\(\int (b d+2 c d x)^5 (a+b x+c x^2)^2 \, dx\) [1121]

Optimal result

Integrand size = 24, antiderivative size = 73 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^2 \, dx=\frac {\left (b^2-4 a c\right )^2 d^5 (b+2 c x)^6}{192 c^3}-\frac {\left (b^2-4 a c\right ) d^5 (b+2 c x)^8}{128 c^3}+\frac {d^5 (b+2 c x)^{10}}{320 c^3} \]

[Out]

1/192*(-4*a*c+b^2)^2*d^5*(2*c*x+b)^6/c^3-1/128*(-4*a*c+b^2)*d^5*(2*c*x+b)^8/c^3+1/320*d^5*(2*c*x+b)^10/c^3

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^2 \, dx=-\frac {d^5 \left (b^2-4 a c\right ) (b+2 c x)^8}{128 c^3}+\frac {d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^6}{192 c^3}+\frac {d^5 (b+2 c x)^{10}}{320 c^3} \]

[In]

Int[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^2,x]

[Out]

((b^2 - 4*a*c)^2*d^5*(b + 2*c*x)^6)/(192*c^3) - ((b^2 - 4*a*c)*d^5*(b + 2*c*x)^8)/(128*c^3) + (d^5*(b + 2*c*x)
^10)/(320*c^3)

Rule 697

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] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^2 (b d+2 c d x)^5}{16 c^2}+\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^7}{8 c^2 d^2}+\frac {(b d+2 c d x)^9}{16 c^2 d^4}\right ) \, dx \\ & = \frac {\left (b^2-4 a c\right )^2 d^5 (b+2 c x)^6}{192 c^3}-\frac {\left (b^2-4 a c\right ) d^5 (b+2 c x)^8}{128 c^3}+\frac {d^5 (b+2 c x)^{10}}{320 c^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(168\) vs. \(2(73)=146\).

Time = 0.06 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.30 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{15} d^5 x (b+c x) \left (5 a^2 \left (3 b^4+12 b^3 c x+28 b^2 c^2 x^2+32 b c^3 x^3+16 c^4 x^4\right )+x^2 (b+c x)^2 \left (5 b^4+30 b^3 c x+78 b^2 c^2 x^2+96 b c^3 x^3+48 c^4 x^4\right )+5 a x \left (3 b^5+19 b^4 c x+56 b^3 c^2 x^2+88 b^2 c^3 x^3+72 b c^4 x^4+24 c^5 x^5\right )\right ) \]

[In]

Integrate[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^2,x]

[Out]

(d^5*x*(b + c*x)*(5*a^2*(3*b^4 + 12*b^3*c*x + 28*b^2*c^2*x^2 + 32*b*c^3*x^3 + 16*c^4*x^4) + x^2*(b + c*x)^2*(5
*b^4 + 30*b^3*c*x + 78*b^2*c^2*x^2 + 96*b*c^3*x^3 + 48*c^4*x^4) + 5*a*x*(3*b^5 + 19*b^4*c*x + 56*b^3*c^2*x^2 +
88*b^2*c^3*x^3 + 72*b*c^4*x^4 + 24*c^5*x^5)))/15

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(226\) vs. \(2(67)=134\).

Time = 2.54 (sec) , antiderivative size = 227, normalized size of antiderivative = 3.11

