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

Optimal. Leaf size=121 \[ \frac {a b}{(a+b x) (b c-a d)^3}+\frac {a d+b c}{(c+d x) (b c-a d)^3}+\frac {c}{2 (c+d x)^2 (b c-a d)^2}+\frac {b (2 a d+b c) \log (a+b x)}{(b c-a d)^4}-\frac {b (2 a d+b c) \log (c+d x)}{(b c-a d)^4} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \frac {a b}{(a+b x) (b c-a d)^3}+\frac {a d+b c}{(c+d x) (b c-a d)^3}+\frac {c}{2 (c+d x)^2 (b c-a d)^2}+\frac {b (2 a d+b c) \log (a+b x)}{(b c-a d)^4}-\frac {b (2 a d+b c) \log (c+d x)}{(b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Int[x/((a + b*x)^2*(c + d*x)^3),x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 111, normalized size = 0.92 \[ \frac {\frac {c (b c-a d)^2}{(c+d x)^2}+\frac {2 a b (b c-a d)}{a+b x}+\frac {2 (a d+b c) (b c-a d)}{c+d x}+2 b (2 a d+b c) \log (a+b x)-2 b (2 a d+b c) \log (c+d x)}{2 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.63, size = 586, normalized size = 4.84 \[ \frac {5 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + 2 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x^{2} + {\left (3 \, b^{3} c^{3} + 4 \, a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x + 2 \, {\left (a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.09, size = 255, normalized size = 2.11 \[ \frac {\frac {2 \, a b^{5}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x + a\right )}} - \frac {2 \, {\left (b^{4} c + 2 \, a b^{3} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac {3 \, b^{3} c d^{2} + 2 \, a b^{2} d^{3} + \frac {2 \, {\left (2 \, b^{5} c^{2} d - a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{{\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2}}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 170, normalized size = 1.40 \[ \frac {2 a b d \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}-\frac {2 a b d \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}+\frac {b^{2} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}-\frac {b^{2} c \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}-\frac {a b}{\left (a d -b c \right )^{3} \left (b x +a \right )}-\frac {a d}{\left (a d -b c \right )^{3} \left (d x +c \right )}-\frac {b c}{\left (a d -b c \right )^{3} \left (d x +c \right )}+\frac {c}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.32, size = 406, normalized size = 3.36 \[ \frac {{\left (b^{2} c + 2 \, a b d\right )} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {{\left (b^{2} c + 2 \, a b d\right )} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {5 \, a b c^{2} + a^{2} c d + 2 \, {\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{2} + {\left (3 \, b^{2} c^{2} + 7 \, a b c d + 2 \, a^{2} d^{2}\right )} x}{2 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.57, size = 377, normalized size = 3.12 \[ \frac {2\,b\,\mathrm {atanh}\left (\frac {b\,\left (2\,a\,d+b\,c\right )\,\left (a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4\right )}{\left (c\,b^2+2\,a\,d\,b\right )\,{\left (a\,d-b\,c\right )}^4}+\frac {2\,b^2\,d\,x\,\left (2\,a\,d+b\,c\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{\left (c\,b^2+2\,a\,d\,b\right )\,{\left (a\,d-b\,c\right )}^4}\right )\,\left (2\,a\,d+b\,c\right )}{{\left (a\,d-b\,c\right )}^4}-\frac {\frac {d\,a^2\,c+5\,b\,a\,c^2}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x\,\left (2\,a\,d+b\,c\right )\,\left (a\,d+3\,b\,c\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b\,d\,x^2\,\left (2\,a\,d+b\,c\right )}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((a + b*x)^2*(c + d*x)^3),x)

[Out]

(2*b*atanh((b*(2*a*d + b*c)*(a^4*d^4 - b^4*c^4 + 2*a*b^3*c^3*d - 2*a^3*b*c*d^3))/((b^2*c + 2*a*b*d)*(a*d - b*c
)^4) + (2*b^2*d*x*(2*a*d + b*c)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/((b^2*c + 2*a*b*d)*(a*d -
b*c)^4))*(2*a*d + b*c))/(a*d - b*c)^4 - ((5*a*b*c^2 + a^2*c*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*(2*a*d + b*c)*(a*d + 3*b*c))/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (b*d*x^2
*(2*a*d + b*c))/(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 = 2.35, size = 774, normalized size = 6.40 \[ - \frac {b \left (2 a d + b c\right ) \log {\left (x + \frac {- \frac {a^{5} b d^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {5 a^{4} b^{2} c d^{4} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{3} b^{3} c^{2} d^{3} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{2} b^{4} c^{3} d^{2} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} - \frac {5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d + \frac {b^{6} c^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {b \left (2 a d + b c\right ) \log {\left (x + \frac {\frac {a^{5} b d^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {5 a^{4} b^{2} c d^{4} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{3} b^{3} c^{2} d^{3} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{2} b^{4} c^{3} d^{2} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} + \frac {5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d - \frac {b^{6} c^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {- a^{2} c d - 5 a b c^{2} + x^{2} \left (- 4 a b d^{2} - 2 b^{2} c d\right ) + x \left (- 2 a^{2} d^{2} - 7 a b c d - 3 b^{2} c^{2}\right )}{2 a^{4} c^{2} d^{3} - 6 a^{3} b c^{3} d^{2} + 6 a^{2} b^{2} c^{4} d - 2 a b^{3} c^{5} + x^{3} \left (2 a^{3} b d^{5} - 6 a^{2} b^{2} c d^{4} + 6 a b^{3} c^{2} d^{3} - 2 b^{4} c^{3} d^{2}\right ) + x^{2} \left (2 a^{4} d^{5} - 2 a^{3} b c d^{4} - 6 a^{2} b^{2} c^{2} d^{3} + 10 a b^{3} c^{3} d^{2} - 4 b^{4} c^{4} d\right ) + x \left (4 a^{4} c d^{4} - 10 a^{3} b c^{2} d^{3} + 6 a^{2} b^{2} c^{3} d^{2} + 2 a b^{3} c^{4} d - 2 b^{4} c^{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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