Integrand size = 26, antiderivative size = 121 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=-\frac {\left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}{64 c^4 d}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac {(b d+2 c d x)^{13/2}}{832 c^4 d^7} \] Output:
-1/64*(-4*a*c+b^2)^3*(2*c*d*x+b*d)^(1/2)/c^4/d+3/320*(-4*a*c+b^2)^2*(2*c*d *x+b*d)^(5/2)/c^4/d^3-1/192*(-4*a*c+b^2)*(2*c*d*x+b*d)^(9/2)/c^4/d^5+1/832 *(2*c*d*x+b*d)^(13/2)/c^4/d^7
Time = 0.11 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=\frac {\sqrt {d (b+2 c x)} \left (-195 b^6+2340 a b^4 c-9360 a^2 b^2 c^2+12480 a^3 c^3+117 b^4 (b+2 c x)^2-936 a b^2 c (b+2 c x)^2+1872 a^2 c^2 (b+2 c x)^2-65 b^2 (b+2 c x)^4+260 a c (b+2 c x)^4+15 (b+2 c x)^6\right )}{12480 c^4 d} \] Input:
Integrate[(a + b*x + c*x^2)^3/Sqrt[b*d + 2*c*d*x],x]
Output:
(Sqrt[d*(b + 2*c*x)]*(-195*b^6 + 2340*a*b^4*c - 9360*a^2*b^2*c^2 + 12480*a ^3*c^3 + 117*b^4*(b + 2*c*x)^2 - 936*a*b^2*c*(b + 2*c*x)^2 + 1872*a^2*c^2* (b + 2*c*x)^2 - 65*b^2*(b + 2*c*x)^4 + 260*a*c*(b + 2*c*x)^4 + 15*(b + 2*c *x)^6))/(12480*c^4*d)
Time = 0.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1107, 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.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx\) |
\(\Big \downarrow \) 1107 |
\(\displaystyle \int \left (\frac {3 \left (4 a c-b^2\right ) (b d+2 c d x)^{7/2}}{64 c^3 d^4}+\frac {3 \left (4 a c-b^2\right )^2 (b d+2 c d x)^{3/2}}{64 c^3 d^2}+\frac {\left (4 a c-b^2\right )^3}{64 c^3 \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{11/2}}{64 c^3 d^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac {\left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}{64 c^4 d}+\frac {(b d+2 c d x)^{13/2}}{832 c^4 d^7}\) |
Input:
Int[(a + b*x + c*x^2)^3/Sqrt[b*d + 2*c*d*x],x]
Output:
-1/64*((b^2 - 4*a*c)^3*Sqrt[b*d + 2*c*d*x])/(c^4*d) + (3*(b^2 - 4*a*c)^2*( b*d + 2*c*d*x)^(5/2))/(320*c^4*d^3) - ((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(9/2) )/(192*c^4*d^5) + (b*d + 2*c*d*x)^(13/2)/(832*c^4*d^7)
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b*e, 0] && IGtQ[p, 0] && !(EqQ[ m, 3] && NeQ[p, 1])
Time = 0.92 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.21
method | result | size |
derivativedivides | \(\frac {\frac {\left (2 c d x +b d \right )^{\frac {13}{2}}}{13}+\frac {\left (12 a \,d^{2} c -3 b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {9}{2}}}{9}+\frac {\left (\left (4 a \,d^{2} c -b^{2} d^{2}\right ) \left (8 a \,d^{2} c -2 b^{2} d^{2}\right )+\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{2}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{3} \sqrt {2 c d x +b d}}{64 d^{7} c^{4}}\) | \(147\) |
default | \(\frac {\frac {\left (2 c d x +b d \right )^{\frac {13}{2}}}{13}+\frac {\left (12 a \,d^{2} c -3 b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {9}{2}}}{9}+\frac {\left (\left (4 a \,d^{2} c -b^{2} d^{2}\right ) \left (8 a \,d^{2} c -2 b^{2} d^{2}\right )+\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{2}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{3} \sqrt {2 c d x +b d}}{64 d^{7} c^{4}}\) | \(147\) |