method	result	size
gosper	\(\frac {x \left (48 c^{7} x^{9}+240 b \,c^{6} x^{8}+120 x^{7} a \,c^{6}+510 x^{7} b^{2} c^{5}+480 x^{6} a b \,c^{5}+600 x^{6} b^{3} c^{4}+80 x^{5} a^{2} c^{5}+800 x^{5} a \,b^{2} c^{4}+425 x^{5} b^{4} c^{3}+240 x^{4} a^{2} b \,c^{4}+720 a \,b^{3} c^{3} x^{4}+183 x^{4} b^{5} c^{2}+300 a^{2} b^{2} c^{3} x^{3}+375 c^{2} x^{3} a \,b^{4}+45 x^{3} b^{6} c +200 a^{2} b^{3} c^{2} x^{2}+110 x^{2} a \,b^{5} c +5 x^{2} b^{7}+75 a^{2} b^{4} c x +15 x a \,b^{6}+15 a^{2} b^{5}\right ) d^{5}}{15}\)	\(227\)
norman	\(\left (\frac {16}{3} c^{5} d^{5} a^{2}+\frac {160}{3} b^{2} d^{5} c^{4} a +\frac {85}{3} b^{4} d^{5} c^{3}\right ) x^{6}+\left (16 b \,d^{5} c^{4} a^{2}+48 b^{3} d^{5} c^{3} a +\frac {61}{5} b^{5} c^{2} d^{5}\right ) x^{5}+\left (\frac {40}{3} b^{3} d^{5} c^{2} a^{2}+\frac {22}{3} a \,b^{5} c \,d^{5}+\frac {1}{3} b^{7} d^{5}\right ) x^{3}+\left (8 a \,c^{6} d^{5}+34 b^{2} d^{5} c^{5}\right ) x^{8}+\left (32 c^{5} d^{5} a b +40 c^{4} b^{3} d^{5}\right ) x^{7}+\left (5 b^{4} d^{5} c \,a^{2}+b^{6} d^{5} a \right ) x^{2}+\left (20 b^{2} d^{5} c^{3} a^{2}+25 b^{4} d^{5} c^{2} a +3 b^{6} d^{5} c \right ) x^{4}+a^{2} b^{5} d^{5} x +\frac {16 c^{7} d^{5} x^{10}}{5}+16 b \,d^{5} c^{6} x^{9}\)	\(268\)
risch	\(\frac {16}{5} c^{7} d^{5} x^{10}+16 b \,d^{5} c^{6} x^{9}+8 d^{5} a \,c^{6} x^{8}+34 d^{5} b^{2} c^{5} x^{8}+32 d^{5} a b \,c^{5} x^{7}+40 d^{5} b^{3} c^{4} x^{7}+\frac {16}{3} d^{5} x^{6} a^{2} c^{5}+\frac {160}{3} d^{5} x^{6} a \,b^{2} c^{4}+\frac {85}{3} d^{5} x^{6} b^{4} c^{3}+16 d^{5} a^{2} b \,c^{4} x^{5}+48 d^{5} x^{5} a \,b^{3} c^{3}+\frac {61}{5} d^{5} x^{5} b^{5} c^{2}+20 d^{5} a^{2} b^{2} c^{3} x^{4}+25 d^{5} c^{2} x^{4} a \,b^{4}+3 d^{5} b^{6} c \,x^{4}+\frac {40}{3} d^{5} x^{3} a^{2} c^{2} b^{3}+\frac {22}{3} d^{5} x^{3} a \,b^{5} c +\frac {1}{3} d^{5} x^{3} b^{7}+5 d^{5} a^{2} b^{4} c \,x^{2}+d^{5} a \,b^{6} x^{2}+a^{2} b^{5} d^{5} x\)	\(287\)
parallelrisch	\(\frac {16}{5} c^{7} d^{5} x^{10}+16 b \,d^{5} c^{6} x^{9}+8 d^{5} a \,c^{6} x^{8}+34 d^{5} b^{2} c^{5} x^{8}+32 d^{5} a b \,c^{5} x^{7}+40 d^{5} b^{3} c^{4} x^{7}+\frac {16}{3} d^{5} x^{6} a^{2} c^{5}+\frac {160}{3} d^{5} x^{6} a \,b^{2} c^{4}+\frac {85}{3} d^{5} x^{6} b^{4} c^{3}+16 d^{5} a^{2} b \,c^{4} x^{5}+48 d^{5} x^{5} a \,b^{3} c^{3}+\frac {61}{5} d^{5} x^{5} b^{5} c^{2}+20 d^{5} a^{2} b^{2} c^{3} x^{4}+25 d^{5} c^{2} x^{4} a \,b^{4}+3 d^{5} b^{6} c \,x^{4}+\frac {40}{3} d^{5} x^{3} a^{2} c^{2} b^{3}+\frac {22}{3} d^{5} x^{3} a \,b^{5} c +\frac {1}{3} d^{5} x^{3} b^{7}+5 d^{5} a^{2} b^{4} c \,x^{2}+d^{5} a \,b^{6} x^{2}+a^{2} b^{5} d^{5} x\)	\(287\)
default	\(\frac {16 c^{7} d^{5} x^{10}}{5}+16 b \,d^{5} c^{6} x^{9}+\frac {\left (240 b^{2} d^{5} c^{5}+32 c^{5} d^{5} \left (2 a c +b^{2}\right )\right ) x^{8}}{8}+\frac {\left (200 c^{4} b^{3} d^{5}+80 b \,d^{5} c^{4} \left (2 a c +b^{2}\right )+64 c^{5} d^{5} a b \right ) x^{7}}{7}+\frac {\left (90 b^{4} d^{5} c^{3}+80 b^{2} d^{5} c^{3} \left (2 a c +b^{2}\right )+160 b^{2} d^{5} c^{4} a +32 c^{5} d^{5} a^{2}\right ) x^{6}}{6}+\frac {\left (21 b^{5} c^{2} d^{5}+40 b^{3} d^{5} c^{2} \left (2 a c +b^{2}\right )+160 b^{3} d^{5} c^{3} a +80 b \,d^{5} c^{4} a^{2}\right ) x^{5}}{5}+\frac {\left (2 b^{6} d^{5} c +10 b^{4} d^{5} c \left (2 a c +b^{2}\right )+80 b^{4} d^{5} c^{2} a +80 b^{2} d^{5} c^{3} a^{2}\right ) x^{4}}{4}+\frac {\left (b^{5} d^{5} \left (2 a c +b^{2}\right )+20 a \,b^{5} c \,d^{5}+40 b^{3} d^{5} c^{2} a^{2}\right ) x^{3}}{3}+\frac {\left (10 b^{4} d^{5} c \,a^{2}+2 b^{6} d^{5} a \right ) x^{2}}{2}+a^{2} b^{5} d^{5} x\)	\(362\)

