3.294 \(\int \frac {x^3}{(a+b x)^2 (c+d x)^3} \, dx\)

Optimal. Leaf size=129 \[ \frac {a^3}{b (a+b x) (b c-a d)^3}+\frac {3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac {3 a^2 c \log (c+d x)}{(b c-a d)^4}+\frac {c^3}{2 d^2 (c+d x)^2 (b c-a d)^2}-\frac {c^2 (b c-3 a d)}{d^2 (c+d x) (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {88} \[ \frac {a^3}{b (a+b x) (b c-a d)^3}+\frac {3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac {3 a^2 c \log (c+d x)}{(b c-a d)^4}-\frac {c^2 (b c-3 a d)}{d^2 (c+d x) (b c-a d)^3}+\frac {c^3}{2 d^2 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((a + b*x)^2*(c + d*x)^3),x]

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 130, normalized size = 1.01 \[ \frac {a^3}{b (a+b x) (b c-a d)^3}+\frac {3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac {3 a^2 c \log (c+d x)}{(b c-a d)^4}+\frac {c^3}{2 d^2 (c+d x)^2 (a d-b c)^2}+\frac {b c^3-3 a c^2 d}{d^2 (c+d x) (a d-b c)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.91, size = 621, normalized size = 4.81 \[ -\frac {a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} + 2 \, a^{4} c^{2} d^{3} + 2 \, {\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} - a^{3} b c d^{4} + a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3} + 4 \, a^{4} c d^{4}\right )} x - 6 \, {\left (a^{2} b^{2} c d^{4} x^{3} + a^{3} b c^{3} d^{2} + {\left (2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4}\right )} x^{2} + {\left (a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (a^{2} b^{2} c d^{4} x^{3} + a^{3} b c^{3} d^{2} + {\left (2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4}\right )} x^{2} + {\left (a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{5} c^{6} d^{2} - 4 \, a^{2} b^{4} c^{5} d^{3} + 6 \, a^{3} b^{3} c^{4} d^{4} - 4 \, a^{4} b^{2} c^{3} d^{5} + a^{5} b c^{2} d^{6} + {\left (b^{6} c^{4} d^{4} - 4 \, a b^{5} c^{3} d^{5} + 6 \, a^{2} b^{4} c^{2} d^{6} - 4 \, a^{3} b^{3} c d^{7} + a^{4} b^{2} d^{8}\right )} x^{3} + {\left (2 \, b^{6} c^{5} d^{3} - 7 \, a b^{5} c^{4} d^{4} + 8 \, a^{2} b^{4} c^{3} d^{5} - 2 \, a^{3} b^{3} c^{2} d^{6} - 2 \, a^{4} b^{2} c d^{7} + a^{5} b d^{8}\right )} x^{2} + {\left (b^{6} c^{6} d^{2} - 2 \, a b^{5} c^{5} d^{3} - 2 \, a^{2} b^{4} c^{4} d^{4} + 8 \, a^{3} b^{3} c^{3} d^{5} - 7 \, a^{4} b^{2} c^{2} d^{6} + 2 \, a^{5} b c d^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*b^3*c^5 - 6*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 + 2*a^4*c^2*d^3 + 2*(b^4*c^4*d - 4*a*b^3*c^3*d^2 + 3*a^2*b
^2*c^2*d^3 - a^3*b*c*d^4 + a^4*d^5)*x^2 + (b^4*c^5 - 4*a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 2*a^3*b*c^2*d^3 + 4*a
^4*c*d^4)*x - 6*(a^2*b^2*c*d^4*x^3 + a^3*b*c^3*d^2 + (2*a^2*b^2*c^2*d^3 + a^3*b*c*d^4)*x^2 + (a^2*b^2*c^3*d^2
+ 2*a^3*b*c^2*d^3)*x)*log(b*x + a) + 6*(a^2*b^2*c*d^4*x^3 + a^3*b*c^3*d^2 + (2*a^2*b^2*c^2*d^3 + a^3*b*c*d^4)*
x^2 + (a^2*b^2*c^3*d^2 + 2*a^3*b*c^2*d^3)*x)*log(d*x + c))/(a*b^5*c^6*d^2 - 4*a^2*b^4*c^5*d^3 + 6*a^3*b^3*c^4*
d^4 - 4*a^4*b^2*c^3*d^5 + a^5*b*c^2*d^6 + (b^6*c^4*d^4 - 4*a*b^5*c^3*d^5 + 6*a^2*b^4*c^2*d^6 - 4*a^3*b^3*c*d^7
+ a^4*b^2*d^8)*x^3 + (2*b^6*c^5*d^3 - 7*a*b^5*c^4*d^4 + 8*a^2*b^4*c^3*d^5 - 2*a^3*b^3*c^2*d^6 - 2*a^4*b^2*c*d
^7 + a^5*b*d^8)*x^2 + (b^6*c^6*d^2 - 2*a*b^5*c^5*d^3 - 2*a^2*b^4*c^4*d^4 + 8*a^3*b^3*c^3*d^5 - 7*a^4*b^2*c^2*d
^6 + 2*a^5*b*c*d^7)*x)

