Integrand size = 44, antiderivative size = 138 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (2 c d-b e) (d+e x)^8}+\frac {2 (9 b e g-2 c (e f+8 d g)) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 e^2 (2 c d-b e)^2 (d+e x)^7} \] Output:
-2/9*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/e^2/(-b*e+2*c*d)/(e *x+d)^8+2/63*(9*b*e*g-2*c*(8*d*g+e*f))*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7 /2)/e^2/(-b*e+2*c*d)^2/(e*x+d)^7
Time = 0.35 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.75 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-b e (7 e f+2 d g+9 e g x)+2 c \left (d^2 g+e^2 f x+8 d e (f+g x)\right )\right )}{63 e^2 (-2 c d+b e)^2 (d+e x)^5} \] Input:
Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x )^8,x]
Output:
(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-(b*e* (7*e*f + 2*d*g + 9*e*g*x)) + 2*c*(d^2*g + e^2*f*x + 8*d*e*(f + g*x))))/(63 *e^2*(-2*c*d + b*e)^2*(d + e*x)^5)
Time = 0.64 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1220, 1123}
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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-9 b e g+16 c d g+2 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^7}dx}{9 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (d+e x)^8 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-9 b e g+16 c d g+2 c e f)}{63 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (d+e x)^8 (2 c d-b e)}\) |
Input:
Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^8,x]
Output:
(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9*e^2*(2*c*d - b*e)*(d + e*x)^8) - (2*(2*c*e*f + 16*c*d*g - 9*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(63*e^2*(2*c*d - b*e)^2*(d + e*x)^7)
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b *e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + 2*p + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Time = 12.24 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93
| method | result | size |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (9 b \,e^{2} g x -16 c d e g x -2 c \,e^{2} f x +2 b d e g +7 b \,e^{2} f -2 c \,d^{2} g -16 c d e f \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{7} e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right )}\) | \(128\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (9 b \,e^{2} g x -16 c d e g x -2 c \,e^{2} f x +2 b d e g +7 b \,e^{2} f -2 c \,d^{2} g -16 c d e f \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{7} e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right )}\) | \(128\) |
| default | \(-\frac {2 g \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7 e^{8} \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{7}}-\frac {\left (d g -e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{e^{9}}\) | \(213\) |
| trager | \(-\frac {2 \left (9 b \,c^{3} e^{5} g \,x^{4}-16 c^{4} d \,e^{4} g \,x^{4}-2 c^{4} e^{5} f \,x^{4}+27 b^{2} c^{2} e^{5} g \,x^{3}-73 b \,c^{3} d \,e^{4} g \,x^{3}+b \,c^{3} e^{5} f \,x^{3}+46 c^{4} d^{2} e^{3} g \,x^{3}-10 c^{4} d \,e^{4} f \,x^{3}+27 b^{3} c \,e^{5} g \,x^{2}-96 b^{2} c^{2} d \,e^{4} g \,x^{2}+15 b^{2} c^{2} e^{5} f \,x^{2}+111 b \,c^{3} d^{2} e^{3} g \,x^{2}-57 b \,c^{3} d \,e^{4} f \,x^{2}-42 c^{4} d^{3} e^{2} g \,x^{2}+42 c^{4} d^{2} e^{3} f \,x^{2}+9 b^{4} e^{5} g x -37 b^{3} c d \,e^{4} g x +19 b^{3} c \,e^{5} f x +57 b^{2} c^{2} d^{2} e^{3} g x -84 b^{2} c^{2} d \,e^{4} f x -39 b \,c^{3} d^{3} e^{2} g x +111 b \,c^{3} d^{2} e^{3} f x +10 c^{4} d^{4} e g x -46 c^{4} d^{3} e^{2} f x +2 b^{4} d \,e^{4} g +7 b^{4} e^{5} f -8 b^{3} c \,d^{2} e^{3} g -37 b^{3} c d \,e^{4} f +12 b^{2} c^{2} d^{3} e^{2} g +69 b^{2} c^{2} d^{2} e^{3} f -8 b \,c^{3} d^{4} e g -55 b \,c^{3} d^{3} e^{2} f +2 c^{4} d^{5} g +16 c^{4} d^{4} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{63 \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) e^{2} \left (e x +d \right )^{5}}\) | \(509\) |
Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^8,x,method=_RET URNVERBOSE)
Output:
-2/63*(c*e*x+b*e-c*d)*(9*b*e^2*g*x-16*c*d*e*g*x-2*c*e^2*f*x+2*b*d*e*g+7*b* e^2*f-2*c*d^2*g-16*c*d*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d) ^7/e^2/(b^2*e^2-4*b*c*d*e+4*c^2*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (130) = 260\).
