3.3.37 \(\int \frac {x^5}{(a+b x) (c+d x)^3} \, dx\)

Optimal. Leaf size=161 \[ -\frac {a^5 \log (a+b x)}{b^3 (b c-a d)^3}+\frac {c^3 \left (10 a^2 d^2-15 a b c d+6 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^3}-\frac {x (a d+3 b c)}{b^2 d^4}-\frac {c^5}{2 d^5 (c+d x)^2 (b c-a d)}+\frac {c^4 (4 b c-5 a d)}{d^5 (c+d x) (b c-a d)^2}+\frac {x^2}{2 b d^3} \]

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Rubi [A]  time = 0.19, antiderivative size = 161, 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} \begin {gather*} \frac {c^3 \left (10 a^2 d^2-15 a b c d+6 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^3}-\frac {a^5 \log (a+b x)}{b^3 (b c-a d)^3}-\frac {x (a d+3 b c)}{b^2 d^4}+\frac {c^4 (4 b c-5 a d)}{d^5 (c+d x) (b c-a d)^2}-\frac {c^5}{2 d^5 (c+d x)^2 (b c-a d)}+\frac {x^2}{2 b d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x)*(c + d*x)^3),x]

[Out]

-(((3*b*c + a*d)*x)/(b^2*d^4)) + x^2/(2*b*d^3) - c^5/(2*d^5*(b*c - a*d)*(c + d*x)^2) + (c^4*(4*b*c - 5*a*d))/(
d^5*(b*c - a*d)^2*(c + d*x)) - (a^5*Log[a + b*x])/(b^3*(b*c - a*d)^3) + (c^3*(6*b^2*c^2 - 15*a*b*c*d + 10*a^2*
d^2)*Log[c + d*x])/(d^5*(b*c - a*d)^3)

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^5}{(a+b x) (c+d x)^3} \, dx &=\int \left (\frac {-3 b c-a d}{b^2 d^4}+\frac {x}{b d^3}-\frac {a^5}{b^2 (b c-a d)^3 (a+b x)}-\frac {c^5}{d^4 (-b c+a d) (c+d x)^3}-\frac {c^4 (4 b c-5 a d)}{d^4 (-b c+a d)^2 (c+d x)^2}-\frac {c^3 \left (6 b^2 c^2-15 a b c d+10 a^2 d^2\right )}{d^4 (-b c+a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac {(3 b c+a d) x}{b^2 d^4}+\frac {x^2}{2 b d^3}-\frac {c^5}{2 d^5 (b c-a d) (c+d x)^2}+\frac {c^4 (4 b c-5 a d)}{d^5 (b c-a d)^2 (c+d x)}-\frac {a^5 \log (a+b x)}{b^3 (b c-a d)^3}+\frac {c^3 \left (6 b^2 c^2-15 a b c d+10 a^2 d^2\right ) \log (c+d x)}{d^5 (b c-a d)^3}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 161, normalized size = 1.00 \begin {gather*} \frac {1}{2} \left (-\frac {2 a^5 \log (a+b x)}{b^3 (b c-a d)^3}-\frac {2 c^3 \left (10 a^2 d^2-15 a b c d+6 b^2 c^2\right ) \log (c+d x)}{d^5 (a d-b c)^3}-\frac {2 x (a d+3 b c)}{b^2 d^4}+\frac {c^5}{d^5 (c+d x)^2 (a d-b c)}+\frac {2 c^4 (4 b c-5 a d)}{d^5 (c+d x) (b c-a d)^2}+\frac {x^2}{b d^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/((a + b*x)*(c + d*x)^3),x]

[Out]

((-2*(3*b*c + a*d)*x)/(b^2*d^4) + x^2/(b*d^3) + c^5/(d^5*(-(b*c) + a*d)*(c + d*x)^2) + (2*c^4*(4*b*c - 5*a*d))
/(d^5*(b*c - a*d)^2*(c + d*x)) - (2*a^5*Log[a + b*x])/(b^3*(b*c - a*d)^3) - (2*c^3*(6*b^2*c^2 - 15*a*b*c*d + 1
0*a^2*d^2)*Log[c + d*x])/(d^5*(-(b*c) + a*d)^3))/2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5}{(a+b x) (c+d x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^5/((a + b*x)*(c + d*x)^3),x]

