### 3.1178 $$\int \frac{(b d+2 c d x)^9}{(a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=123 $96 c^2 d^9 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+96 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^9 (b+2 c x)^4$

[Out]

96*c^2*(b^2 - 4*a*c)*d^9*(b + 2*c*x)^2 + 48*c^2*d^9*(b + 2*c*x)^4 - (d^9*(b + 2*c*x)^8)/(2*(a + b*x + c*x^2)^2
) - (8*c*d^9*(b + 2*c*x)^6)/(a + b*x + c*x^2) + 96*c^2*(b^2 - 4*a*c)^2*d^9*Log[a + b*x + c*x^2]

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Rubi [A]  time = 0.0848583, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {686, 692, 628} $96 c^2 d^9 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+96 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^9 (b+2 c x)^4$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^9/(a + b*x + c*x^2)^3,x]

[Out]

96*c^2*(b^2 - 4*a*c)*d^9*(b + 2*c*x)^2 + 48*c^2*d^9*(b + 2*c*x)^4 - (d^9*(b + 2*c*x)^8)/(2*(a + b*x + c*x^2)^2
) - (8*c*d^9*(b + 2*c*x)^6)/(a + b*x + c*x^2) + 96*c^2*(b^2 - 4*a*c)^2*d^9*Log[a + b*x + c*x^2]

Rule 686

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

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(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] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{(b d+2 c d x)^9}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}+\left (8 c d^2\right ) \int \frac{(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+\left (96 c^2 d^4\right ) \int \frac{(b d+2 c d x)^5}{a+b x+c x^2} \, dx\\ &=48 c^2 d^9 (b+2 c x)^4-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+\left (96 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=96 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^2+48 c^2 d^9 (b+2 c x)^4-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+\left (96 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=96 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^2+48 c^2 d^9 (b+2 c x)^4-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+96 c^2 \left (b^2-4 a c\right )^2 d^9 \log \left (a+b x+c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0610094, size = 131, normalized size = 1.07 $d^9 \left (-384 c^4 x^2 \left (2 a c-b^2\right )+256 b c^3 x \left (b^2-3 a c\right )+96 c^2 \left (b^2-4 a c\right )^2 \log (a+x (b+c x))+\frac{16 c \left (4 a c-b^2\right )^3}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^4}{2 (a+x (b+c x))^2}+256 b c^5 x^3+128 c^6 x^4\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^9/(a + b*x + c*x^2)^3,x]

[Out]

d^9*(256*b*c^3*(b^2 - 3*a*c)*x - 384*c^4*(-b^2 + 2*a*c)*x^2 + 256*b*c^5*x^3 + 128*c^6*x^4 - (b^2 - 4*a*c)^4/(2
*(a + x*(b + c*x))^2) + (16*c*(-b^2 + 4*a*c)^3)/(a + x*(b + c*x)) + 96*c^2*(b^2 - 4*a*c)^2*Log[a + x*(b + c*x)
])

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Maple [B]  time = 0.052, size = 465, normalized size = 3.8 \begin{align*} 128\,{d}^{9}{c}^{6}{x}^{4}+256\,{d}^{9}b{c}^{5}{x}^{3}-768\,{d}^{9}{x}^{2}a{c}^{5}+384\,{d}^{9}{x}^{2}{b}^{2}{c}^{4}-768\,{d}^{9}ab{c}^{4}x+256\,{d}^{9}{b}^{3}{c}^{3}x+1024\,{\frac{{d}^{9}{x}^{2}{a}^{3}{c}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-768\,{\frac{{d}^{9}{x}^{2}{a}^{2}{b}^{2}{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+192\,{\frac{{d}^{9}{x}^{2}a{b}^{4}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-16\,{\frac{{d}^{9}{x}^{2}{b}^{6}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+1024\,{\frac{{d}^{9}x{a}^{3}b{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-768\,{\frac{{d}^{9}x{a}^{2}{b}^{3}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+192\,{\frac{{d}^{9}xa{b}^{5}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-16\,{\frac{{d}^{9}x{b}^{7}c}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+896\,{\frac{{d}^{9}{a}^{4}{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-640\,{\frac{{d}^{9}{a}^{3}{b}^{2}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+144\,{\frac{{d}^{9}{a}^{2}{b}^{4}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-8\,{\frac{{d}^{9}a{b}^{6}c}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{9}{b}^{8}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}+1536\,{d}^{9}\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}{c}^{4}-768\,{d}^{9}\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{2}{c}^{3}+96\,{d}^{9}\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}{c}^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^9/(c*x^2+b*x+a)^3,x)