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giac [A]  time = 1.09, size = 230, normalized size = 1.78 \[ -\frac {3 \, a^{2} b c \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac {a^{3} b^{2}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x + a\right )}} + \frac {b^{2} c^{3} - 6 \, a b c^{2} d - \frac {6 \, {\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a^2*b*c*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^
2*c*d^3 + a^4*b*d^4) + a^3*b^2/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*(b*x + a)) + 1/2*(b^
2*c^3 - 6*a*b*c^2*d - 6*(a*b^3*c^3 - a^2*b^2*c^2*d)/((b*x + a)*b))/((b*c - a*d)^4*(b*c/(b*x + a) - a*d/(b*x +
a) + d)^2)

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maple [A]  time = 0.01, size = 147, normalized size = 1.14 \[ \frac {3 a^{2} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}-\frac {3 a^{2} c \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}-\frac {a^{3}}{\left (a d -b c \right )^{3} \left (b x +a \right ) b}-\frac {3 a \,c^{2}}{\left (a d -b c \right )^{3} \left (d x +c \right ) d}+\frac {b \,c^{3}}{\left (a d -b c \right )^{3} \left (d x +c \right ) d^{2}}+\frac {c^{3}}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.28, size = 463, normalized size = 3.59 \[ \frac {3 \, a^{2} c \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {3 \, a^{2} c \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {a b^{2} c^{4} - 5 \, a^{2} b c^{3} d - 2 \, a^{3} c^{2} d^{2} + 2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - a^{3} d^{4}\right )} x^{2} + {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 6 \, a^{2} b c^{2} d^{2} - 4 \, a^{3} c d^{3}\right )} x}{2 \, {\left (a b^{4} c^{5} d^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} + 3 \, a^{3} b^{2} c^{3} d^{4} - a^{4} b c^{2} d^{5} + {\left (b^{5} c^{3} d^{4} - 3 \, a b^{4} c^{2} d^{5} + 3 \, a^{2} b^{3} c d^{6} - a^{3} b^{2} d^{7}\right )} x^{3} + {\left (2 \, b^{5} c^{4} d^{3} - 5 \, a b^{4} c^{3} d^{4} + 3 \, a^{2} b^{3} c^{2} d^{5} + a^{3} b^{2} c d^{6} - a^{4} b d^{7}\right )} x^{2} + {\left (b^{5} c^{5} d^{2} - a b^{4} c^{4} d^{3} - 3 \, a^{2} b^{3} c^{3} d^{4} + 5 \, a^{3} b^{2} c^{2} d^{5} - 2 \, a^{4} b c d^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a^2*c*log(b*x + a)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) - 3*a^2*c*log(d*x
+ c)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) - 1/2*(a*b^2*c^4 - 5*a^2*b*c^3*d
- 2*a^3*c^2*d^2 + 2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 - a^3*d^4)*x^2 + (b^3*c^4 - 3*a*b^2*c^3*d - 6*a^2*b*c^2*d^2
- 4*a^3*c*d^3)*x)/(a*b^4*c^5*d^2 - 3*a^2*b^3*c^4*d^3 + 3*a^3*b^2*c^3*d^4 - a^4*b*c^2*d^5 + (b^5*c^3*d^4 - 3*a*
b^4*c^2*d^5 + 3*a^2*b^3*c*d^6 - a^3*b^2*d^7)*x^3 + (2*b^5*c^4*d^3 - 5*a*b^4*c^3*d^4 + 3*a^2*b^3*c^2*d^5 + a^3*
b^2*c*d^6 - a^4*b*d^7)*x^2 + (b^5*c^5*d^2 - a*b^4*c^4*d^3 - 3*a^2*b^3*c^3*d^4 + 5*a^3*b^2*c^2*d^5 - 2*a^4*b*c*
d^6)*x)

