Integrand size = 46, antiderivative size = 171 \[ \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c^3 d^3 (d+e x)^{5/2}}+\frac {4 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c^3 d^3 (d+e x)^{7/2}}+\frac {2 g^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{9 c^3 d^3 (d+e x)^{9/2}} \] Output:
2/5*(-a*e*g+c*d*f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^3/d^3/(e*x+ d)^(5/2)+4/7*g*(-a*e*g+c*d*f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^3/ d^3/(e*x+d)^(7/2)+2/9*g^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/c^3/d^3/ (e*x+d)^(9/2)
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.53 \[ \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{5/2} \left (8 a^2 e^2 g^2-4 a c d e g (9 f+5 g x)+c^2 d^2 \left (63 f^2+90 f g x+35 g^2 x^2\right )\right )}{315 c^3 d^3 (d+e x)^{5/2}} \] Input:
Integrate[((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]
Output:
(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(9*f + 5*g *x) + c^2*d^2*(63*f^2 + 90*f*g*x + 35*g^2*x^2)))/(315*c^3*d^3*(d + e*x)^(5 /2))
Time = 0.68 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1253, 1221, 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.
| \(\displaystyle \int \frac {(f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1253 |
| \(\displaystyle \frac {4 (c d f-a e g) \int \frac {(f+g x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2}}dx}{9 c d}+\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {4 (c d f-a e g) \left (\frac {1}{7} \left (-\frac {2 a e g}{c d}-\frac {5 d g}{e}+7 f\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2}}dx+\frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d e (d+e x)^{3/2}}\right )}{9 c d}+\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}}+\frac {4 (c d f-a e g) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \left (-\frac {2 a e g}{c d}-\frac {5 d g}{e}+7 f\right )}{35 c d (d+e x)^{5/2}}+\frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d e (d+e x)^{3/2}}\right )}{9 c d}\) |
Input:
Int[((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^ (3/2),x]
Output:
(2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*c*d*(d + e*x)^(5/2)) + (4*(c*d*f - a*e*g)*((2*(7*f - (5*d*g)/e - (2*a*e*g)/(c*d))*( a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(35*c*d*(d + e*x)^(5/2)) + ( 2*g*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*c*d*e*(d + e*x)^(3/2 ))))/(9*c*d)
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_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 )/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c *f - b*g))/(c*e*(m + 2*p + 2)) Int[(d + e*x)^m*(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] && NeQ[m + 2*p + 2, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^(m - 1)*(f + g*x)^n* ((a + b*x + c*x^2)^(p + 1)/(c*(m - n - 1))), x] - Simp[n*((c*e*f + c*d*g - b*e*g)/(c*e*(m - n - 1))) Int[(d + e*x)^m*(f + g*x)^(n - 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] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (Intege rQ[2*p] || IntegerQ[n])
Time = 2.90 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.63
| method | result | size |
| default | \(\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (c d x +a e \right )^{2} \left (35 g^{2} x^{2} d^{2} c^{2}-20 a c d e \,g^{2} x +90 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-36 a c d e f g +63 f^{2} c^{2} d^{2}\right )}{315 \sqrt {e x +d}\, d^{3} c^{3}}\) | \(108\) |