[In]

int((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/15*x*(48*c^7*x^9+240*b*c^6*x^8+120*a*c^6*x^7+510*b^2*c^5*x^7+480*a*b*c^5*x^6+600*b^3*c^4*x^6+80*a^2*c^5*x^5+
800*a*b^2*c^4*x^5+425*b^4*c^3*x^5+240*a^2*b*c^4*x^4+720*a*b^3*c^3*x^4+183*b^5*c^2*x^4+300*a^2*b^2*c^3*x^3+375*
a*b^4*c^2*x^3+45*b^6*c*x^3+200*a^2*b^3*c^2*x^2+110*a*b^5*c*x^2+5*b^7*x^2+75*a^2*b^4*c*x+15*a*b^6*x+15*a^2*b^5)
*d^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (67) = 134\).

Time = 0.27 (sec) , antiderivative size = 237, normalized size of antiderivative = 3.25 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^2 \, dx=\frac {16}{5} \, c^{7} d^{5} x^{10} + 16 \, b c^{6} d^{5} x^{9} + 2 \, {\left (17 \, b^{2} c^{5} + 4 \, a c^{6}\right )} d^{5} x^{8} + a^{2} b^{5} d^{5} x + 8 \, {\left (5 \, b^{3} c^{4} + 4 \, a b c^{5}\right )} d^{5} x^{7} + \frac {1}{3} \, {\left (85 \, b^{4} c^{3} + 160 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{5} x^{6} + \frac {1}{5} \, {\left (61 \, b^{5} c^{2} + 240 \, a b^{3} c^{3} + 80 \, a^{2} b c^{4}\right )} d^{5} x^{5} + {\left (3 \, b^{6} c + 25 \, a b^{4} c^{2} + 20 \, a^{2} b^{2} c^{3}\right )} d^{5} x^{4} + \frac {1}{3} \, {\left (b^{7} + 22 \, a b^{5} c + 40 \, a^{2} b^{3} c^{2}\right )} d^{5} x^{3} + {\left (a b^{6} + 5 \, a^{2} b^{4} c\right )} d^{5} x^{2} \]

[In]

integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

16/5*c^7*d^5*x^10 + 16*b*c^6*d^5*x^9 + 2*(17*b^2*c^5 + 4*a*c^6)*d^5*x^8 + a^2*b^5*d^5*x + 8*(5*b^3*c^4 + 4*a*b
*c^5)*d^5*x^7 + 1/3*(85*b^4*c^3 + 160*a*b^2*c^4 + 16*a^2*c^5)*d^5*x^6 + 1/5*(61*b^5*c^2 + 240*a*b^3*c^3 + 80*a
^2*b*c^4)*d^5*x^5 + (3*b^6*c + 25*a*b^4*c^2 + 20*a^2*b^2*c^3)*d^5*x^4 + 1/3*(b^7 + 22*a*b^5*c + 40*a^2*b^3*c^2
)*d^5*x^3 + (a*b^6 + 5*a^2*b^4*c)*d^5*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (68) = 136\).