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mupad [B]  time = 0.55, size = 396, normalized size = 3.07 \[ \frac {6\,a^2\,c\,\mathrm {atanh}\left (\frac {a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4}{{\left (a\,d-b\,c\right )}^4}+\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{{\left (a\,d-b\,c\right )}^4}-\frac {\frac {x^2\,\left (a^3\,d^3+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{b\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {c\,x\,\left (4\,a^3\,d^3+6\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,b\,d^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {a\,c^2\,\left (2\,a^2\,d^2+5\,a\,b\,c\,d-b^2\,c^2\right )}{2\,b\,d^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + b*x)^2*(c + d*x)^3),x)

[Out]

(6*a^2*c*atanh((a^4*d^4 - b^4*c^4 + 2*a*b^3*c^3*d - 2*a^3*b*c*d^3)/(a*d - b*c)^4 + (2*b*d*x*(a^3*d^3 - b^3*c^3
+ 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*d - b*c)^4))/(a*d - b*c)^4 - ((x^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d))/
(b*d*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (c*x*(4*a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*
b*c*d^2))/(2*b*d^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (a*c^2*(2*a^2*d^2 - b^2*c^2 + 5*a*b*
c*d))/(2*b*d^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/(x*(b*c^2 + 2*a*c*d) + a*c^2 + x^2*(a*d^2
+ 2*b*c*d) + b*d^2*x^3)

