Integrand size = 73, antiderivative size = 71 \[ \int (d+e x)^2 (f+g x)^2 \left (44 e^2 f^2-40 d e f g+\frac {25 d^2 g^2}{2}-6 e g \left (-8 e f+\frac {5 d g}{2}\right ) x+\frac {33}{2} e^2 g^2 x^2\right )^{3/2} \, dx=\frac {(d+e x)^3 \left (88 e^2 f^2-80 d e f g+25 d^2 g^2+6 e g (16 e f-5 d g) x+33 e^2 g^2 x^2\right )^{5/2}}{528 \sqrt {2} e^3} \] Output:
1/1056*(e*x+d)^3*(88*e^2*f^2-80*d*e*f*g+25*d^2*g^2+6*e*g*(-5*d*g+16*e*f)*x +33*e^2*g^2*x^2)^(5/2)*2^(1/2)/e^3
Time = 10.91 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96 \[ \int (d+e x)^2 (f+g x)^2 \left (44 e^2 f^2-40 d e f g+\frac {25 d^2 g^2}{2}-6 e g \left (-8 e f+\frac {5 d g}{2}\right ) x+\frac {33}{2} e^2 g^2 x^2\right )^{3/2} \, dx=\frac {(d+e x)^3 \left (25 d^2 g^2-10 d e g (8 f+3 g x)+e^2 \left (88 f^2+96 f g x+33 g^2 x^2\right )\right )^{5/2}}{528 \sqrt {2} e^3} \] Input:
Integrate[(d + e*x)^2*(f + g*x)^2*(44*e^2*f^2 - 40*d*e*f*g + (25*d^2*g^2)/ 2 - 6*e*g*(-8*e*f + (5*d*g)/2)*x + (33*e^2*g^2*x^2)/2)^(3/2),x]
Output:
((d + e*x)^3*(25*d^2*g^2 - 10*d*e*g*(8*f + 3*g*x) + e^2*(88*f^2 + 96*f*g*x + 33*g^2*x^2))^(5/2))/(528*Sqrt[2]*e^3)
Time = 0.44 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {1208}
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.
| \(\displaystyle \int (d+e x)^2 (f+g x)^2 \left (\frac {25 d^2 g^2}{2}-6 e g x \left (\frac {5 d g}{2}-8 e f\right )-40 d e f g+44 e^2 f^2+\frac {33}{2} e^2 g^2 x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1208 |
| \(\displaystyle \frac {(d+e x)^3 \left (25 d^2 g^2+6 e g x (16 e f-5 d g)-80 d e f g+88 e^2 f^2+33 e^2 g^2 x^2\right )^{5/2}}{528 \sqrt {2} e^3}\) |
Input:
Int[(d + e*x)^2*(f + g*x)^2*(44*e^2*f^2 - 40*d*e*f*g + (25*d^2*g^2)/2 - 6* e*g*(-8*e*f + (5*d*g)/2)*x + (33*e^2*g^2*x^2)/2)^(3/2),x]
Output:
((d + e*x)^3*(88*e^2*f^2 - 80*d*e*f*g + 25*d^2*g^2 + 6*e*g*(16*e*f - 5*d*g )*x + 33*e^2*g^2*x^2)^(5/2))/(528*Sqrt[2]*e^3)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^2*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g^2*(d + e*x)^(m + 1)*((a + b*x + c *x^2)^(p + 1)/(c*e*(m + 2*p + 3))), x] /; FreeQ[{a, b, c, d, e, f, g, m, p} , x] && EqQ[b*e*g*(m + p + 2) + 2*c*(d*g*(p + 1) - e*f*(m + 2*p + 3)), 0] & & EqQ[e*(c*f^2 - b*f*g + a*g^2)*(m + 1) + (2*c*f - b*g)*(e*f - d*g)*(p + 1) , 0] && NeQ[m + 2*p + 3, 0]
