{5} \, {\left (1750 \, e^{7} f^{5} g + 6925 \, d e^{6} f^{4} g^{2} + 200 \, d^{2} e^{5} f^{3} g^{3} - 1130 \, d^{3} e^{4} f^{2} g^{4} + 490 \, d^{4} e^{3} f g^{5} - 3 \, d^{5} e^{2} g^{6}\right )} x^{5} + {\left (1225 \, e^{7} f^{6} + 13650 \, d e^{6} f^{5} g + 7425 \, d^{2} e^{5} f^{4} g^{2} - 9100 \, d^{3} e^{4} f^{3} g^{3} + 3435 \, d^{4} e^{3} f^{2} g^{4} - 198 \, d^{5} e^{2} f g^{5} + 27 \, d^{6} e g^{6}\right )} x^{4} + 4 \, {\left (1225 \, d e^{6} f^{6} + 3150 \, d^{2} e^{5} f^{5} g - 2775 \, d^{3} e^{4} f^{4} g^{2} + 500 \, d^{4} e^{3} f^{3} g^{3} + 387 \, d^{5} e^{2} f^{2} g^{4} - 162 \, d^{6} e f g^{5} + 27 \, d^{7} g^{6}\right )} x^{3} + 6 \, {\left (1225 \, d^{2} e^{5} f^{6} - 350 \, d^{3} e^{4} f^{5} g - 950 \, d^{4} e^{3} f^{4} g^{2} + 960 \, d^{5} e^{2} f^{3} g^{3} - 351 \, d^{6} e f^{2} g^{4} + 54 \, d^{7} f g^{5}\right )} x^{2} + 4 \, {\left (1225 \, d^{3} e^{4} f^{6} - 2100 \, d^{4} e^{3} f^{5} g + 1530 \, d^{5} e^{2} f^{4} g^{2} - 540 \, d^{6} e f^{3} g^{3} + 81 \, d^{7} f^{2} g^{4}\right )} x \] Input:

integrate((e*x+d)^3*(g*x+f)^2*(70*e^2*f^2-60*d*e*f*g+18*d^2*g^2-8*e*g*(3*d 
*g-10*e*f)*x+28*e^2*g^2*x^2)^2,x, algorithm="fricas")
 

Output:

392/5*e^7*g^6*x^10 + 112*(6*e^7*f*g^5 + d*e^6*g^6)*x^9 + 12*(209*e^7*f^2*g 
^4 + 86*d*e^6*f*g^5 - d^2*e^5*g^6)*x^8 + 32/7*(1135*e^7*f^3*g^3 + 984*d*e^ 
6*f^2*g^4 - 81*d^2*e^5*f*g^5 + 20*d^3*e^4*g^6)*x^7 + 2*(3135*e^7*f^4*g^2 + 
 5620*d*e^6*f^3*g^3 - 558*d^2*e^5*f^2*g^4 - 60*d^3*e^4*f*g^5 + 95*d^4*e^3* 
g^6)*x^6 + 12/5*(1750*e^7*f^5*g + 6925*d*e^6*f^4*g^2 + 200*d^2*e^5*f^3*g^3 
 - 1130*d^3*e^4*f^2*g^4 + 490*d^4*e^3*f*g^5 - 3*d^5*e^2*g^6)*x^5 + (1225*e 
^7*f^6 + 13650*d*e^6*f^5*g + 7425*d^2*e^5*f^4*g^2 - 9100*d^3*e^4*f^3*g^3 + 
 3435*d^4*e^3*f^2*g^4 - 198*d^5*e^2*f*g^5 + 27*d^6*e*g^6)*x^4 + 4*(1225*d* 
e^6*f^6 + 3150*d^2*e^5*f^5*g - 2775*d^3*e^4*f^4*g^2 + 500*d^4*e^3*f^3*g^3 
+ 387*d^5*e^2*f^2*g^4 - 162*d^6*e*f*g^5 + 27*d^7*g^6)*x^3 + 6*(1225*d^2*e^ 
5*f^6 - 350*d^3*e^4*f^5*g - 950*d^4*e^3*f^4*g^2 + 960*d^5*e^2*f^3*g^3 - 35 
1*d^6*e*f^2*g^4 + 54*d^7*f*g^5)*x^2 + 4*(1225*d^3*e^4*f^6 - 2100*d^4*e^3*f 
^5*g + 1530*d^5*e^2*f^4*g^2 - 540*d^6*e*f^3*g^3 + 81*d^7*f^2*g^4)*x
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (66) = 132\).