[Out]

IntegrateAlgebraic[x^5/((a + b*x)*(c + d*x)^3), x]

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fricas [B]  time = 1.11, size = 579, normalized size = 3.60 \begin {gather*} \frac {7 \, b^{5} c^{7} - 16 \, a b^{4} c^{6} d + 9 \, a^{2} b^{3} c^{5} d^{2} + {\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^{4} - 2 \, {\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^{3} - {\left (11 \, b^{5} c^{5} d^{2} - 29 \, a b^{4} c^{4} d^{3} + 21 \, a^{2} b^{3} c^{3} d^{4} + a^{3} b^{2} c^{2} d^{5} - 4 \, a^{4} b c d^{6}\right )} x^{2} + 2 \, {\left (b^{5} c^{6} d - a b^{4} c^{5} d^{2} - a^{2} b^{3} c^{4} d^{3} + a^{4} b c^{2} d^{5}\right )} x - 2 \, {\left (a^{5} d^{7} x^{2} + 2 \, a^{5} c d^{6} x + a^{5} c^{2} d^{5}\right )} \log \left (b x + a\right ) + 2 \, {\left (6 \, b^{5} c^{7} - 15 \, a b^{4} c^{6} d + 10 \, a^{2} b^{3} c^{5} d^{2} + {\left (6 \, b^{5} c^{5} d^{2} - 15 \, a b^{4} c^{4} d^{3} + 10 \, a^{2} b^{3} c^{3} d^{4}\right )} x^{2} + 2 \, {\left (6 \, b^{5} c^{6} d - 15 \, a b^{4} c^{5} d^{2} + 10 \, a^{2} b^{3} c^{4} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (b^{6} c^{5} d^{5} - 3 \, a b^{5} c^{4} d^{6} + 3 \, a^{2} b^{4} c^{3} d^{7} - a^{3} b^{3} c^{2} d^{8} + {\left (b^{6} c^{3} d^{7} - 3 \, a b^{5} c^{2} d^{8} + 3 \, a^{2} b^{4} c d^{9} - a^{3} b^{3} d^{10}\right )} x^{2} + 2 \, {\left (b^{6} c^{4} d^{6} - 3 \, a b^{5} c^{3} d^{7} + 3 \, a^{2} b^{4} c^{2} d^{8} - a^{3} b^{3} c d^{9}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)/(d*x+c)^3,x, algorithm="fricas")

[Out]

1/2*(7*b^5*c^7 - 16*a*b^4*c^6*d + 9*a^2*b^3*c^5*d^2 + (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^4 - 2*(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^3 - (11*b^
5*c^5*d^2 - 29*a*b^4*c^4*d^3 + 21*a^2*b^3*c^3*d^4 + a^3*b^2*c^2*d^5 - 4*a^4*b*c*d^6)*x^2 + 2*(b^5*c^6*d - a*b^
4*c^5*d^2 - a^2*b^3*c^4*d^3 + a^4*b*c^2*d^5)*x - 2*(a^5*d^7*x^2 + 2*a^5*c*d^6*x + a^5*c^2*d^5)*log(b*x + a) +
2*(6*b^5*c^7 - 15*a*b^4*c^6*d + 10*a^2*b^3*c^5*d^2 + (6*b^5*c^5*d^2 - 15*a*b^4*c^4*d^3 + 10*a^2*b^3*c^3*d^4)*x
^2 + 2*(6*b^5*c^6*d - 15*a*b^4*c^5*d^2 + 10*a^2*b^3*c^4*d^3)*x)*log(d*x + c))/(b^6*c^5*d^5 - 3*a*b^5*c^4*d^6 +
3*a^2*b^4*c^3*d^7 - a^3*b^3*c^2*d^8 + (b^6*c^3*d^7 - 3*a*b^5*c^2*d^8 + 3*a^2*b^4*c*d^9 - a^3*b^3*d^10)*x^2 +
2*(b^6*c^4*d^6 - 3*a*b^5*c^3*d^7 + 3*a^2*b^4*c^2*d^8 - a^3*b^3*c*d^9)*x)

