3.1352 \(\int \frac{(a+b x)^6}{(c+d x)^3} \, dx\)

Optimal. Leaf size=158 \[ -\frac{2 b^5 (c+d x)^3 (b c-a d)}{d^7}+\frac{15 b^4 (c+d x)^2 (b c-a d)^2}{2 d^7}-\frac{20 b^3 x (b c-a d)^3}{d^6}+\frac{15 b^2 (b c-a d)^4 \log (c+d x)}{d^7}+\frac{6 b (b c-a d)^5}{d^7 (c+d x)}-\frac{(b c-a d)^6}{2 d^7 (c+d x)^2}+\frac{b^6 (c+d x)^4}{4 d^7} \]

[Out]

(-20*b^3*(b*c - a*d)^3*x)/d^6 - (b*c - a*d)^6/(2*d^7*(c + d*x)^2) + (6*b*(b*c -
a*d)^5)/(d^7*(c + d*x)) + (15*b^4*(b*c - a*d)^2*(c + d*x)^2)/(2*d^7) - (2*b^5*(b
*c - a*d)*(c + d*x)^3)/d^7 + (b^6*(c + d*x)^4)/(4*d^7) + (15*b^2*(b*c - a*d)^4*L
og[c + d*x])/d^7

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Rubi [A]  time = 0.38961, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{2 b^5 (c+d x)^3 (b c-a d)}{d^7}+\frac{15 b^4 (c+d x)^2 (b c-a d)^2}{2 d^7}-\frac{20 b^3 x (b c-a d)^3}{d^6}+\frac{15 b^2 (b c-a d)^4 \log (c+d x)}{d^7}+\frac{6 b (b c-a d)^5}{d^7 (c+d x)}-\frac{(b c-a d)^6}{2 d^7 (c+d x)^2}+\frac{b^6 (c+d x)^4}{4 d^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^6/(c + d*x)^3,x]

[Out]

(-20*b^3*(b*c - a*d)^3*x)/d^6 - (b*c - a*d)^6/(2*d^7*(c + d*x)^2) + (6*b*(b*c -
a*d)^5)/(d^7*(c + d*x)) + (15*b^4*(b*c - a*d)^2*(c + d*x)^2)/(2*d^7) - (2*b^5*(b
*c - a*d)*(c + d*x)^3)/d^7 + (b^6*(c + d*x)^4)/(4*d^7) + (15*b^2*(b*c - a*d)^4*L
og[c + d*x])/d^7

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Rubi in Sympy [A]  time = 56.8351, size = 144, normalized size = 0.91 \[ \frac{b^{6} \left (c + d x\right )^{4}}{4 d^{7}} + \frac{2 b^{5} \left (c + d x\right )^{3} \left (a d - b c\right )}{d^{7}} + \frac{15 b^{4} \left (c + d x\right )^{2} \left (a d - b c\right )^{2}}{2 d^{7}} + \frac{20 b^{3} x \left (a d - b c\right )^{3}}{d^{6}} + \frac{15 b^{2} \left (a d - b c\right )^{4} \log{\left (c + d x \right )}}{d^{7}} - \frac{6 b \left (a d - b c\right )^{5}}{d^{7} \left (c + d x\right )} - \frac{\left (a d - b c\right )^{6}}{2 d^{7} \left (c + d x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**6/(d*x+c)**3,x)

[Out]

b**6*(c + d*x)**4/(4*d**7) + 2*b**5*(c + d*x)**3*(a*d - b*c)/d**7 + 15*b**4*(c +
d*x)**2*(a*d - b*c)**2/(2*d**7) + 20*b**3*x*(a*d - b*c)**3/d**6 + 15*b**2*(a*d
- b*c)**4*log(c + d*x)/d**7 - 6*b*(a*d - b*c)**5/(d**7*(c + d*x)) - (a*d - b*c)*
*6/(2*d**7*(c + d*x)**2)

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Mathematica [A]  time = 0.187322, size = 303, normalized size = 1.92 \[ \frac{-2 a^6 d^6-12 a^5 b d^5 (c+2 d x)+30 a^4 b^2 c d^4 (3 c+4 d x)+40 a^3 b^3 d^3 \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )+30 a^2 b^4 d^2 \left (7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right )+4 a b^5 d \left (-27 c^5+6 c^4 d x+63 c^3 d^2 x^2+20 c^2 d^3 x^3-5 c d^4 x^4+2 d^5 x^5\right )+60 b^2 (c+d x)^2 (b c-a d)^4 \log (c+d x)+b^6 \left (22 c^6-16 c^5 d x-68 c^4 d^2 x^2-20 c^3 d^3 x^3+5 c^2 d^4 x^4-2 c d^5 x^5+d^6 x^6\right )}{4 d^7 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^6/(c + d*x)^3,x]