pseudoelliptic | \(\frac {\left (15 x^{6} c^{6}+45 x^{5} b \,c^{5}+65 a \,c^{5} x^{4}+40 x^{4} b^{2} c^{4}+130 a b \,c^{4} x^{3}+5 b^{3} c^{3} x^{3}+117 a^{2} c^{4} x^{2}+39 a \,b^{2} c^{3} x^{2}-3 c^{2} x^{2} b^{4}+117 a^{2} b \,c^{3} x -26 x a \,b^{3} c^{2}+2 x c \,b^{5}+195 a^{3} c^{3}-117 a^{2} b^{2} c^{2}+26 a \,b^{4} c -2 b^{6}\right ) \sqrt {d \left (2 c x +b \right )}}{195 d \,c^{4}}\) | \(170\) |
trager | \(\frac {\left (15 x^{6} c^{6}+45 x^{5} b \,c^{5}+65 a \,c^{5} x^{4}+40 x^{4} b^{2} c^{4}+130 a b \,c^{4} x^{3}+5 b^{3} c^{3} x^{3}+117 a^{2} c^{4} x^{2}+39 a \,b^{2} c^{3} x^{2}-3 c^{2} x^{2} b^{4}+117 a^{2} b \,c^{3} x -26 x a \,b^{3} c^{2}+2 x c \,b^{5}+195 a^{3} c^{3}-117 a^{2} b^{2} c^{2}+26 a \,b^{4} c -2 b^{6}\right ) \sqrt {2 c d x +b d}}{195 d \,c^{4}}\) | \(171\) |
gosper | \(\frac {\left (2 c x +b \right ) \left (15 x^{6} c^{6}+45 x^{5} b \,c^{5}+65 a \,c^{5} x^{4}+40 x^{4} b^{2} c^{4}+130 a b \,c^{4} x^{3}+5 b^{3} c^{3} x^{3}+117 a^{2} c^{4} x^{2}+39 a \,b^{2} c^{3} x^{2}-3 c^{2} x^{2} b^{4}+117 a^{2} b \,c^{3} x -26 x a \,b^{3} c^{2}+2 x c \,b^{5}+195 a^{3} c^{3}-117 a^{2} b^{2} c^{2}+26 a \,b^{4} c -2 b^{6}\right )}{195 c^{4} \sqrt {2 c d x +b d}}\) | \(174\) |
orering | \(\frac {\left (2 c x +b \right ) \left (15 x^{6} c^{6}+45 x^{5} b \,c^{5}+65 a \,c^{5} x^{4}+40 x^{4} b^{2} c^{4}+130 a b \,c^{4} x^{3}+5 b^{3} c^{3} x^{3}+117 a^{2} c^{4} x^{2}+39 a \,b^{2} c^{3} x^{2}-3 c^{2} x^{2} b^{4}+117 a^{2} b \,c^{3} x -26 x a \,b^{3} c^{2}+2 x c \,b^{5}+195 a^{3} c^{3}-117 a^{2} b^{2} c^{2}+26 a \,b^{4} c -2 b^{6}\right )}{195 c^{4} \sqrt {2 c d x +b d}}\) | \(174\) |
Input:
int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/64/d^7/c^4*(1/13*(2*c*d*x+b*d)^(13/2)+1/9*(12*a*c*d^2-3*b^2*d^2)*(2*c*d* x+b*d)^(9/2)+1/5*((4*a*c*d^2-b^2*d^2)*(8*a*c*d^2-2*b^2*d^2)+(4*a*c*d^2-b^2 *d^2)^2)*(2*c*d*x+b*d)^(5/2)+(4*a*c*d^2-b^2*d^2)^3*(2*c*d*x+b*d)^(1/2))
Time = 0.09 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=\frac {{\left (15 \, c^{6} x^{6} + 45 \, b c^{5} x^{5} - 2 \, b^{6} + 26 \, a b^{4} c - 117 \, a^{2} b^{2} c^{2} + 195 \, a^{3} c^{3} + 5 \, {\left (8 \, b^{2} c^{4} + 13 \, a c^{5}\right )} x^{4} + 5 \, {\left (b^{3} c^{3} + 26 \, a b c^{4}\right )} x^{3} - 3 \, {\left (b^{4} c^{2} - 13 \, a b^{2} c^{3} - 39 \, a^{2} c^{4}\right )} x^{2} + {\left (2 \, b^{5} c - 26 \, a b^{3} c^{2} + 117 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{195 \, c^{4} d} \] Input:
integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(1/2),x, algorithm="fricas")
Output:
1/195*(15*c^6*x^6 + 45*b*c^5*x^5 - 2*b^6 + 26*a*b^4*c - 117*a^2*b^2*c^2 + 195*a^3*c^3 + 5*(8*b^2*c^4 + 13*a*c^5)*x^4 + 5*(b^3*c^3 + 26*a*b*c^4)*x^3 - 3*(b^4*c^2 - 13*a*b^2*c^3 - 39*a^2*c^4)*x^2 + (2*b^5*c - 26*a*b^3*c^2 + 117*a^2*b*c^3)*x)*sqrt(2*c*d*x + b*d)/(c^4*d)
Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (116) = 232\).