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giac [A]  time = 1.00, size = 251, normalized size = 1.56 \begin {gather*} -\frac {a^{5} \log \left ({\left | b x + a \right |}\right )}{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}} + \frac {{\left (6 \, b^{2} c^{5} - 15 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}} + \frac {b d^{3} x^{2} - 6 \, b c d^{2} x - 2 \, a d^{3} x}{2 \, b^{2} d^{6}} + \frac {7 \, b^{2} c^{7} - 16 \, a b c^{6} d + 9 \, a^{2} c^{5} d^{2} + 2 \, {\left (4 \, b^{2} c^{6} d - 9 \, a b c^{5} d^{2} + 5 \, a^{2} c^{4} d^{3}\right )} x}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)/(d*x+c)^3,x, algorithm="giac")

[Out]

-a^5*log(abs(b*x + a))/(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) + (6*b^2*c^5 - 15*a*b*c^4*d +
10*a^2*c^3*d^2)*log(abs(d*x + c))/(b^3*c^3*d^5 - 3*a*b^2*c^2*d^6 + 3*a^2*b*c*d^7 - a^3*d^8) + 1/2*(b*d^3*x^2
- 6*b*c*d^2*x - 2*a*d^3*x)/(b^2*d^6) + 1/2*(7*b^2*c^7 - 16*a*b*c^6*d + 9*a^2*c^5*d^2 + 2*(4*b^2*c^6*d - 9*a*b*
c^5*d^2 + 5*a^2*c^4*d^3)*x)/((b*c - a*d)^3*(d*x + c)^2*d^5)

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maple [A]  time = 0.01, size = 213, normalized size = 1.32 \begin {gather*} \frac {a^{5} \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} b^{3}}-\frac {10 a^{2} c^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} d^{3}}+\frac {15 a b \,c^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} d^{4}}-\frac {6 b^{2} c^{5} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} d^{5}}-\frac {5 a \,c^{4}}{\left (a d -b c \right )^{2} \left (d x +c \right ) d^{4}}+\frac {4 b \,c^{5}}{\left (a d -b c \right )^{2} \left (d x +c \right ) d^{5}}+\frac {c^{5}}{2 \left (a d -b c \right ) \left (d x +c \right )^{2} d^{5}}+\frac {x^{2}}{2 b \,d^{3}}-\frac {a x}{b^{2} d^{3}}-\frac {3 c x}{b \,d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x+a)/(d*x+c)^3,x)

[Out]

1/2*x^2/b/d^3-1/b^2/d^3*x*a-3/b/d^4*x*c-5/d^4*c^4/(a*d-b*c)^2/(d*x+c)*a+4/d^5*c^5/(a*d-b*c)^2/(d*x+c)*b+1/2/d^
5*c^5/(a*d-b*c)/(d*x+c)^2-10/d^3*c^3/(a*d-b*c)^3*ln(d*x+c)*a^2+15/d^4*c^4/(a*d-b*c)^3*ln(d*x+c)*a*b-6/d^5*c^5/
(a*d-b*c)^3*ln(d*x+c)*b^2+1/b^3*a^5/(a*d-b*c)^3*ln(b*x+a)

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maxima [A]  time = 1.09, size = 290, normalized size = 1.80 \begin {gather*} -\frac {a^{5} \log \left (b x + a\right )}{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}} + \frac {{\left (6 \, b^{2} c^{5} - 15 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}} + \frac {7 \, b c^{6} - 9 \, a c^{5} d + 2 \, {\left (4 \, b c^{5} d - 5 \, a c^{4} d^{2}\right )} x}{2 \, {\left (b^{2} c^{4} d^{5} - 2 \, a b c^{3} d^{6} + a^{2} c^{2} d^{7} + {\left (b^{2} c^{2} d^{7} - 2 \, a b c d^{8} + a^{2} d^{9}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d^{6} - 2 \, a b c^{2} d^{7} + a^{2} c d^{8}\right )} x\right )}} + \frac {b d x^{2} - 2 \, {\left (3 \, b c + a d\right )} x}{2 \, b^{2} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)/(d*x+c)^3,x, algorithm="maxima")