[Out]

(-2*a^6*d^6 - 12*a^5*b*d^5*(c + 2*d*x) + 30*a^4*b^2*c*d^4*(3*c + 4*d*x) + 40*a^3
*b^3*d^3*(-5*c^3 - 4*c^2*d*x + 4*c*d^2*x^2 + 2*d^3*x^3) + 30*a^2*b^4*d^2*(7*c^4
+ 2*c^3*d*x - 11*c^2*d^2*x^2 - 4*c*d^3*x^3 + d^4*x^4) + 4*a*b^5*d*(-27*c^5 + 6*c
^4*d*x + 63*c^3*d^2*x^2 + 20*c^2*d^3*x^3 - 5*c*d^4*x^4 + 2*d^5*x^5) + b^6*(22*c^
6 - 16*c^5*d*x - 68*c^4*d^2*x^2 - 20*c^3*d^3*x^3 + 5*c^2*d^4*x^4 - 2*c*d^5*x^5 +
d^6*x^6) + 60*b^2*(b*c - a*d)^4*(c + d*x)^2*Log[c + d*x])/(4*d^7*(c + d*x)^2)

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Maple [B]  time = 0.014, size = 464, normalized size = 2.9 \[ 2\,{\frac{{b}^{5}{x}^{3}a}{{d}^{3}}}-{\frac{{b}^{6}{x}^{3}c}{{d}^{4}}}+{\frac{15\,{b}^{4}{x}^{2}{a}^{2}}{2\,{d}^{3}}}+3\,{\frac{{b}^{6}{x}^{2}{c}^{2}}{{d}^{5}}}+20\,{\frac{{a}^{3}{b}^{3}x}{{d}^{3}}}-10\,{\frac{{b}^{6}{c}^{3}x}{{d}^{6}}}+15\,{\frac{{b}^{2}\ln \left ( dx+c \right ){a}^{4}}{{d}^{3}}}+15\,{\frac{{b}^{6}\ln \left ( dx+c \right ){c}^{4}}{{d}^{7}}}-{\frac{{b}^{6}{c}^{6}}{2\,{d}^{7} \left ( dx+c \right ) ^{2}}}-6\,{\frac{{a}^{5}b}{{d}^{2} \left ( dx+c \right ) }}+6\,{\frac{{b}^{6}{c}^{5}}{{d}^{7} \left ( dx+c \right ) }}+90\,{\frac{{b}^{4}\ln \left ( dx+c \right ){a}^{2}{c}^{2}}{{d}^{5}}}-60\,{\frac{{b}^{5}\ln \left ( dx+c \right ) a{c}^{3}}{{d}^{6}}}+3\,{\frac{{a}^{5}bc}{{d}^{2} \left ( dx+c \right ) ^{2}}}+36\,{\frac{a{b}^{5}{c}^{2}x}{{d}^{5}}}-9\,{\frac{{b}^{5}{x}^{2}ac}{{d}^{4}}}-45\,{\frac{{a}^{2}{b}^{4}cx}{{d}^{4}}}-{\frac{15\,{a}^{4}{b}^{2}{c}^{2}}{2\,{d}^{3} \left ( dx+c \right ) ^{2}}}+10\,{\frac{{a}^{3}{b}^{3}{c}^{3}}{{d}^{4} \left ( dx+c \right ) ^{2}}}-{\frac{15\,{a}^{2}{b}^{4}{c}^{4}}{2\,{d}^{5} \left ( dx+c \right ) ^{2}}}+3\,{\frac{a{b}^{5}{c}^{5}}{{d}^{6} \left ( dx+c \right ) ^{2}}}+60\,{\frac{{a}^{2}{b}^{4}{c}^{3}}{{d}^{5} \left ( dx+c \right ) }}-30\,{\frac{a{b}^{5}{c}^{4}}{{d}^{6} \left ( dx+c \right ) }}-60\,{\frac{{b}^{3}\ln \left ( dx+c \right ){a}^{3}c}{{d}^{4}}}-{\frac{{a}^{6}}{2\,d \left ( dx+c \right ) ^{2}}}+{\frac{{b}^{6}{x}^{4}}{4\,{d}^{3}}}+30\,{\frac{{a}^{4}{b}^{2}c}{{d}^{3} \left ( dx+c \right ) }}-60\,{\frac{{a}^{3}{b}^{3}{c}^{2}}{{d}^{4} \left ( dx+c \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6/(d*x+c)^3,x)

