Integrand size = 63, antiderivative size = 63 \[ \int (d+e x)^3 (f+g x)^2 \left (48 e^2 f^2-32 d e f g+8 d^2 g^2-8 e g (-8 e f+2 d g) x+24 e^2 g^2 x^2\right ) \, dx=\frac {(d+e x)^4 \left (6 e^2 f^2-4 d e f g+d^2 g^2+2 e g (4 e f-d g) x+3 e^2 g^2 x^2\right )^2}{3 e^3} \] Output:
1/3*(e*x+d)^4*(6*e^2*f^2-4*d*e*f*g+d^2*g^2+2*e*g*(-d*g+4*e*f)*x+3*e^2*g^2* x^2)^2/e^3
Leaf count is larger than twice the leaf count of optimal. \(236\) vs. \(2(63)=126\).
Time = 0.10 (sec) , antiderivative size = 236, normalized size of antiderivative = 3.75 \[ \int (d+e x)^3 (f+g x)^2 \left (48 e^2 f^2-32 d e f g+8 d^2 g^2-8 e g (-8 e f+2 d g) x+24 e^2 g^2 x^2\right ) \, dx=\frac {1}{3} x \left (8 d^5 g^2 \left (3 f^2+3 f g x+g^2 x^2\right )+e^5 x^3 \left (6 f^2+8 f g x+3 g^2 x^2\right )^2+8 d^3 e^2 f \left (18 f^3+12 f^2 g x-2 f g^2 x^2-3 g^3 x^3\right )-2 d^4 e g \left (48 f^3+42 f^2 g x+8 f g^2 x^2-3 g^3 x^3\right )+8 d e^4 x^2 \left (18 f^4+42 f^3 g x+39 f^2 g^2 x^2+17 f g^3 x^3+3 g^4 x^4\right )+4 d^2 e^3 x \left (54 f^4+96 f^3 g x+69 f^2 g^2 x^2+24 f g^3 x^3+4 g^4 x^4\right )\right ) \] Input:
Integrate[(d + e*x)^3*(f + g*x)^2*(48*e^2*f^2 - 32*d*e*f*g + 8*d^2*g^2 - 8 *e*g*(-8*e*f + 2*d*g)*x + 24*e^2*g^2*x^2),x]
Output:
(x*(8*d^5*g^2*(3*f^2 + 3*f*g*x + g^2*x^2) + e^5*x^3*(6*f^2 + 8*f*g*x + 3*g ^2*x^2)^2 + 8*d^3*e^2*f*(18*f^3 + 12*f^2*g*x - 2*f*g^2*x^2 - 3*g^3*x^3) - 2*d^4*e*g*(48*f^3 + 42*f^2*g*x + 8*f*g^2*x^2 - 3*g^3*x^3) + 8*d*e^4*x^2*(1 8*f^4 + 42*f^3*g*x + 39*f^2*g^2*x^2 + 17*f*g^3*x^3 + 3*g^4*x^4) + 4*d^2*e^ 3*x*(54*f^4 + 96*f^3*g*x + 69*f^2*g^2*x^2 + 24*f*g^3*x^3 + 4*g^4*x^4)))/3
Time = 0.72 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.76, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, 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.
| \(\displaystyle \int (d+e x)^3 (f+g x)^2 \left (8 d^2 g^2-8 e g x (2 d g-8 e f)-32 d e f g+48 e^2 f^2+24 e^2 g^2 x^2\right ) \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {112 g^3 (d+e x)^6 (e f-d g)}{e^2}+\frac {200 g^2 (d+e x)^5 (e f-d g)^2}{e^2}+\frac {160 g (d+e x)^4 (e f-d g)^3}{e^2}+\frac {48 (d+e x)^3 (e f-d g)^4}{e^2}+\frac {24 g^4 (d+e x)^7}{e^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {16 g^3 (d+e x)^7 (e f-d g)}{e^3}+\frac {100 g^2 (d+e x)^6 (e f-d g)^2}{3 e^3}+\frac {32 g (d+e x)^5 (e f-d g)^3}{e^3}+\frac {12 (d+e x)^4 (e f-d g)^4}{e^3}+\frac {3 g^4 (d+e x)^8}{e^3}\) |
Input:
Int[(d + e*x)^3*(f + g*x)^2*(48*e^2*f^2 - 32*d*e*f*g + 8*d^2*g^2 - 8*e*g*( -8*e*f + 2*d*g)*x + 24*e^2*g^2*x^2),x]
Output:
(12*(e*f - d*g)^4*(d + e*x)^4)/e^3 + (32*g*(e*f - d*g)^3*(d + e*x)^5)/e^3 + (100*g^2*(e*f - d*g)^2*(d + e*x)^6)/(3*e^3) + (16*g^3*(e*f - d*g)*(d + e *x)^7)/e^3 + (3*g^4*(d + e*x)^8)/e^3
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]
Leaf count of result is larger than twice the leaf count of optimal. \(310\) vs. \(2(61)=122\).
