3.3.94 \(\int \frac {(c+d x)^3}{x^4 (a+b x)^3} \, dx\)

Optimal. Leaf size=193 \[ -\frac {3 c (b c-a d) (2 b c-a d)}{a^5 x}-\frac {(b c-a d)^2 (4 b c-a d)}{a^5 (a+b x)}+\frac {3 c^2 (b c-a d)}{2 a^4 x^2}-\frac {(b c-a d)^3}{2 a^4 (a+b x)^2}-\frac {c^3}{3 a^3 x^3}-\frac {\log (x) (b c-a d) \left (a^2 d^2-8 a b c d+10 b^2 c^2\right )}{a^6}+\frac {(b c-a d) \left (a^2 d^2-8 a b c d+10 b^2 c^2\right ) \log (a+b x)}{a^6} \]

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Rubi [A]  time = 0.18, antiderivative size = 193, 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 {\log (x) (b c-a d) \left (a^2 d^2-8 a b c d+10 b^2 c^2\right )}{a^6}+\frac {(b c-a d) \left (a^2 d^2-8 a b c d+10 b^2 c^2\right ) \log (a+b x)}{a^6}+\frac {3 c^2 (b c-a d)}{2 a^4 x^2}-\frac {3 c (b c-a d) (2 b c-a d)}{a^5 x}-\frac {(b c-a d)^2 (4 b c-a d)}{a^5 (a+b x)}-\frac {(b c-a d)^3}{2 a^4 (a+b x)^2}-\frac {c^3}{3 a^3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^4*(a + b*x)^3),x]

[Out]

-c^3/(3*a^3*x^3) + (3*c^2*(b*c - a*d))/(2*a^4*x^2) - (3*c*(b*c - a*d)*(2*b*c - a*d))/(a^5*x) - (b*c - a*d)^3/(
2*a^4*(a + b*x)^2) - ((b*c - a*d)^2*(4*b*c - a*d))/(a^5*(a + b*x)) - ((b*c - a*d)*(10*b^2*c^2 - 8*a*b*c*d + a^
2*d^2)*Log[x])/a^6 + ((b*c - a*d)*(10*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*Log[a + b*x])/a^6

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 {(c+d x)^3}{x^4 (a+b x)^3} \, dx &=\int \left (\frac {c^3}{a^3 x^4}+\frac {3 c^2 (-b c+a d)}{a^4 x^3}+\frac {3 c (b c-a d) (2 b c-a d)}{a^5 x^2}+\frac {(b c-a d) \left (-10 b^2 c^2+8 a b c d-a^2 d^2\right )}{a^6 x}-\frac {b (-b c+a d)^3}{a^4 (a+b x)^3}-\frac {b (-4 b c+a d) (-b c+a d)^2}{a^5 (a+b x)^2}+\frac {b (b c-a d) \left (10 b^2 c^2-8 a b c d+a^2 d^2\right )}{a^6 (a+b x)}\right ) \, dx\\ &=-\frac {c^3}{3 a^3 x^3}+\frac {3 c^2 (b c-a d)}{2 a^4 x^2}-\frac {3 c (b c-a d) (2 b c-a d)}{a^5 x}-\frac {(b c-a d)^3}{2 a^4 (a+b x)^2}-\frac {(b c-a d)^2 (4 b c-a d)}{a^5 (a+b x)}-\frac {(b c-a d) \left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) \log (x)}{a^6}+\frac {(b c-a d) \left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) \log (a+b x)}{a^6}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 202, normalized size = 1.05 \begin {gather*} \frac {-\frac {2 a^3 c^3}{x^3}-\frac {18 a c \left (a^2 d^2-3 a b c d+2 b^2 c^2\right )}{x}-\frac {9 a^2 c^2 (a d-b c)}{x^2}+\frac {3 a^2 (a d-b c)^3}{(a+b x)^2}+6 \log (x) \left (a^3 d^3-9 a^2 b c d^2+18 a b^2 c^2 d-10 b^3 c^3\right )+6 \left (-a^3 d^3+9 a^2 b c d^2-18 a b^2 c^2 d+10 b^3 c^3\right ) \log (a+b x)+\frac {6 a (b c-a d)^2 (a d-4 b c)}{a+b x}}{6 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(x^4*(a + b*x)^3),x]

[Out]