Time = 0.97 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=\begin {cases} \frac {\frac {\sqrt {b d + 2 c d x} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}{64 c^{3}} + \frac {\left (b d + 2 c d x\right )^{\frac {5}{2}} \cdot \left (48 a^{2} c^{2} - 24 a b^{2} c + 3 b^{4}\right )}{320 c^{3} d^{2}} + \frac {\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac {9}{2}}}{576 c^{3} d^{4}} + \frac {\left (b d + 2 c d x\right )^{\frac {13}{2}}}{832 c^{3} d^{6}}}{c d} & \text {for}\: c d \neq 0 \\\frac {a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}}{\sqrt {b d}} & \text {otherwise} \end {cases} \] Input:
integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**(1/2),x)
Output:
Piecewise(((sqrt(b*d + 2*c*d*x)*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b **4*c - b**6)/(64*c**3) + (b*d + 2*c*d*x)**(5/2)*(48*a**2*c**2 - 24*a*b**2 *c + 3*b**4)/(320*c**3*d**2) + (12*a*c - 3*b**2)*(b*d + 2*c*d*x)**(9/2)/(5 76*c**3*d**4) + (b*d + 2*c*d*x)**(13/2)/(832*c**3*d**6))/(c*d), Ne(c*d, 0) ), ((a**3*x + 3*a**2*b*x**2/2 + b*c**2*x**6/2 + c**3*x**7/7 + x**5*(3*a*c* *2 + 3*b**2*c)/5 + x**4*(6*a*b*c + b**3)/4 + x**3*(3*a**2*c + 3*a*b**2)/3) /sqrt(b*d), True))
Leaf count of result is larger than twice the leaf count of optimal. 778 vs. \(2 (105) = 210\).
Time = 0.05 (sec) , antiderivative size = 778, normalized size of antiderivative = 6.43 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(1/2),x, algorithm="maxima")
Output:
1/960960*(960960*sqrt(2*c*d*x + b*d)*a^3 - 48048*a^2*(10*(3*sqrt(2*c*d*x + b*d)*b*d - (2*c*d*x + b*d)^(3/2))*b/(c*d) - (15*sqrt(2*c*d*x + b*d)*b^2*d ^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))/(c*d^2)) + 57 2*a*(84*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3 *(2*c*d*x + b*d)^(5/2))*b^2/(c^2*d^2) - 36*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2* c*d*x + b*d)^(7/2))*b/(c^2*d^3) + (315*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*( 2*c*d*x + b*d)^(3/2)*b^3*d^3 + 378*(2*c*d*x + b*d)^(5/2)*b^2*d^2 - 180*(2* c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d)^(9/2))/(c^2*d^4)) - 3432*(35*s qrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d* x + b*d)^(5/2)*b*d - 5*(2*c*d*x + b*d)^(7/2))*b^3/(c^3*d^3) + 572*(315*sqr t(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 378*(2*c*d* x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d )^(9/2))*b^2/(c^3*d^4) - 130*(693*sqrt(2*c*d*x + b*d)*b^5*d^5 - 1155*(2*c* d*x + b*d)^(3/2)*b^4*d^4 + 1386*(2*c*d*x + b*d)^(5/2)*b^3*d^3 - 990*(2*c*d *x + b*d)^(7/2)*b^2*d^2 + 385*(2*c*d*x + b*d)^(9/2)*b*d - 63*(2*c*d*x + b* d)^(11/2))*b/(c^3*d^5) + 5*(3003*sqrt(2*c*d*x + b*d)*b^6*d^6 - 6006*(2*c*d *x + b*d)^(3/2)*b^5*d^5 + 9009*(2*c*d*x + b*d)^(5/2)*b^4*d^4 - 8580*(2*c*d *x + b*d)^(7/2)*b^3*d^3 + 5005*(2*c*d*x + b*d)^(9/2)*b^2*d^2 - 1638*(2*c*d *x + b*d)^(11/2)*b*d + 231*(2*c*d*x + b*d)^(13/2))/(c^3*d^6))/(c*d)
Leaf count of result is larger than twice the leaf count of optimal. 778 vs. \(2 (105) = 210\).