[Out]

-a^5*log(b*x + a)/(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) + (6*b^2*c^5 - 15*a*b*c^4*d + 10*a
^2*c^3*d^2)*log(d*x + c)/(b^3*c^3*d^5 - 3*a*b^2*c^2*d^6 + 3*a^2*b*c*d^7 - a^3*d^8) + 1/2*(7*b*c^6 - 9*a*c^5*d
+ 2*(4*b*c^5*d - 5*a*c^4*d^2)*x)/(b^2*c^4*d^5 - 2*a*b*c^3*d^6 + a^2*c^2*d^7 + (b^2*c^2*d^7 - 2*a*b*c*d^8 + a^2
*d^9)*x^2 + 2*(b^2*c^3*d^6 - 2*a*b*c^2*d^7 + a^2*c*d^8)*x) + 1/2*(b*d*x^2 - 2*(3*b*c + a*d)*x)/(b^2*d^4)

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mupad [B]  time = 0.66, size = 293, normalized size = 1.82 \begin {gather*} \frac {\frac {x\,\left (4\,b^3\,c^5-5\,a\,b^2\,c^4\,d\right )}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}+\frac {7\,b^3\,c^6-9\,a\,b^2\,c^5\,d}{2\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b^2\,c^2\,d^4+2\,b^2\,c\,d^5\,x+b^2\,d^6\,x^2}-\frac {a^5\,\ln \left (a+b\,x\right )}{-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}+\frac {x^2}{2\,b\,d^3}-\frac {\ln \left (c+d\,x\right )\,\left (10\,a^2\,c^3\,d^2-15\,a\,b\,c^4\,d+6\,b^2\,c^5\right )}{a^3\,d^8-3\,a^2\,b\,c\,d^7+3\,a\,b^2\,c^2\,d^6-b^3\,c^3\,d^5}-\frac {x\,\left (a\,d^3+3\,b\,c\,d^2\right )}{b^2\,d^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/((a + b*x)*(c + d*x)^3),x)

[Out]

((x*(4*b^3*c^5 - 5*a*b^2*c^4*d))/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d) + (7*b^3*c^6 - 9*a*b^2*c^5*d)/(2*d*(a^2*d^2 +
b^2*c^2 - 2*a*b*c*d)))/(b^2*c^2*d^4 + b^2*d^6*x^2 + 2*b^2*c*d^5*x) - (a^5*log(a + b*x))/(b^6*c^3 - a^3*b^3*d^
3 + 3*a^2*b^4*c*d^2 - 3*a*b^5*c^2*d) + x^2/(2*b*d^3) - (log(c + d*x)*(6*b^2*c^5 + 10*a^2*c^3*d^2 - 15*a*b*c^4*
d))/(a^3*d^8 - b^3*c^3*d^5 + 3*a*b^2*c^2*d^6 - 3*a^2*b*c*d^7) - (x*(a*d^3 + 3*b*c*d^2))/(b^2*d^6)

