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

Optimal. Leaf size=124 \[ -\frac {a^2}{(a+b x) (b c-a d)^3}-\frac {c^2}{2 d (c+d x)^2 (b c-a d)^2}-\frac {2 a c}{(c+d x) (b c-a d)^3}-\frac {a (a d+2 b c) \log (a+b x)}{(b c-a d)^4}+\frac {a (a d+2 b c) \log (c+d x)}{(b c-a d)^4} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 124, 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} \[ -\frac {a^2}{(a+b x) (b c-a d)^3}-\frac {c^2}{2 d (c+d x)^2 (b c-a d)^2}-\frac {2 a c}{(c+d x) (b c-a d)^3}-\frac {a (a d+2 b c) \log (a+b x)}{(b c-a d)^4}+\frac {a (a d+2 b c) \log (c+d x)}{(b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((a + b*x)^2*(c + d*x)^3),x]

[Out]

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

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Mathematica [A]  time = 0.17, size = 123, normalized size = 0.99 \[ \frac {a^2}{(a+b x) (a d-b c)^3}-\frac {c^2}{2 d (c+d x)^2 (b c-a d)^2}-\frac {2 a c}{(c+d x) (b c-a d)^3}-\frac {a (a d+2 b c) \log (a+b x)}{(b c-a d)^4}+\frac {a (a d+2 b c) \log (c+d x)}{(b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.85, size = 617, normalized size = 4.98 \[ -\frac {a b^{2} c^{4} + 4 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2} + 2 \, {\left (2 \, a b^{2} c^{2} d^{2} - a^{2} b c d^{3} - a^{3} d^{4}\right )} x^{2} + {\left (b^{3} c^{4} + 2 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 8 \, a^{3} c d^{3}\right )} x + 2 \, {\left (2 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2} + {\left (2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{3} + {\left (4 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x^{2} + {\left (2 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (2 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2} + {\left (2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{3} + {\left (4 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x^{2} + {\left (2 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{4} c^{6} d - 4 \, a^{2} b^{3} c^{5} d^{2} + 6 \, a^{3} b^{2} c^{4} d^{3} - 4 \, a^{4} b c^{3} d^{4} + a^{5} c^{2} d^{5} + {\left (b^{5} c^{4} d^{3} - 4 \, a b^{4} c^{3} d^{4} + 6 \, a^{2} b^{3} c^{2} d^{5} - 4 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}\right )} x^{3} + {\left (2 \, b^{5} c^{5} d^{2} - 7 \, a b^{4} c^{4} d^{3} + 8 \, a^{2} b^{3} c^{3} d^{4} - 2 \, a^{3} b^{2} c^{2} d^{5} - 2 \, a^{4} b c d^{6} + a^{5} d^{7}\right )} x^{2} + {\left (b^{5} c^{6} d - 2 \, a b^{4} c^{5} d^{2} - 2 \, a^{2} b^{3} c^{4} d^{3} + 8 \, a^{3} b^{2} c^{3} d^{4} - 7 \, a^{4} b c^{2} d^{5} + 2 \, a^{5} c d^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.00, size = 248, normalized size = 2.00 \[ -\frac {a^{2} b^{3}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x + a\right )}} + \frac {{\left (2 \, a b^{2} c + a^{2} b d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac {b^{2} c^{2} d + 4 \, a b c d^{2} + \frac {2 \, {\left (b^{4} c^{3} + a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2}\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2*b^3/((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)) + (2*a*b^2*c + a^2*b*d)*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) + 1/2*(b^2*c^2*d + 4*a*b*c*d^2 + 2*(b^4*c^3 + a*b^3*c^2*d - 2*a^2*b^2*c*d^2)/((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 = 154, normalized size = 1.24 \[ -\frac {a^{2} d \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}+\frac {a^{2} d \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}-\frac {2 a b c \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}+\frac {2 a b c \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}+\frac {a^{2}}{\left (a d -b c \right )^{3} \left (b x +a \right )}+\frac {2 a c}{\left (a d -b c \right )^{3} \left (d x +c \right )}-\frac {c^{2}}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.38, size = 420, normalized size = 3.39 \[ -\frac {{\left (2 \, a b c + a^{2} d\right )} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {{\left (2 \, a b c + a^{2} d\right )} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {a b c^{3} + 5 \, a^{2} c^{2} d + 2 \, {\left (2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2} + {\left (b^{2} c^{3} + 3 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} x}{2 \, {\left (a b^{3} c^{5} d - 3 \, a^{2} b^{2} c^{4} d^{2} + 3 \, a^{3} b c^{3} d^{3} - a^{4} c^{2} d^{4} + {\left (b^{4} c^{3} d^{3} - 3 \, a b^{3} c^{2} d^{4} + 3 \, a^{2} b^{2} c d^{5} - a^{3} b d^{6}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d^{2} - 5 \, a b^{3} c^{3} d^{3} + 3 \, a^{2} b^{2} c^{2} d^{4} + a^{3} b c d^{5} - a^{4} d^{6}\right )} x^{2} + {\left (b^{4} c^{5} d - a b^{3} c^{4} d^{2} - 3 \, a^{2} b^{2} c^{3} d^{3} + 5 \, a^{3} b c^{2} d^{4} - 2 \, a^{4} c d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.56, size = 386, normalized size = 3.11 \[ \frac {\frac {a\,d\,x^2\,\left (a\,d+2\,b\,c\right )}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}+\frac {a\,c^2\,\left (5\,a\,d+b\,c\right )}{2\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {c\,x\,\left (8\,a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{2\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3}-\frac {2\,a\,\mathrm {atanh}\left (\frac {a\,\left (a\,d+2\,b\,c\right )\,\left (a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4\right )}{\left (d\,a^2+2\,b\,c\,a\right )\,{\left (a\,d-b\,c\right )}^4}+\frac {2\,a\,b\,d\,x\,\left (a\,d+2\,b\,c\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{\left (d\,a^2+2\,b\,c\,a\right )\,{\left (a\,d-b\,c\right )}^4}\right )\,\left (a\,d+2\,b\,c\right )}{{\left (a\,d-b\,c\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((a + b*x)^2*(c + d*x)^3),x)