Time = 0.75 (sec) , antiderivative size = 311, normalized size of antiderivative = 4.94
| method | result | size |
| norman | \(\left (\frac {16}{3} d^{2} e^{3} g^{4}+\frac {136}{3} d \,e^{4} f \,g^{3}+\frac {100}{3} e^{5} f^{2} g^{2}\right ) x^{6}+\left (\frac {8}{3} d^{5} g^{4}-\frac {16}{3} d^{4} e f \,g^{3}-\frac {16}{3} d^{3} f^{2} e^{2} g^{2}+128 d^{2} e^{3} f^{3} g +48 d \,e^{4} f^{4}\right ) x^{3}+\left (8 d \,e^{4} g^{4}+16 e^{5} f \,g^{3}\right ) x^{7}+\left (32 d^{2} e^{3} f \,g^{3}+104 d \,e^{4} f^{2} g^{2}+32 e^{5} f^{3} g \right ) x^{5}+\left (8 d^{5} f^{2} g^{2}-32 d^{4} e \,f^{3} g +48 d^{3} e^{2} f^{4}\right ) x +\left (8 d^{5} f \,g^{3}-28 d^{4} e \,f^{2} g^{2}+32 d^{3} e^{2} f^{3} g +72 d^{2} e^{3} f^{4}\right ) x^{2}+\left (2 d^{4} e \,g^{4}-8 d^{3} e^{2} f \,g^{3}+92 d^{2} e^{3} f^{2} g^{2}+112 d \,e^{4} f^{3} g +12 e^{5} f^{4}\right ) x^{4}+3 e^{5} g^{4} x^{8}\) | \(311\) |
| gosper | \(\frac {x \left (9 e^{5} g^{4} x^{7}+24 d \,e^{4} g^{4} x^{6}+48 e^{5} f \,g^{3} x^{6}+16 x^{5} d^{2} e^{3} g^{4}+136 x^{5} d \,e^{4} f \,g^{3}+100 x^{5} e^{5} f^{2} g^{2}+96 d^{2} e^{3} f \,g^{3} x^{4}+312 d \,e^{4} f^{2} g^{2} x^{4}+96 e^{5} f^{3} g \,x^{4}+6 x^{3} d^{4} e \,g^{4}-24 x^{3} d^{3} e^{2} f \,g^{3}+276 x^{3} d^{2} e^{3} f^{2} g^{2}+336 x^{3} d \,e^{4} f^{3} g +36 x^{3} e^{5} f^{4}+8 x^{2} d^{5} g^{4}-16 x^{2} d^{4} e f \,g^{3}-16 x^{2} d^{3} f^{2} e^{2} g^{2}+384 x^{2} d^{2} e^{3} f^{3} g +144 x^{2} d \,e^{4} f^{4}+24 x \,d^{5} f \,g^{3}-84 x \,d^{4} e \,f^{2} g^{2}+96 x \,d^{3} e^{2} f^{3} g +216 x \,d^{2} e^{3} f^{4}+24 d^{5} f^{2} g^{2}-96 d^{4} e \,f^{3} g +144 d^{3} e^{2} f^{4}\right )}{3}\) | \(339\) |
| risch | \(12 e^{5} f^{4} x^{4}-\frac {16}{3} x^{3} d^{4} e f \,g^{3}-\frac {16}{3} x^{3} d^{3} f^{2} e^{2} g^{2}+32 d^{2} e^{3} f \,g^{3} x^{5}+\frac {136}{3} x^{6} d \,e^{4} f \,g^{3}+\frac {16}{3} x^{6} d^{2} e^{3} g^{4}+\frac {100}{3} x^{6} e^{5} f^{2} g^{2}+48 x^{3} d \,e^{4} f^{4}+8 d \,e^{4} g^{4} x^{7}+112 d \,e^{4} f^{3} g \,x^{4}-28 d^{4} e \,f^{2} g^{2} x^{2}+32 d^{3} e^{2} f^{3} g \,x^{2}-32 d^{4} e \,f^{3} g x -8 d^{3} e^{2} f \,g^{3} x^{4}+92 