Time = 106.89 (sec) , antiderivative size = 646, normalized size of antiderivative = 4.68 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (2 \, c^{4} e^{5} f + {\left (16 \, c^{4} d e^{4} - 9 \, b c^{3} e^{5}\right )} g\right )} x^{4} + {\left ({\left (10 \, c^{4} d e^{4} - b c^{3} e^{5}\right )} f - {\left (46 \, c^{4} d^{2} e^{3} - 73 \, b c^{3} d e^{4} + 27 \, b^{2} c^{2} e^{5}\right )} g\right )} x^{3} - 3 \, {\left ({\left (14 \, c^{4} d^{2} e^{3} - 19 \, b c^{3} d e^{4} + 5 \, b^{2} c^{2} e^{5}\right )} f - {\left (14 \, c^{4} d^{3} e^{2} - 37 \, b c^{3} d^{2} e^{3} + 32 \, b^{2} c^{2} d e^{4} - 9 \, b^{3} c e^{5}\right )} g\right )} x^{2} - {\left (16 \, c^{4} d^{4} e - 55 \, b c^{3} d^{3} e^{2} + 69 \, b^{2} c^{2} d^{2} e^{3} - 37 \, b^{3} c d e^{4} + 7 \, b^{4} e^{5}\right )} f - 2 \, {\left (c^{4} d^{5} - 4 \, b c^{3} d^{4} e + 6 \, b^{2} c^{2} d^{3} e^{2} - 4 \, b^{3} c d^{2} e^{3} + b^{4} d e^{4}\right )} g + {\left ({\left (46 \, c^{4} d^{3} e^{2} - 111 \, b c^{3} d^{2} e^{3} + 84 \, b^{2} c^{2} d e^{4} - 19 \, b^{3} c e^{5}\right )} f - {\left (10 \, c^{4} d^{4} e - 39 \, b c^{3} d^{3} e^{2} + 57 \, b^{2} c^{2} d^{2} e^{3} - 37 \, b^{3} c d e^{4} + 9 \, b^{4} e^{5}\right )} g\right )} x\right )}}{63 \, {\left (4 \, c^{2} d^{7} e^{2} - 4 \, b c d^{6} e^{3} + b^{2} d^{5} e^{4} + {\left (4 \, c^{2} d^{2} e^{7} - 4 \, b c d e^{8} + b^{2} e^{9}\right )} x^{5} + 5 \, {\left (4 \, c^{2} d^{3} e^{6} - 4 \, b c d^{2} e^{7} + b^{2} d e^{8}\right )} x^{4} + 10 \, {\left (4 \, c^{2} d^{4} e^{5} - 4 \, b c d^{3} e^{6} + b^{2} d^{2} e^{7}\right )} x^{3} + 10 \, {\left (4 \, c^{2} d^{5} e^{4} - 4 \, b c d^{4} e^{5} + b^{2} d^{3} e^{6}\right )} x^{2} + 5 \, {\left (4 \, c^{2} d^{6} e^{3} - 4 \, b c d^{5} e^{4} + b^{2} d^{4} e^{5}\right )} x\right )}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^8,x, algo rithm="fricas")
Output:
2/63*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((2*c^4*e^5*f + (16*c^4*d* e^4 - 9*b*c^3*e^5)*g)*x^4 + ((10*c^4*d*e^4 - b*c^3*e^5)*f - (46*c^4*d^2*e^ 3 - 73*b*c^3*d*e^4 + 27*b^2*c^2*e^5)*g)*x^3 - 3*((14*c^4*d^2*e^3 - 19*b*c^ 3*d*e^4 + 5*b^2*c^2*e^5)*f - (14*c^4*d^3*e^2 - 37*b*c^3*d^2*e^3 + 32*b^2*c ^2*d*e^4 - 9*b^3*c*e^5)*g)*x^2 - (16*c^4*d^4*e - 55*b*c^3*d^3*e^2 + 69*b^2 *c^2*d^2*e^3 - 37*b^3*c*d*e^4 + 7*b^4*e^5)*f - 2*(c^4*d^5 - 4*b*c^3*d^4*e + 6*b^2*c^2*d^3*e^2 - 4*b^3*c*d^2*e^3 + b^4*d*e^4)*g + ((46*c^4*d^3*e^2 - 111*b*c^3*d^2*e^3 + 84*b^2*c^2*d*e^4 - 19*b^3*c*e^5)*f - (10*c^4*d^4*e - 3 9*b*c^3*d^3*e^2 + 57*b^2*c^2*d^2*e^3 - 37*b^3*c*d*e^4 + 9*b^4*e^5)*g)*x)/( 4*c^2*d^7*e^2 - 4*b*c*d^6*e^3 + b^2*d^5*e^4 + (4*c^2*d^2*e^7 - 4*b*c*d*e^8 + b^2*e^9)*x^5 + 5*(4*c^2*d^3*e^6 - 4*b*c*d^2*e^7 + b^2*d*e^8)*x^4 + 10*( 4*c^2*d^4*e^5 - 4*b*c*d^3*e^6 + b^2*d^2*e^7)*x^3 + 10*(4*c^2*d^5*e^4 - 4*b *c*d^4*e^5 + b^2*d^3*e^6)*x^2 + 5*(4*c^2*d^6*e^3 - 4*b*c*d^5*e^4 + b^2*d^4 *e^5)*x)
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**8,x )
Output:
Timed out
Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^8,x, algo rithm="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(b*e-2*c*d>0)', see `assume?` for more deta
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^8,x, algo rithm="giac")