Time = 0.12 (sec) , antiderivative size = 778, normalized size of antiderivative = 6.43 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(1/2),x, algorithm="giac")
Output:
1/960960*(960960*sqrt(2*c*d*x + b*d)*a^3 - 480480*(3*sqrt(2*c*d*x + b*d)*b *d - (2*c*d*x + b*d)^(3/2))*a^2*b/(c*d) + 48048*(15*sqrt(2*c*d*x + b*d)*b^ 2*d^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*a*b^2/(c^2 *d^2) + 48048*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c*d*x + b*d)^(3/2)*b *d + 3*(2*c*d*x + b*d)^(5/2))*a^2/(c*d^2) - 3432*(35*sqrt(2*c*d*x + b*d)*b ^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*d*x + b*d)^(7/2))*b^3/(c^3*d^3) - 20592*(35*sqrt(2*c*d*x + b*d)*b^ 3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*d*x + b*d)^(7/2))*a*b/(c^2*d^3) + 572*(315*sqrt(2*c*d*x + b*d)*b^4* d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 378*(2*c*d*x + b*d)^(5/2)*b^2*d^ 2 - 180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d)^(9/2))*b^2/(c^3*d^4 ) + 572*(315*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d ^3 + 378*(2*c*d*x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*b*d + 3 5*(2*c*d*x + b*d)^(9/2))*a/(c^2*d^4) - 130*(693*sqrt(2*c*d*x + b*d)*b^5*d^ 5 - 1155*(2*c*d*x + b*d)^(3/2)*b^4*d^4 + 1386*(2*c*d*x + b*d)^(5/2)*b^3*d^ 3 - 990*(2*c*d*x + b*d)^(7/2)*b^2*d^2 + 385*(2*c*d*x + b*d)^(9/2)*b*d - 63 *(2*c*d*x + b*d)^(11/2))*b/(c^3*d^5) + 5*(3003*sqrt(2*c*d*x + b*d)*b^6*d^6 - 6006*(2*c*d*x + b*d)^(3/2)*b^5*d^5 + 9009*(2*c*d*x + b*d)^(5/2)*b^4*d^4 - 8580*(2*c*d*x + b*d)^(7/2)*b^3*d^3 + 5005*(2*c*d*x + b*d)^(9/2)*b^2*d^2 - 1638*(2*c*d*x + b*d)^(11/2)*b*d + 231*(2*c*d*x + b*d)^(13/2))/(c^3*d...
Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=\frac {{\left (b\,d+2\,c\,d\,x\right )}^{13/2}}{832\,c^4\,d^7}+\frac {{\left (b\,d+2\,c\,d\,x\right )}^{9/2}\,\left (4\,a\,c-b^2\right )}{192\,c^4\,d^5}+\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (4\,a\,c-b^2\right )}^3}{64\,c^4\,d}+\frac {3\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,{\left (4\,a\,c-b^2\right )}^2}{320\,c^4\,d^3} \] Input:
int((a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(1/2),x)
Output:
(b*d + 2*c*d*x)^(13/2)/(832*c^4*d^7) + ((b*d + 2*c*d*x)^(9/2)*(4*a*c - b^2 ))/(192*c^4*d^5) + ((b*d + 2*c*d*x)^(1/2)*(4*a*c - b^2)^3)/(64*c^4*d) + (3 *(b*d + 2*c*d*x)^(5/2)*(4*a*c - b^2)^2)/(320*c^4*d^3)
Time = 0.22 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=\frac {\sqrt {d}\, \sqrt {2 c x +b}\, \left (15 c^{6} x^{6}+45 b \,c^{5} x^{5}+65 a \,c^{5} x^{4}+40 b^{2} c^{4} x^{4}+130 a b \,c^{4} x^{3}+5 b^{3} c^{3} x^{3}+117 a^{2} c^{4} x^{2}+39 a \,b^{2} c^{3} x^{2}-3 b^{4} c^{2} x^{2}+117 a^{2} b \,c^{3} x -26 a \,b^{3} c^{2} x +2 b^{5} c x +195 a^{3} c^{3}-117 a^{2} b^{2} c^{2}+26 a \,b^{4} c -2 b^{6}\right )}{195 c^{4} d} \] Input:
int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(1/2),x)
Output:
(sqrt(d)*sqrt(b + 2*c*x)*(195*a**3*c**3 - 117*a**2*b**2*c**2 + 117*a**2*b* c**3*x + 117*a**2*c**4*x**2 + 26*a*b**4*c - 26*a*b**3*c**2*x + 39*a*b**2*c **3*x**2 + 130*a*b*c**4*x**3 + 65*a*c**5*x**4 - 2*b**6 + 2*b**5*c*x - 3*b* *4*c**2*x**2 + 5*b**3*c**3*x**3 + 40*b**2*c**4*x**4 + 45*b*c**5*x**5 + 15* c**6*x**6))/(195*c**4*d)