[Out]

128*d^9*c^6*x^4+256*d^9*b*c^5*x^3-768*d^9*x^2*a*c^5+384*d^9*x^2*b^2*c^4-768*d^9*a*b*c^4*x+256*d^9*b^3*c^3*x+10
24*d^9/(c*x^2+b*x+a)^2*x^2*a^3*c^5-768*d^9/(c*x^2+b*x+a)^2*x^2*a^2*b^2*c^4+192*d^9/(c*x^2+b*x+a)^2*x^2*a*b^4*c
^3-16*d^9/(c*x^2+b*x+a)^2*x^2*b^6*c^2+1024*d^9/(c*x^2+b*x+a)^2*x*a^3*b*c^4-768*d^9/(c*x^2+b*x+a)^2*x*a^2*b^3*c
^3+192*d^9/(c*x^2+b*x+a)^2*x*a*b^5*c^2-16*d^9/(c*x^2+b*x+a)^2*x*b^7*c+896*d^9/(c*x^2+b*x+a)^2*a^4*c^4-640*d^9/
(c*x^2+b*x+a)^2*a^3*b^2*c^3+144*d^9/(c*x^2+b*x+a)^2*a^2*b^4*c^2-8*d^9/(c*x^2+b*x+a)^2*a*b^6*c-1/2*d^9/(c*x^2+b
*x+a)^2*b^8+1536*d^9*ln(c*x^2+b*x+a)*a^2*c^4-768*d^9*ln(c*x^2+b*x+a)*a*b^2*c^3+96*d^9*ln(c*x^2+b*x+a)*b^4*c^2

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Maxima [B]  time = 1.17008, size = 375, normalized size = 3.05 \begin{align*} 128 \, c^{6} d^{9} x^{4} + 256 \, b c^{5} d^{9} x^{3} + 384 \,{\left (b^{2} c^{4} - 2 \, a c^{5}\right )} d^{9} x^{2} + 256 \,{\left (b^{3} c^{3} - 3 \, a b c^{4}\right )} d^{9} x + 96 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{9} \log \left (c x^{2} + b x + a\right ) - \frac{32 \,{\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{9} x^{2} + 32 \,{\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d^{9} x +{\left (b^{8} + 16 \, a b^{6} c - 288 \, a^{2} b^{4} c^{2} + 1280 \, a^{3} b^{2} c^{3} - 1792 \, a^{4} c^{4}\right )} d^{9}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^9/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

128*c^6*d^9*x^4 + 256*b*c^5*d^9*x^3 + 384*(b^2*c^4 - 2*a*c^5)*d^9*x^2 + 256*(b^3*c^3 - 3*a*b*c^4)*d^9*x + 96*(
b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^9*log(c*x^2 + b*x + a) - 1/2*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^
4 - 64*a^3*c^5)*d^9*x^2 + 32*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^9*x + (b^8 + 16*a*b^6*c
- 288*a^2*b^4*c^2 + 1280*a^3*b^2*c^3 - 1792*a^4*c^4)*d^9)/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 +
a^2)