Time = 5.36 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.59
| method | result | size |
| orering | \(\frac {\left (e x +d \right )^{3} \left (33 e^{2} g^{2} x^{2}-30 d e \,g^{2} x +96 e^{2} f g x +25 d^{2} g^{2}-80 d e f g +88 e^{2} f^{2}\right ) \left (44 e^{2} f^{2}-40 d e f g +\frac {25 d^{2} g^{2}}{2}-6 e g \left (-8 e f +\frac {5 d g}{2}\right ) x +\frac {33 e^{2} g^{2} x^{2}}{2}\right )^{\frac {3}{2}}}{264 e^{3}}\) | \(113\) |
| gosper | \(\frac {\left (33 e^{2} g^{2} x^{2}-30 d e \,g^{2} x +96 e^{2} f g x +25 d^{2} g^{2}-80 d e f g +88 e^{2} f^{2}\right ) \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right ) \left (66 e^{2} g^{2} x^{2}-60 d e \,g^{2} x +192 e^{2} f g x +50 d^{2} g^{2}-160 d e f g +176 e^{2} f^{2}\right )^{\frac {3}{2}}}{2112 e^{3}}\) | \(135\) |
| trager | \(\frac {\left (1089 e^{7} g^{4} x^{7}+1287 d \,e^{6} g^{4} x^{6}+6336 e^{7} f \,g^{3} x^{6}-123 d^{2} e^{5} g^{4} x^{5}+7968 d \,e^{6} f \,g^{3} x^{5}+15024 e^{7} f^{2} g^{2} x^{5}+1299 d^{3} e^{4} g^{4} x^{4}-4512 d^{2} e^{5} f \,g^{3} x^{4}+24432 d \,e^{6} f^{2} g^{2} x^{4}+16896 e^{7} f^{3} g \,x^{4}+1795 d^{4} e^{3} g^{4} x^{3}-1984 d^{3} e^{4} f \,g^{3} x^{3}-6048 d^{2} e^{5} f^{2} g^{2} x^{3}+36608 d \,e^{6} f^{3} g \,x^{3}+7744 e^{7} f^{4} x^{3}-75 d^{5} e^{2} g^{4} x^{2}+5760 d^{4} e^{3} f \,g^{3} x^{2}-14496 d^{3} e^{4} f^{2} g^{2} x^{2}+8448 d^{2} e^{5} f^{3} g \,x^{2}+23232 d \,e^{6} f^{4} x^{2}+375 d^{6} e \,g^{4} x -2400 d^{5} e^{2} f \,g^{3} x +11760 d^{4} e^{3} f^{2} g^{2} x -25344 d^{3} e^{4} f^{3} g x +23232 d^{2} e^{5} f^{4} x +625 d^{7} g^{4}-4000 d^{6} e f \,g^{3}+10800 d^{5} e^{2} f^{2} g^{2}-14080 d^{4} e^{3} f^{3} g +7744 d^{3} e^{4} f^{4}\right ) \sqrt {44 e^{2} f^{2}-40 d e f g +\frac {25}{2} d^{2} g^{2}+48 e^{2} f g x -15 d e \,g^{2} x +\frac {33}{2} e^{2} g^{2} x^{2}}}{528 e^{3}}\) | \(452\) |
| default | \(\text {Expression too large to display}\) | \(6223\) |
Input:
int((e*x+d)^2*(g*x+f)^2*(44*e^2*f^2-40*d*e*f*g+25/2*d^2*g^2-6*e*g*(-8*e*f+ 5/2*d*g)*x+33/2*e^2*g^2*x^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/264*(e*x+d)^3*(33*e^2*g^2*x^2-30*d*e*g^2*x+96*e^2*f*g*x+25*d^2*g^2-80*d* e*f*g+88*e^2*f^2)/e^3*(44*e^2*f^2-40*d*e*f*g+25/2*d^2*g^2-6*e*g*(-8*e*f+5/ 2*d*g)*x+33/2*e^2*g^2*x^2)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (65) = 130\).