Output:
Timed out
Time = 20.37 (sec) , antiderivative size = 12294, normalized size of antiderivative = 89.09 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Too large to display} \] Input:
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^8,x)
Output:
(((d*((336*b^2*c^5*e^3*f - 2016*c^7*d^3*g + 384*b^3*c^4*e^3*g + 800*c^7*d^ 2*e*f - 1024*b*c^6*d*e^2*f + 3424*b*c^6*d^2*e*g - 1968*b^2*c^5*d*e^2*g)/(9 45*e*(b*e - 2*c*d)^5) - (d*((16*c^5*(21*b^2*e^2*g + 50*c^2*d^2*g + 10*b*c* e^2*f - 14*c^2*d*e*f - 64*b*c*d*e*g))/(945*(b*e - 2*c*d)^5) - (d*((32*c^6* e*(5*b*e*g - 7*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(945 *(b*e - 2*c*d)^5)))/e))/e))/e - (132*b^3*c^4*e^3*f + 126*b^4*c^3*e^3*g - 1 008*b*c^6*d^3*g + 400*b*c^6*d^2*e*f - 456*b^2*c^5*d*e^2*f + 1512*b^2*c^5*d ^2*e*g - 756*b^3*c^4*d*e^2*g)/(945*e*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((696*b^2*c^5*e^3*f - 6016*c^7* d^3*g + 1044*b^3*c^4*e^3*g + 1920*c^7*d^2*e*f - 2304*b*c^6*d*e^2*f + 9984* b*c^6*d^2*e*g - 5568*b^2*c^5*d*e^2*g)/(945*e*(b*e - 2*c*d)^5) - (d*((8*c^5 *(29*b^2*e^2*g + 80*c^2*d^2*g + 10*b*c*e^2*f - 16*c^2*d*e*f - 96*b*c*d*e*g ))/(315*(b*e - 2*c*d)^5) - (d*((16*c^6*e*(15*b*e*g - 24*c*d*g + 2*c*e*f))/ (945*(b*e - 2*c*d)^5) - (32*c^7*d*e*g)/(945*(b*e - 2*c*d)^5)))/e))/e))/e - (292*b^3*c^4*e^3*f + 376*b^4*c^3*e^3*g - 3008*b*c^6*d^3*g + 960*b*c^6*d^2 *e*f - 1056*b^2*c^5*d*e^2*f + 4512*b^2*c^5*d^2*e*g - 2256*b^3*c^4*d*e^2*g) /(945*e*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((840*b^2*c^5*e^3*f - 7616*c^7*d^3*g + 1308*b^3*c^4*e^3*g + 2368*c^7*d^2*e*f - 2816*b*c^6*d*e^2*f + 12608*b*c^6*d^2*e*g - 7008*b^2*c^5 *d*e^2*g)/(945*e*(b*e - 2*c*d)^5) - (d*((8*c^5*(105*b^2*e^2*g + 296*c^2...
Time = 1.88 (sec) , antiderivative size = 2583, normalized size of antiderivative = 18.72 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx =\text {Too large to display} \] Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^8,x)
Output:
(2*i*( - 2*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b* e + c*d - c*e*x)*b**4*d*e**4*g - 7*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**4*e**5*f - 9*sqrt(d + e*x)*sqr t(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**4*e**5*g *x + 8*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**3*c*d**2*e**3*g + 37*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**3*c*d*e**4*f + 37*sqrt(d + e *x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**3 *c*d*e**4*g*x - 19*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sq rt( - b*e + c*d - c*e*x)*b**3*c*e**5*f*x - 27*sqrt(d + e*x)*sqrt(b*e - 2*c *d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**3*c*e**5*g*x**2 - 1 2*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**3*e**2*g - 69*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**2*e**3*f - 57*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)* b**2*c**2*d**2*e**3*g*x + 84*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d*e**4*f*x + 96*sqrt(d + e*x) *sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c* *2*d*e**4*g*x**2 - 15*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d) *sqrt( - b*e + c*d - c*e*x)*b**2*c**2*e**5*f*x**2 - 27*sqrt(d + e*x)*sq...