d^{2} e^{3} f^{2} g^{2} x^{4}+16 e^{5} f \,g^{3} x^{7}+32 e^{5} f^{3} g \,x^{5}+2 d^{4} e \,g^{4} x^{4}+8 d^{5} f \,g^{3} x^{2}+72 d^{2} e^{3} f^{4} x^{2}+8 d^{5} f^{2} g^{2} x +48 d^{3} e^{2} f^{4} x +128 x^{3} d^{2} e^{3} f^{3} g +104 d \,e^{4} f^{2} g^{2} x^{5}+\frac {8}{3} x^{3} d^{5} g^{4}+3 e^{5} g^{4} x^{8}\) | \(347\) |
| parallelrisch | \(12 e^{5} f^{4} x^{4}-\frac {16}{3} x^{3} d^{4} e f \,g^{3}-\frac {16}{3} x^{3} d^{3} f^{2} e^{2} g^{2}+32 d^{2} e^{3} f \,g^{3} x^{5}+\frac {136}{3} x^{6} d \,e^{4} f \,g^{3}+\frac {16}{3} x^{6} d^{2} e^{3} g^{4}+\frac {100}{3} x^{6} e^{5} f^{2} g^{2}+48 x^{3} d \,e^{4} f^{4}+8 d \,e^{4} g^{4} x^{7}+112 d \,e^{4} f^{3} g \,x^{4}-28 d^{4} e \,f^{2} g^{2} x^{2}+32 d^{3} e^{2} f^{3} g \,x^{2}-32 d^{4} e \,f^{3} g x -8 d^{3} e^{2} f \,g^{3} x^{4}+92 d^{2} e^{3} f^{2} g^{2} x^{4}+16 e^{5} f \,g^{3} x^{7}+32 e^{5} f^{3} g \,x^{5}+2 d^{4} e \,g^{4} x^{4}+8 d^{5} f \,g^{3} x^{2}+72 d^{2} e^{3} f^{4} x^{2}+8 d^{5} f^{2} g^{2} x +48 d^{3} e^{2} f^{4} x +128 x^{3} d^{2} e^{3} f^{3} g +104 d \,e^{4} f^{2} g^{2} x^{5}+\frac {8}{3} x^{3} d^{5} g^{4}+3 e^{5} g^{4} x^{8}\) | \(347\) |
| orering | \(\frac {x \left (9 e^{5} g^{4} x^{7}+24 d \,e^{4} g^{4} x^{6}+48 e^{5} f \,g^{3} x^{6}+16 x^{5} d^{2} e^{3} g^{4}+136 x^{5} d \,e^{4} f \,g^{3}+100 x^{5} e^{5} f^{2} g^{2}+96 d^{2} e^{3} f \,g^{3} x^{4}+312 d \,e^{4} f^{2} g^{2} x^{4}+96 e^{5} f^{3} g \,x^{4}+6 x^{3} d^{4} e \,g^{4}-24 x^{3} d^{3} e^{2} f \,g^{3}+276 x^{3} d^{2} e^{3} f^{2} g^{2}+336 x^{3} d \,e^{4} f^{3} g +36 x^{3} e^{5} f^{4}+8 x^{2} d^{5} g^{4}-16 x^{2} d^{4} e f \,g^{3}-16 x^{2} d^{3} f^{2} e^{2} g^{2}+384 x^{2} d^{2} e^{3} f^{3} g +144 x^{2} d \,e^{4} f^{4}+24 x \,d^{5} f \,g^{3}-84 x \,d^{4} e \,f^{2} g^{2}+96 x \,d^{3} e^{2} f^{3} g +216 x \,d^{2} e^{3} f^{4}+24 d^{5} f^{2} g^{2}-96 d^{4} e \,f^{3} g +144 d^{3} e^{2} f^{4}\right ) \left (48 e^{2} f^{2}-32 d e f g +8 d^{2} g^{2}-8 e g \left (2 d g -8 e f \right ) x +24 e^{2} g^{2} x^{2}\right )}{72 e^{2} g^{2} x^{2}-48 d e \,g^{2} x +192 e^{2} f g x +24 d^{2} g^{2}-96 d e f g +144 e^{2} f^{2}}\) | \(438\) |