[Out]

2*b^5/d^3*x^3*a-b^6/d^4*x^3*c+15/2*b^4/d^3*x^2*a^2+3*b^6/d^5*x^2*c^2+20*b^3/d^3*
a^3*x-10*b^6/d^6*c^3*x+15*b^2/d^3*ln(d*x+c)*a^4+15*b^6/d^7*ln(d*x+c)*c^4-1/2/d^7
/(d*x+c)^2*b^6*c^6-6*b/d^2/(d*x+c)*a^5+6*b^6/d^7/(d*x+c)*c^5+90*b^4/d^5*ln(d*x+c
)*a^2*c^2-60*b^5/d^6*ln(d*x+c)*a*c^3+3/d^2/(d*x+c)^2*a^5*b*c+36*b^5/d^5*a*c^2*x-
9*b^5/d^4*x^2*a*c-45*b^4/d^4*a^2*c*x-15/2/d^3/(d*x+c)^2*a^4*b^2*c^2+10/d^4/(d*x+
c)^2*a^3*b^3*c^3-15/2/d^5/(d*x+c)^2*a^2*b^4*c^4+3/d^6/(d*x+c)^2*a*b^5*c^5+60*b^4
/d^5/(d*x+c)*a^2*c^3-30*b^5/d^6/(d*x+c)*a*c^4-60*b^3/d^4*ln(d*x+c)*a^3*c-1/2/d/(
d*x+c)^2*a^6+1/4*b^6/d^3*x^4+30*b^2/d^3/(d*x+c)*a^4*c-60*b^3/d^4/(d*x+c)*a^3*c^2

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Maxima [A]  time = 1.36438, size = 491, normalized size = 3.11 \[ \frac{11 \, b^{6} c^{6} - 54 \, a b^{5} c^{5} d + 105 \, a^{2} b^{4} c^{4} d^{2} - 100 \, a^{3} b^{3} c^{3} d^{3} + 45 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} - a^{6} d^{6} + 12 \,{\left (b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 10 \, a^{2} b^{4} c^{3} d^{3} - 10 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - a^{5} b d^{6}\right )} x}{2 \,{\left (d^{9} x^{2} + 2 \, c d^{8} x + c^{2} d^{7}\right )}} + \frac{b^{6} d^{3} x^{4} - 4 \,{\left (b^{6} c d^{2} - 2 \, a b^{5} d^{3}\right )} x^{3} + 6 \,{\left (2 \, b^{6} c^{2} d - 6 \, a b^{5} c d^{2} + 5 \, a^{2} b^{4} d^{3}\right )} x^{2} - 4 \,{\left (10 \, b^{6} c^{3} - 36 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} - 20 \, a^{3} b^{3} d^{3}\right )} x}{4 \, d^{6}} + \frac{15 \,{\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} \log \left (d x + c\right )}{d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^6/(d*x + c)^3,x, algorithm="maxima")

[Out]

1/2*(11*b^6*c^6 - 54*a*b^5*c^5*d + 105*a^2*b^4*c^4*d^2 - 100*a^3*b^3*c^3*d^3 + 4
5*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 - a^6*d^6 + 12*(b^6*c^5*d - 5*a*b^5*c^4*d^2 +
10*a^2*b^4*c^3*d^3 - 10*a^3*b^3*c^2*d^4 + 5*a^4*b^2*c*d^5 - a^5*b*d^6)*x)/(d^9*x
^2 + 2*c*d^8*x + c^2*d^7) + 1/4*(b^6*d^3*x^4 - 4*(b^6*c*d^2 - 2*a*b^5*d^3)*x^3 +
6*(2*b^6*c^2*d - 6*a*b^5*c*d^2 + 5*a^2*b^4*d^3)*x^2 - 4*(10*b^6*c^3 - 36*a*b^5*
c^2*d + 45*a^2*b^4*c*d^2 - 20*a^3*b^3*d^3)*x)/d^6 + 15*(b^6*c^4 - 4*a*b^5*c^3*d
+ 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*log(d*x + c)/d^7

