Integrand size = 46, antiderivative size = 417 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (2 c d-b e)^4 (c e f+c d g-b e g) (d+e x)^{3/2}}{3 c^6 e^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (2 c d-b e)^3 (4 c e f+6 c d g-5 b e g) \sqrt {d+e x}}{c^6 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {4 (2 c d-b e)^2 (3 c e f+7 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^6 e^2 \sqrt {d+e x}}+\frac {4 (2 c d-b e) (2 c e f+8 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c^6 e^2 (d+e x)^{3/2}}-\frac {2 (c e f+9 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c^6 e^2 (d+e x)^{5/2}}+\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c^6 e^2 (d+e x)^{7/2}} \] Output:
2/3*(-b*e+2*c*d)^4*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(3/2)/c^6/e^2/(d*(-b*e+c*d )-b*e^2*x-c*e^2*x^2)^(3/2)-2*(-b*e+2*c*d)^3*(-5*b*e*g+6*c*d*g+4*c*e*f)*(e* x+d)^(1/2)/c^6/e^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)-4*(-b*e+2*c*d)^2 *(-5*b*e*g+7*c*d*g+3*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c^6/e^2 /(e*x+d)^(1/2)+4/3*(-b*e+2*c*d)*(-5*b*e*g+8*c*d*g+2*c*e*f)*(d*(-b*e+c*d)-b *e^2*x-c*e^2*x^2)^(3/2)/c^6/e^2/(e*x+d)^(3/2)-2/5*(-5*b*e*g+9*c*d*g+c*e*f) *(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^6/e^2/(e*x+d)^(5/2)+2/7*g*(d*(-b *e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^6/e^2/(e*x+d)^(7/2)
Time = 0.49 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {d+e x} \left (-1280 b^5 e^5 g+128 b^4 c e^4 (7 e f+78 d g-15 e g x)-32 b^3 c^2 e^3 \left (953 d^2 g+2 d e (91 f-204 g x)+3 e^2 x (-14 f+5 g x)\right )+16 b^2 c^3 e^2 \left (2844 d^3 g+3 d^2 e (287 f-681 g x)+e^3 x^2 (21 f+5 g x)+6 d e^2 x (-77 f+29 g x)\right )+c^5 \left (9414 d^5 g+3 d^4 e (1687 f-4707 g x)+3 e^5 x^4 (7 f+5 g x)+2 d e^4 x^3 (98 f+57 g x)+2 d^2 e^3 x^2 (903 f+257 g x)+12 d^3 e^2 x (-637 f+292 g x)\right )-2 b c^4 e \left (16563 d^4 g+12 d^3 e (581 f-1482 g x)+e^4 x^3 (28 f+15 g x)+12 d e^3 x^2 (63 f+16 g x)+6 d^2 e^2 x (-1106 f+449 g x)\right )\right )}{105 c^6 e^2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[((d + e*x)^(13/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^ 2)^(5/2),x]
Output:
(2*Sqrt[d + e*x]*(-1280*b^5*e^5*g + 128*b^4*c*e^4*(7*e*f + 78*d*g - 15*e*g *x) - 32*b^3*c^2*e^3*(953*d^2*g + 2*d*e*(91*f - 204*g*x) + 3*e^2*x*(-14*f + 5*g*x)) + 16*b^2*c^3*e^2*(2844*d^3*g + 3*d^2*e*(287*f - 681*g*x) + e^3*x ^2*(21*f + 5*g*x) + 6*d*e^2*x*(-77*f + 29*g*x)) + c^5*(9414*d^5*g + 3*d^4* e*(1687*f - 4707*g*x) + 3*e^5*x^4*(7*f + 5*g*x) + 2*d*e^4*x^3*(98*f + 57*g *x) + 2*d^2*e^3*x^2*(903*f + 257*g*x) + 12*d^3*e^2*x*(-637*f + 292*g*x)) - 2*b*c^4*e*(16563*d^4*g + 12*d^3*e*(581*f - 1482*g*x) + e^4*x^3*(28*f + 15 *g*x) + 12*d*e^3*x^2*(63*f + 16*g*x) + 6*d^2*e^2*x*(-1106*f + 449*g*x))))/ (105*c^6*e^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))] )
Time = 1.19 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1218, 1128, 1128, 1128, 1128, 1122}
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.