| default | \(3 e^{5} g^{4} x^{8}+\frac {8 \left (3 \left (3 d \,e^{2} g^{2}+2 e^{3} f g \right ) e^{2} g^{2}-e^{4} g^{3} \left (2 d g -8 e f \right )\right ) x^{7}}{7}+\frac {4 \left (3 \left (3 d^{2} e \,g^{2}+6 d \,e^{2} f g +e^{3} f^{2}\right ) e^{2} g^{2}-\left (3 d \,e^{2} g^{2}+2 e^{3} f g \right ) e g \left (2 d g -8 e f \right )+e^{3} g^{2} \left (d^{2} g^{2}-4 d e f g +6 e^{2} f^{2}\right )\right ) x^{6}}{3}+\frac {8 \left (3 \left (d^{3} g^{2}+6 d^{2} e f g +3 d \,e^{2} f^{2}\right ) e^{2} g^{2}-\left (3 d^{2} e \,g^{2}+6 d \,e^{2} f g +e^{3} f^{2}\right ) e g \left (2 d g -8 e f \right )+\left (3 d \,e^{2} g^{2}+2 e^{3} f g \right ) \left (d^{2} g^{2}-4 d e f g +6 e^{2} f^{2}\right )\right ) x^{5}}{5}+2 \left (3 \left (2 d^{3} f g +3 d^{2} e \,f^{2}\right ) e^{2} g^{2}-\left (d^{3} g^{2}+6 d^{2} e f g +3 d \,e^{2} f^{2}\right ) e g \left (2 d g -8 e f \right )+\left (3 d^{2} e \,g^{2}+6 d \,e^{2} f g +e^{3} f^{2}\right ) \left (d^{2} g^{2}-4 d e f g +6 e^{2} f^{2}\right )\right ) x^{4}+\frac {8 \left (3 d^{3} f^{2} e^{2} g^{2}-\left (2 d^{3} f g +3 d^{2} e \,f^{2}\right ) e g \left (2 d g -8 e f \right )+\left (d^{3} g^{2}+6 d^{2} e f g +3 d \,e^{2} f^{2}\right ) \left (d^{2} g^{2}-4 d e f g +6 e^{2} f^{2}\right )\right ) x^{3}}{3}+4 \left (-d^{3} f^{2} e g \left (2 d g -8 e f \right )+\left (2 d^{3} f g +3 d^{2} e \,f^{2}\right ) \left (d^{2} g^{2}-4 d e f g +6 e^{2} f^{2}\right )\right ) x^{2}+8 d^{3} f^{2} \left (d^{2} g^{2}-4 d e f g +6 e^{2} f^{2}\right ) x\) | \(587\) |
Input:
int((e*x+d)^3*(g*x+f)^2*(48*e^2*f^2-32*d*e*f*g+8*d^2*g^2-8*e*g*(2*d*g-8*e* f)*x+24*e^2*g^2*x^2),x,method=_RETURNVERBOSE)
Output:
(16/3*d^2*e^3*g^4+136/3*d*e^4*f*g^3+100/3*e^5*f^2*g^2)*x^6+(8/3*d^5*g^4-16 /3*d^4*e*f*g^3-16/3*d^3*f^2*e^2*g^2+128*d^2*e^3*f^3*g+48*d*e^4*f^4)*x^3+(8 *d*e^4*g^4+16*e^5*f*g^3)*x^7+(32*d^2*e^3*f*g^3+104*d*e^4*f^2*g^2+32*e^5*f^ 3*g)*x^5+(8*d^5*f^2*g^2-32*d^4*e*f^3*g+48*d^3*e^2*f^4)*x+(8*d^5*f*g^3-28*d ^4*e*f^2*g^2+32*d^3*e^2*f^3*g+72*d^2*e^3*f^4)*x^2+(2*d^4*e*g^4-8*d^3*e^2*f *g^3+92*d^2*e^3*f^2*g^2+112*d*e^4*f^3*g+12*e^5*f^4)*x^4+3*e^5*g^4*x^8
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (61) = 122\).