((-2*a^3*c^3)/x^3 - (9*a^2*c^2*(-(b*c) + a*d))/x^2 - (18*a*c*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2))/x + (3*a^2*(-(
b*c) + a*d)^3)/(a + b*x)^2 + (6*a*(b*c - a*d)^2*(-4*b*c + a*d))/(a + b*x) + 6*(-10*b^3*c^3 + 18*a*b^2*c^2*d -
9*a^2*b*c*d^2 + a^3*d^3)*Log[x] + 6*(10*b^3*c^3 - 18*a*b^2*c^2*d + 9*a^2*b*c*d^2 - a^3*d^3)*Log[a + b*x])/(6*a
^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^3}{x^4 (a+b x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^3/(x^4*(a + b*x)^3),x]

[Out]

IntegrateAlgebraic[(c + d*x)^3/(x^4*(a + b*x)^3), x]

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fricas [B]  time = 1.71, size = 487, normalized size = 2.52 \begin {gather*} -\frac {2 \, a^{5} c^{3} + 6 \, {\left (10 \, a b^{4} c^{3} - 18 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + 9 \, {\left (10 \, a^{2} b^{3} c^{3} - 18 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 2 \, {\left (10 \, a^{3} b^{2} c^{3} - 18 \, a^{4} b c^{2} d + 9 \, a^{5} c d^{2}\right )} x^{2} - {\left (5 \, a^{4} b c^{3} - 9 \, a^{5} c^{2} d\right )} x - 6 \, {\left ({\left (10 \, b^{5} c^{3} - 18 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 2 \, {\left (10 \, a b^{4} c^{3} - 18 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + {\left (10 \, a^{2} b^{3} c^{3} - 18 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) + 6 \, {\left ({\left (10 \, b^{5} c^{3} - 18 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 2 \, {\left (10 \, a b^{4} c^{3} - 18 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + {\left (10 \, a^{2} b^{3} c^{3} - 18 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3}\right )} \log \relax (x)}{6 \, {\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^4/(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/6*(2*a^5*c^3 + 6*(10*a*b^4*c^3 - 18*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - a^4*b*d^3)*x^4 + 9*(10*a^2*b^3*c^3 -
18*a^3*b^2*c^2*d + 9*a^4*b*c*d^2 - a^5*d^3)*x^3 + 2*(10*a^3*b^2*c^3 - 18*a^4*b*c^2*d + 9*a^5*c*d^2)*x^2 - (5*a
^4*b*c^3 - 9*a^5*c^2*d)*x - 6*((10*b^5*c^3 - 18*a*b^4*c^2*d + 9*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^5 + 2*(10*a*b^4
*c^3 - 18*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - a^4*b*d^3)*x^4 + (10*a^2*b^3*c^3 - 18*a^3*b^2*c^2*d + 9*a^4*b*c*d^
2 - a^5*d^3)*x^3)*log(b*x + a) + 6*((10*b^5*c^3 - 18*a*b^4*c^2*d + 9*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^5 + 2*(10*
a*b^4*c^3 - 18*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - a^4*b*d^3)*x^4 + (10*a^2*b^3*c^3 - 18*a^3*b^2*c^2*d + 9*a^4*b
*c*d^2 - a^5*d^3)*x^3)*log(x))/(a^6*b^2*x^5 + 2*a^7*b*x^4 + a^8*x^3)

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giac [A]  time = 1.01, size = 277, normalized size = 1.44 \begin {gather*} -\frac {{\left (10 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {{\left (10 \, b^{4} c^{3} - 18 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac {2 \, a^{5} c^{3} + 6 \, {\left (10 \, a b^{4} c^{3} - 18 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + 9 \, {\left (10 \, a^{2} b^{3} c^{3} - 18 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 2 \, {\left (10 \, a^{3} b^{2} c^{3} - 18 \, a^{4} b c^{2} d + 9 \, a^{5} c d^{2}\right )} x^{2} - {\left (5 \, a^{4} b c^{3} - 9 \, a^{5} c^{2} d\right )} x}{6 \, {\left (b x + a\right )}^{2} a^{6} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^4/(b*x+a)^3,x, algorithm="giac")

[Out]

-(10*b^3*c^3 - 18*a*b^2*c^2*d + 9*a^2*b*c*d^2 - a^3*d^3)*log(abs(x))/a^6 + (10*b^4*c^3 - 18*a*b^3*c^2*d + 9*a^
2*b^2*c*d^2 - a^3*b*d^3)*log(abs(b*x + a))/(a^6*b) - 1/6*(2*a^5*c^3 + 6*(10*a*b^4*c^3 - 18*a^2*b^3*c^2*d + 9*a
^3*b^2*c*d^2 - a^4*b*d^3)*x^4 + 9*(10*a^2*b^3*c^3 - 18*a^3*b^2*c^2*d + 9*a^4*b*c*d^2 - a^5*d^3)*x^3 + 2*(10*a^
3*b^2*c^3 - 18*a^4*b*c^2*d + 9*a^5*c*d^2)*x^2 - (5*a^4*b*c^3 - 9*a^5*c^2*d)*x)/((b*x + a)^2*a^6*x^3)