| \(\displaystyle \int \frac {(d+e x)^{13/2} (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1218 |
| \(\displaystyle \frac {2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-10 b e g+13 c d g+7 c e f) \int \frac {(d+e x)^{11/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-10 b e g+13 c d g+7 c e f) \left (\frac {8 (2 c d-b e) \int \frac {(d+e x)^{9/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{7 c}-\frac {2 (d+e x)^{9/2}}{7 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-10 b e g+13 c d g+7 c e f) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \int \frac {(d+e x)^{7/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{5 c}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{7 c}-\frac {2 (d+e x)^{9/2}}{7 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-10 b e g+13 c d g+7 c e f) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \int \frac {(d+e x)^{5/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 c}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{5 c}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{7 c}-\frac {2 (d+e x)^{9/2}}{7 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-10 b e g+13 c d g+7 c e f) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {(d+e x)^{3/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{c}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{5 c}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{7 c}-\frac {2 (d+e x)^{9/2}}{7 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {\left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (\frac {4 \sqrt {d+e x} (2 c d-b e)}{c^2 e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{5 c}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{7 c}-\frac {2 (d+e x)^{9/2}}{7 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-10 b e g+13 c d g+7 c e f)}{3 c e (2 c d-b e)}\) |
Input:
Int[((d + e*x)^(13/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/ 2),x]
Output:
(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(13/2))/(3*c*e^2*(2*c*d - b*e)*(d*(c* d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - ((7*c*e*f + 13*c*d*g - 10*b*e*g)* ((-2*(d + e*x)^(9/2))/(7*c*e*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (8*(2*c*d - b*e)*((-2*(d + e*x)^(7/2))/(5*c*e*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (6*(2*c*d - b*e)*((-2*(d + e*x)^(5/2))/(3*c*e*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (4*(2*c*d - b*e)*((4*(2*c*d - b*e)*Sqrt[ d + e*x])/(c^2*e*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(d + e*x) ^(3/2))/(c*e*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])))/(3*c)))/(5*c)))/ (7*c)))/(3*c*e*(2*c*d - b*e))
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[Simplify[m + p], 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(( a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Simp[e*((m*(g*(c* d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))) I nt[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d , e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 1.58 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.27
| method | result | size |
