3.13.59 \(\int \frac {1}{(b d+2 c d x)^4 (a+b x+c x^2)^{5/2}} \, dx\) [1259]

Optimal. Leaf size=162 \[ -\frac {2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}+\frac {16 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}+\frac {256 c^2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac {512 c^2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x)} \]

[Out]

-2/3/(-4*a*c+b^2)/d^4/(2*c*x+b)^3/(c*x^2+b*x+a)^(3/2)+16*c/(-4*a*c+b^2)^2/d^4/(2*c*x+b)^3/(c*x^2+b*x+a)^(1/2)+
256/3*c^2*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^3/d^4/(2*c*x+b)^3+512/3*c^2*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^4/d^4/
(2*c*x+b)

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Rubi [A]
time = 0.06, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {701, 707, 696} \begin {gather*} \frac {512 c^2 \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^4 (b+2 c x)}+\frac {256 c^2 \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3}+\frac {16 c}{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}-\frac {2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2)),x]

[Out]

-2/(3*(b^2 - 4*a*c)*d^4*(b + 2*c*x)^3*(a + b*x + c*x^2)^(3/2)) + (16*c)/((b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^3*Sqr
t[a + b*x + c*x^2]) + (256*c^2*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^3) + (512*c^2*Sqrt[a
+ b*x + c*x^2])/(3*(b^2 - 4*a*c)^4*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*} \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}-\frac {(8 c) \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx}{b^2-4 a c}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}+\frac {16 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}+\frac {\left (128 c^2\right ) \int \frac {1}{(b d+2 c d x)^4 \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}+\frac {16 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}+\frac {256 c^2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac {\left (256 c^2\right ) \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 )^3 d^2}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}+\frac {16 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}+\frac {256 c^2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac {512 c^2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x)}\\ \end {align*}

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Mathematica [A]
time = 0.78, size = 178, normalized size = 1.10 \begin {gather*} \frac {2 \left (-b^6+24 b^5 c x+64 b^3 c^2 x \left (9 a+28 c x^2\right )+12 b^4 c \left (3 a+34 c x^2\right )+384 b c^3 x \left (a^2+8 a c x^2+8 c^2 x^4\right )+48 b^2 c^2 \left (3 a^2+44 a c x^2+72 c^2 x^4\right )+64 c^3 \left (-a^3+6 a^2 c x^2+24 a c^2 x^4+16 c^3 x^6\right )\right )}{3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x)^3 (a+x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*(-b^6 + 24*b^5*c*x + 64*b^3*c^2*x*(9*a + 28*c*x^2) + 12*b^4*c*(3*a + 34*c*x^2) + 384*b*c^3*x*(a^2 + 8*a*c*x
^2 + 8*c^2*x^4) + 48*b^2*c^2*(3*a^2 + 44*a*c*x^2 + 72*c^2*x^4) + 64*c^3*(-a^3 + 6*a^2*c*x^2 + 24*a*c^2*x^4 + 1
6*c^3*x^6)))/(3*(b^2 - 4*a*c)^4*d^4*(b + 2*c*x)^3*(a + x*(b + c*x))^(3/2))

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Maple [A]
time = 0.74, size = 264, normalized size = 1.63