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maple [A]  time = 0.01, size = 326, normalized size = 1.69 \begin {gather*} \frac {d^{3}}{2 \left (b x +a \right )^{2} a}-\frac {3 b c \,d^{2}}{2 \left (b x +a \right )^{2} a^{2}}+\frac {3 b^{2} c^{2} d}{2 \left (b x +a \right )^{2} a^{3}}-\frac {b^{3} c^{3}}{2 \left (b x +a \right )^{2} a^{4}}+\frac {d^{3}}{\left (b x +a \right ) a^{2}}-\frac {6 b c \,d^{2}}{\left (b x +a \right ) a^{3}}+\frac {d^{3} \ln \relax (x )}{a^{3}}-\frac {d^{3} \ln \left (b x +a \right )}{a^{3}}+\frac {9 b^{2} c^{2} d}{\left (b x +a \right ) a^{4}}-\frac {9 b c \,d^{2} \ln \relax (x )}{a^{4}}+\frac {9 b c \,d^{2} \ln \left (b x +a \right )}{a^{4}}-\frac {4 b^{3} c^{3}}{\left (b x +a \right ) a^{5}}+\frac {18 b^{2} c^{2} d \ln \relax (x )}{a^{5}}-\frac {18 b^{2} c^{2} d \ln \left (b x +a \right )}{a^{5}}-\frac {10 b^{3} c^{3} \ln \relax (x )}{a^{6}}+\frac {10 b^{3} c^{3} \ln \left (b x +a \right )}{a^{6}}-\frac {3 c \,d^{2}}{a^{3} x}+\frac {9 b \,c^{2} d}{a^{4} x}-\frac {6 b^{2} c^{3}}{a^{5} x}-\frac {3 c^{2} d}{2 a^{3} x^{2}}+\frac {3 b \,c^{3}}{2 a^{4} x^{2}}-\frac {c^{3}}{3 a^{3} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/x^4/(b*x+a)^3,x)

[Out]

-1/3*c^3/a^3/x^3+1/a^3*ln(x)*d^3-9/a^4*ln(x)*b*c*d^2+18/a^5*ln(x)*b^2*c^2*d-10/a^6*ln(x)*b^3*c^3-3*c/a^3/x*d^2
+9*c^2/a^4/x*b*d-6*c^3/a^5/x*b^2-3/2*c^2/a^3/x^2*d+3/2*c^3/a^4/x^2*b-1/a^3*ln(b*x+a)*d^3+9/a^4*ln(b*x+a)*b*c*d
^2-18/a^5*ln(b*x+a)*b^2*c^2*d+10/a^6*ln(b*x+a)*b^3*c^3+1/a^2/(b*x+a)*d^3-6/a^3/(b*x+a)*b*c*d^2+9/a^4/(b*x+a)*b
^2*c^2*d-4/a^5/(b*x+a)*b^3*c^3+1/2/a/(b*x+a)^2*d^3-3/2/a^2/(b*x+a)^2*b*c*d^2+3/2/a^3/(b*x+a)^2*b^2*c^2*d-1/2/a
^4/(b*x+a)^2*b^3*c^3

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maxima [A]  time = 0.96, size = 280, normalized size = 1.45 \begin {gather*} -\frac {2 \, a^{4} c^{3} + 6 \, {\left (10 \, b^{4} c^{3} - 18 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} + 9 \, {\left (10 \, a b^{3} c^{3} - 18 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3} + 2 \, {\left (10 \, a^{2} b^{2} c^{3} - 18 \, a^{3} b c^{2} d + 9 \, a^{4} c d^{2}\right )} x^{2} - {\left (5 \, a^{3} b c^{3} - 9 \, a^{4} c^{2} d\right )} x}{6 \, {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} + \frac {{\left (10 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{6}} - \frac {{\left (10 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \relax (x)}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^4/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/6*(2*a^4*c^3 + 6*(10*b^4*c^3 - 18*a*b^3*c^2*d + 9*a^2*b^2*c*d^2 - a^3*b*d^3)*x^4 + 9*(10*a*b^3*c^3 - 18*a^2
*b^2*c^2*d + 9*a^3*b*c*d^2 - a^4*d^3)*x^3 + 2*(10*a^2*b^2*c^3 - 18*a^3*b*c^2*d + 9*a^4*c*d^2)*x^2 - (5*a^3*b*c
^3 - 9*a^4*c^2*d)*x)/(a^5*b^2*x^5 + 2*a^6*b*x^4 + a^7*x^3) + (10*b^3*c^3 - 18*a*b^2*c^2*d + 9*a^2*b*c*d^2 - a^
3*d^3)*log(b*x + a)/a^6 - (10*b^3*c^3 - 18*a*b^2*c^2*d + 9*a^2*b*c*d^2 - a^3*d^3)*log(x)/a^6