Time = 0.08 (sec) , antiderivative size = 629, normalized size of antiderivative = 9.83 \[ \int (d+e x)^3 (f+g x)^2 \left (70 e^2 f^2-60 d e f g+18 d^2 g^2-8 e g (-10 e f+3 d g) x+28 e^2 g^2 x^2\right )^2 \, dx=\frac {392 e^{7} g^{6} x^{10}}{5} + x^{9} \cdot \left (112 d e^{6} g^{6} + 672 e^{7} f g^{5}\right ) + x^{8} \left (- 12 d^{2} e^{5} g^{6} + 1032 d e^{6} f g^{5} + 2508 e^{7} f^{2} g^{4}\right ) + x^{7} \cdot \left (\frac {640 d^{3} e^{4} g^{6}}{7} - \frac {2592 d^{2} e^{5} f g^{5}}{7} + \frac {31488 d e^{6} f^{2} g^{4}}{7} + \frac {36320 e^{7} f^{3} g^{3}}{7}\right ) + x^{6} \cdot \left (190 d^{4} e^{3} g^{6} - 120 d^{3} e^{4} f g^{5} - 1116 d^{2} e^{5} f^{2} g^{4} + 11240 d e^{6} f^{3} g^{3} + 6270 e^{7} f^{4} g^{2}\right ) + x^{5} \left (- \frac {36 d^{5} e^{2} g^{6}}{5} + 1176 d^{4} e^{3} f g^{5} - 2712 d^{3} e^{4} f^{2} g^{4} + 480 d^{2} e^{5} f^{3} g^{3} + 16620 d e^{6} f^{4} g^{2} + 4200 e^{7} f^{5} g\right ) + x^{4} \cdot \left (27 d^{6} e g^{6} - 198 d^{5} e^{2} f g^{5} + 3435 d^{4} e^{3} f^{2} g^{4} - 9100 d^{3} e^{4} f^{3} g^{3} + 7425 d^{2} e^{5} f^{4} g^{2} + 13650 d e^{6} f^{5} g + 1225 e^{7} f^{6}\right ) + x^{3} \cdot \left (108 d^{7} g^{6} - 648 d^{6} e f g^{5} + 1548 d^{5} e^{2} f^{2} g^{4} + 2000 d^{4} e^{3} f^{3} g^{3} - 11100 d^{3} e^{4} f^{4} g^{2} + 12600 d^{2} e^{5} f^{5} g + 4900 d e^{6} f^{6}\right ) + x^{2} \cdot \left (324 d^{7} f g^{5} - 2106 d^{6} e f^{2} g^{4} + 5760 d^{5} e^{2} f^{3} g^{3} - 5700 d^{4} e^{3} f^{4} g^{2} - 2100 d^{3} e^{4} f^{5} g + 7350 d^{2} e^{5} f^{6}\right ) + x \left (324 d^{7} f^{2} g^{4} - 2160 d^{6} e f^{3} g^{3} + 6120 d^{5} e^{2} f^{4} g^{2} - 8400 d^{4} e^{3} f^{5} g + 4900 d^{3} e^{4} f^{6}\right ) \] Input:

integrate((e*x+d)**3*(g*x+f)**2*(70*e**2*f**2-60*d*e*f*g+18*d**2*g**2-8*e* 
g*(3*d*g-10*e*f)*x+28*e**2*g**2*x**2)**2,x)
 

Output:

392*e**7*g**6*x**10/5 + x**9*(112*d*e**6*g**6 + 672*e**7*f*g**5) + x**8*(- 
12*d**2*e**5*g**6 + 1032*d*e**6*f*g**5 + 2508*e**7*f**2*g**4) + x**7*(640* 
d**3*e**4*g**6/7 - 2592*d**2*e**5*f*g**5/7 + 31488*d*e**6*f**2*g**4/7 + 36 
320*e**7*f**3*g**3/7) + x**6*(190*d**4*e**3*g**6 - 120*d**3*e**4*f*g**5 - 
1116*d**2*e**5*f**2*g**4 + 11240*d*e**6*f**3*g**3 + 6270*e**7*f**4*g**2) + 
 x**5*(-36*d**5*e**2*g**6/5 + 1176*d**4*e**3*f*g**5 - 2712*d**3*e**4*f**2* 
g**4 + 480*d**2*e**5*f**3*g**3 + 16620*d*e**6*f**4*g**2 + 4200*e**7*f**5*g 
) + x**4*(27*d**6*e*g**6 - 198*d**5*e**2*f*g**5 + 3435*d**4*e**3*f**2*g**4 
 - 9100*d**3*e**4*f**3*g**3 + 7425*d**2*e**5*f**4*g**2 + 13650*d*e**6*f**5 
*g + 1225*e**7*f**6) + x**3*(108*d**7*g**6 - 648*d**6*e*f*g**5 + 1548*d**5 
*e**2*f**2*g**4 + 2000*d**4*e**3*f**3*g**3 - 11100*d**3*e**4*f**4*g**2 + 1 
2600*d**2*e**5*f**5*g + 4900*d*e**6*f**6) + x**2*(324*d**7*f*g**5 - 2106*d 
**6*e*f**2*g**4 + 5760*d**5*e**2*f**3*g**3 - 5700*d**4*e**3*f**4*g**2 - 21 
00*d**3*e**4*f**5*g + 7350*d**2*e**5*f**6) + x*(324*d**7*f**2*g**4 - 2160* 
d**6*e*f**3*g**3 + 6120*d**5*e**2*f**4*g**2 - 8400*d**4*e**3*f**5*g + 4900 
*d**3*e**4*f**6)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (62) = 124\).

Time = 0.04 (sec) , antiderivative size = 584, normalized size of antiderivative = 9.12 \[ \int (d+e x)^3 (f+g x)^2 \left (70 e^2 f^2-60 d e f g+18 d^2 g^2-8 e g (-10 e f+3 d g) x+28 e^2 g^2 x^2\right )^2 \, dx=\frac {392}{5} \, e^{7} g^{6} x^{10} + 112 \, {\left (6 \, e^{7} f g^{5} + d e^{6} g^{6}\right )} x^{9} + 12 \, {\left (209 \, e^{7} f^{2} g^{4} + 86 \, d e^{6} f g^{5} - d^{2} e^{5} g^{6}\right )} x^{8} + \frac {32}{7} \, {\left (1135 \, e^{7} f^{3} g^{3} + 984 \, d e^{6} f^{2} g^{4} - 81 \, d^{2} e^{5} f g^{5} + 20 \, d^{3} e^{4} g^{6}\right )} x^{7} + 2 \, {\left (3135 \, e^{7} f^{4} g^{2} + 5620 \, d e^{6} f^{3} g^{3} - 558 \, d^{2} e^{5} f^{2} g^{4} - 60 \, d^{3} e^{4} f g^{5} + 95 \, d^{4} e^{3} g^{6}\right )} x^{6} + \frac {12}{5} \, {\left (1750 \, e^{7} f^{5} g + 6925 \, d e^{6} f^{4} g^{2} + 200 \, d^{2} e^{5} f^{3} g^{3} - 1130 \, d^{3} e^{4} f^{2} g^{4} + 490 \, d^{4} e^{3} f g^{5} - 3 \, d^{5} e^{2} g^{6}\right )} x^{5} + {\left (1225 \, e^{7} f^{6} + 13650 \, d e^{6} f^{5} g + 7425 \, d^{2} e^{5} f^{4} g^{2} - 9100 \, d^{3} e^{4} f^{3} g^{3} + 3435 \, d^{4} e^{3} f^{2} g^{4} - 198 \, d^{5} e^{2} f g^{5} + 27 \, d^{6} e g^{6}\right )} x^{4} + 4 \, {\left (1225 \, d e^{6} f^{6} + 3150 \, d^{2} e^{5} f^{5} g - 2775 \, d^{3} e^{4} f^{4} g^{2} + 500 \, d^{4} e^{3} f^{3} g^{3} + 387 \, d^{5} e^{2} f^{2} g^{4} - 162 \, d^{6} e f g^{5} + 27 \, d^{7} g^{6}\right )} x^{3} + 6 \, {\left (1225 \, d^{2} e^{5} f^{6} - 350 \, d^{3} e^{4} f^{5} g - 950 \, d^{4} e^{3} f^{4} g^{2} + 960 \, d^{5} e^{2} f^{3} g^{3} - 351 \, d^{6} e f^{2} g^{4} + 54 \, d^{7} f g^{5}\right )} x^{2} + 4 \, {\left (1225 \, d^{3} e^{4} f^{6} - 2100 \, d^{4} e^{3} f^{5} g + 1530 \, d^{5} e^{2} f^{4} g^{2} - 540 \, d^{6} e f^{3} g^{3} + 81 \, d^{7} f^{2} g^{4}\right )} x \] Input:

integrate((e*x+d)^3*(g*x+f)^2*(70*e^2*f^2-60*d*e*f*g+18*d^2*g^2-8*e*g*(3*d 
*g-10*e*f)*x+28*e^2*g^2*x^2)^2,x, algorithm="maxima")
 

Output:

392/5*e^7*g^6*x^10 + 112*(6*e^7*f*g^5 + d*e^6*g^6)*x^9 + 12*(209*e^7*f^2*g 
^4 + 86*d*e^6*f*g^5 - d^2*e^5*g^6)*x^8 + 32/7*(1135*e^7*f^3*g^3 + 984*d*e^ 
6*f^2*g^4 - 81*d^2*e^5*f*g^5 + 20*d^3*e^4*g^6)*x^7 + 2*(3135*e^7*f^4*g^2 + 
 5620*d*e^6*f^3*g^3 - 558*d^2*e^5*f^2*g^4 - 60*d^3*e^4*f*g^5 + 95*d^4*e^3* 
g^6)*x^6 + 12/5*(1750*e^7*f^5*g + 6925*d*e^6*f^4*g^2 + 200*d^2*e^5*f^3*g^3 
 - 1130*d^3*e^4*f^2*g^4 + 490*d^4*e^3*f*g^5 - 3*d^5*e^2*g^6)*x^5 + (1225*e 
^7*f^6 + 13650*d*e^6*f^5*g + 7425*d^2*e^5*f^4*g^2 - 9100*d^3*e^4*f^3*g^3 + 
 3435*d^4*e^3*f^2*g^4 - 198*d^5*e^2*f*g^5 + 27*d^6*e*g^6)*x^4 + 4*(1225*d* 
e^6*f^6 + 3150*d^2*e^5*f^5*g - 2775*d^3*e^4*f^4*g^2 + 500*d^4*e^3*f^3*g^3 
+ 387*d^5*e^2*f^2*g^4 - 162*d^6*e*f*g^5 + 27*d^7*g^6)*x^3 + 6*(1225*d^2*e^ 
5*f^6 - 350*d^3*e^4*f^5*g - 950*d^4*e^3*f^4*g^2 + 960*d^5*e^2*f^3*g^3 - 35 
1*d^6*e*f^2*g^4 + 54*d^7*f*g^5)*x^2 + 4*(1225*d^3*e^4*f^6 - 2100*d^4*e^3*f 
^5*g + 1530*d^5*e^2*f^4*g^2 - 540*d^6*e*f^3*g^3 + 81*d^7*f^2*g^4)*x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (62) = 124\).