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Fricas [A]  time = 0.203701, size = 740, normalized size = 4.68 \[ \frac{b^{6} d^{6} x^{6} + 22 \, b^{6} c^{6} - 108 \, a b^{5} c^{5} d + 210 \, a^{2} b^{4} c^{4} d^{2} - 200 \, a^{3} b^{3} c^{3} d^{3} + 90 \, a^{4} b^{2} c^{2} d^{4} - 12 \, a^{5} b c d^{5} - 2 \, a^{6} d^{6} - 2 \,{\left (b^{6} c d^{5} - 4 \, a b^{5} d^{6}\right )} x^{5} + 5 \,{\left (b^{6} c^{2} d^{4} - 4 \, a b^{5} c d^{5} + 6 \, a^{2} b^{4} d^{6}\right )} x^{4} - 20 \,{\left (b^{6} c^{3} d^{3} - 4 \, a b^{5} c^{2} d^{4} + 6 \, a^{2} b^{4} c d^{5} - 4 \, a^{3} b^{3} d^{6}\right )} x^{3} - 2 \,{\left (34 \, b^{6} c^{4} d^{2} - 126 \, a b^{5} c^{3} d^{3} + 165 \, a^{2} b^{4} c^{2} d^{4} - 80 \, a^{3} b^{3} c d^{5}\right )} x^{2} - 4 \,{\left (4 \, b^{6} c^{5} d - 6 \, a b^{5} c^{4} d^{2} - 15 \, a^{2} b^{4} c^{3} d^{3} + 40 \, a^{3} b^{3} c^{2} d^{4} - 30 \, a^{4} b^{2} c d^{5} + 6 \, a^{5} b d^{6}\right )} x + 60 \,{\left (b^{6} c^{6} - 4 \, a b^{5} c^{5} d + 6 \, a^{2} b^{4} c^{4} d^{2} - 4 \, a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 2 \,{\left (b^{6} c^{5} d - 4 \, a b^{5} c^{4} d^{2} + 6 \, a^{2} b^{4} c^{3} d^{3} - 4 \, a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5}\right )} x\right )} \log \left (d x + c\right )}{4 \,{\left (d^{9} x^{2} + 2 \, c d^{8} x + c^{2} d^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^6/(d*x + c)^3,x, algorithm="fricas")

[Out]

1/4*(b^6*d^6*x^6 + 22*b^6*c^6 - 108*a*b^5*c^5*d + 210*a^2*b^4*c^4*d^2 - 200*a^3*
b^3*c^3*d^3 + 90*a^4*b^2*c^2*d^4 - 12*a^5*b*c*d^5 - 2*a^6*d^6 - 2*(b^6*c*d^5 - 4
*a*b^5*d^6)*x^5 + 5*(b^6*c^2*d^4 - 4*a*b^5*c*d^5 + 6*a^2*b^4*d^6)*x^4 - 20*(b^6*
c^3*d^3 - 4*a*b^5*c^2*d^4 + 6*a^2*b^4*c*d^5 - 4*a^3*b^3*d^6)*x^3 - 2*(34*b^6*c^4
*d^2 - 126*a*b^5*c^3*d^3 + 165*a^2*b^4*c^2*d^4 - 80*a^3*b^3*c*d^5)*x^2 - 4*(4*b^
6*c^5*d - 6*a*b^5*c^4*d^2 - 15*a^2*b^4*c^3*d^3 + 40*a^3*b^3*c^2*d^4 - 30*a^4*b^2
*c*d^5 + 6*a^5*b*d^6)*x + 60*(b^6*c^6 - 4*a*b^5*c^5*d + 6*a^2*b^4*c^4*d^2 - 4*a^
3*b^3*c^3*d^3 + a^4*b^2*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2
*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^2 + 2*(b^6*c^5*d - 4*a*b^5*c^4*d^2 + 6*a
^2*b^4*c^3*d^3 - 4*a^3*b^3*c^2*d^4 + a^4*b^2*c*d^5)*x)*log(d*x + c))/(d^9*x^2 +
2*c*d^8*x + c^2*d^7)