method	result	size
trager	\(-\frac {2 \left (-1024 c^{6} x^{6}-3072 b \,c^{5} x^{5}-1536 a \,c^{5} x^{4}-3456 b^{2} c^{4} x^{4}-3072 a b \,c^{4} x^{3}-1792 b^{3} x^{3} c^{3}-384 a^{2} c^{4} x^{2}-2112 a \,b^{2} c^{3} x^{2}-408 b^{4} c^{2} x^{2}-384 a^{2} b \,c^{3} x -576 a \,b^{3} c^{2} x -24 b^{5} c x +64 a^{3} c^{3}-144 a^{2} b^{2} c^{2}-36 a \,b^{4} c +b^{6}\right )}{3 d^{4} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )^{4} \left (2 c x +b \right )^{3}}\)	\(187\)
gosper	\(-\frac {2 \left (-1024 c^{6} x^{6}-3072 b \,c^{5} x^{5}-1536 a \,c^{5} x^{4}-3456 b^{2} c^{4} x^{4}-3072 a b \,c^{4} x^{3}-1792 b^{3} x^{3} c^{3}-384 a^{2} c^{4} x^{2}-2112 a \,b^{2} c^{3} x^{2}-408 b^{4} c^{2} x^{2}-384 a^{2} b \,c^{3} x -576 a \,b^{3} c^{2} x -24 b^{5} c x +64 a^{3} c^{3}-144 a^{2} b^{2} c^{2}-36 a \,b^{4} c +b^{6}\right )}{3 \left (2 c x +b \right )^{3} d^{4} \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\)	\(218\)
default	\(\frac {-\frac {4 c}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3} \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}-\frac {8 c^{2} \left (-\frac {4 c}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}-\frac {16 c^{2} \left (\frac {4 c \left (x +\frac {b}{2 c}\right )}{3 \left (4 a c -b^{2}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}+\frac {32 c^{2} \left (x +\frac {b}{2 c}\right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}}{16 d^{4} c^{4}}\)	\(264\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/16/d^4/c^4*(-4/3/(4*a*c-b^2)*c/(x+1/2*b/c)^3/((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)-8*c^2/(4*a*c-b^2)*(-4
/(4*a*c-b^2)*c/(x+1/2*b/c)/((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)-16*c^2/(4*a*c-b^2)*(4/3*c*(x+1/2*b/c)/(4*
a*c-b^2)/((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)+32/3*c^2/(4*a*c-b^2)^2*(x+1/2*b/c)/((x+1/2*b/c)^2*c+1/4*(4*
a*c-b^2)/c)^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 677 vs. \(2 (148) = 296\).
time = 14.82, size = 677, normalized size = 4.18 \begin {gather*} \frac {2 \, {\left (1024 \, c^{6} x^{6} + 3072 \, b c^{5} x^{5} - b^{6} + 36 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 384 \, {\left (9 \, b^{2} c^{4} + 4 \, a c^{5}\right )} x^{4} + 256 \, {\left (7 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} x^{3} + 24 \, {\left (17 \, b^{4} c^{2} + 88 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 24 \, {\left (b^{5} c + 24 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (8 \, {\left (b^{8} c^{5} - 16 \, a b^{6} c^{6} + 96 \, a^{2} b^{4} c^{7} - 256 \, a^{3} b^{2} c^{8} + 256 \, a^{4} c^{9}\right )} d^{4} x^{7} + 28 \, {\left (b^{9} c^{4} - 16 \, a b^{7} c^{5} + 96 \, a^{2} b^{5} c^{6} - 256 \, a^{3} b^{3} c^{7} + 256 \, a^{4} b c^{8}\right )} d^{4} x^{6} + 2 \, {\left (19 \, b^{10} c^{3} - 296 \, a b^{8} c^{4} + 1696 \, a^{2} b^{6} c^{5} - 4096 \, a^{3} b^{4} c^{6} + 2816 \, a^{4} b^{2} c^{7} + 2048 \, a^{5} c^{8}\right )} d^{4} x^{5} + 5 \, {\left (5 \, b^{11} c^{2} - 72 \, a b^{9} c^{3} + 352 \, a^{2} b^{7} c^{4} - 512 \, a^{3} b^{5} c^{5} - 768 \, a^{4} b^{3} c^{6} + 2048 \, a^{5} b c^{7}\right )} d^{4} x^{4} + 4 \, {\left (2 \, b^{12} c - 23 \, a b^{10} c^{2} + 50 \, a^{2} b^{8} c^{3} + 320 \, a^{3} b^{6} c^{4} - 1600 \, a^{4} b^{4} c^{5} + 1792 \, a^{5} b^{2} c^{6} + 512 \, a^{6} c^{7}\right )} d^{4} x^{3} + {\left (b^{13} - 2 \, a b^{11} c - 116 \, a^{2} b^{9} c^{2} + 896 \, a^{3} b^{7} c^{3} - 2176 \, a^{4} b^{5} c^{4} + 512 \, a^{5} b^{3} c^{5} + 3072 \, a^{6} b c^{6}\right )} d^{4} x^{2} + 2 \, {\left (a b^{12} - 13 \, a^{2} b^{10} c + 48 \, a^{3} b^{8} c^{2} + 32 \, a^{4} b^{6} c^{3} - 512 \, a^{5} b^{4} c^{4} + 768 \, a^{6} b^{2} c^{5}\right )} d^{4} x + {\left (a^{2} b^{11} - 16 \, a^{3} b^{9} c + 96 \, a^{4} b^{7} c^{2} - 256 \, a^{5} b^{5} c^{3} + 256 \, a^{6} b^{3} c^{4}\right )} d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