Time = 10.45 (sec) , antiderivative size = 424, normalized size of antiderivative = 5.97 \[ \int (d+e x)^2 (f+g x)^2 \left (44 e^2 f^2-40 d e f g+\frac {25 d^2 g^2}{2}-6 e g \left (-8 e f+\frac {5 d g}{2}\right ) x+\frac {33}{2} e^2 g^2 x^2\right )^{3/2} \, dx=\frac {{\left (1089 \, e^{7} g^{4} x^{7} + 7744 \, d^{3} e^{4} f^{4} - 14080 \, d^{4} e^{3} f^{3} g + 10800 \, d^{5} e^{2} f^{2} g^{2} - 4000 \, d^{6} e f g^{3} + 625 \, d^{7} g^{4} + 99 \, {\left (64 \, e^{7} f g^{3} + 13 \, d e^{6} g^{4}\right )} x^{6} + 3 \, {\left (5008 \, e^{7} f^{2} g^{2} + 2656 \, d e^{6} f g^{3} - 41 \, d^{2} e^{5} g^{4}\right )} x^{5} + 3 \, {\left (5632 \, e^{7} f^{3} g + 8144 \, d e^{6} f^{2} g^{2} - 1504 \, d^{2} e^{5} f g^{3} + 433 \, d^{3} e^{4} g^{4}\right )} x^{4} + {\left (7744 \, e^{7} f^{4} + 36608 \, d e^{6} f^{3} g - 6048 \, d^{2} e^{5} f^{2} g^{2} - 1984 \, d^{3} e^{4} f g^{3} + 1795 \, d^{4} e^{3} g^{4}\right )} x^{3} + 3 \, {\left (7744 \, d e^{6} f^{4} + 2816 \, d^{2} e^{5} f^{3} g - 4832 \, d^{3} e^{4} f^{2} g^{2} + 1920 \, d^{4} e^{3} f g^{3} - 25 \, d^{5} e^{2} g^{4}\right )} x^{2} + 3 \, {\left (7744 \, d^{2} e^{5} f^{4} - 8448 \, d^{3} e^{4} f^{3} g + 3920 \, d^{4} e^{3} f^{2} g^{2} - 800 \, d^{5} e^{2} f g^{3} + 125 \, d^{6} e g^{4}\right )} x\right )} \sqrt {\frac {33}{2} \, e^{2} g^{2} x^{2} + 44 \, e^{2} f^{2} - 40 \, d e f g + \frac {25}{2} \, d^{2} g^{2} + 3 \, {\left (16 \, e^{2} f g - 5 \, d e g^{2}\right )} x}}{528 \, e^{3}} \] Input:
integrate((e*x+d)^2*(g*x+f)^2*(44*e^2*f^2-40*d*e*f*g+25/2*d^2*g^2-6*e*g*(- 8*e*f+5/2*d*g)*x+33/2*e^2*g^2*x^2)^(3/2),x, algorithm="fricas")
Output:
1/528*(1089*e^7*g^4*x^7 + 7744*d^3*e^4*f^4 - 14080*d^4*e^3*f^3*g + 10800*d ^5*e^2*f^2*g^2 - 4000*d^6*e*f*g^3 + 625*d^7*g^4 + 99*(64*e^7*f*g^3 + 13*d* e^6*g^4)*x^6 + 3*(5008*e^7*f^2*g^2 + 2656*d*e^6*f*g^3 - 41*d^2*e^5*g^4)*x^ 5 + 3*(5632*e^7*f^3*g + 8144*d*e^6*f^2*g^2 - 1504*d^2*e^5*f*g^3 + 433*d^3* e^4*g^4)*x^4 + (7744*e^7*f^4 + 36608*d*e^6*f^3*g - 6048*d^2*e^5*f^2*g^2 - 1984*d^3*e^4*f*g^3 + 1795*d^4*e^3*g^4)*x^3 + 3*(7744*d*e^6*f^4 + 2816*d^2* e^5*f^3*g - 4832*d^3*e^4*f^2*g^2 + 1920*d^4*e^3*f*g^3 - 25*d^5*e^2*g^4)*x^ 2 + 3*(7744*d^2*e^5*f^4 - 8448*d^3*e^4*f^3*g + 3920*d^4*e^3*f^2*g^2 - 800* d^5*e^2*f*g^3 + 125*d^6*e*g^4)*x)*sqrt(33/2*e^2*g^2*x^2 + 44*e^2*f^2 - 40* d*e*f*g + 25/2*d^2*g^2 + 3*(16*e^2*f*g - 5*d*e*g^2)*x)/e^3