| default | \(\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (-15 g \,e^{5} x^{5} c^{5}+30 b \,c^{4} e^{5} g \,x^{4}-114 c^{5} d \,e^{4} g \,x^{4}-21 c^{5} e^{5} f \,x^{4}-80 b^{2} c^{3} e^{5} g \,x^{3}+384 b \,c^{4} d \,e^{4} g \,x^{3}+56 b \,c^{4} e^{5} f \,x^{3}-514 c^{5} d^{2} e^{3} g \,x^{3}-196 c^{5} d \,e^{4} f \,x^{3}+480 b^{3} c^{2} e^{5} g \,x^{2}-2784 b^{2} c^{3} d \,e^{4} g \,x^{2}-336 b^{2} c^{3} e^{5} f \,x^{2}+5388 b \,c^{4} d^{2} e^{3} g \,x^{2}+1512 b \,c^{4} d \,e^{4} f \,x^{2}-3504 c^{5} d^{3} e^{2} g \,x^{2}-1806 c^{5} d^{2} e^{3} f \,x^{2}+1920 b^{4} c \,e^{5} g x -13056 b^{3} c^{2} d \,e^{4} g x -1344 b^{3} c^{2} e^{5} f x +32688 b^{2} c^{3} d^{2} e^{3} g x +7392 b^{2} c^{3} d \,e^{4} f x -35568 b \,c^{4} d^{3} e^{2} g x -13272 b \,c^{4} d^{2} e^{3} f x +14121 c^{5} d^{4} e g x +7644 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -9984 b^{4} c d \,e^{4} g -896 b^{4} c \,e^{5} f +30496 b^{3} c^{2} d^{2} e^{3} g +5824 b^{3} c^{2} d \,e^{4} f -45504 b^{2} c^{3} d^{3} e^{2} g -13776 b^{2} c^{3} d^{2} e^{3} f +33126 b \,c^{4} d^{4} e g +13944 b \,c^{4} d^{3} e^{2} f -9414 c^{5} d^{5} g -5061 f \,d^{4} c^{5} e \right )}{105 \sqrt {e x +d}\, \left (c e x +b e -c d \right )^{2} c^{6} e^{2}}\) | \(529\) |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-15 g \,e^{5} x^{5} c^{5}+30 b \,c^{4} e^{5} g \,x^{4}-114 c^{5} d \,e^{4} g \,x^{4}-21 c^{5} e^{5} f \,x^{4}-80 b^{2} c^{3} e^{5} g \,x^{3}+384 b \,c^{4} d \,e^{4} g \,x^{3}+56 b \,c^{4} e^{5} f \,x^{3}-514 c^{5} d^{2} e^{3} g \,x^{3}-196 c^{5} d \,e^{4} f \,x^{3}+480 b^{3} c^{2} e^{5} g \,x^{2}-2784 b^{2} c^{3} d \,e^{4} g \,x^{2}-336 b^{2} c^{3} e^{5} f \,x^{2}+5388 b \,c^{4} d^{2} e^{3} g \,x^{2}+1512 b \,c^{4} d \,e^{4} f \,x^{2}-3504 c^{5} d^{3} e^{2} g \,x^{2}-1806 c^{5} d^{2} e^{3} f \,x^{2}+1920 b^{4} c \,e^{5} g x -13056 b^{3} c^{2} d \,e^{4} g x -1344 b^{3} c^{2} e^{5} f x +32688 b^{2} c^{3} d^{2} e^{3} g x +7392 b^{2} c^{3} d \,e^{4} f x -35568 b \,c^{4} d^{3} e^{2} g x -13272 b \,c^{4} d^{2} e^{3} f x +14121 c^{5} d^{4} e g x +7644 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -9984 b^{4} c d \,e^{4} g -896 b^{4} c \,e^{5} f +30496 b^{3} c^{2} d^{2} e^{3} g +5824 b^{3} c^{2} d \,e^{4} f -45504 b^{2} c^{3} d^{3} e^{2} g -13776 b^{2} c^{3} d^{2} e^{3} f +33126 b \,c^{4} d^{4} e g +13944 b \,c^{4} d^{3} e^{2} f -9414 c^{5} d^{5} g -5061 f \,d^{4} c^{5} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{105 c^{6} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(535\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-15 g \,e^{5} x^{5} c^{5}+30 b \,c^{4} e^{5} g \,x^{4}-114 c^{5} d \,e^{4} g \,x^{4}-21 c^{5} e^{5} f \,x^{4}-80 b^{2} c^{3} e^{5} g \,x^{3}+384 b \,c^{4} d \,e^{4} g \,x^{3}+56 b \,c^{4} e^{5} f \,x^{3}-514 c^{5} d^{2} e^{3} g \,x^{3}-196 c^{5} d \,e^{4} f \,x^{3}+480 b^{3} c^{2} e^{5} g \,x^{2}-2784 b^{2} c^{3} d \,e^{4} g \,x^{2}-336 b^{2} c^{3} e^{5} f \,x^{2}+5388 b \,c^{4} d^{2} e^{3} g \,x^{2}+1512 b \,c^{4} d \,e^{4} f \,x^{2}-3504 c^{5} d^{3} e^{2} g \,x^{2}-1806 c^{5} d^{2} e^{3} f \,x^{2}+1920 b^{4} c \,e^{5} g x -13056 b^{3} c^{2} d \,e^{4} g x -1344 b^{3} c^{2} e^{5} f x +32688 b^{2} c^{3} d^{2} e^{3} g x +7392 b^{2} c^{3} d \,e^{4} f x -35568 b \,c^{4} d^{3} e^{2} g x -13272 b \,c^{4} d^{2} e^{3} f x +14121 c^{5} d^{4} e g x +7644 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -9984 b^{4} c d \,e^{4} g -896 b^{4} c \,e^{5} f +30496 b^{3} c^{2} d^{2} e^{3} g +5824 b^{3} c^{2} d \,e^{4} f -45504 b^{2} c^{3} d^{3} e^{2} g -13776 b^{2} c^{3} d^{2} e^{3} f +33126 b \,c^{4} d^{4} e g +13944 b \,c^{4} d^{3} e^{2} f -9414 c^{5} d^{5} g -5061 f \,d^{4} c^{5} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{105 c^{6} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(535\) |