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sympy [B]  time = 15.40, size = 748, normalized size = 4.65 \begin {gather*} \frac {a^{5} \log {\left (x + \frac {\frac {a^{9} d^{8}}{b \left (a d - b c\right )^{3}} - \frac {4 a^{8} c d^{7}}{\left (a d - b c\right )^{3}} + \frac {6 a^{7} b c^{2} d^{6}}{\left (a d - b c\right )^{3}} - \frac {4 a^{6} b^{2} c^{3} d^{5}}{\left (a d - b c\right )^{3}} + \frac {a^{5} b^{3} c^{4} d^{4}}{\left (a d - b c\right )^{3}} + a^{5} c d^{4} + 10 a^{3} b^{2} c^{3} d^{2} - 15 a^{2} b^{3} c^{4} d + 6 a b^{4} c^{5}}{a^{5} d^{5} + 10 a^{2} b^{3} c^{3} d^{2} - 15 a b^{4} c^{4} d + 6 b^{5} c^{5}} \right )}}{b^{3} \left (a d - b c\right )^{3}} - \frac {c^{3} \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right ) \log {\left (x + \frac {a^{5} c d^{4} - \frac {a^{4} b^{2} c^{3} d^{3} \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b^{3} c^{4} d^{2} \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 10 a^{3} b^{2} c^{3} d^{2} - \frac {6 a^{2} b^{4} c^{5} d \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} - 15 a^{2} b^{3} c^{4} d + \frac {4 a b^{5} c^{6} \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 6 a b^{4} c^{5} - \frac {b^{6} c^{7} \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{d \left (a d - b c\right )^{3}}}{a^{5} d^{5} + 10 a^{2} b^{3} c^{3} d^{2} - 15 a b^{4} c^{4} d + 6 b^{5} c^{5}} \right )}}{d^{5} \left (a d - b c\right )^{3}} + x \left (- \frac {a}{b^{2} d^{3}} - \frac {3 c}{b d^{4}}\right ) + \frac {- 9 a c^{5} d + 7 b c^{6} + x \left (- 10 a c^{4} d^{2} + 8 b c^{5} d\right )}{2 a^{2} c^{2} d^{7} - 4 a b c^{3} d^{6} + 2 b^{2} c^{4} d^{5} + x^{2} \left (2 a^{2} d^{9} - 4 a b c d^{8} + 2 b^{2} c^{2} d^{7}\right ) + x \left (4 a^{2} c d^{8} - 8 a b c^{2} d^{7} + 4 b^{2} c^{3} d^{6}\right )} + \frac {x^{2}}{2 b d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x+a)/(d*x+c)**3,x)

[Out]

a**5*log(x + (a**9*d**8/(b*(a*d - b*c)**3) - 4*a**8*c*d**7/(a*d - b*c)**3 + 6*a**7*b*c**2*d**6/(a*d - b*c)**3
- 4*a**6*b**2*c**3*d**5/(a*d - b*c)**3 + a**5*b**3*c**4*d**4/(a*d - b*c)**3 + a**5*c*d**4 + 10*a**3*b**2*c**3*
d**2 - 15*a**2*b**3*c**4*d + 6*a*b**4*c**5)/(a**5*d**5 + 10*a**2*b**3*c**3*d**2 - 15*a*b**4*c**4*d + 6*b**5*c*
*5))/(b**3*(a*d - b*c)**3) - c**3*(10*a**2*d**2 - 15*a*b*c*d + 6*b**2*c**2)*log(x + (a**5*c*d**4 - a**4*b**2*c
**3*d**3*(10*a**2*d**2 - 15*a*b*c*d + 6*b**2*c**2)/(a*d - b*c)**3 + 4*a**3*b**3*c**4*d**2*(10*a**2*d**2 - 15*a
*b*c*d + 6*b**2*c**2)/(a*d - b*c)**3 + 10*a**3*b**2*c**3*d**2 - 6*a**2*b**4*c**5*d*(10*a**2*d**2 - 15*a*b*c*d
+ 6*b**2*c**2)/(a*d - b*c)**3 - 15*a**2*b**3*c**4*d + 4*a*b**5*c**6*(10*a**2*d**2 - 15*a*b*c*d + 6*b**2*c**2)/
(a*d - b*c)**3 + 6*a*b**4*c**5 - b**6*c**7*(10*a**2*d**2 - 15*a*b*c*d + 6*b**2*c**2)/(d*(a*d - b*c)**3))/(a**5
*d**5 + 10*a**2*b**3*c**3*d**2 - 15*a*b**4*c**4*d + 6*b**5*c**5))/(d**5*(a*d - b*c)**3) + x*(-a/(b**2*d**3) -
3*c/(b*d**4)) + (-9*a*c**5*d + 7*b*c**6 + x*(-10*a*c**4*d**2 + 8*b*c**5*d))/(2*a**2*c**2*d**7 - 4*a*b*c**3*d**
6 + 2*b**2*c**4*d**5 + x**2*(2*a**2*d**9 - 4*a*b*c*d**8 + 2*b**2*c**2*d**7) + x*(4*a**2*c*d**8 - 8*a*b*c**2*d*
*7 + 4*b**2*c**3*d**6)) + x**2/(2*b*d**3)

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