\(\int \frac {1}{(b d+2 c d x)^4 (a+b x+c x^2)^{3/2}} \, dx\) [1249]

Optimal result

Integrand size = 26, antiderivative size = 118 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2}{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}-\frac {32 c \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3}-\frac {64 c \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)} \]

[Out]

-2/(-4*a*c+b^2)/d^4/(2*c*x+b)^3/(c*x^2+b*x+a)^(1/2)-32/3*c*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^2/d^4/(2*c*x+b)^3-
64/3*c*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^3/d^4/(2*c*x+b)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {701, 707, 696} \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {64 c \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)}-\frac {32 c \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3}-\frac {2}{d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}} \]

[In]

Int[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2)),x]

[Out]

-2/((b^2 - 4*a*c)*d^4*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2]) - (32*c*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^2*d
^4*(b + 2*c*x)^3) - (64*c*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x))

Rule 696

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*c*(d + e*x)^(m + 1
)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*
a*c, 0] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]

Rule 701

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*c*(d + e*x)^(m + 1
)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Dist[2*c*e*((m + 2*p + 3)/(e*(p + 1)*(b^2 - 4*a*
c))), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && RationalQ[m] && IntegerQ[2*p]

Rule 707

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[-2*b*d*(d + e*x)^(m
+ 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m + 1)*(b^2 - 4*a*c))), x] + Dist[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 -
4*a*c))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*
c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && Rationa
lQ[p]) || IntegerQ[(m + 2*p + 3)/2])

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}-\frac {(16 c) \int \frac {1}{(b d+2 c d x)^4 \sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c} \\ & = -\frac {2}{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}-\frac {32 c \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3}-\frac {(32 c) \int \frac {1}{(b d+2 c d x)^2 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 d^2} \\ & = -\frac {2}{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}-\frac {32 c \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3}-\frac {64 c \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \left (3 b^4+48 b^3 c x+64 b c^2 x \left (a+4 c x^2\right )+8 b^2 c \left (3 a+22 c x^2\right )+16 c^2 \left (-a^2+4 a c x^2+8 c^2 x^4\right )\right )}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt {a+x (b+c x)}} \]

[In]

Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(3*b^4 + 48*b^3*c*x + 64*b*c^2*x*(a + 4*c*x^2) + 8*b^2*c*(3*a + 22*c*x^2) + 16*c^2*(-a^2 + 4*a*c*x^2 + 8*c
^2*x^4)))/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^3*Sqrt[a + x*(b + c*x)])

Maple [A] (verified)

Time = 2.93 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.94

method	result	size
trager	\(-\frac {2 \left (-128 c^{4} x^{4}-256 b \,c^{3} x^{3}-64 x^{2} c^{3} a -176 b^{2} c^{2} x^{2}-64 a b \,c^{2} x -48 b^{3} c x +16 a^{2} c^{2}-24 a \,b^{2} c -3 b^{4}\right )}{3 d^{4} \sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right )^{3} \left (2 c x +b \right )^{3}}\)	\(111\)
gosper	\(-\frac {2 \left (-128 c^{4} x^{4}-256 b \,c^{3} x^{3}-64 x^{2} c^{3} a -176 b^{2} c^{2} x^{2}-64 a b \,c^{2} x -48 b^{3} c x +16 a^{2} c^{2}-24 a \,b^{2} c -3 b^{4}\right )}{3 \left (2 c x +b \right )^{3} d^{4} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {c \,x^{2}+b x +a}}\)	\(133\)
default	\(\frac {-\frac {4 c}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3} \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}-\frac {16 c^{2} \left (-\frac {4 c}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}-\frac {32 c^{3} \left (x +\frac {b}{2 c}\right )}{\left (4 a c -b^{2}\right )^{2} \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}\right )}{3 \left (4 a c -b^{2}\right )}}{16 d^{4} c^{4}}\)	\(193\)

[In]

int(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/3/d^4*(-128*c^4*x^4-256*b*c^3*x^3-64*a*c^3*x^2-176*b^2*c^2*x^2-64*a*b*c^2*x-48*b^3*c*x+16*a^2*c^2-24*a*b^2*
c-3*b^4)/(c*x^2+b*x+a)^(1/2)/(4*a*c-b^2)^3/(2*c*x+b)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (108) = 216\).