2/3*(1024*c^6*x^6 + 3072*b*c^5*x^5 - b^6 + 36*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3 + 384*(9*b^2*c^4 + 4*a*c^
5)*x^4 + 256*(7*b^3*c^3 + 12*a*b*c^4)*x^3 + 24*(17*b^4*c^2 + 88*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 24*(b^5*c + 24*a
*b^3*c^2 + 16*a^2*b*c^3)*x)*sqrt(c*x^2 + b*x + a)/(8*(b^8*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^
8 + 256*a^4*c^9)*d^4*x^7 + 28*(b^9*c^4 - 16*a*b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d^4*
x^6 + 2*(19*b^10*c^3 - 296*a*b^8*c^4 + 1696*a^2*b^6*c^5 - 4096*a^3*b^4*c^6 + 2816*a^4*b^2*c^7 + 2048*a^5*c^8)*
d^4*x^5 + 5*(5*b^11*c^2 - 72*a*b^9*c^3 + 352*a^2*b^7*c^4 - 512*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)
*d^4*x^4 + 4*(2*b^12*c - 23*a*b^10*c^2 + 50*a^2*b^8*c^3 + 320*a^3*b^6*c^4 - 1600*a^4*b^4*c^5 + 1792*a^5*b^2*c^
6 + 512*a^6*c^7)*d^4*x^3 + (b^13 - 2*a*b^11*c - 116*a^2*b^9*c^2 + 896*a^3*b^7*c^3 - 2176*a^4*b^5*c^4 + 512*a^5
*b^3*c^5 + 3072*a^6*b*c^6)*d^4*x^2 + 2*(a*b^12 - 13*a^2*b^10*c + 48*a^3*b^8*c^2 + 32*a^4*b^6*c^3 - 512*a^5*b^4
*c^4 + 768*a^6*b^2*c^5)*d^4*x + (a^2*b^11 - 16*a^3*b^9*c + 96*a^4*b^7*c^2 - 256*a^5*b^5*c^3 + 256*a^6*b^3*c^4)
*d^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a^{2} b^{4} \sqrt {a + b x + c x^{2}} + 8 a^{2} b^{3} c x \sqrt {a + b x + c x^{2}} + 24 a^{2} b^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 32 a^{2} b c^{3} x^{3} \sqrt {a + b x + c x^{2}} + 16 a^{2} c^{4} x^{4} \sqrt {a + b x + c x^{2}} + 2 a b^{5} x \sqrt {a + b x + c x^{2}} + 18 a b^{4} c x^{2} \sqrt {a + b x + c x^{2}} + 64 a b^{3} c^{2} x^{3} \sqrt {a + b x + c x^{2}} + 112 a b^{2} c^{3} x^{4} \sqrt {a + b x + c x^{2}} + 96 a b c^{4} x^{5} \sqrt {a + b x + c x^{2}} + 32 a c^{5} x^{6} \sqrt {a + b x + c x^{2}} + b^{6} x^{2} \sqrt {a + b x + c x^{2}} + 10 b^{5} c x^{3} \sqrt {a + b x + c x^{2}} + 41 b^{4} c^{2} x^{4} \sqrt {a + b x + c x^{2}} + 88 b^{3} c^{3} x^{5} \sqrt {a + b x + c x^{2}} + 104 b^{2} c^{4} x^{6} \sqrt {a + b x + c x^{2}} + 64 b c^{5} x^{7} \sqrt {a + b x + c x^{2}} + 16 c^{6} x^{8} \sqrt {a + b x + c x^{2}}}\, dx}{d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/(a**2*b**4*sqrt(a + b*x + c*x**2) + 8*a**2*b**3*c*x*sqrt(a + b*x + c*x**2) + 24*a**2*b**2*c**2*x**2
*sqrt(a + b*x + c*x**2) + 32*a**2*b*c**3*x**3*sqrt(a + b*x + c*x**2) + 16*a**2*c**4*x**4*sqrt(a + b*x + c*x**2
) + 2*a*b**5*x*sqrt(a + b*x + c*x**2) + 18*a*b**4*c*x**2*sqrt(a + b*x + c*x**2) + 64*a*b**3*c**2*x**3*sqrt(a +
b*x + c*x**2) + 112*a*b**2*c**3*x**4*sqrt(a + b*x + c*x**2) + 96*a*b*c**4*x**5*sqrt(a + b*x + c*x**2) + 32*a*
c**5*x**6*sqrt(a + b*x + c*x**2) + b**6*x**2*sqrt(a + b*x + c*x**2) + 10*b**5*c*x**3*sqrt(a + b*x + c*x**2) +
41*b**4*c**2*x**4*sqrt(a + b*x + c*x**2) + 88*b**3*c**3*x**5*sqrt(a + b*x + c*x**2) + 104*b**2*c**4*x**6*sqrt(
a + b*x + c*x**2) + 64*b*c**5*x**7*sqrt(a + b*x + c*x**2) + 16*c**6*x**8*sqrt(a + b*x + c*x**2)), x)/d**4