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Fricas [B]  time = 1.98246, size = 1035, normalized size = 8.41 \begin{align*} \frac{256 \, c^{8} d^{9} x^{8} + 1024 \, b c^{7} d^{9} x^{7} + 1024 \,{\left (2 \, b^{2} c^{6} - a c^{7}\right )} d^{9} x^{6} + 512 \,{\left (5 \, b^{3} c^{5} - 6 \, a b c^{6}\right )} d^{9} x^{5} + 256 \,{\left (7 \, b^{4} c^{4} - 8 \, a b^{2} c^{5} - 11 \, a^{2} c^{6}\right )} d^{9} x^{4} + 512 \,{\left (b^{5} c^{3} + 2 \, a b^{3} c^{4} - 11 \, a^{2} b c^{5}\right )} d^{9} x^{3} - 32 \,{\left (b^{6} c^{2} - 44 \, a b^{4} c^{3} + 120 \, a^{2} b^{2} c^{4} - 16 \, a^{3} c^{5}\right )} d^{9} x^{2} - 32 \,{\left (b^{7} c - 12 \, a b^{5} c^{2} + 32 \, a^{2} b^{3} c^{3} - 16 \, a^{3} b c^{4}\right )} d^{9} x -{\left (b^{8} + 16 \, a b^{6} c - 288 \, a^{2} b^{4} c^{2} + 1280 \, a^{3} b^{2} c^{3} - 1792 \, a^{4} c^{4}\right )} d^{9} + 192 \,{\left ({\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{9} x^{4} + 2 \,{\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{9} x^{3} +{\left (b^{6} c^{2} - 6 \, a b^{4} c^{3} + 32 \, a^{3} c^{5}\right )} d^{9} x^{2} + 2 \,{\left (a b^{5} c^{2} - 8 \, a^{2} b^{3} c^{3} + 16 \, a^{3} b c^{4}\right )} d^{9} x +{\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}\right )} d^{9}\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^9/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/2*(256*c^8*d^9*x^8 + 1024*b*c^7*d^9*x^7 + 1024*(2*b^2*c^6 - a*c^7)*d^9*x^6 + 512*(5*b^3*c^5 - 6*a*b*c^6)*d^9
*x^5 + 256*(7*b^4*c^4 - 8*a*b^2*c^5 - 11*a^2*c^6)*d^9*x^4 + 512*(b^5*c^3 + 2*a*b^3*c^4 - 11*a^2*b*c^5)*d^9*x^3
- 32*(b^6*c^2 - 44*a*b^4*c^3 + 120*a^2*b^2*c^4 - 16*a^3*c^5)*d^9*x^2 - 32*(b^7*c - 12*a*b^5*c^2 + 32*a^2*b^3*
c^3 - 16*a^3*b*c^4)*d^9*x - (b^8 + 16*a*b^6*c - 288*a^2*b^4*c^2 + 1280*a^3*b^2*c^3 - 1792*a^4*c^4)*d^9 + 192*(
(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^9*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^9*x^3 + (b^6*c^2 - 6
*a*b^4*c^3 + 32*a^3*c^5)*d^9*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d^9*x + (a^2*b^4*c^2 - 8*a^3*b
^2*c^3 + 16*a^4*c^4)*d^9)*log(c*x^2 + b*x + a))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)

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Sympy [B]  time = 31.3321, size = 320, normalized size = 2.6 \begin{align*} 256 b c^{5} d^{9} x^{3} + 128 c^{6} d^{9} x^{4} + 96 c^{2} d^{9} \left (4 a c - b^{2}\right )^{2} \log{\left (a + b x + c x^{2} \right )} + x^{2} \left (- 768 a c^{5} d^{9} + 384 b^{2} c^{4} d^{9}\right ) + x \left (- 768 a b c^{4} d^{9} + 256 b^{3} c^{3} d^{9}\right ) + \frac{1792 a^{4} c^{4} d^{9} - 1280 a^{3} b^{2} c^{3} d^{9} + 288 a^{2} b^{4} c^{2} d^{9} - 16 a b^{6} c d^{9} - b^{8} d^{9} + x^{2} \left (2048 a^{3} c^{5} d^{9} - 1536 a^{2} b^{2} c^{4} d^{9} + 384 a b^{4} c^{3} d^{9} - 32 b^{6} c^{2} d^{9}\right ) + x \left (2048 a^{3} b c^{4} d^{9} - 1536 a^{2} b^{3} c^{3} d^{9} + 384 a b^{5} c^{2} d^{9} - 32 b^{7} c d^{9}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**9/(c*x**2+b*x+a)**3,x)