Time = 0.07 (sec) , antiderivative size = 313, normalized size of antiderivative = 4.97 \[ \int (d+e x)^3 (f+g x)^2 \left (48 e^2 f^2-32 d e f g+8 d^2 g^2-8 e g (-8 e f+2 d g) x+24 e^2 g^2 x^2\right ) \, dx=3 \, e^{5} g^{4} x^{8} + 8 \, {\left (2 \, e^{5} f g^{3} + d e^{4} g^{4}\right )} x^{7} + \frac {4}{3} \, {\left (25 \, e^{5} f^{2} g^{2} + 34 \, d e^{4} f g^{3} + 4 \, d^{2} e^{3} g^{4}\right )} x^{6} + 8 \, {\left (4 \, e^{5} f^{3} g + 13 \, d e^{4} f^{2} g^{2} + 4 \, d^{2} e^{3} f g^{3}\right )} x^{5} + 2 \, {\left (6 \, e^{5} f^{4} + 56 \, d e^{4} f^{3} g + 46 \, d^{2} e^{3} f^{2} g^{2} - 4 \, d^{3} e^{2} f g^{3} + d^{4} e g^{4}\right )} x^{4} + \frac {8}{3} \, {\left (18 \, d e^{4} f^{4} + 48 \, d^{2} e^{3} f^{3} g - 2 \, d^{3} e^{2} f^{2} g^{2} - 2 \, d^{4} e f g^{3} + d^{5} g^{4}\right )} x^{3} + 4 \, {\left (18 \, d^{2} e^{3} f^{4} + 8 \, d^{3} e^{2} f^{3} g - 7 \, d^{4} e f^{2} g^{2} + 2 \, d^{5} f g^{3}\right )} x^{2} + 8 \, {\left (6 \, d^{3} e^{2} f^{4} - 4 \, d^{4} e f^{3} g + d^{5} f^{2} g^{2}\right )} x \] Input:
integrate((e*x+d)^3*(g*x+f)^2*(48*e^2*f^2-32*d*e*f*g+8*d^2*g^2-8*e*g*(2*d* g-8*e*f)*x+24*e^2*g^2*x^2),x, algorithm="fricas")
Output:
3*e^5*g^4*x^8 + 8*(2*e^5*f*g^3 + d*e^4*g^4)*x^7 + 4/3*(25*e^5*f^2*g^2 + 34 *d*e^4*f*g^3 + 4*d^2*e^3*g^4)*x^6 + 8*(4*e^5*f^3*g + 13*d*e^4*f^2*g^2 + 4* d^2*e^3*f*g^3)*x^5 + 2*(6*e^5*f^4 + 56*d*e^4*f^3*g + 46*d^2*e^3*f^2*g^2 - 4*d^3*e^2*f*g^3 + d^4*e*g^4)*x^4 + 8/3*(18*d*e^4*f^4 + 48*d^2*e^3*f^3*g - 2*d^3*e^2*f^2*g^2 - 2*d^4*e*f*g^3 + d^5*g^4)*x^3 + 4*(18*d^2*e^3*f^4 + 8*d ^3*e^2*f^3*g - 7*d^4*e*f^2*g^2 + 2*d^5*f*g^3)*x^2 + 8*(6*d^3*e^2*f^4 - 4*d ^4*e*f^3*g + d^5*f^2*g^2)*x
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (63) = 126\).
Time = 0.04 (sec) , antiderivative size = 340, normalized size of antiderivative = 5.40 \[ \int (d+e x)^3 (f+g x)^2 \left (48 e^2 f^2-32 d e f g+8 d^2 g^2-8 e g (-8 e f+2 d g) x+24 e^2 g^2 x^2\right ) \, dx=3 e^{5} g^{4} x^{8} + x^{7} \cdot \left (8 d e^{4} g^{4} + 16 e^{5} f g^{3}\right ) + x^{6} \cdot \left (\frac {16 d^{2} e^{3} g^{4}}{3} + \frac {136 d e^{4} f g^{3}}{3} + \frac {100 e^{5} f^{2} g^{2}}{3}\right ) + x^{5} \cdot \left (32 d^{2} e^{3} f g^{3} + 104 d e^{4} f^{2} g^{2} + 32 e^{5} f^{3} g\right ) + x^{4} \cdot \left (2 d^{4} e g^{4} - 8 d^{3} e^{2} f g^{3} + 92 d^{2} e^{3} f^{2} g^{2} + 112 d e^{4} f^{3} g + 12 e^{5} f^{4}\right ) + x^{3} \cdot \left (\frac {8 d^{5} g^{4}}{3} - \frac {16 d^{4} e f g^{3}}{3} - \frac {16 d^{3} e^{2} f^{2} g^{2}}{3} + 128 d^{2} e^{3} f^{3} g + 48 d e^{4} f^{4}\right ) + x^{2} \cdot \left (8 d^{5} f g^{3} - 28 d^{4} e f^{2} g^{2} + 32 d^{3} e^{2} f^{3} g + 72 d^{2} e^{3} f^{4}\right ) + x \left (8 d^{5} f^{2} g^{2} - 32 d^{4} e f^{3} g + 48 d^{3} e^{2} f^{4}\right ) \] Input:
integrate((e*x+d)**3*(g*x+f)**2*(48*e**2*f**2-32*d*e*f*g+8*d**2*g**2-8*e*g *(2*d*g-8*e*f)*x+24*e**2*g**2*x**2),x)
Output:
3*e**5*g**4*x**8 + x**7*(8*d*e**4*g**4 + 16*e**5*f*g**3) + x**6*(16*d**2*e **3*g**4/3 + 136*d*e**4*f*g**3/3 + 100*e**5*f**2*g**2/3) + x**5*(32*d**2*e **3*f*g**3 + 104*d*e**4*f**2*g**2 + 32*e**5*f**3*g) + x**4*(2*d**4*e*g**4 - 8*d**3*e**2*f*g**3 + 92*d**2*e**3*f**2*g**2 + 112*d*e**4*f**3*g + 12*e** 5*f**4) + x**3*(8*d**5*g**4/3 - 16*d**4*e*f*g**3/3 - 16*d**3*e**2*f**2*g** 2/3 + 128*d**2*e**3*f**3*g + 48*d*e**4*f**4) + x**2*(8*d**5*f*g**3 - 28*d* *4*e*f**2*g**2 + 32*d**3*e**2*f**3*g + 72*d**2*e**3*f**4) + x*(8*d**5*f**2 *g**2 - 32*d**4*e*f**3*g + 48*d**3*e**2*f**4)
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (61) = 122\).