| gosper | \(\frac {2 \left (c d x +a e \right ) \left (35 g^{2} x^{2} d^{2} c^{2}-20 a c d e \,g^{2} x +90 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-36 a c d e f g +63 f^{2} c^{2} d^{2}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{315 d^{3} c^{3} \left (e x +d \right )^{\frac {3}{2}}}\) | \(116\) |
| orering | \(\frac {2 \left (35 g^{2} x^{2} d^{2} c^{2}-20 a c d e \,g^{2} x +90 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-36 a c d e f g +63 f^{2} c^{2} d^{2}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{315 d^{3} c^{3} \left (e x +d \right )^{\frac {3}{2}}}\) | \(117\) |
Input:
int((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d)^(3/2),x,meth od=_RETURNVERBOSE)
Output:
2/315*((e*x+d)*(c*d*x+a*e))^(1/2)/(e*x+d)^(1/2)*(c*d*x+a*e)^2*(35*c^2*d^2* g^2*x^2-20*a*c*d*e*g^2*x+90*c^2*d^2*f*g*x+8*a^2*e^2*g^2-36*a*c*d*e*f*g+63* c^2*d^2*f^2)/d^3/c^3
Time = 0.10 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.35 \[ \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (35 \, c^{4} d^{4} g^{2} x^{4} + 63 \, a^{2} c^{2} d^{2} e^{2} f^{2} - 36 \, a^{3} c d e^{3} f g + 8 \, a^{4} e^{4} g^{2} + 10 \, {\left (9 \, c^{4} d^{4} f g + 5 \, a c^{3} d^{3} e g^{2}\right )} x^{3} + 3 \, {\left (21 \, c^{4} d^{4} f^{2} + 48 \, a c^{3} d^{3} e f g + a^{2} c^{2} d^{2} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (63 \, a c^{3} d^{3} e f^{2} + 9 \, a^{2} c^{2} d^{2} e^{2} f g - 2 \, a^{3} c d e^{3} g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{315 \, {\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \] Input:
integrate((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2), x, algorithm="fricas")
Output:
2/315*(35*c^4*d^4*g^2*x^4 + 63*a^2*c^2*d^2*e^2*f^2 - 36*a^3*c*d*e^3*f*g + 8*a^4*e^4*g^2 + 10*(9*c^4*d^4*f*g + 5*a*c^3*d^3*e*g^2)*x^3 + 3*(21*c^4*d^4 *f^2 + 48*a*c^3*d^3*e*f*g + a^2*c^2*d^2*e^2*g^2)*x^2 + 2*(63*a*c^3*d^3*e*f ^2 + 9*a^2*c^2*d^2*e^2*f*g - 2*a^3*c*d*e^3*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^3*d^3*e*x + c^3*d^4)
\[ \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (f + g x\right )^{2}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((g*x+f)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)** (3/2),x)
Output:
Integral(((d + e*x)*(a*e + c*d*x))**(3/2)*(f + g*x)**2/(d + e*x)**(3/2), x )
Time = 0.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.12 \[ \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt {c d x + a e} f^{2}}{5 \, c d} + \frac {4 \, {\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} e x^{2} + a^{2} c d e^{2} x - 2 \, a^{3} e^{3}\right )} \sqrt {c d x + a e} f g}{35 \, c^{2} d^{2}} + \frac {2 \, {\left (35 \, c^{4} d^{4} x^{4} + 50 \, a c^{3} d^{3} e x^{3} + 3 \, a^{2} c^{2} d^{2} e^{2} x^{2} - 4 \, a^{3} c d e^{3} x + 8 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} g^{2}}{315 \, c^{3} d^{3}} \] Input:
integrate((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2), x, algorithm="maxima")
Output:
2/5*(c^2*d^2*x^2 + 2*a*c*d*e*x + a^2*e^2)*sqrt(c*d*x + a*e)*f^2/(c*d) + 4/ 35*(5*c^3*d^3*x^3 + 8*a*c^2*d^2*e*x^2 + a^2*c*d*e^2*x - 2*a^3*e^3)*sqrt(c* d*x + a*e)*f*g/(c^2*d^2) + 2/315*(35*c^4*d^4*x^4 + 50*a*c^3*d^3*e*x^3 + 3* a^2*c^2*d^2*e^2*x^2 - 4*a^3*c*d*e^3*x + 8*a^4*e^4)*sqrt(c*d*x + a*e)*g^2/( c^3*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (153) = 306\).