Input:
int((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method =_RETURNVERBOSE)
Output:
2/105/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-15*c^5*e^5*g*x^5+30 *b*c^4*e^5*g*x^4-114*c^5*d*e^4*g*x^4-21*c^5*e^5*f*x^4-80*b^2*c^3*e^5*g*x^3 +384*b*c^4*d*e^4*g*x^3+56*b*c^4*e^5*f*x^3-514*c^5*d^2*e^3*g*x^3-196*c^5*d* e^4*f*x^3+480*b^3*c^2*e^5*g*x^2-2784*b^2*c^3*d*e^4*g*x^2-336*b^2*c^3*e^5*f *x^2+5388*b*c^4*d^2*e^3*g*x^2+1512*b*c^4*d*e^4*f*x^2-3504*c^5*d^3*e^2*g*x^ 2-1806*c^5*d^2*e^3*f*x^2+1920*b^4*c*e^5*g*x-13056*b^3*c^2*d*e^4*g*x-1344*b ^3*c^2*e^5*f*x+32688*b^2*c^3*d^2*e^3*g*x+7392*b^2*c^3*d*e^4*f*x-35568*b*c^ 4*d^3*e^2*g*x-13272*b*c^4*d^2*e^3*f*x+14121*c^5*d^4*e*g*x+7644*c^5*d^3*e^2 *f*x+1280*b^5*e^5*g-9984*b^4*c*d*e^4*g-896*b^4*c*e^5*f+30496*b^3*c^2*d^2*e ^3*g+5824*b^3*c^2*d*e^4*f-45504*b^2*c^3*d^3*e^2*g-13776*b^2*c^3*d^2*e^3*f+ 33126*b*c^4*d^4*e*g+13944*b*c^4*d^3*e^2*f-9414*c^5*d^5*g-5061*c^5*d^4*e*f) /(c*e*x+b*e-c*d)^2/c^6/e^2
Time = 0.26 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.39 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (15 \, c^{5} e^{5} g x^{5} + 3 \, {\left (7 \, c^{5} e^{5} f + 2 \, {\left (19 \, c^{5} d e^{4} - 5 \, b c^{4} e^{5}\right )} g\right )} x^{4} + 2 \, {\left (14 \, {\left (7 \, c^{5} d e^{4} - 2 \, b c^{4} e^{5}\right )} f + {\left (257 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} + 40 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} + 6 \, {\left (7 \, {\left (43 \, c^{5} d^{2} e^{3} - 36 \, b c^{4} d e^{4} + 8 \, b^{2} c^{3} e^{5}\right )} f + 2 \, {\left (292 \, c^{5} d^{3} e^{2} - 449 \, b c^{4} d^{2} e^{3} + 232 \, b^{2} c^{3} d e^{4} - 40 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} + 7 \, {\left (723 \, c^{5} d^{4} e - 1992 \, b c^{4} d^{3} e^{2} + 1968 \, b^{2} c^{3} d^{2} e^{3} - 832 \, b^{3} c^{2} d e^{4} + 128 \, b^{4} c e^{5}\right )} f + 2 \, {\left (4707 \, c^{5} d^{5} - 16563 \, b c^{4} d^{4} e + 22752 \, b^{2} c^{3} d^{3} e^{2} - 15248 \, b^{3} c^{2} d^{2} e^{3} + 4992 \, b^{4} c d e^{4} - 640 \, b^{5} e^{5}\right )} g - 3 \, {\left (28 \, {\left (91 \, c^{5} d^{3} e^{2} - 158 \, b c^{4} d^{2} e^{3} + 88 \, b^{2} c^{3} d e^{4} - 16 \, b^{3} c^{2} e^{5}\right )} f + {\left (4707 \, c^{5} d^{4} e - 11856 \, b c^{4} d^{3} e^{2} + 10896 \, b^{2} c^{3} d^{2} e^{3} - 4352 \, b^{3} c^{2} d e^{4} + 640 \, b^{4} c e^{5}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{105 \, {\left (c^{8} e^{5} x^{3} + c^{8} d^{3} e^{2} - 2 \, b c^{7} d^{2} e^{3} + b^{2} c^{6} d e^{4} - {\left (c^{8} d e^{4} - 2 \, b c^{7} e^{5}\right )} x^{2} - {\left (c^{8} d^{2} e^{3} - b^{2} c^{6} e^{5}\right )} x\right )}} \] Input:
integrate((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")
Output:
-2/105*(15*c^5*e^5*g*x^5 + 3*(7*c^5*e^5*f + 2*(19*c^5*d*e^4 - 5*b*c^4*e^5) *g)*x^4 + 2*(14*(7*c^5*d*e^4 - 2*b*c^4*e^5)*f + (257*c^5*d^2*e^3 - 192*b*c ^4*d*e^4 + 40*b^2*c^3*e^5)*g)*x^3 + 6*(7*(43*c^5*d^2*e^3 - 36*b*c^4*d*e^4 + 8*b^2*c^3*e^5)*f + 2*(292*c^5*d^3*e^2 - 449*b*c^4*d^2*e^3 + 232*b^2*c^3* d*e^4 - 40*b^3*c^2*e^5)*g)*x^2 + 7*(723*c^5*d^4*e - 1992*b*c^4*d^3*e^2 + 1 968*b^2*c^3*d^2*e^3 - 832*b^3*c^2*d*e^4 + 128*b^4*c*e^5)*f + 2*(4707*c^5*d ^5 - 16563*b*c^4*d^4*e + 22752*b^2*c^3*d^3*e^2 - 15248*b^3*c^2*d^2*e^3 + 4 992*b^4*c*d*e^4 - 640*b^5*e^5)*g - 3*(28*(91*c^5*d^3*e^2 - 158*b*c^4*d^2*e ^3 + 88*b^2*c^3*d*e^4 - 16*b^3*c^2*e^5)*f + (4707*c^5*d^4*e - 11856*b*c^4* d^3*e^2 + 10896*b^2*c^3*d^2*e^3 - 4352*b^3*c^2*d*e^4 + 640*b^4*c*e^5)*g)*x )*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^8*e^5*x^3 + c^8*d^3*e^2 - 2*b*c^7*d^2*e^3 + b^2*c^6*d*e^4 - (c^8*d*e^4 - 2*b*c^7*e^5)* x^2 - (c^8*d^2*e^3 - b^2*c^6*e^5)*x)
Timed out. \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(13/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5 /2),x)
Output:
Timed out
Time = 0.10 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (3 \, c^{4} e^{4} x^{4} + 723 \, c^{4} d^{4} - 1992 \, b c^{3} d^{3} e + 1968 \, b^{2} c^{2} d^{2} e^{2} - 832 \, b^{3} c d e^{3} + 128 \, b^{4} e^{4} + 4 \, {\left (7 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} x^{3} + 6 \, {\left (43 \, c^{4} d^{2} e^{2} - 36 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} x^{2} - 12 \, {\left (91 \, c^{4} d^{3} e - 158 \, b c^{3} d^{2} e^{2} + 88 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} x\right )} f}{15 \, {\left (c^{6} e^{2} x - c^{6} d e + b c^{5} e^{2}\right )} \sqrt {-c e x + c d - b e}} + \frac {2 \, {\left (15 \, c^{5} e^{5} x^{5} + 9414 \, c^{5} d^{5} - 33126 \, b c^{4} d^{4} e + 45504 \, b^{2} c^{3} d^{3} e^{2} - 30496 \, b^{3} c^{2} d^{2} e^{3} + 9984 \, b^{4} c d e^{4} - 1280 \, b^{5} e^{5} + 6 \, {\left (19 \, c^{5} d e^{4} - 5 \, b c^{4} e^{5}\right )} x^{4} + 2 \, {\left (257 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} + 40 \, b^{2} c^{3} e^{5}\right )} x^{3} + 12 \, {\left (292 \, c^{5} d^{3} e^{2} - 449 \, b c^{4} d^{2} e^{3} + 232 \, b^{2} c^{3} d e^{4} - 40 \, b^{3} c^{2} e^{5}\right )} x^{2} - 3 \, {\left (4707 \, c^{5} d^{4} e - 11856 \, b c^{4} d^{3} e^{2} + 10896 \, b^{2} c^{3} d^{2} e^{3} - 4352 \, b^{3} c^{2} d e^{4} + 640 \, b^{4} c e^{5}\right )} x\right )} g}{105 \, {\left (c^{7} e^{3} x - c^{7} d e^{2} + b c^{6} e^{3}\right )} \sqrt {-c e x + c d - b e}} \] Input:
integrate((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")
Output:
2/15*(3*c^4*e^4*x^4 + 723*c^4*d^4 - 1992*b*c^3*d^3*e + 1968*b^2*c^2*d^2*e^ 2 - 832*b^3*c*d*e^3 + 128*b^4*e^4 + 4*(7*c^4*d*e^3 - 2*b*c^3*e^4)*x^3 + 6* (43*c^4*d^2*e^2 - 36*b*c^3*d*e^3 + 8*b^2*c^2*e^4)*x^2 - 12*(91*c^4*d^3*e - 158*b*c^3*d^2*e^2 + 88*b^2*c^2*d*e^3 - 16*b^3*c*e^4)*x)*f/((c^6*e^2*x - c ^6*d*e + b*c^5*e^2)*sqrt(-c*e*x + c*d - b*e)) + 2/105*(15*c^5*e^5*x^5 + 94 14*c^5*d^5 - 33126*b*c^4*d^4*e + 45504*b^2*c^3*d^3*e^2 - 30496*b^3*c^2*d^2 *e^3 + 9984*b^4*c*d*e^4 - 1280*b^5*e^5 + 6*(19*c^5*d*e^4 - 5*b*c^4*e^5)*x^ 4 + 2*(257*c^5*d^2*e^3 - 192*b*c^4*d*e^4 + 40*b^2*c^3*e^5)*x^3 + 12*(292*c ^5*d^3*e^2 - 449*b*c^4*d^2*e^3 + 232*b^2*c^3*d*e^4 - 40*b^3*c^2*e^5)*x^2 - 3*(4707*c^5*d^4*e - 11856*b*c^4*d^3*e^2 + 10896*b^2*c^3*d^2*e^3 - 4352*b^ 3*c^2*d*e^4 + 640*b^4*c*e^5)*x)*g/((c^7*e^3*x - c^7*d*e^2 + b*c^6*e^3)*sqr t(-c*e*x + c*d - b*e))
Leaf count of result is larger than twice the leaf count of optimal. 975 vs. \(2 (385) = 770\).