Timed out. \[ \int (d+e x)^2 (f+g x)^2 \left (44 e^2 f^2-40 d e f g+\frac {25 d^2 g^2}{2}-6 e g \left (-8 e f+\frac {5 d g}{2}\right ) x+\frac {33}{2} e^2 g^2 x^2\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**2*(g*x+f)**2*(44*e**2*f**2-40*d*e*f*g+25/2*d**2*g**2-6* e*g*(-8*e*f+5/2*d*g)*x+33/2*e**2*g**2*x**2)**(3/2),x)
Output:
Timed out
Exception generated. \[ \int (d+e x)^2 (f+g x)^2 \left (44 e^2 f^2-40 d e f g+\frac {25 d^2 g^2}{2}-6 e g \left (-8 e f+\frac {5 d g}{2}\right ) x+\frac {33}{2} e^2 g^2 x^2\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^2*(g*x+f)^2*(44*e^2*f^2-40*d*e*f*g+25/2*d^2*g^2-6*e*g*(- 8*e*f+5/2*d*g)*x+33/2*e^2*g^2*x^2)^(3/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (65) = 130\).
Time = 0.29 (sec) , antiderivative size = 494, normalized size of antiderivative = 6.96 \[ \int (d+e x)^2 (f+g x)^2 \left (44 e^2 f^2-40 d e f g+\frac {25 d^2 g^2}{2}-6 e g \left (-8 e f+\frac {5 d g}{2}\right ) x+\frac {33}{2} e^2 g^2 x^2\right )^{3/2} \, dx=\frac {1}{1056} \, \sqrt {2} \sqrt {33 \, e^{2} g^{2} x^{2} + 96 \, e^{2} f g x - 30 \, d e g^{2} x + 88 \, e^{2} f^{2} - 80 \, d e f g + 25 \, d^{2} g^{2}} {\left ({\left ({\left ({\left (3 \, {\left ({\left (33 \, {\left (11 \, e^{4} g^{4} x + \frac {64 \, e^{17} f g^{16} + 13 \, d e^{16} g^{17}}{e^{13} g^{13}}\right )} x + \frac {5008 \, e^{17} f^{2} g^{15} + 2656 \, d e^{16} f g^{16} - 41 \, d^{2} e^{15} g^{17}}{e^{13} g^{13}}\right )} x + \frac {5632 \, e^{17} f^{3} g^{14} + 8144 \, d e^{16} f^{2} g^{15} - 1504 \, d^{2} e^{15} f g^{16} + 433 \, d^{3} e^{14} g^{17}}{e^{13} g^{13}}\right )} x + \frac {7744 \, e^{17} f^{4} g^{13} + 36608 \, d e^{16} f^{3} g^{14} - 6048 \, d^{2} e^{15} f^{2} g^{15} - 1984 \, d^{3} e^{14} f g^{16} + 1795 \, d^{4} e^{13} g^{17}}{e^{13} g^{13}}\right )} x + \frac {3 \, {\left (7744 \, d