Time = 2.24 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.23 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} + 3 \, b^{4} + 24 \, a b^{2} c - 16 \, a^{2} c^{2} + 16 \, {\left (11 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 16 \, {\left (3 \, b^{3} c + 4 \, a b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (8 \, {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{4} x^{5} + 20 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{4} x^{4} + 2 \, {\left (9 \, b^{8} c^{2} - 104 \, a b^{6} c^{3} + 384 \, a^{2} b^{4} c^{4} - 384 \, a^{3} b^{2} c^{5} - 256 \, a^{4} c^{6}\right )} d^{4} x^{3} + {\left (7 \, b^{9} c - 72 \, a b^{7} c^{2} + 192 \, a^{2} b^{5} c^{3} + 128 \, a^{3} b^{3} c^{4} - 768 \, a^{4} b c^{5}\right )} d^{4} x^{2} + {\left (b^{10} - 6 \, a b^{8} c - 24 \, a^{2} b^{6} c^{2} + 224 \, a^{3} b^{4} c^{3} - 384 \, a^{4} b^{2} c^{4}\right )} d^{4} x + {\left (a b^{9} - 12 \, a^{2} b^{7} c + 48 \, a^{3} b^{5} c^{2} - 64 \, a^{4} b^{3} c^{3}\right )} d^{4}\right )}} \]

[In]

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2/3*(128*c^4*x^4 + 256*b*c^3*x^3 + 3*b^4 + 24*a*b^2*c - 16*a^2*c^2 + 16*(11*b^2*c^2 + 4*a*c^3)*x^2 + 16*(3*b^
3*c + 4*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a)/(8*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^4*x^5 +
20*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^4*x^4 + 2*(9*b^8*c^2 - 104*a*b^6*c^3 + 384*a^2*b
^4*c^4 - 384*a^3*b^2*c^5 - 256*a^4*c^6)*d^4*x^3 + (7*b^9*c - 72*a*b^7*c^2 + 192*a^2*b^5*c^3 + 128*a^3*b^3*c^4
- 768*a^4*b*c^5)*d^4*x^2 + (b^10 - 6*a*b^8*c - 24*a^2*b^6*c^2 + 224*a^3*b^4*c^3 - 384*a^4*b^2*c^4)*d^4*x + (a*
b^9 - 12*a^2*b^7*c + 48*a^3*b^5*c^2 - 64*a^4*b^3*c^3)*d^4)

Sympy [F]

\[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\int \frac {1}{a b^{4} \sqrt {a + b x + c x^{2}} + 8 a b^{3} c x \sqrt {a + b x + c x^{2}} + 24 a b^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 32 a b c^{3} x^{3} \sqrt {a + b x + c x^{2}} + 16 a c^{4} x^{4} \sqrt {a + b x + c x^{2}} + b^{5} x \sqrt {a + b x + c x^{2}} + 9 b^{4} c x^{2} \sqrt {a + b x + c x^{2}} + 32 b^{3} c^{2} x^{3} \sqrt {a + b x + c x^{2}} + 56 b^{2} c^{3} x^{4} \sqrt {a + b x + c x^{2}} + 48 b c^{4} x^{5} \sqrt {a + b x + c x^{2}} + 16 c^{5} x^{6} \sqrt {a + b x + c x^{2}}}\, dx}{d^{4}} \]

[In]

integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral(1/(a*b**4*sqrt(a + b*x + c*x**2) + 8*a*b**3*c*x*sqrt(a + b*x + c*x**2) + 24*a*b**2*c**2*x**2*sqrt(a +
b*x + c*x**2) + 32*a*b*c**3*x**3*sqrt(a + b*x + c*x**2) + 16*a*c**4*x**4*sqrt(a + b*x + c*x**2) + b**5*x*sqrt
(a + b*x + c*x**2) + 9*b**4*c*x**2*sqrt(a + b*x + c*x**2) + 32*b**3*c**2*x**3*sqrt(a + b*x + c*x**2) + 56*b**2
*c**3*x**4*sqrt(a + b*x + c*x**2) + 48*b*c**4*x**5*sqrt(a + b*x + c*x**2) + 16*c**5*x**6*sqrt(a + b*x + c*x**2
)), x)/d**4

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (108) = 216\).