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%%{1024,[7]%%%},[8,3,4,0]%%%}+%%%{%%%{-768,[6]%%%},[8,2
,4,2]%%%}+%

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Mupad [B]
time = 2.38, size = 2500, normalized size = 15.43 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2)),x)

[Out]

(128*c^6*(a + b*x + c*x^2)^(1/2))/(12*b^9*c^4*d^4 - 144*a*b^7*c^5*d^4 + 72*b^8*c^5*d^4*x + 576*a^2*b^5*c^6*d^4
- 768*a^3*b^3*c^7*d^4 - 6144*a^3*c^10*d^4*x^3 + 144*b^7*c^6*d^4*x^2 + 96*b^6*c^7*d^4*x^3 + 6912*a^2*b^3*c^8*d
^4*x^2 + 4608*a^2*b^2*c^9*d^4*x^3 - 864*a*b^6*c^6*d^4*x + 3456*a^2*b^4*c^7*d^4*x - 4608*a^3*b^2*c^8*d^4*x - 17
28*a*b^5*c^7*d^4*x^2 - 9216*a^3*b*c^9*d^4*x^2 - 1152*a*b^4*c^8*d^4*x^3) - (800*b^7*c^5)/((a + b*x + c*x^2)^(3/
2)*(480*b^12*c^5*d^4 - 9600*a*b^10*c^6*d^4 + 1920*b^11*c^6*d^4*x + 76800*a^2*b^8*c^7*d^4 - 307200*a^3*b^6*c^8*
d^4 + 614400*a^4*b^4*c^9*d^4 - 491520*a^5*b^2*c^10*d^4 - 1966080*a^5*c^12*d^4*x^2 + 1920*b^10*c^7*d^4*x^2 + 30
7200*a^2*b^6*c^9*d^4*x^2 - 1228800*a^3*b^4*c^10*d^4*x^2 + 2457600*a^4*b^2*c^11*d^4*x^2 - 38400*a*b^9*c^7*d^4*x
- 1966080*a^5*b*c^11*d^4*x + 307200*a^2*b^7*c^8*d^4*x - 1228800*a^3*b^5*c^9*d^4*x + 2457600*a^4*b^3*c^10*d^4*
x - 38400*a*b^8*c^8*d^4*x^2)) + (384*a*c^4)/((a + b*x + c*x^2)^(1/2)*(4*b^9*c^2*d^4 - 64*a*b^7*c^3*d^4 + 1024*
a^4*b*c^6*d^4 + 2048*a^4*c^7*d^4*x + 8*b^8*c^3*d^4*x + 384*a^2*b^5*c^4*d^4 - 1024*a^3*b^3*c^5*d^4 - 128*a*b^6*
c^4*d^4*x + 768*a^2*b^4*c^5*d^4*x - 2048*a^3*b^2*c^6*d^4*x)) - (224*b^2*c^3)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^9
*c^2*d^4 - 64*a*b^7*c^3*d^4 + 1024*a^4*b*c^6*d^4 + 2048*a^4*c^7*d^4*x + 8*b^8*c^3*d^4*x + 384*a^2*b^5*c^4*d^4
- 1024*a^3*b^3*c^5*d^4 - 128*a*b^6*c^4*d^4*x + 768*a^2*b^4*c^5*d^4*x - 2048*a^3*b^2*c^6*d^4*x)) + (256*c^5*x^2