e^{16} f^{4} g^{13} + 2816 \, d^{2} e^{15} f^{3} g^{14} - 4832 \, d^{3} e^{14} f^{2} g^{15} + 1920 \, d^{4} e^{13} f g^{16} - 25 \, d^{5} e^{12} g^{17}\right )}}{e^{13} g^{13}}\right )} x + \frac {3 \, {\left (7744 \, d^{2} e^{15} f^{4} g^{13} - 8448 \, d^{3} e^{14} f^{3} g^{14} + 3920 \, d^{4} e^{13} f^{2} g^{15} - 800 \, d^{5} e^{12} f g^{16} + 125 \, d^{6} e^{11} g^{17}\right )}}{e^{13} g^{13}}\right )} x + \frac {7744 \, d^{3} e^{14} f^{4} g^{13} - 14080 \, d^{4} e^{13} f^{3} g^{14} + 10800 \, d^{5} e^{12} f^{2} g^{15} - 4000 \, d^{6} e^{11} f g^{16} + 625 \, d^{7} e^{10} g^{17}}{e^{13} g^{13}}\right )} \] Input:
integrate((e*x+d)^2*(g*x+f)^2*(44*e^2*f^2-40*d*e*f*g+25/2*d^2*g^2-6*e*g*(- 8*e*f+5/2*d*g)*x+33/2*e^2*g^2*x^2)^(3/2),x, algorithm="giac")
Output:
1/1056*sqrt(2)*sqrt(33*e^2*g^2*x^2 + 96*e^2*f*g*x - 30*d*e*g^2*x + 88*e^2* f^2 - 80*d*e*f*g + 25*d^2*g^2)*((((3*((33*(11*e^4*g^4*x + (64*e^17*f*g^16 + 13*d*e^16*g^17)/(e^13*g^13))*x + (5008*e^17*f^2*g^15 + 2656*d*e^16*f*g^1 6 - 41*d^2*e^15*g^17)/(e^13*g^13))*x + (5632*e^17*f^3*g^14 + 8144*d*e^16*f ^2*g^15 - 1504*d^2*e^15*f*g^16 + 433*d^3*e^14*g^17)/(e^13*g^13))*x + (7744 *e^17*f^4*g^13 + 36608*d*e^16*f^3*g^14 - 6048*d^2*e^15*f^2*g^15 - 1984*d^3 *e^14*f*g^16 + 1795*d^4*e^13*g^17)/(e^13*g^13))*x + 3*(7744*d*e^16*f^4*g^1 3 + 2816*d^2*e^15*f^3*g^14 - 4832*d^3*e^14*f^2*g^15 + 1920*d^4*e^13*f*g^16 - 25*d^5*e^12*g^17)/(e^13*g^13))*x + 3*(7744*d^2*e^15*f^4*g^13 - 8448*d^3 *e^14*f^3*g^14 + 3920*d^4*e^13*f^2*g^15 - 800*d^5*e^12*f*g^16 + 125*d^6*e^ 11*g^17)/(e^13*g^13))*x + (7744*d^3*e^14*f^4*g^13 - 14080*d^4*e^13*f^3*g^1 4 + 10800*d^5*e^12*f^2*g^15 - 4000*d^6*e^11*f*g^16 + 625*d^7*e^10*g^17)/(e ^13*g^13))
Time = 14.50 (sec) , antiderivative size = 398, normalized size of antiderivative = 5.61 \[ \int (d+e x)^2 (f+g x)^2 \left (44 e^2 f^2-40 d e f g+\frac {25 d^2 g^2}{2}-6 e g \left (-8 e f+\frac {5 d g}{2}\right ) x+\frac {33}{2} e^2 g^2 x^2\right )^{3/2} \, dx=\frac {\sqrt {2}\,\sqrt {25\,d^2\,g^2+88\,e^2\,f^2+33\,e^2\,g^2\,x^2-12\,e\,g\,x\,\left (\frac {5\,d\,g}{2}-8\,e\,f\right )-80\,d\,e\,f\,g}\,\left (\frac {x^3\,\left (\frac {1795\,d^4\,e^3\,g^4}{528}-\frac {124\,d^3\,e^4\,f\,g^3}{33}-\frac {126\,d^2\,e^5\,f^2\,g^2}{11}+\frac {208\,d\,e^6\,f^3\,g}{3}+\frac {44\,e^7\,f^4}{3}\right )}{e^3}+\frac {d^3\,{\left (25\,d^2\,g^2-80\,d\,e\,f\,g+88\,e^2\,f^2\right )}^2}{528\,e^3}+\frac {x^4\,\left (\frac {433\,d^3\,e^4\,g^4}{176}-\frac {94\,d^2\,e^5\,f\,g^3}{11}+\frac {509\,d\,e^6\,f^2\,g^2}{11}+32\,e^7\,f^3\,g\right )}{e^3}+\frac {x\,\left (375\,d^6\,e\,g^4-2400\,d^5\,e^2\,f\,g^3+11760\,d^4\,e^3\,f^2\,g^2-25344\,d^3\,e^4\,f^3\,g+23232\,d^2\,e^5\,f^4\right )}{528\,e^3}+\frac {33\,e^4\,g^4\,x^7}{16}+\frac {x^2\,\left (-\frac {25\,d^5\,e^2\,g^4}{176}+\frac {120\,d^4\,e^3\,f\,g^3}{11}-\frac {302\,d^3\,e^4\,f^2\,g^2}{11}+16\,d^2\,e^5\,f^3\,g+44\,d\,e^6\,f^4\right )}{e^3}+e^3\,g^3\,x^6\,\left (\frac {39\,d\,g}{16}+12\,e\,f\right )+e^2\,g^2\,x^5\,\left (-\frac {41\,d^2\,g^2}{176}+\frac {166\,d\,e\,f\,g}{11}+\frac {313\,e^2\,f^2}{11}\right )\right )}{2} \] Input:
int((f + g*x)^2*(d + e*x)^2*((25*d^2*g^2)/2 + 44*e^2*f^2 + (33*e^2*g^2*x^2 )/2 - 6*e*g*x*((5*d*g)/2 - 8*e*f) - 40*d*e*f*g)^(3/2),x)
Output:
(2^(1/2)*(25*d^2*g^2 + 88*e^2*f^2 + 33*e^2*g^2*x^2 - 12*e*g*x*((5*d*g)/2 - 8*e*f) - 80*d*e*f*g)^(1/2)*((x^3*((44*e^7*f^4)/3 + (1795*d^4*e^3*g^4)/528 - (124*d^3*e^4*f*g^3)/33 - (126*d^2*e^5*f^2*g^2)/11 + (208*d*e^6*f^3*g)/3 ))/e^3 + (d^3*(25*d^2*g^2 + 88*e^2*f^2 - 80*d*e*f*g)^2)/(528*e^3) + (x^4*( 32*e^7*f^3*g + (433*d^3*e^4*g^4)/176 + (509*d*e^6*f^2*g^2)/11 - (94*d^2*e^ 5*f*g^3)/11))/e^3 + (x*(375*d^6*e*g^4 + 23232*d^2*e^5*f^4 - 25344*d^3*e^4* f^3*g - 2400*d^5*e^2*f*g^3 + 11760*d^4*e^3*f^2*g^2))/(528*e^3) + (33*e^4*g ^4*x^7)/16 + (x^2*(44*d*e^6*f^4 - (25*d^5*e^2*g^4)/176 + 16*d^2*e^5*f^3*g + (120*d^4*e^3*f*g^3)/11 - (302*d^3*e^4*f^2*g^2)/11))/e^3 + e^3*g^3*x^6*(( 39*d*g)/16 + 12*e*f) + e^2*g^2*x^5*((313*e^2*f^2)/11 - (41*d^2*g^2)/176 + (166*d*e*f*g)/11)))/2