Time = 0.04 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.99 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^2 \, dx=a^{2} b^{5} d^{5} x + 16 b c^{6} d^{5} x^{9} + \frac {16 c^{7} d^{5} x^{10}}{5} + x^{8} \cdot \left (8 a c^{6} d^{5} + 34 b^{2} c^{5} d^{5}\right ) + x^{7} \cdot \left (32 a b c^{5} d^{5} + 40 b^{3} c^{4} d^{5}\right ) + x^{6} \cdot \left (\frac {16 a^{2} c^{5} d^{5}}{3} + \frac {160 a b^{2} c^{4} d^{5}}{3} + \frac {85 b^{4} c^{3} d^{5}}{3}\right ) + x^{5} \cdot \left (16 a^{2} b c^{4} d^{5} + 48 a b^{3} c^{3} d^{5} + \frac {61 b^{5} c^{2} d^{5}}{5}\right ) + x^{4} \cdot \left (20 a^{2} b^{2} c^{3} d^{5} + 25 a b^{4} c^{2} d^{5} + 3 b^{6} c d^{5}\right ) + x^{3} \cdot \left (\frac {40 a^{2} b^{3} c^{2} d^{5}}{3} + \frac {22 a b^{5} c d^{5}}{3} + \frac {b^{7} d^{5}}{3}\right ) + x^{2} \cdot \left (5 a^{2} b^{4} c d^{5} + a b^{6} d^{5}\right ) \]

[In]

integrate((2*c*d*x+b*d)**5*(c*x**2+b*x+a)**2,x)

[Out]

a**2*b**5*d**5*x + 16*b*c**6*d**5*x**9 + 16*c**7*d**5*x**10/5 + x**8*(8*a*c**6*d**5 + 34*b**2*c**5*d**5) + x**
7*(32*a*b*c**5*d**5 + 40*b**3*c**4*d**5) + x**6*(16*a**2*c**5*d**5/3 + 160*a*b**2*c**4*d**5/3 + 85*b**4*c**3*d
**5/3) + x**5*(16*a**2*b*c**4*d**5 + 48*a*b**3*c**3*d**5 + 61*b**5*c**2*d**5/5) + x**4*(20*a**2*b**2*c**3*d**5
+ 25*a*b**4*c**2*d**5 + 3*b**6*c*d**5) + x**3*(40*a**2*b**3*c**2*d**5/3 + 22*a*b**5*c*d**5/3 + b**7*d**5/3) +
x**2*(5*a**2*b**4*c*d**5 + a*b**6*d**5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (67) = 134\).

Time = 0.20 (sec) , antiderivative size = 237, normalized size of antiderivative = 3.25 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^2 \, dx=\frac {16}{5} \, c^{7} d^{5} x^{10} + 16 \, b c^{6} d^{5} x^{9} + 2 \, {\left (17 \, b^{2} c^{5} + 4 \, a c^{6}\right )} d^{5} x^{8} + a^{2} b^{5} d^{5} x + 8 \, {\left (5 \, b^{3} c^{4} + 4 \, a b c^{5}\right )} d^{5} x^{7} + \frac {1}{3} \, {\left (85 \, b^{4} c^{3} + 160 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{5} x^{6} + \frac {1}{5} \, {\left (61 \, b^{5} c^{2} + 240 \, a b^{3} c^{3} + 80 \, a^{2} b c^{4}\right )} d^{5} x^{5} + {\left (3 \, b^{6} c + 25 \, a b^{4} c^{2} + 20 \, a^{2} b^{2} c^{3}\right )} d^{5} x^{4} + \frac {1}{3} \, {\left (b^{7} + 22 \, a b^{5} c + 40 \, a^{2} b^{3} c^{2}\right )} d^{5} x^{3} + {\left (a b^{6} + 5 \, a^{2} b^{4} c\right )} d^{5} x^{2} \]

[In]

integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

16/5*c^7*d^5*x^10 + 16*b*c^6*d^5*x^9 + 2*(17*b^2*c^5 + 4*a*c^6)*d^5*x^8 + a^2*b^5*d^5*x + 8*(5*b^3*c^4 + 4*a*b
*c^5)*d^5*x^7 + 1/3*(85*b^4*c^3 + 160*a*b^2*c^4 + 16*a^2*c^5)*d^5*x^6 + 1/5*(61*b^5*c^2 + 240*a*b^3*c^3 + 80*a
^2*b*c^4)*d^5*x^5 + (3*b^6*c + 25*a*b^4*c^2 + 20*a^2*b^2*c^3)*d^5*x^4 + 1/3*(b^7 + 22*a*b^5*c + 40*a^2*b^3*c^2
)*d^5*x^3 + (a*b^6 + 5*a^2*b^4*c)*d^5*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (67) = 134\).