)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^9*c^2*d^4 - 64*a*b^7*c^3*d^4 + 1024*a^4*b*c^6*d^4 + 2048*a^4*c^7*d^4*x + 8*b
^8*c^3*d^4*x + 384*a^2*b^5*c^4*d^4 - 1024*a^3*b^3*c^5*d^4 - 128*a*b^6*c^4*d^4*x + 768*a^2*b^4*c^5*d^4*x - 2048
*a^3*b^2*c^6*d^4*x)) + (8*b^5*c^2)/((a + b*x + c*x^2)^(3/2)*(8*b^10*c^2*d^4 - 128*a*b^8*c^3*d^4 + 32*b^9*c^3*d
^4*x + 768*a^2*b^6*c^4*d^4 - 2048*a^3*b^4*c^5*d^4 + 2048*a^4*b^2*c^6*d^4 + 8192*a^4*c^8*d^4*x^2 + 32*b^8*c^4*d
^4*x^2 + 3072*a^2*b^4*c^6*d^4*x^2 - 8192*a^3*b^2*c^7*d^4*x^2 - 512*a*b^7*c^4*d^4*x + 8192*a^4*b*c^7*d^4*x + 30
72*a^2*b^5*c^5*d^4*x - 8192*a^3*b^3*c^6*d^4*x - 512*a*b^6*c^5*d^4*x^2)) + (3584*b^4*c^6)/(3*(a + b*x + c*x^2)^
(1/2)*(32*b^11*c^5*d^4 - 640*a*b^9*c^6*d^4 - 32768*a^5*b*c^10*d^4 - 65536*a^5*c^11*d^4*x + 64*b^10*c^6*d^4*x +
5120*a^2*b^7*c^7*d^4 - 20480*a^3*b^5*c^8*d^4 + 40960*a^4*b^3*c^9*d^4 - 1280*a*b^8*c^7*d^4*x + 10240*a^2*b^6*c
^8*d^4*x - 40960*a^3*b^4*c^9*d^4*x + 81920*a^4*b^2*c^10*d^4*x)) - (64*a*b^3*c^3)/((a + b*x + c*x^2)^(3/2)*(8*b
^10*c^2*d^4 - 128*a*b^8*c^3*d^4 + 32*b^9*c^3*d^4*x + 768*a^2*b^6*c^4*d^4 - 2048*a^3*b^4*c^5*d^4 + 2048*a^4*b^2
*c^6*d^4 + 8192*a^4*c^8*d^4*x^2 + 32*b^8*c^4*d^4*x^2 + 3072*a^2*b^4*c^6*d^4*x^2 - 8192*a^3*b^2*c^7*d^4*x^2 - 5
12*a*b^7*c^4*d^4*x + 8192*a^4*b*c^7*d^4*x + 3072*a^2*b^5*c^5*d^4*x - 8192*a^3*b^3*c^6*d^4*x - 512*a*b^6*c^5*d^
4*x^2)) + (128*a^2*b*c^4)/((a + b*x + c*x^2)^(3/2)*(8*b^10*c^2*d^4 - 128*a*b^8*c^3*d^4 + 32*b^9*c^3*d^4*x + 76
8*a^2*b^6*c^4*d^4 - 2048*a^3*b^4*c^5*d^4 + 2048*a^4*b^2*c^6*d^4 + 8192*a^4*c^8*d^4*x^2 + 32*b^8*c^4*d^4*x^2 +
3072*a^2*b^4*c^6*d^4*x^2 - 8192*a^3*b^2*c^7*d^4*x^2 - 512*a*b^7*c^4*d^4*x + 8192*a^4*b*c^7*d^4*x + 3072*a^2*b^
5*c^5*d^4*x - 8192*a^3*b^3*c^6*d^4*x - 512*a*b^6*c^5*d^4*x^2)) + (256*a^2*c^5*x)/((a + b*x + c*x^2)^(3/2)*(8*b
^10*c^2*d^4 - 128*a*b^8*c^3*d^4 + 32*b^9*c^3*d^4*x + 768*a^2*b^6*c^4*d^4 - 2048*a^3*b^4*c^5*d^4 + 2048*a^4*b^2