Time = 0.04 (sec) , antiderivative size = 313, normalized size of antiderivative = 4.97 \[ \int (d+e x)^3 (f+g x)^2 \left (48 e^2 f^2-32 d e f g+8 d^2 g^2-8 e g (-8 e f+2 d g) x+24 e^2 g^2 x^2\right ) \, dx=3 \, e^{5} g^{4} x^{8} + 8 \, {\left (2 \, e^{5} f g^{3} + d e^{4} g^{4}\right )} x^{7} + \frac {4}{3} \, {\left (25 \, e^{5} f^{2} g^{2} + 34 \, d e^{4} f g^{3} + 4 \, d^{2} e^{3} g^{4}\right )} x^{6} + 8 \, {\left (4 \, e^{5} f^{3} g + 13 \, d e^{4} f^{2} g^{2} + 4 \, d^{2} e^{3} f g^{3}\right )} x^{5} + 2 \, {\left (6 \, e^{5} f^{4} + 56 \, d e^{4} f^{3} g + 46 \, d^{2} e^{3} f^{2} g^{2} - 4 \, d^{3} e^{2} f g^{3} + d^{4} e g^{4}\right )} x^{4} + \frac {8}{3} \, {\left (18 \, d e^{4} f^{4} + 48 \, d^{2} e^{3} f^{3} g - 2 \, d^{3} e^{2} f^{2} g^{2} - 2 \, d^{4} e f g^{3} + d^{5} g^{4}\right )} x^{3} + 4 \, {\left (18 \, d^{2} e^{3} f^{4} + 8 \, d^{3} e^{2} f^{3} g - 7 \, d^{4} e f^{2} g^{2} + 2 \, d^{5} f g^{3}\right )} x^{2} + 8 \, {\left (6 \, d^{3} e^{2} f^{4} - 4 \, d^{4} e f^{3} g + d^{5} f^{2} g^{2}\right )} x \] Input:
integrate((e*x+d)^3*(g*x+f)^2*(48*e^2*f^2-32*d*e*f*g+8*d^2*g^2-8*e*g*(2*d* g-8*e*f)*x+24*e^2*g^2*x^2),x, algorithm="maxima")
Output:
3*e^5*g^4*x^8 + 8*(2*e^5*f*g^3 + d*e^4*g^4)*x^7 + 4/3*(25*e^5*f^2*g^2 + 34 *d*e^4*f*g^3 + 4*d^2*e^3*g^4)*x^6 + 8*(4*e^5*f^3*g + 13*d*e^4*f^2*g^2 + 4* d^2*e^3*f*g^3)*x^5 + 2*(6*e^5*f^4 + 56*d*e^4*f^3*g + 46*d^2*e^3*f^2*g^2 - 4*d^3*e^2*f*g^3 + d^4*e*g^4)*x^4 + 8/3*(18*d*e^4*f^4 + 48*d^2*e^3*f^3*g - 2*d^3*e^2*f^2*g^2 - 2*d^4*e*f*g^3 + d^5*g^4)*x^3 + 4*(18*d^2*e^3*f^4 + 8*d ^3*e^2*f^3*g - 7*d^4*e*f^2*g^2 + 2*d^5*f*g^3)*x^2 + 8*(6*d^3*e^2*f^4 - 4*d ^4*e*f^3*g + d^5*f^2*g^2)*x
Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (61) = 122\).