Time = 0.14 (sec) , antiderivative size = 540, normalized size of antiderivative = 3.16 \[ \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {105 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a f^{2} {\left | e \right |}}{c d e^{2}} - \frac {21 \, {\left (5 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}\right )} f^{2} {\left | e \right |}}{c d e^{5}} - \frac {42 \, {\left (5 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}\right )} a f g {\left | e \right |}}{c^{2} d^{2} e^{4}} + \frac {6 \, {\left (35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} f g {\left | e \right |}}{c^{2} d^{2} e^{7}} + \frac {3 \, {\left (35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} a g^{2} {\left | e \right |}}{c^{3} d^{3} e^{6}} - \frac {{\left (105 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} e^{9} - 189 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} e^{6} + 135 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a e^{3} - 35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}}\right )} g^{2} {\left | e \right |}}{c^{3} d^{3} e^{9}}\right )}}{315 \, e} \] Input:
integrate((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2), x, algorithm="giac")
Output:
2/315*(105*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*f^2*abs(e)/(c*d*e^2 ) - 21*(5*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((e*x + d)*c *d*e - c*d^2*e + a*e^3)^(5/2))*f^2*abs(e)/(c*d*e^5) - 42*(5*((e*x + d)*c*d *e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^ (5/2))*a*f*g*abs(e)/(c^2*d^2*e^4) + 6*(35*((e*x + d)*c*d*e - c*d^2*e + a*e ^3)^(3/2)*a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 1 5*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*f*g*abs(e)/(c^2*d^2*e^7) + 3* (35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((e*x + d)*c*d* e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^ (7/2))*a*g^2*abs(e)/(c^3*d^3*e^6) - (105*((e*x + d)*c*d*e - c*d^2*e + a*e^ 3)^(3/2)*a^3*e^9 - 189*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*g^2*abs(e)/(c^3*d^3*e^9))/e
Time = 6.39 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.20 \[ \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,g\,x^3\,\left (5\,a\,e\,g+9\,c\,d\,f\right )}{63}+\frac {16\,a^4\,e^4\,g^2-72\,a^3\,c\,d\,e^3\,f\,g+126\,a^2\,c^2\,d^2\,e^2\,f^2}{315\,c^3\,d^3}+\frac {x^2\,\left (6\,a^2\,c^2\,d^2\,e^2\,g^2+288\,a\,c^3\,d^3\,e\,f\,g+126\,c^4\,d^4\,f^2\right )}{315\,c^3\,d^3}+\frac {2\,c\,d\,g^2\,x^4}{9}+\frac {4\,a\,e\,x\,\left (-2\,a^2\,e^2\,g^2+9\,a\,c\,d\,e\,f\,g+63\,c^2\,d^2\,f^2\right )}{315\,c^2\,d^2}\right )}{\sqrt {d+e\,x}} \] Input:
int(((f + g*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/(d + e*x)^ (3/2),x)
Output:
((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((4*g*x^3*(5*a*e*g + 9*c*d* f))/63 + (16*a^4*e^4*g^2 + 126*a^2*c^2*d^2*e^2*f^2 - 72*a^3*c*d*e^3*f*g)/( 315*c^3*d^3) + (x^2*(126*c^4*d^4*f^2 + 6*a^2*c^2*d^2*e^2*g^2 + 288*a*c^3*d ^3*e*f*g))/(315*c^3*d^3) + (2*c*d*g^2*x^4)/9 + (4*a*e*x*(63*c^2*d^2*f^2 - 2*a^2*e^2*g^2 + 9*a*c*d*e*f*g))/(315*c^2*d^2)))/(d + e*x)^(1/2)
Time = 0.36 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.14 \[ \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (35 c^{4} d^{4} g^{2} x^{4}+50 a \,c^{3} d^{3} e \,g^{2} x^{3}+90 c^{4} d^{4} f g \,x^{3}+3 a^{2} c^{2} d^{2} e^{2} g^{2} x^{2}+144 a \,c^{3} d^{3} e f g \,x^{2}+63 c^{4} d^{4} f^{2} x^{2}-4 a^{3} c d \,e^{3} g^{2} x +18 a^{2} c^{2} d^{2} e^{2} f g x +126 a \,c^{3} d^{3} e \,f^{2} x +8 a^{4} e^{4} g^{2}-36 a^{3} c d \,e^{3} f g +63 a^{2} c^{2} d^{2} e^{2} f^{2}\right )}{315 c^{3} d^{3}} \] Input:
int((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x)
Output:
(2*sqrt(a*e + c*d*x)*(8*a**4*e**4*g**2 - 36*a**3*c*d*e**3*f*g - 4*a**3*c*d *e**3*g**2*x + 63*a**2*c**2*d**2*e**2*f**2 + 18*a**2*c**2*d**2*e**2*f*g*x + 3*a**2*c**2*d**2*e**2*g**2*x**2 + 126*a*c**3*d**3*e*f**2*x + 144*a*c**3* d**3*e*f*g*x**2 + 50*a*c**3*d**3*e*g**2*x**3 + 63*c**4*d**4*f**2*x**2 + 90 *c**4*d**4*f*g*x**3 + 35*c**4*d**4*g**2*x**4))/(315*c**3*d**3)