[Out]

256*b*c**5*d**9*x**3 + 128*c**6*d**9*x**4 + 96*c**2*d**9*(4*a*c - b**2)**2*log(a + b*x + c*x**2) + x**2*(-768*
a*c**5*d**9 + 384*b**2*c**4*d**9) + x*(-768*a*b*c**4*d**9 + 256*b**3*c**3*d**9) + (1792*a**4*c**4*d**9 - 1280*
a**3*b**2*c**3*d**9 + 288*a**2*b**4*c**2*d**9 - 16*a*b**6*c*d**9 - b**8*d**9 + x**2*(2048*a**3*c**5*d**9 - 153
6*a**2*b**2*c**4*d**9 + 384*a*b**4*c**3*d**9 - 32*b**6*c**2*d**9) + x*(2048*a**3*b*c**4*d**9 - 1536*a**2*b**3*
c**3*d**9 + 384*a*b**5*c**2*d**9 - 32*b**7*c*d**9))/(2*a**2 + 4*a*b*x + 4*b*c*x**3 + 2*c**2*x**4 + x**2*(4*a*c
+ 2*b**2))

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Giac [B]  time = 1.27181, size = 404, normalized size = 3.28 \begin{align*} 96 \,{\left (b^{4} c^{2} d^{9} - 8 \, a b^{2} c^{3} d^{9} + 16 \, a^{2} c^{4} d^{9}\right )} \log \left (c x^{2} + b x + a\right ) - \frac{b^{8} d^{9} + 16 \, a b^{6} c d^{9} - 288 \, a^{2} b^{4} c^{2} d^{9} + 1280 \, a^{3} b^{2} c^{3} d^{9} - 1792 \, a^{4} c^{4} d^{9} + 32 \,{\left (b^{6} c^{2} d^{9} - 12 \, a b^{4} c^{3} d^{9} + 48 \, a^{2} b^{2} c^{4} d^{9} - 64 \, a^{3} c^{5} d^{9}\right )} x^{2} + 32 \,{\left (b^{7} c d^{9} - 12 \, a b^{5} c^{2} d^{9} + 48 \, a^{2} b^{3} c^{3} d^{9} - 64 \, a^{3} b c^{4} d^{9}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} + \frac{128 \,{\left (c^{18} d^{9} x^{4} + 2 \, b c^{17} d^{9} x^{3} + 3 \, b^{2} c^{16} d^{9} x^{2} - 6 \, a c^{17} d^{9} x^{2} + 2 \, b^{3} c^{15} d^{9} x - 6 \, a b c^{16} d^{9} x\right )}}{c^{12}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^9/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

96*(b^4*c^2*d^9 - 8*a*b^2*c^3*d^9 + 16*a^2*c^4*d^9)*log(c*x^2 + b*x + a) - 1/2*(b^8*d^9 + 16*a*b^6*c*d^9 - 288
*a^2*b^4*c^2*d^9 + 1280*a^3*b^2*c^3*d^9 - 1792*a^4*c^4*d^9 + 32*(b^6*c^2*d^9 - 12*a*b^4*c^3*d^9 + 48*a^2*b^2*c
^4*d^9 - 64*a^3*c^5*d^9)*x^2 + 32*(b^7*c*d^9 - 12*a*b^5*c^2*d^9 + 48*a^2*b^3*c^3*d^9 - 64*a^3*b*c^4*d^9)*x)/(c
*x^2 + b*x + a)^2 + 128*(c^18*d^9*x^4 + 2*b*c^17*d^9*x^3 + 3*b^2*c^16*d^9*x^2 - 6*a*c^17*d^9*x^2 + 2*b^3*c^15*
d^9*x - 6*a*b*c^16*d^9*x)/c^12