Time = 0.39 (sec) , antiderivative size = 975, normalized size of antiderivative = 2.34 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")
Output:
-2/105*(35*(16*c^5*d^4*e*f - 32*b*c^4*d^3*e^2*f + 24*b^2*c^3*d^2*e^3*f - 8 *b^3*c^2*d*e^4*f + b^4*c*e^5*f + 16*c^5*d^5*g - 48*b*c^4*d^4*e*g + 56*b^2* c^3*d^3*e^2*g - 32*b^3*c^2*d^2*e^3*g + 9*b^4*c*d*e^4*g - b^5*e^5*g + 96*(( e*x + d)*c - 2*c*d + b*e)*c^4*d^3*e*f - 144*((e*x + d)*c - 2*c*d + b*e)*b* c^3*d^2*e^2*f + 72*((e*x + d)*c - 2*c*d + b*e)*b^2*c^2*d*e^3*f - 12*((e*x + d)*c - 2*c*d + b*e)*b^3*c*e^4*f + 144*((e*x + d)*c - 2*c*d + b*e)*c^4*d^ 4*g - 336*((e*x + d)*c - 2*c*d + b*e)*b*c^3*d^3*e*g + 288*((e*x + d)*c - 2 *c*d + b*e)*b^2*c^2*d^2*e^2*g - 108*((e*x + d)*c - 2*c*d + b*e)*b^3*c*d*e^ 3*g + 15*((e*x + d)*c - 2*c*d + b*e)*b^4*e^4*g)/(((e*x + d)*c - 2*c*d + b* e)*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^6*e) + (2520*sqrt(-(e*x + d)*c + 2*c *d - b*e)*c^39*d^2*e^7*f - 2520*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^38*d* e^8*f + 630*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^37*e^9*f + 5880*sqrt(-( e*x + d)*c + 2*c*d - b*e)*c^39*d^3*e^6*g - 10080*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^38*d^2*e^7*g + 5670*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^37* d*e^8*g - 1050*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^3*c^36*e^9*g - 280*(-(e* x + d)*c + 2*c*d - b*e)^(3/2)*c^38*d*e^7*f + 140*(-(e*x + d)*c + 2*c*d - b *e)^(3/2)*b*c^37*e^8*f - 1120*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^38*d^2* e^6*g + 1260*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^37*d*e^7*g - 350*(-(e* x + d)*c + 2*c*d - b*e)^(3/2)*b^2*c^36*e^8*g + 21*((e*x + d)*c - 2*c*d + b *e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^37*e^7*f + 189*((e*x + d)*c - ...