Time = 0.31 (sec) , antiderivative size = 398, normalized size of antiderivative = 3.37 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (\frac {2 \, c x}{b^{6} d^{4} - 12 \, a b^{4} c d^{4} + 48 \, a^{2} b^{2} c^{2} d^{4} - 64 \, a^{3} c^{3} d^{4}} + \frac {b}{b^{6} d^{4} - 12 \, a b^{4} c d^{4} + 48 \, a^{2} b^{2} c^{2} d^{4} - 64 \, a^{3} c^{3} d^{4}}\right )}}{\sqrt {c x^{2} + b x + a}} - \frac {4 \, {\left (12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} c^{\frac {5}{2}} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b c^{2} + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} c^{\frac {3}{2}} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{\frac {5}{2}} + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} c - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c^{2} + 5 \, b^{4} \sqrt {c} - 22 \, a b^{2} c^{\frac {3}{2}} + 20 \, a^{2} c^{\frac {5}{2}}\right )}}{3 \, {\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )} {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{3}} \]

[In]

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

-2*(2*c*x/(b^6*d^4 - 12*a*b^4*c*d^4 + 48*a^2*b^2*c^2*d^4 - 64*a^3*c^3*d^4) + b/(b^6*d^4 - 12*a*b^4*c*d^4 + 48*
a^2*b^2*c^2*d^4 - 64*a^3*c^3*d^4))/sqrt(c*x^2 + b*x + a) - 4/3*(12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(5/
2) + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^2 + 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(3/2) - 4
8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(5/2) + 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c - 48*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))*a*b*c^2 + 5*b^4*sqrt(c) - 22*a*b^2*c^(3/2) + 20*a^2*c^(5/2))/((b^4*d^4 - 8*a*b^2*c
*d^4 + 16*a^2*c^2*d^4)*(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sq
rt(c) + b^2 - 2*a*c)^3)

Mupad [B] (verification not implemented)

Time = 11.30 (sec) , antiderivative size = 4588, normalized size of antiderivative = 38.88 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

int(1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2)),x)

[Out]

(8*b^6*c^2)/((a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^
4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4
*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (128*b^8*c^5)/((a + b*x + c
*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^
6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 15
36*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a
^5*b^2*c^11*d^4*x)) - (2560*a^3*c^5)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*
b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*
c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) -
(32*c^4*(a + b*x + c*x^2)^(1/2))/(12*b^7*c^3*d^4 - 96*a*b^5*c^4*d^4 + 72*b^6*c^4*d^4*x + 192*a^2*b^3*c^5*d^4 +
1536*a^2*c^8*d^4*x^3 + 144*b^5*c^5*d^4*x^2 + 96*b^4*c^6*d^4*x^3 - 576*a*b^4*c^5*d^4*x + 1152*a^2*b^2*c^6*d^4*
x - 1152*a*b^3*c^6*d^4*x^2 + 2304*a^2*b*c^7*d^4*x^2 - 768*a*b^2*c^7*d^4*x^3) - (352*a*b^4*c^3)/(3*(a + b*x + c
*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x +
640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4
*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (64*b^5*c^3*x)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^
2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 -
2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*
d^4*x + 10240*a^4*b^2*c^7*d^4*x)) + (1536*a*b^6*c^6)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^
6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b
^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*
x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*d^4*x)) - (512*b^7*c^6*x)/((a +
b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*
b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d
^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 3
93216*a^5*b^2*c^11*d^4*x)) + (1664*a^2*b^2*c^4)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4
- 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 51
20*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7
*d^4*x)) - (1024*a^2*c^6*x^2)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d
^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4
- 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (64*b^4
*c^4*x^2)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^
4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4
*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (6144*a^2*b^4*c^7)/((a + b*
x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^
12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4
- 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393
216*a^5*b^2*c^11*d^4*x)) + (8192*a^3*b^2*c^8)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 +
131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*
d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 819
20*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*d^4*x)) + (32768*a^3*c^10*x^2)/((a + b*
x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^
12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4
- 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393
216*a^5*b^2*c^11*d^4*x)) - (512*b^6*c^7*x^2)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 +
131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d
^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 8192
0*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*d^4*x)) + (512*a*b^2*c^5*x^2)/(3*(a + b*
x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4
*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^
5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (24576*a^2*b^3*c^8*x)/((a + b*x + c*x^2)^(1/2)*
(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 76
80*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^
7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*
d^4*x)) + (6144*a*b^4*c^8*x^2)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c
^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^
4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*
d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*d^4*x)) + (512*a*b^3*c^4*x)/(3*(a + b*x + c*x^2)^(1/2)
*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7
*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a
^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (1024*a^2*b*c^5*x)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 -
80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^
3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x +
10240*a^4*b^2*c^7*d^4*x)) - (24576*a^2*b^2*c^9*x^2)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^
6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b
^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*
x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*d^4*x)) + (6144*a*b^5*c^7*x)/((a
+ b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x +
64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^1
0*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x
- 393216*a^5*b^2*c^11*d^4*x)) + (32768*a^3*b*c^9*x)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6
*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^
7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x
- 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*d^4*x))