Time = 1.82 (sec) , antiderivative size = 452, normalized size of antiderivative = 6.37 \[ \int (d+e x)^2 (f+g x)^2 \left (44 e^2 f^2-40 d e f g+\frac {25 d^2 g^2}{2}-6 e g \left (-8 e f+\frac {5 d g}{2}\right ) x+\frac {33}{2} e^2 g^2 x^2\right )^{3/2} \, dx=\frac {\sqrt {33 e^{2} g^{2} x^{2}-30 d e \,g^{2} x +96 e^{2} f g x +25 d^{2} g^{2}-80 d e f g +88 e^{2} f^{2}}\, \sqrt {2}\, \left (1089 e^{7} g^{4} x^{7}+1287 d \,e^{6} g^{4} x^{6}+6336 e^{7} f \,g^{3} x^{6}-123 d^{2} e^{5} g^{4} x^{5}+7968 d \,e^{6} f \,g^{3} x^{5}+15024 e^{7} f^{2} g^{2} x^{5}+1299 d^{3} e^{4} g^{4} x^{4}-4512 d^{2} e^{5} f \,g^{3} x^{4}+24432 d \,e^{6} f^{2} g^{2} x^{4}+16896 e^{7} f^{3} g \,x^{4}+1795 d^{4} e^{3} g^{4} x^{3}-1984 d^{3} e^{4} f \,g^{3} x^{3}-6048 d^{2} e^{5} f^{2} g^{2} x^{3}+36608 d \,e^{6} f^{3} g \,x^{3}+7744 e^{7} f^{4} x^{3}-75 d^{5} e^{2} g^{4} x^{2}+5760 d^{4} e^{3} f \,g^{3} x^{2}-14496 d^{3} e^{4} f^{2} g^{2} x^{2}+8448 d^{2} e^{5} f^{3} g \,x^{2}+23232 d \,e^{6} f^{4} x^{2}+375 d^{6} e \,g^{4} x -2400 d^{5} e^{2} f \,g^{3} x +11760 d^{4} e^{3} f^{2} g^{2} x -25344 d^{3} e^{4} f^{3} g x +23232 d^{2} e^{5} f^{4} x +625 d^{7} g^{4}-4000 d^{6} e f \,g^{3}+10800 d^{5} e^{2} f^{2} g^{2}-14080 d^{4} e^{3} f^{3} g +7744 d^{3} e^{4} f^{4}\right )}{1056 e^{3}} \] Input:
int((e*x+d)^2*(g*x+f)^2*(44*e^2*f^2-40*d*e*f*g+25/2*d^2*g^2-6*e*g*(-8*e*f+ 5/2*d*g)*x+33/2*e^2*g^2*x^2)^(3/2),x)
Output:
(sqrt(25*d**2*g**2 - 80*d*e*f*g - 30*d*e*g**2*x + 88*e**2*f**2 + 96*e**2*f *g*x + 33*e**2*g**2*x**2)*sqrt(2)*(625*d**7*g**4 - 4000*d**6*e*f*g**3 + 37 5*d**6*e*g**4*x + 10800*d**5*e**2*f**2*g**2 - 2400*d**5*e**2*f*g**3*x - 75 *d**5*e**2*g**4*x**2 - 14080*d**4*e**3*f**3*g + 11760*d**4*e**3*f**2*g**2* x + 5760*d**4*e**3*f*g**3*x**2 + 1795*d**4*e**3*g**4*x**3 + 7744*d**3*e**4 *f**4 - 25344*d**3*e**4*f**3*g*x - 14496*d**3*e**4*f**2*g**2*x**2 - 1984*d **3*e**4*f*g**3*x**3 + 1299*d**3*e**4*g**4*x**4 + 23232*d**2*e**5*f**4*x + 8448*d**2*e**5*f**3*g*x**2 - 6048*d**2*e**5*f**2*g**2*x**3 - 4512*d**2*e* *5*f*g**3*x**4 - 123*d**2*e**5*g**4*x**5 + 23232*d*e**6*f**4*x**2 + 36608* d*e**6*f**3*g*x**3 + 24432*d*e**6*f**2*g**2*x**4 + 7968*d*e**6*f*g**3*x**5 + 1287*d*e**6*g**4*x**6 + 7744*e**7*f**4*x**3 + 16896*e**7*f**3*g*x**4 + 15024*e**7*f**2*g**2*x**5 + 6336*e**7*f*g**3*x**6 + 1089*e**7*g**4*x**7))/ (1056*e**3)