Time = 0.17 (sec) , antiderivative size = 659, normalized size of antiderivative = 10.30 \[ \int (d+e x)^3 (f+g x)^2 \left (70 e^2 f^2-60 d e f g+18 d^2 g^2-8 e g (-10 e f+3 d g) x+28 e^2 g^2 x^2\right )^2 \, dx=\frac {392}{5} \, e^{7} g^{6} x^{10} + 672 \, e^{7} f g^{5} x^{9} + 112 \, d e^{6} g^{6} x^{9} + 2508 \, e^{7} f^{2} g^{4} x^{8} + 1032 \, d e^{6} f g^{5} x^{8} - 12 \, d^{2} e^{5} g^{6} x^{8} + \frac {36320}{7} \, e^{7} f^{3} g^{3} x^{7} + \frac {31488}{7} \, d e^{6} f^{2} g^{4} x^{7} - \frac {2592}{7} \, d^{2} e^{5} f g^{5} x^{7} + \frac {640}{7} \, d^{3} e^{4} g^{6} x^{7} + 6270 \, e^{7} f^{4} g^{2} x^{6} + 11240 \, d e^{6} f^{3} g^{3} x^{6} - 1116 \, d^{2} e^{5} f^{2} g^{4} x^{6} - 120 \, d^{3} e^{4} f g^{5} x^{6} + 190 \, d^{4} e^{3} g^{6} x^{6} + 4200 \, e^{7} f^{5} g x^{5} + 16620 \, d e^{6} f^{4} g^{2} x^{5} + 480 \, d^{2} e^{5} f^{3} g^{3} x^{5} - 2712 \, d^{3} e^{4} f^{2} g^{4} x^{5} + 1176 \, d^{4} e^{3} f g^{5} x^{5} - \frac {36}{5} \, d^{5} e^{2} g^{6} x^{5} + 1225 \, e^{7} f^{6} x^{4} + 13650 \, d e^{6} f^{5} g x^{4} + 7425 \, d^{2} e^{5} f^{4} g^{2} x^{4} - 9100 \, d^{3} e^{4} f^{3} g^{3} x^{4} + 3435 \, d^{4} e^{3} f^{2} g^{4} x^{4} - 198 \, d^{5} e^{2} f g^{5} x^{4} + 27 \, d^{6} e g^{6} x^{4} + 4900 \, d e^{6} f^{6} x^{3} + 12600 \, d^{2} e^{5} f^{5} g x^{3} - 11100 \, d^{3} e^{4} f^{4} g^{2} x^{3} + 2000 \, d^{4} e^{3} f^{3} g^{3} x^{3} + 1548 \, d^{5} e^{2} f^{2} g^{4} x^{3} - 648 \, d^{6} e f g^{5} x^{3} + 108 \, d^{7} g^{6} x^{3} + 7350 \, d^{2} e^{5} f^{6} x^{2} - 2100 \, d^{3} e^{4} f^{5} g x^{2} - 5700 \, d^{4} e^{3} f^{4} g^{2} x^{2} + 5760 \, d^{5} e^{2} f^{3} g^{3} x^{2} - 2106 \, d^{6} e f^{2} g^{4} x^{2} + 324 \, d^{7} f g^{5} x^{2} + 4900 \, d^{3} e^{4} f^{6} x - 8400 \, d^{4} e^{3} f^{5} g x + 6120 \, d^{5} e^{2} f^{4} g^{2} x - 2160 \, d^{6} e f^{3} g^{3} x + 324 \, d^{7} f^{2} g^{4} x \] Input:

integrate((e*x+d)^3*(g*x+f)^2*(70*e^2*f^2-60*d*e*f*g+18*d^2*g^2-8*e*g*(3*d 
*g-10*e*f)*x+28*e^2*g^2*x^2)^2,x, algorithm="giac")
 

Output:

392/5*e^7*g^6*x^10 + 672*e^7*f*g^5*x^9 + 112*d*e^6*g^6*x^9 + 2508*e^7*f^2* 
g^4*x^8 + 1032*d*e^6*f*g^5*x^8 - 12*d^2*e^5*g^6*x^8 + 36320/7*e^7*f^3*g^3* 
x^7 + 31488/7*d*e^6*f^2*g^4*x^7 - 2592/7*d^2*e^5*f*g^5*x^7 + 640/7*d^3*e^4 
*g^6*x^7 + 6270*e^7*f^4*g^2*x^6 + 11240*d*e^6*f^3*g^3*x^6 - 1116*d^2*e^5*f 
^2*g^4*x^6 - 120*d^3*e^4*f*g^5*x^6 + 190*d^4*e^3*g^6*x^6 + 4200*e^7*f^5*g* 
x^5 + 16620*d*e^6*f^4*g^2*x^5 + 480*d^2*e^5*f^3*g^3*x^5 - 2712*d^3*e^4*f^2 
*g^4*x^5 + 1176*d^4*e^3*f*g^5*x^5 - 36/5*d^5*e^2*g^6*x^5 + 1225*e^7*f^6*x^ 
4 + 13650*d*e^6*f^5*g*x^4 + 7425*d^2*e^5*f^4*g^2*x^4 - 9100*d^3*e^4*f^3*g^ 
3*x^4 + 3435*d^4*e^3*f^2*g^4*x^4 - 198*d^5*e^2*f*g^5*x^4 + 27*d^6*e*g^6*x^ 
4 + 4900*d*e^6*f^6*x^3 + 12600*d^2*e^5*f^5*g*x^3 - 11100*d^3*e^4*f^4*g^2*x 
^3 + 2000*d^4*e^3*f^3*g^3*x^3 + 1548*d^5*e^2*f^2*g^4*x^3 - 648*d^6*e*f*g^5 
*x^3 + 108*d^7*g^6*x^3 + 7350*d^2*e^5*f^6*x^2 - 2100*d^3*e^4*f^5*g*x^2 - 5 
700*d^4*e^3*f^4*g^2*x^2 + 5760*d^5*e^2*f^3*g^3*x^2 - 2106*d^6*e*f^2*g^4*x^ 
2 + 324*d^7*f*g^5*x^2 + 4900*d^3*e^4*f^6*x - 8400*d^4*e^3*f^5*g*x + 6120*d 
^5*e^2*f^4*g^2*x - 2160*d^6*e*f^3*g^3*x + 324*d^7*f^2*g^4*x
 

Mupad [B] (verification not implemented)

Time = 13.18 (sec) , antiderivative size = 538, normalized size of antiderivative = 8.41 \[ \int (d+e x)^3 (f+g x)^2 \left (70 e^2 f^2-60 d e f g+18 d^2 g^2-8 e g (-10 e f+3 d g) x+28 e^2 g^2 x^2\right )^2 \, dx=x^6\,\left (190\,d^4\,e^3\,g^6-120\,d^3\,e^4\,f\,g^5-1116\,d^2\,e^5\,f^2\,g^4+11240\,d\,e^6\,f^3\,g^3+6270\,e^7\,f^4\,g^2\right )+x^2\,\left (324\,d^7\,f\,g^5-2106\,d^6\,e\,f^2\,g^4+5760\,d^5\,e^2\,f^3\,g^3-5700\,d^4\,e^3\,f^4\,g^2-2100\,d^3\,e^4\,f^5\,g+7350\,d^2\,e^5\,f^6\right )+x^5\,\left (-\frac {36\,d^5\,e^2\,g^6}{5}+1176\,d^4\,e^3\,f\,g^5-2712\,d^3\,e^4\,f^2\,g^4+480\,d^2\,e^5\,f^3\,g^3+16620\,d\,e^6\,f^4\,g^2+4200\,e^7\,f^5\,g\right )+x^3\,\left (108\,d^7\,g^6-648\,d^6\,e\,f\,g^5+1548\,d^5\,e^2\,f^2\,g^4+2000\,d^4\,e^3\,f^3\,g^3-11100\,d^3\,e^4\,f^4\,g^2+12600\,d^2\,e^5\,f^5\,g+4900\,d\,e^6\,f^6\right )+x^4\,\left (27\,d^6\,e\,g^6-198\,d^5\,e^2\,f\,g^5+3435\,d^4\,e^3\,f^2\,g^4-9100\,d^3\,e^4\,f^3\,g^3+7425\,d^2\,e^5\,f^4\,g^2+13650\,d\,e^6\,f^5\,g+1225\,e^7\,f^6\right )+\frac {392\,e^7\,g^6\,x^{10}}{5}+\frac {32\,e^4\,g^3\,x^7\,\left (20\,d^3\,g^3-81\,d^2\,e\,f\,g^2+984\,d\,e^2\,f^2\,g+1135\,e^3\,f^3\right )}{7}+112\,e^6\,g^5\,x^9\,\left (d\,g+6\,e\,f\right )+4\,d^3\,f^2\,x\,{\left (9\,d^2\,g^2-30\,d\,e\,f\,g+35\,e^2\,f^2\right )}^2+12\,e^5\,g^4\,x^8\,\left (-d^2\,g^2+86\,d\,e\,f\,g+209\,e^2\,f^2\right ) \] Input:

int((f + g*x)^2*(d + e*x)^3*(18*d^2*g^2 + 70*e^2*f^2 + 28*e^2*g^2*x^2 - 8* 
e*g*x*(3*d*g - 10*e*f) - 60*d*e*f*g)^2,x)
 