Time = 12.57 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.43 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {\sqrt {d+e\,x}\,\left (-2560\,g\,b^5\,e^5+19968\,g\,b^4\,c\,d\,e^4+1792\,f\,b^4\,c\,e^5-60992\,g\,b^3\,c^2\,d^2\,e^3-11648\,f\,b^3\,c^2\,d\,e^4+91008\,g\,b^2\,c^3\,d^3\,e^2+27552\,f\,b^2\,c^3\,d^2\,e^3-66252\,g\,b\,c^4\,d^4\,e-27888\,f\,b\,c^4\,d^3\,e^2+18828\,g\,c^5\,d^5+10122\,f\,c^5\,d^4\,e\right )}{105\,c^8\,e^5}+\frac {2\,g\,x^5\,\sqrt {d+e\,x}}{7\,c^3}+\frac {4\,x^3\,\sqrt {d+e\,x}\,\left (40\,g\,b^2\,e^2-192\,g\,b\,c\,d\,e-28\,f\,b\,c\,e^2+257\,g\,c^2\,d^2+98\,f\,c^2\,d\,e\right )}{105\,c^5\,e^2}+\frac {2\,x^4\,\sqrt {d+e\,x}\,\left (38\,c\,d\,g-10\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^4\,e}-\frac {x\,\sqrt {d+e\,x}\,\left (3840\,g\,b^4\,c\,e^5-26112\,g\,b^3\,c^2\,d\,e^4-2688\,f\,b^3\,c^2\,e^5+65376\,g\,b^2\,c^3\,d^2\,e^3+14784\,f\,b^2\,c^3\,d\,e^4-71136\,g\,b\,c^4\,d^3\,e^2-26544\,f\,b\,c^4\,d^2\,e^3+28242\,g\,c^5\,d^4\,e+15288\,f\,c^5\,d^3\,e^2\right )}{105\,c^8\,e^5}+\frac {x^2\,\sqrt {d+e\,x}\,\left (-960\,g\,b^3\,c^2\,e^5+5568\,g\,b^2\,c^3\,d\,e^4+672\,f\,b^2\,c^3\,e^5-10776\,g\,b\,c^4\,d^2\,e^3-3024\,f\,b\,c^4\,d\,e^4+7008\,g\,c^5\,d^3\,e^2+3612\,f\,c^5\,d^2\,e^3\right )}{105\,c^8\,e^5}\right )}{x^3+\frac {x\,\left (105\,b^2\,c^6\,e^5-105\,c^8\,d^2\,e^3\right )}{105\,c^8\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \] Input:
int(((f + g*x)*(d + e*x)^(13/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/ 2),x)
Output:
-((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*(((d + e*x)^(1/2)*(18828*c^5 *d^5*g - 2560*b^5*e^5*g + 1792*b^4*c*e^5*f + 10122*c^5*d^4*e*f - 66252*b*c ^4*d^4*e*g + 19968*b^4*c*d*e^4*g - 27888*b*c^4*d^3*e^2*f - 11648*b^3*c^2*d *e^4*f + 27552*b^2*c^3*d^2*e^3*f + 91008*b^2*c^3*d^3*e^2*g - 60992*b^3*c^2 *d^2*e^3*g))/(105*c^8*e^5) + (2*g*x^5*(d + e*x)^(1/2))/(7*c^3) + (4*x^3*(d + e*x)^(1/2)*(40*b^2*e^2*g + 257*c^2*d^2*g - 28*b*c*e^2*f + 98*c^2*d*e*f - 192*b*c*d*e*g))/(105*c^5*e^2) + (2*x^4*(d + e*x)^(1/2)*(38*c*d*g - 10*b* e*g + 7*c*e*f))/(35*c^4*e) - (x*(d + e*x)^(1/2)*(15288*c^5*d^3*e^2*f - 268 8*b^3*c^2*e^5*f + 3840*b^4*c*e^5*g + 28242*c^5*d^4*e*g - 26544*b*c^4*d^2*e ^3*f + 14784*b^2*c^3*d*e^4*f - 71136*b*c^4*d^3*e^2*g - 26112*b^3*c^2*d*e^4 *g + 65376*b^2*c^3*d^2*e^3*g))/(105*c^8*e^5) + (x^2*(d + e*x)^(1/2)*(672*b ^2*c^3*e^5*f - 960*b^3*c^2*e^5*g + 3612*c^5*d^2*e^3*f + 7008*c^5*d^3*e^2*g - 3024*b*c^4*d*e^4*f - 10776*b*c^4*d^2*e^3*g + 5568*b^2*c^3*d*e^4*g))/(10 5*c^8*e^5)))/(x^3 + (x*(105*b^2*c^6*e^5 - 105*c^8*d^2*e^3))/(105*c^8*e^5) + (d*(b*e - c*d)^2)/(c^2*e^3) + (x^2*(2*b*e - c*d))/(c*e))