Time = 0.17 (sec) , antiderivative size = 346, normalized size of antiderivative = 5.49 \[ \int (d+e x)^3 (f+g x)^2 \left (48 e^2 f^2-32 d e f g+8 d^2 g^2-8 e g (-8 e f+2 d g) x+24 e^2 g^2 x^2\right ) \, dx=3 \, e^{5} g^{4} x^{8} + 16 \, e^{5} f g^{3} x^{7} + 8 \, d e^{4} g^{4} x^{7} + \frac {100}{3} \, e^{5} f^{2} g^{2} x^{6} + \frac {136}{3} \, d e^{4} f g^{3} x^{6} + \frac {16}{3} \, d^{2} e^{3} g^{4} x^{6} + 32 \, e^{5} f^{3} g x^{5} + 104 \, d e^{4} f^{2} g^{2} x^{5} + 32 \, d^{2} e^{3} f g^{3} x^{5} + 12 \, e^{5} f^{4} x^{4} + 112 \, d e^{4} f^{3} g x^{4} + 92 \, d^{2} e^{3} f^{2} g^{2} x^{4} - 8 \, d^{3} e^{2} f g^{3} x^{4} + 2 \, d^{4} e g^{4} x^{4} + 48 \, d e^{4} f^{4} x^{3} + 128 \, d^{2} e^{3} f^{3} g x^{3} - \frac {16}{3} \, d^{3} e^{2} f^{2} g^{2} x^{3} - \frac {16}{3} \, d^{4} e f g^{3} x^{3} + \frac {8}{3} \, d^{5} g^{4} x^{3} + 72 \, d^{2} e^{3} f^{4} x^{2} + 32 \, d^{3} e^{2} f^{3} g x^{2} - 28 \, d^{4} e f^{2} g^{2} x^{2} + 8 \, d^{5} f g^{3} x^{2} + 48 \, d^{3} e^{2} f^{4} x - 32 \, d^{4} e f^{3} g x + 8 \, d^{5} f^{2} g^{2} x \] Input:
integrate((e*x+d)^3*(g*x+f)^2*(48*e^2*f^2-32*d*e*f*g+8*d^2*g^2-8*e*g*(2*d* g-8*e*f)*x+24*e^2*g^2*x^2),x, algorithm="giac")
Output:
3*e^5*g^4*x^8 + 16*e^5*f*g^3*x^7 + 8*d*e^4*g^4*x^7 + 100/3*e^5*f^2*g^2*x^6 + 136/3*d*e^4*f*g^3*x^6 + 16/3*d^2*e^3*g^4*x^6 + 32*e^5*f^3*g*x^5 + 104*d *e^4*f^2*g^2*x^5 + 32*d^2*e^3*f*g^3*x^5 + 12*e^5*f^4*x^4 + 112*d*e^4*f^3*g *x^4 + 92*d^2*e^3*f^2*g^2*x^4 - 8*d^3*e^2*f*g^3*x^4 + 2*d^4*e*g^4*x^4 + 48 *d*e^4*f^4*x^3 + 128*d^2*e^3*f^3*g*x^3 - 16/3*d^3*e^2*f^2*g^2*x^3 - 16/3*d ^4*e*f*g^3*x^3 + 8/3*d^5*g^4*x^3 + 72*d^2*e^3*f^4*x^2 + 32*d^3*e^2*f^3*g*x ^2 - 28*d^4*e*f^2*g^2*x^2 + 8*d^5*f*g^3*x^2 + 48*d^3*e^2*f^4*x - 32*d^4*e* f^3*g*x + 8*d^5*f^2*g^2*x
Time = 0.14 (sec) , antiderivative size = 294, normalized size of antiderivative = 4.67 \[ \int (d+e x)^3 (f+g x)^2 \left (48 e^2 f^2-32 d e f g+8 d^2 g^2-8 e g (-8 e f+2 d g) x+24 e^2 g^2 x^2\right ) \, dx=x^3\,\left (\frac {8\,d^5\,g^4}{3}-\frac {16\,d^4\,e\,f\,g^3}{3}-\frac {16\,d^3\,e^2\,f^2\,g^2}{3}+128\,d^2\,e^3\,f^3\,g+48\,d\,e^4\,f^4\right )+x^4\,\left (2\,d^4\,e\,g^4-8\,d^3\,e^2\,f\,g^3+92\,d^2\,e^3\,f^2\,g^2+112\,d\,e^4\,f^3\,g+12\,e^5\,f^4\right )+x^2\,\left (8\,d^5\,f\,g^3-28\,d^4\,e\,f^2\,g^2+32\,d^3\,e^2\,f^3\,g+72\,d^2\,e^3\,f^4\right )+3\,e^5\,g^4\,x^8+8\,e^4\,g^3\,x^7\,\left (d\,g+2\,e\,f\right )+8\,d^3\,f^2\,x\,\left (d^2\,g^2-4\,d\,e\,f\,g+6\,e^2\,f^2\right )+\frac {4\,e^3\,g^2\,x^6\,\left (4\,d^2\,g^2+34\,d\,e\,f\,g+25\,e^2\,f^2\right )}{3}+8\,e^3\,f\,g\,x^5\,\left (4\,d^2\,g^2+13\,d\,e\,f\,g+4\,e^2\,f^2\right ) \] Input:
int((f + g*x)^2*(d + e*x)^3*(8*d^2*g^2 + 48*e^2*f^2 + 24*e^2*g^2*x^2 - 8*e *g*x*(2*d*g - 8*e*f) - 32*d*e*f*g),x)
Output:
x^3*((8*d^5*g^4)/3 + 48*d*e^4*f^4 + 128*d^2*e^3*f^3*g - (16*d^3*e^2*f^2*g^ 2)/3 - (16*d^4*e*f*g^3)/3) + x^4*(12*e^5*f^4 + 2*d^4*e*g^4 - 8*d^3*e^2*f*g ^3 + 92*d^2*e^3*f^2*g^2 + 112*d*e^4*f^3*g) + x^2*(8*d^5*f*g^3 + 72*d^2*e^3 *f^4 + 32*d^3*e^2*f^3*g - 28*d^4*e*f^2*g^2) + 3*e^5*g^4*x^8 + 8*e^4*g^3*x^ 7*(d*g + 2*e*f) + 8*d^3*f^2*x*(d^2*g^2 + 6*e^2*f^2 - 4*d*e*f*g) + (4*e^3*g ^2*x^6*(4*d^2*g^2 + 25*e^2*f^2 + 34*d*e*f*g))/3 + 8*e^3*f*g*x^5*(4*d^2*g^2 + 4*e^2*f^2 + 13*d*e*f*g)
Time = 0.17 (sec) , antiderivative size = 338, normalized size of antiderivative = 5.37 \[ \int (d+e x)^3 (f+g x)^2 \left (48 e^2 f^2-32 d e f g+8 d^2 g^2-8 e g (-8 e f+2 d g) x+24 e^2 g^2 x^2\right ) \, dx=\frac {x \left (9 e^{5} g^{4} x^{7}+24 d \,e^{4} g^{4} x^{6}+48 e^{5} f \,g^{3} x^{6}+16 d^{2} e^{3} g^{4} x^{5}+136 d \,e^{4} f \,g^{3} x^{5}+100 e^{5} f^{2} g^{2} x^{5}+96 d^{2} e^{3} f \,g^{3} x^{4}+312 d \,e^{4} f^{2} g^{2} x^{4}+96 e^{5} f^{3} g \,x^{4}+6 d^{4} e \,g^{4} x^{3}-24 d^{3} e^{2} f \,g^{3} x^{3}+276 d^{2} e^{3} f^{2} g^{2} x^{3}+336 d \,e^{4} f^{3} g \,x^{3}+36 e^{5} f^{4} x^{3}+8 d^{5} g^{4} x^{2}-16 d^{4} e f \,g^{3} x^{2}-16 d^{3} e^{2} f^{2} g^{2} x^{2}+384 d^{2} e^{3} f^{3} g \,x^{2}+144 d \,e^{4} f^{4} x^{2}+24 d^{5} f \,g^{3} x -84 d^{4} e \,f^{2} g^{2} x +96 d^{3} e^{2} f^{3} g x +216 d^{2} e^{3} f^{4} x +24 d^{5} f^{2} g^{2}-96 d^{4} e \,f^{3} g +144 d^{3} e^{2} f^{4}\right )}{3} \] Input:
int((e*x+d)^3*(g*x+f)^2*(48*e^2*f^2-32*d*e*f*g+8*d^2*g^2-8*e*g*(2*d*g-8*e* f)*x+24*e^2*g^2*x^2),x)
Output:
(x*(24*d**5*f**2*g**2 + 24*d**5*f*g**3*x + 8*d**5*g**4*x**2 - 96*d**4*e*f* *3*g - 84*d**4*e*f**2*g**2*x - 16*d**4*e*f*g**3*x**2 + 6*d**4*e*g**4*x**3 + 144*d**3*e**2*f**4 + 96*d**3*e**2*f**3*g*x - 16*d**3*e**2*f**2*g**2*x**2 - 24*d**3*e**2*f*g**3*x**3 + 216*d**2*e**3*f**4*x + 384*d**2*e**3*f**3*g* x**2 + 276*d**2*e**3*f**2*g**2*x**3 + 96*d**2*e**3*f*g**3*x**4 + 16*d**2*e **3*g**4*x**5 + 144*d*e**4*f**4*x**2 + 336*d*e**4*f**3*g*x**3 + 312*d*e**4 *f**2*g**2*x**4 + 136*d*e**4*f*g**3*x**5 + 24*d*e**4*g**4*x**6 + 36*e**5*f **4*x**3 + 96*e**5*f**3*g*x**4 + 100*e**5*f**2*g**2*x**5 + 48*e**5*f*g**3* x**6 + 9*e**5*g**4*x**7))/3