*c^6*d^4 + 8192*a^4*c^8*d^4*x^2 + 32*b^8*c^4*d^4*x^2 + 3072*a^2*b^4*c^6*d^4*x^2 - 8192*a^3*b^2*c^7*d^4*x^2 - 5
12*a*b^7*c^4*d^4*x + 8192*a^4*b*c^7*d^4*x + 3072*a^2*b^5*c^5*d^4*x - 8192*a^3*b^3*c^6*d^4*x - 512*a*b^6*c^5*d^
4*x^2)) + (16*b^4*c^3*x)/((a + b*x + c*x^2)^(3/2)*(8*b^10*c^2*d^4 - 128*a*b^8*c^3*d^4 + 32*b^9*c^3*d^4*x + 768
*a^2*b^6*c^4*d^4 - 2048*a^3*b^4*c^5*d^4 + 2048*a^4*b^2*c^6*d^4 + 8192*a^4*c^8*d^4*x^2 + 32*b^8*c^4*d^4*x^2 + 3
072*a^2*b^4*c^6*d^4*x^2 - 8192*a^3*b^2*c^7*d^4*x^2 - 512*a*b^7*c^4*d^4*x + 8192*a^4*b*c^7*d^4*x + 3072*a^2*b^5
*c^5*d^4*x - 8192*a^3*b^3*c^6*d^4*x - 512*a*b^6*c^5*d^4*x^2)) - (14336*a*b^2*c^7)/(3*(a + b*x + c*x^2)^(1/2)*(
32*b^11*c^5*d^4 - 640*a*b^9*c^6*d^4 - 32768*a^5*b*c^10*d^4 - 65536*a^5*c^11*d^4*x + 64*b^10*c^6*d^4*x + 5120*a
^2*b^7*c^7*d^4 - 20480*a^3*b^5*c^8*d^4 + 40960*a^4*b^3*c^9*d^4 - 1280*a*b^8*c^7*d^4*x + 10240*a^2*b^6*c^8*d^4*
x - 40960*a^3*b^4*c^9*d^4*x + 81920*a^4*b^2*c^10*d^4*x)) - (57344*a*c^9*x^2)/(3*(a + b*x + c*x^2)^(1/2)*(32*b^
11*c^5*d^4 - 640*a*b^9*c^6*d^4 - 32768*a^5*b*c^10*d^4 - 65536*a^5*c^11*d^4*x + 64*b^10*c^6*d^4*x + 5120*a^2*b^
7*c^7*d^4 - 20480*a^3*b^5*c^8*d^4 + 40960*a^4*b^3*c^9*d^4 - 1280*a*b^8*c^7*d^4*x + 10240*a^2*b^6*c^8*d^4*x - 4
0960*a^3*b^4*c^9*d^4*x + 81920*a^4*b^2*c^10*d^4*x)) + (14336*b^3*c^7*x)/(3*(a + b*x + c*x^2)^(1/2)*(32*b^11*c^
5*d^4 - 640*a*b^9*c^6*d^4 - 32768*a^5*b*c^10*d^4 - 65536*a^5*c^11*d^4*x + 64*b^10*c^6*d^4*x + 5120*a^2*b^7*c^7
*d^4 - 20480*a^3*b^5*c^8*d^4 + 40960*a^4*b^3*c^9*d^4 - 1280*a*b^8*c^7*d^4*x + 10240*a^2*b^6*c^8*d^4*x - 40960*
a^3*b^4*c^9*d^4*x + 81920*a^4*b^2*c^10*d^4*x)) + (9600*a*b^5*c^6)/((a + b*x + c*x^2)^(3/2)*(480*b^12*c^5*d^4 -
9600*a*b^10*c^6*d^4 + 1920*b^11*c^6*d^4*x + 76...

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