Time = 0.29 (sec) , antiderivative size = 286, normalized size of antiderivative = 3.92 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^2 \, dx=\frac {16}{5} \, c^{7} d^{5} x^{10} + 16 \, b c^{6} d^{5} x^{9} + 34 \, b^{2} c^{5} d^{5} x^{8} + 8 \, a c^{6} d^{5} x^{8} + 40 \, b^{3} c^{4} d^{5} x^{7} + 32 \, a b c^{5} d^{5} x^{7} + \frac {85}{3} \, b^{4} c^{3} d^{5} x^{6} + \frac {160}{3} \, a b^{2} c^{4} d^{5} x^{6} + \frac {16}{3} \, a^{2} c^{5} d^{5} x^{6} + \frac {61}{5} \, b^{5} c^{2} d^{5} x^{5} + 48 \, a b^{3} c^{3} d^{5} x^{5} + 16 \, a^{2} b c^{4} d^{5} x^{5} + 3 \, b^{6} c d^{5} x^{4} + 25 \, a b^{4} c^{2} d^{5} x^{4} + 20 \, a^{2} b^{2} c^{3} d^{5} x^{4} + \frac {1}{3} \, b^{7} d^{5} x^{3} + \frac {22}{3} \, a b^{5} c d^{5} x^{3} + \frac {40}{3} \, a^{2} b^{3} c^{2} d^{5} x^{3} + a b^{6} d^{5} x^{2} + 5 \, a^{2} b^{4} c d^{5} x^{2} + a^{2} b^{5} d^{5} x \]

[In]

integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

16/5*c^7*d^5*x^10 + 16*b*c^6*d^5*x^9 + 34*b^2*c^5*d^5*x^8 + 8*a*c^6*d^5*x^8 + 40*b^3*c^4*d^5*x^7 + 32*a*b*c^5*
d^5*x^7 + 85/3*b^4*c^3*d^5*x^6 + 160/3*a*b^2*c^4*d^5*x^6 + 16/3*a^2*c^5*d^5*x^6 + 61/5*b^5*c^2*d^5*x^5 + 48*a*
b^3*c^3*d^5*x^5 + 16*a^2*b*c^4*d^5*x^5 + 3*b^6*c*d^5*x^4 + 25*a*b^4*c^2*d^5*x^4 + 20*a^2*b^2*c^3*d^5*x^4 + 1/3
*b^7*d^5*x^3 + 22/3*a*b^5*c*d^5*x^3 + 40/3*a^2*b^3*c^2*d^5*x^3 + a*b^6*d^5*x^2 + 5*a^2*b^4*c*d^5*x^2 + a^2*b^5
*d^5*x

Mupad [B] (verification not implemented)

Time = 10.05 (sec) , antiderivative size = 224, normalized size of antiderivative = 3.07 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^2 \, dx=\frac {16\,c^7\,d^5\,x^{10}}{5}+\frac {c^3\,d^5\,x^6\,\left (16\,a^2\,c^2+160\,a\,b^2\,c+85\,b^4\right )}{3}+a^2\,b^5\,d^5\,x+16\,b\,c^6\,d^5\,x^9+2\,c^5\,d^5\,x^8\,\left (17\,b^2+4\,a\,c\right )+\frac {b^3\,d^5\,x^3\,\left (40\,a^2\,c^2+22\,a\,b^2\,c+b^4\right )}{3}+a\,b^4\,d^5\,x^2\,\left (b^2+5\,a\,c\right )+b^2\,c\,d^5\,x^4\,\left (20\,a^2\,c^2+25\,a\,b^2\,c+3\,b^4\right )+\frac {b\,c^2\,d^5\,x^5\,\left (80\,a^2\,c^2+240\,a\,b^2\,c+61\,b^4\right )}{5}+8\,b\,c^4\,d^5\,x^7\,\left (5\,b^2+4\,a\,c\right ) \]

[In]

int((b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^2,x)

[Out]

(16*c^7*d^5*x^10)/5 + (c^3*d^5*x^6*(85*b^4 + 16*a^2*c^2 + 160*a*b^2*c))/3 + a^2*b^5*d^5*x + 16*b*c^6*d^5*x^9 +
2*c^5*d^5*x^8*(4*a*c + 17*b^2) + (b^3*d^5*x^3*(b^4 + 40*a^2*c^2 + 22*a*b^2*c))/3 + a*b^4*d^5*x^2*(5*a*c + b^2
) + b^2*c*d^5*x^4*(3*b^4 + 20*a^2*c^2 + 25*a*b^2*c) + (b*c^2*d^5*x^5*(61*b^4 + 80*a^2*c^2 + 240*a*b^2*c))/5 +
8*b*c^4*d^5*x^7*(4*a*c + 5*b^2)