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sympy [B]  time = 2.09, size = 717, normalized size = 5.56 \[ - \frac {3 a^{2} c \log {\left (x + \frac {- \frac {3 a^{7} c d^{5}}{\left (a d - b c\right )^{4}} + \frac {15 a^{6} b c^{2} d^{4}}{\left (a d - b c\right )^{4}} - \frac {30 a^{5} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{4}} + \frac {30 a^{4} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{4}} - \frac {15 a^{3} b^{4} c^{5} d}{\left (a d - b c\right )^{4}} + 3 a^{3} c d + \frac {3 a^{2} b^{5} c^{6}}{\left (a d - b c\right )^{4}} + 3 a^{2} b c^{2}}{6 a^{2} b c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {3 a^{2} c \log {\left (x + \frac {\frac {3 a^{7} c d^{5}}{\left (a d - b c\right )^{4}} - \frac {15 a^{6} b c^{2} d^{4}}{\left (a d - b c\right )^{4}} + \frac {30 a^{5} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{4}} - \frac {30 a^{4} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + \frac {15 a^{3} b^{4} c^{5} d}{\left (a d - b c\right )^{4}} + 3 a^{3} c d - \frac {3 a^{2} b^{5} c^{6}}{\left (a d - b c\right )^{4}} + 3 a^{2} b c^{2}}{6 a^{2} b c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {- 2 a^{3} c^{2} d^{2} - 5 a^{2} b c^{3} d + a b^{2} c^{4} + x^{2} \left (- 2 a^{3} d^{4} - 6 a b^{2} c^{2} d^{2} + 2 b^{3} c^{3} d\right ) + x \left (- 4 a^{3} c d^{3} - 6 a^{2} b c^{2} d^{2} - 3 a b^{2} c^{3} d + b^{3} c^{4}\right )}{2 a^{4} b c^{2} d^{5} - 6 a^{3} b^{2} c^{3} d^{4} + 6 a^{2} b^{3} c^{4} d^{3} - 2 a b^{4} c^{5} d^{2} + x^{3} \left (2 a^{3} b^{2} d^{7} - 6 a^{2} b^{3} c d^{6} + 6 a b^{4} c^{2} d^{5} - 2 b^{5} c^{3} d^{4}\right ) + x^{2} \left (2 a^{4} b d^{7} - 2 a^{3} b^{2} c d^{6} - 6 a^{2} b^{3} c^{2} d^{5} + 10 a b^{4} c^{3} d^{4} - 4 b^{5} c^{4} d^{3}\right ) + x \left (4 a^{4} b c d^{6} - 10 a^{3} b^{2} c^{2} d^{5} + 6 a^{2} b^{3} c^{3} d^{4} + 2 a b^{4} c^{4} d^{3} - 2 b^{5} c^{5} d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a**2*c*log(x + (-3*a**7*c*d**5/(a*d - b*c)**4 + 15*a**6*b*c**2*d**4/(a*d - b*c)**4 - 30*a**5*b**2*c**3*d**3
/(a*d - b*c)**4 + 30*a**4*b**3*c**4*d**2/(a*d - b*c)**4 - 15*a**3*b**4*c**5*d/(a*d - b*c)**4 + 3*a**3*c*d + 3*
a**2*b**5*c**6/(a*d - b*c)**4 + 3*a**2*b*c**2)/(6*a**2*b*c*d))/(a*d - b*c)**4 + 3*a**2*c*log(x + (3*a**7*c*d**
5/(a*d - b*c)**4 - 15*a**6*b*c**2*d**4/(a*d - b*c)**4 + 30*a**5*b**2*c**3*d**3/(a*d - b*c)**4 - 30*a**4*b**3*c
**4*d**2/(a*d - b*c)**4 + 15*a**3*b**4*c**5*d/(a*d - b*c)**4 + 3*a**3*c*d - 3*a**2*b**5*c**6/(a*d - b*c)**4 +
3*a**2*b*c**2)/(6*a**2*b*c*d))/(a*d - b*c)**4 + (-2*a**3*c**2*d**2 - 5*a**2*b*c**3*d + a*b**2*c**4 + x**2*(-2*
a**3*d**4 - 6*a*b**2*c**2*d**2 + 2*b**3*c**3*d) + x*(-4*a**3*c*d**3 - 6*a**2*b*c**2*d**2 - 3*a*b**2*c**3*d + b
**3*c**4))/(2*a**4*b*c**2*d**5 - 6*a**3*b**2*c**3*d**4 + 6*a**2*b**3*c**4*d**3 - 2*a*b**4*c**5*d**2 + x**3*(2*
a**3*b**2*d**7 - 6*a**2*b**3*c*d**6 + 6*a*b**4*c**2*d**5 - 2*b**5*c**3*d**4) + x**2*(2*a**4*b*d**7 - 2*a**3*b*
*2*c*d**6 - 6*a**2*b**3*c**2*d**5 + 10*a*b**4*c**3*d**4 - 4*b**5*c**4*d**3) + x*(4*a**4*b*c*d**6 - 10*a**3*b**
2*c**2*d**5 + 6*a**2*b**3*c**3*d**4 + 2*a*b**4*c**4*d**3 - 2*b**5*c**5*d**2))

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