Time = 0.22 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.24 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {\frac {256}{15} b^{4} c \,e^{5} f +\frac {2}{5} c^{5} e^{5} f \,x^{4}+\frac {2}{7} c^{5} e^{5} g \,x^{5}+\frac {128}{5} b^{3} c^{2} e^{5} f x -\frac {64}{7} b^{3} c^{2} e^{5} g \,x^{2}+\frac {1312}{5} b^{2} c^{3} d^{2} e^{3} f +\frac {32}{5} b^{2} c^{3} e^{5} f \,x^{2}-\frac {4}{7} b \,c^{4} e^{5} g \,x^{4}-\frac {9414}{35} c^{5} d^{4} e g x -\frac {728}{5} c^{5} d^{3} e^{2} f x +\frac {2336}{35} c^{5} d^{3} e^{2} g \,x^{2}+\frac {172}{5} c^{5} d^{2} e^{3} f \,x^{2}+\frac {1028}{105} c^{5} d^{2} e^{3} g \,x^{3}+\frac {56}{15} c^{5} d \,e^{4} f \,x^{3}+\frac {76}{35} c^{5} d \,e^{4} g \,x^{4}-\frac {256}{35} b \,c^{4} d \,e^{4} g \,x^{3}+\frac {8704}{35} b^{3} c^{2} d \,e^{4} g x -\frac {21792}{35} b^{2} c^{3} d^{2} e^{3} g x -\frac {704}{5} b^{2} c^{3} d \,e^{4} f x +\frac {1856}{35} b^{2} c^{3} d \,e^{4} g \,x^{2}+\frac {23712}{35} b \,c^{4} d^{3} e^{2} g x +\frac {1264}{5} b \,c^{4} d^{2} e^{3} f x -\frac {3592}{35} b \,c^{4} d^{2} e^{3} g \,x^{2}+\frac {6656}{35} b^{4} c d \,e^{4} g -\frac {256}{7} b^{4} c \,e^{5} g x -\frac {60992}{105} b^{3} c^{2} d^{2} e^{3} g -\frac {1664}{15} b^{3} c^{2} d \,e^{4} f +\frac {30336}{35} b^{2} c^{3} d^{3} e^{2} g +\frac {32}{21} b^{2} c^{3} e^{5} g \,x^{3}-\frac {22084}{35} b \,c^{4} d^{4} e g -\frac {1328}{5} b \,c^{4} d^{3} e^{2} f -\frac {16}{15} b \,c^{4} e^{5} f \,x^{3}-\frac {512}{21} b^{5} e^{5} g +\frac {6276}{35} c^{5} d^{5} g +\frac {482}{5} c^{5} d^{4} e f -\frac {144}{5} b \,c^{4} d \,e^{4} f \,x^{2}}{\sqrt {-c e x -b e +c d}\, c^{6} e^{2} \left (c e x +b e -c d \right )} \] Input:
int((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
Output:
(2*( - 1280*b**5*e**5*g + 9984*b**4*c*d*e**4*g + 896*b**4*c*e**5*f - 1920* b**4*c*e**5*g*x - 30496*b**3*c**2*d**2*e**3*g - 5824*b**3*c**2*d*e**4*f + 13056*b**3*c**2*d*e**4*g*x + 1344*b**3*c**2*e**5*f*x - 480*b**3*c**2*e**5* g*x**2 + 45504*b**2*c**3*d**3*e**2*g + 13776*b**2*c**3*d**2*e**3*f - 32688 *b**2*c**3*d**2*e**3*g*x - 7392*b**2*c**3*d*e**4*f*x + 2784*b**2*c**3*d*e* *4*g*x**2 + 336*b**2*c**3*e**5*f*x**2 + 80*b**2*c**3*e**5*g*x**3 - 33126*b *c**4*d**4*e*g - 13944*b*c**4*d**3*e**2*f + 35568*b*c**4*d**3*e**2*g*x + 1 3272*b*c**4*d**2*e**3*f*x - 5388*b*c**4*d**2*e**3*g*x**2 - 1512*b*c**4*d*e **4*f*x**2 - 384*b*c**4*d*e**4*g*x**3 - 56*b*c**4*e**5*f*x**3 - 30*b*c**4* e**5*g*x**4 + 9414*c**5*d**5*g + 5061*c**5*d**4*e*f - 14121*c**5*d**4*e*g* x - 7644*c**5*d**3*e**2*f*x + 3504*c**5*d**3*e**2*g*x**2 + 1806*c**5*d**2* e**3*f*x**2 + 514*c**5*d**2*e**3*g*x**3 + 196*c**5*d*e**4*f*x**3 + 114*c** 5*d*e**4*g*x**4 + 21*c**5*e**5*f*x**4 + 15*c**5*e**5*g*x**5))/(105*sqrt( - b*e + c*d - c*e*x)*c**6*e**2*(b*e - c*d + c*e*x))