Output:

x^6*(190*d^4*e^3*g^6 + 6270*e^7*f^4*g^2 + 11240*d*e^6*f^3*g^3 - 120*d^3*e^ 
4*f*g^5 - 1116*d^2*e^5*f^2*g^4) + x^2*(324*d^7*f*g^5 + 7350*d^2*e^5*f^6 - 
2100*d^3*e^4*f^5*g - 2106*d^6*e*f^2*g^4 - 5700*d^4*e^3*f^4*g^2 + 5760*d^5* 
e^2*f^3*g^3) + x^5*(4200*e^7*f^5*g - (36*d^5*e^2*g^6)/5 + 16620*d*e^6*f^4* 
g^2 + 1176*d^4*e^3*f*g^5 + 480*d^2*e^5*f^3*g^3 - 2712*d^3*e^4*f^2*g^4) + x 
^3*(108*d^7*g^6 + 4900*d*e^6*f^6 + 12600*d^2*e^5*f^5*g - 11100*d^3*e^4*f^4 
*g^2 + 2000*d^4*e^3*f^3*g^3 + 1548*d^5*e^2*f^2*g^4 - 648*d^6*e*f*g^5) + x^ 
4*(1225*e^7*f^6 + 27*d^6*e*g^6 - 198*d^5*e^2*f*g^5 + 7425*d^2*e^5*f^4*g^2 
- 9100*d^3*e^4*f^3*g^3 + 3435*d^4*e^3*f^2*g^4 + 13650*d*e^6*f^5*g) + (392* 
e^7*g^6*x^10)/5 + (32*e^4*g^3*x^7*(20*d^3*g^3 + 1135*e^3*f^3 + 984*d*e^2*f 
^2*g - 81*d^2*e*f*g^2))/7 + 112*e^6*g^5*x^9*(d*g + 6*e*f) + 4*d^3*f^2*x*(9 
*d^2*g^2 + 35*e^2*f^2 - 30*d*e*f*g)^2 + 12*e^5*g^4*x^8*(209*e^2*f^2 - d^2* 
g^2 + 86*d*e*f*g)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 645, normalized size of antiderivative = 10.08 \[ \int (d+e x)^3 (f+g x)^2 \left (70 e^2 f^2-60 d e f g+18 d^2 g^2-8 e g (-10 e f+3 d g) x+28 e^2 g^2 x^2\right )^2 \, dx=\frac {x \left (2744 e^{7} g^{6} x^{9}+3920 d \,e^{6} g^{6} x^{8}+23520 e^{7} f \,g^{5} x^{8}-420 d^{2} e^{5} g^{6} x^{7}+36120 d \,e^{6} f \,g^{5} x^{7}+87780 e^{7} f^{2} g^{4} x^{7}+3200 d^{3} e^{4} g^{6} x^{6}-12960 d^{2} e^{5} f \,g^{5} x^{6}+157440 d \,e^{6} f^{2} g^{4} x^{6}+181600 e^{7} f^{3} g^{3} x^{6}+6650 d^{4} e^{3} g^{6} x^{5}-4200 d^{3} e^{4} f \,g^{5} x^{5}-39060 d^{2} e^{5} f^{2} g^{4} x^{5}+393400 d \,e^{6} f^{3} g^{3} x^{5}+219450 e^{7} f^{4} g^{2} x^{5}-252 d^{5} e^{2} g^{6} x^{4}+41160 d^{4} e^{3} f \,g^{5} x^{4}-94920 d^{3} e^{4} f^{2} g^{4} x^{4}+16800 d^{2} e^{5} f^{3} g^{3} x^{4}+581700 d \,e^{6} f^{4} g^{2} x^{4}+147000 e^{7} f^{5} g \,x^{4}+945 d^{6} e \,g^{6} x^{3}-6930 d^{5} e^{2} f \,g^{5} x^{3}+120225 d^{4} e^{3} f^{2} g^{4} x^{3}-318500 d^{3} e^{4} f^{3} g^{3} x^{3}+259875 d^{2} e^{5} f^{4} g^{2} x^{3}+477750 d \,e^{6} f^{5} g \,x^{3}+42875 e^{7} f^{6} x^{3}+3780 d^{7} g^{6} x^{2}-22680 d^{6} e f \,g^{5} x^{2}+54180 d^{5} e^{2} f^{2} g^{4} x^{2}+70000 d^{4} e^{3} f^{3} g^{3} x^{2}-388500 d^{3} e^{4} f^{4} g^{2} x^{2}+441000 d^{2} e^{5} f^{5} g \,x^{2}+171500 d \,e^{6} f^{6} x^{2}+11340 d^{7} f \,g^{5} x -73710 d^{6} e \,f^{2} g^{4} x +201600 d^{5} e^{2} f^{3} g^{3} x -199500 d^{4} e^{3} f^{4} g^{2} x -73500 d^{3} e^{4} f^{5} g x +257250 d^{2} e^{5} f^{6} x +11340 d^{7} f^{2} g^{4}-75600 d^{6} e \,f^{3} g^{3}+214200 d^{5} e^{2} f^{4} g^{2}-294000 d^{4} e^{3} f^{5} g +171500 d^{3} e^{4} f^{6}\right )}{35} \] Input:

int((e*x+d)^3*(g*x+f)^2*(70*e^2*f^2-60*d*e*f*g+18*d^2*g^2-8*e*g*(3*d*g-10* 
e*f)*x+28*e^2*g^2*x^2)^2,x)
 

Output:

(x*(11340*d**7*f**2*g**4 + 11340*d**7*f*g**5*x + 3780*d**7*g**6*x**2 - 756 
00*d**6*e*f**3*g**3 - 73710*d**6*e*f**2*g**4*x - 22680*d**6*e*f*g**5*x**2 
+ 945*d**6*e*g**6*x**3 + 214200*d**5*e**2*f**4*g**2 + 201600*d**5*e**2*f** 
3*g**3*x + 54180*d**5*e**2*f**2*g**4*x**2 - 6930*d**5*e**2*f*g**5*x**3 - 2 
52*d**5*e**2*g**6*x**4 - 294000*d**4*e**3*f**5*g - 199500*d**4*e**3*f**4*g 
**2*x + 70000*d**4*e**3*f**3*g**3*x**2 + 120225*d**4*e**3*f**2*g**4*x**3 + 
 41160*d**4*e**3*f*g**5*x**4 + 6650*d**4*e**3*g**6*x**5 + 171500*d**3*e**4 
*f**6 - 73500*d**3*e**4*f**5*g*x - 388500*d**3*e**4*f**4*g**2*x**2 - 31850 
0*d**3*e**4*f**3*g**3*x**3 - 94920*d**3*e**4*f**2*g**4*x**4 - 4200*d**3*e* 
*4*f*g**5*x**5 + 3200*d**3*e**4*g**6*x**6 + 257250*d**2*e**5*f**6*x + 4410 
00*d**2*e**5*f**5*g*x**2 + 259875*d**2*e**5*f**4*g**2*x**3 + 16800*d**2*e* 
*5*f**3*g**3*x**4 - 39060*d**2*e**5*f**2*g**4*x**5 - 12960*d**2*e**5*f*g** 
5*x**6 - 420*d**2*e**5*g**6*x**7 + 171500*d*e**6*f**6*x**2 + 477750*d*e**6 
*f**5*g*x**3 + 581700*d*e**6*f**4*g**2*x**4 + 393400*d*e**6*f**3*g**3*x**5 
 + 157440*d*e**6*f**2*g**4*x**6 + 36120*d*e**6*f*g**5*x**7 + 3920*d*e**6*g 
**6*x**8 + 42875*e**7*f**6*x**3 + 147000*e**7*f**5*g*x**4 + 219450*e**7*f* 
*4*g**2*x**5 + 181600*e**7*f**3*g**3*x**6 + 87780*e**7*f**2*g**4*x**7 + 23 
520*e**7*f*g**5*x**8 + 2744*e**7*g**6*x**9))/35