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Sympy [A]  time = 4.62533, size = 335, normalized size = 2.12 \[ \frac{b^{6} x^{4}}{4 d^{3}} + \frac{15 b^{2} \left (a d - b c\right )^{4} \log{\left (c + d x \right )}}{d^{7}} - \frac{a^{6} d^{6} + 6 a^{5} b c d^{5} - 45 a^{4} b^{2} c^{2} d^{4} + 100 a^{3} b^{3} c^{3} d^{3} - 105 a^{2} b^{4} c^{4} d^{2} + 54 a b^{5} c^{5} d - 11 b^{6} c^{6} + x \left (12 a^{5} b d^{6} - 60 a^{4} b^{2} c d^{5} + 120 a^{3} b^{3} c^{2} d^{4} - 120 a^{2} b^{4} c^{3} d^{3} + 60 a b^{5} c^{4} d^{2} - 12 b^{6} c^{5} d\right )}{2 c^{2} d^{7} + 4 c d^{8} x + 2 d^{9} x^{2}} + \frac{x^{3} \left (2 a b^{5} d - b^{6} c\right )}{d^{4}} + \frac{x^{2} \left (15 a^{2} b^{4} d^{2} - 18 a b^{5} c d + 6 b^{6} c^{2}\right )}{2 d^{5}} + \frac{x \left (20 a^{3} b^{3} d^{3} - 45 a^{2} b^{4} c d^{2} + 36 a b^{5} c^{2} d - 10 b^{6} c^{3}\right )}{d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**6/(d*x+c)**3,x)

[Out]

b**6*x**4/(4*d**3) + 15*b**2*(a*d - b*c)**4*log(c + d*x)/d**7 - (a**6*d**6 + 6*a
**5*b*c*d**5 - 45*a**4*b**2*c**2*d**4 + 100*a**3*b**3*c**3*d**3 - 105*a**2*b**4*
c**4*d**2 + 54*a*b**5*c**5*d - 11*b**6*c**6 + x*(12*a**5*b*d**6 - 60*a**4*b**2*c
*d**5 + 120*a**3*b**3*c**2*d**4 - 120*a**2*b**4*c**3*d**3 + 60*a*b**5*c**4*d**2
- 12*b**6*c**5*d))/(2*c**2*d**7 + 4*c*d**8*x + 2*d**9*x**2) + x**3*(2*a*b**5*d -
b**6*c)/d**4 + x**2*(15*a**2*b**4*d**2 - 18*a*b**5*c*d + 6*b**6*c**2)/(2*d**5)
+ x*(20*a**3*b**3*d**3 - 45*a**2*b**4*c*d**2 + 36*a*b**5*c**2*d - 10*b**6*c**3)/
d**6

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GIAC/XCAS [A]  time = 0.218269, size = 489, normalized size = 3.09 \[ \frac{15 \,{\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{7}} + \frac{11 \, b^{6} c^{6} - 54 \, a b^{5} c^{5} d + 105 \, a^{2} b^{4} c^{4} d^{2} - 100 \, a^{3} b^{3} c^{3} d^{3} + 45 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} - a^{6} d^{6} + 12 \,{\left (b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 10 \, a^{2} b^{4} c^{3} d^{3} - 10 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - a^{5} b d^{6}\right )} x}{2 \,{\left (d x + c\right )}^{2} d^{7}} + \frac{b^{6} d^{9} x^{4} - 4 \, b^{6} c d^{8} x^{3} + 8 \, a b^{5} d^{9} x^{3} + 12 \, b^{6} c^{2} d^{7} x^{2} - 36 \, a b^{5} c d^{8} x^{2} + 30 \, a^{2} b^{4} d^{9} x^{2} - 40 \, b^{6} c^{3} d^{6} x + 144 \, a b^{5} c^{2} d^{7} x - 180 \, a^{2} b^{4} c d^{8} x + 80 \, a^{3} b^{3} d^{9} x}{4 \, d^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^6/(d*x + c)^3,x, algorithm="giac")

[Out]

15*(b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)
*ln(abs(d*x + c))/d^7 + 1/2*(11*b^6*c^6 - 54*a*b^5*c^5*d + 105*a^2*b^4*c^4*d^2 -
100*a^3*b^3*c^3*d^3 + 45*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 - a^6*d^6 + 12*(b^6*c^
5*d - 5*a*b^5*c^4*d^2 + 10*a^2*b^4*c^3*d^3 - 10*a^3*b^3*c^2*d^4 + 5*a^4*b^2*c*d^
5 - a^5*b*d^6)*x)/((d*x + c)^2*d^7) + 1/4*(b^6*d^9*x^4 - 4*b^6*c*d^8*x^3 + 8*a*b
^5*d^9*x^3 + 12*b^6*c^2*d^7*x^2 - 36*a*b^5*c*d^8*x^2 + 30*a^2*b^4*d^9*x^2 - 40*b
^6*c^3*d^6*x + 144*a*b^5*c^2*d^7*x - 180*a^2*b^4*c*d^8*x + 80*a^3*b^3*d^9*x)/d^1
2