[Out]

((a*d*x^2*(a*d + 2*b*c))/(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*(5*a*d + b*c))/(2*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*(8*a^2*d^2 + b^2*c^2 + 3*a*b*c*d))/(2*d*(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) - (2
*a*atanh((a*(a*d + 2*b*c)*(a^4*d^4 - b^4*c^4 + 2*a*b^3*c^3*d - 2*a^3*b*c*d^3))/((a^2*d + 2*a*b*c)*(a*d - b*c)^
4) + (2*a*b*d*x*(a*d + 2*b*c)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/((a^2*d + 2*a*b*c)*(a*d - b
*c)^4))*(a*d + 2*b*c))/(a*d - b*c)^4

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sympy [B]  time = 2.26, size = 787, normalized size = 6.35 \[ \frac {a \left (a d + 2 b c\right ) \log {\left (x + \frac {- \frac {a^{6} d^{5} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + \frac {5 a^{5} b c d^{4} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{4} b^{2} c^{2} d^{3} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{3} b^{3} c^{3} d^{2} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + a^{3} d^{2} - \frac {5 a^{2} b^{4} c^{4} d \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + 3 a^{2} b c d + \frac {a b^{5} c^{5} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + 2 a b^{2} c^{2}}{2 a^{2} b d^{2} + 4 a b^{2} c d} \right )}}{\left (a d - b c\right )^{4}} - \frac {a \left (a d + 2 b c\right ) \log {\left (x + \frac {\frac {a^{6} d^{5} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} - \frac {5 a^{5} b c d^{4} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{4} b^{2} c^{2} d^{3} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{3} b^{3} c^{3} d^{2} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + a^{3} d^{2} + \frac {5 a^{2} b^{4} c^{4} d \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + 3 a^{2} b c d - \frac {a b^{5} c^{5} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + 2 a b^{2} c^{2}}{2 a^{2} b d^{2} + 4 a b^{2} c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {5 a^{2} c^{2} d + a b c^{3} + x^{2} \left (2 a^{2} d^{3} + 4 a b c d^{2}\right ) + x \left (8 a^{2} c d^{2} + 3 a b c^{2} d + b^{2} c^{3}\right )}{2 a^{4} c^{2} d^{4} - 6 a^{3} b c^{3} d^{3} + 6 a^{2} b^{2} c^{4} d^{2} - 2 a b^{3} c^{5} d + x^{3} \left (2 a^{3} b d^{6} - 6 a^{2} b^{2} c d^{5} + 6 a b^{3} c^{2} d^{4} - 2 b^{4} c^{3} d^{3}\right ) + x^{2} \left (2 a^{4} d^{6} - 2 a^{3} b c d^{5} - 6 a^{2} b^{2} c^{2} d^{4} + 10 a b^{3} c^{3} d^{3} - 4 b^{4} c^{4} d^{2}\right ) + x \left (4 a^{4} c d^{5} - 10 a^{3} b c^{2} d^{4} + 6 a^{2} b^{2} c^{3} d^{3} + 2 a b^{3} c^{4} d^{2} - 2 b^{4} c^{5} d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(a*d + 2*b*c)*log(x + (-a**6*d**5*(a*d + 2*b*c)/(a*d - b*c)**4 + 5*a**5*b*c*d**4*(a*d + 2*b*c)/(a*d - b*c)**
4 - 10*a**4*b**2*c**2*d**3*(a*d + 2*b*c)/(a*d - b*c)**4 + 10*a**3*b**3*c**3*d**2*(a*d + 2*b*c)/(a*d - b*c)**4
+ a**3*d**2 - 5*a**2*b**4*c**4*d*(a*d + 2*b*c)/(a*d - b*c)**4 + 3*a**2*b*c*d + a*b**5*c**5*(a*d + 2*b*c)/(a*d
- b*c)**4 + 2*a*b**2*c**2)/(2*a**2*b*d**2 + 4*a*b**2*c*d))/(a*d - b*c)**4 - a*(a*d + 2*b*c)*log(x + (a**6*d**5
*(a*d + 2*b*c)/(a*d - b*c)**4 - 5*a**5*b*c*d**4*(a*d + 2*b*c)/(a*d - b*c)**4 + 10*a**4*b**2*c**2*d**3*(a*d + 2
*b*c)/(a*d - b*c)**4 - 10*a**3*b**3*c**3*d**2*(a*d + 2*b*c)/(a*d - b*c)**4 + a**3*d**2 + 5*a**2*b**4*c**4*d*(a
*d + 2*b*c)/(a*d - b*c)**4 + 3*a**2*b*c*d - a*b**5*c**5*(a*d + 2*b*c)/(a*d - b*c)**4 + 2*a*b**2*c**2)/(2*a**2*
b*d**2 + 4*a*b**2*c*d))/(a*d - b*c)**4 + (5*a**2*c**2*d + a*b*c**3 + x**2*(2*a**2*d**3 + 4*a*b*c*d**2) + x*(8*
a**2*c*d**2 + 3*a*b*c**2*d + b**2*c**3))/(2*a**4*c**2*d**4 - 6*a**3*b*c**3*d**3 + 6*a**2*b**2*c**4*d**2 - 2*a*
b**3*c**5*d + x**3*(2*a**3*b*d**6 - 6*a**2*b**2*c*d**5 + 6*a*b**3*c**2*d**4 - 2*b**4*c**3*d**3) + x**2*(2*a**4
*d**6 - 2*a**3*b*c*d**5 - 6*a**2*b**2*c**2*d**4 + 10*a*b**3*c**3*d**3 - 4*b**4*c**4*d**2) + x*(4*a**4*c*d**5 -
10*a**3*b*c**2*d**4 + 6*a**2*b**2*c**3*d**3 + 2*a*b**3*c**4*d**2 - 2*b**4*c**5*d))

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