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mupad [B]  time = 0.53, size = 289, normalized size = 1.50 \begin {gather*} -\frac {\frac {c^3}{3\,a}-\frac {3\,x^3\,\left (a^3\,d^3-9\,a^2\,b\,c\,d^2+18\,a\,b^2\,c^2\,d-10\,b^3\,c^3\right )}{2\,a^4}+\frac {c^2\,x\,\left (9\,a\,d-5\,b\,c\right )}{6\,a^2}+\frac {c\,x^2\,\left (9\,a^2\,d^2-18\,a\,b\,c\,d+10\,b^2\,c^2\right )}{3\,a^3}-\frac {b\,x^4\,\left (a^3\,d^3-9\,a^2\,b\,c\,d^2+18\,a\,b^2\,c^2\,d-10\,b^3\,c^3\right )}{a^5}}{a^2\,x^3+2\,a\,b\,x^4+b^2\,x^5}-\frac {2\,\mathrm {atanh}\left (\frac {\left (a\,d-b\,c\right )\,\left (a+2\,b\,x\right )\,\left (a^2\,d^2-8\,a\,b\,c\,d+10\,b^2\,c^2\right )}{a\,\left (a^3\,d^3-9\,a^2\,b\,c\,d^2+18\,a\,b^2\,c^2\,d-10\,b^3\,c^3\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-8\,a\,b\,c\,d+10\,b^2\,c^2\right )}{a^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(x^4*(a + b*x)^3),x)

[Out]

- (c^3/(3*a) - (3*x^3*(a^3*d^3 - 10*b^3*c^3 + 18*a*b^2*c^2*d - 9*a^2*b*c*d^2))/(2*a^4) + (c^2*x*(9*a*d - 5*b*c
))/(6*a^2) + (c*x^2*(9*a^2*d^2 + 10*b^2*c^2 - 18*a*b*c*d))/(3*a^3) - (b*x^4*(a^3*d^3 - 10*b^3*c^3 + 18*a*b^2*c
^2*d - 9*a^2*b*c*d^2))/a^5)/(a^2*x^3 + b^2*x^5 + 2*a*b*x^4) - (2*atanh(((a*d - b*c)*(a + 2*b*x)*(a^2*d^2 + 10*
b^2*c^2 - 8*a*b*c*d))/(a*(a^3*d^3 - 10*b^3*c^3 + 18*a*b^2*c^2*d - 9*a^2*b*c*d^2)))*(a*d - b*c)*(a^2*d^2 + 10*b
^2*c^2 - 8*a*b*c*d))/a^6

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sympy [B]  time = 2.31, size = 505, normalized size = 2.62 \begin {gather*} \frac {- 2 a^{4} c^{3} + x^{4} \left (6 a^{3} b d^{3} - 54 a^{2} b^{2} c d^{2} + 108 a b^{3} c^{2} d - 60 b^{4} c^{3}\right ) + x^{3} \left (9 a^{4} d^{3} - 81 a^{3} b c d^{2} + 162 a^{2} b^{2} c^{2} d - 90 a b^{3} c^{3}\right ) + x^{2} \left (- 18 a^{4} c d^{2} + 36 a^{3} b c^{2} d - 20 a^{2} b^{2} c^{3}\right ) + x \left (- 9 a^{4} c^{2} d + 5 a^{3} b c^{3}\right )}{6 a^{7} x^{3} + 12 a^{6} b x^{4} + 6 a^{5} b^{2} x^{5}} + \frac {\left (a d - b c\right ) \left (a^{2} d^{2} - 8 a b c d + 10 b^{2} c^{2}\right ) \log {\left (x + \frac {a^{4} d^{3} - 9 a^{3} b c d^{2} + 18 a^{2} b^{2} c^{2} d - 10 a b^{3} c^{3} - a \left (a d - b c\right ) \left (a^{2} d^{2} - 8 a b c d + 10 b^{2} c^{2}\right )}{2 a^{3} b d^{3} - 18 a^{2} b^{2} c d^{2} + 36 a b^{3} c^{2} d - 20 b^{4} c^{3}} \right )}}{a^{6}} - \frac {\left (a d - b c\right ) \left (a^{2} d^{2} - 8 a b c d + 10 b^{2} c^{2}\right ) \log {\left (x + \frac {a^{4} d^{3} - 9 a^{3} b c d^{2} + 18 a^{2} b^{2} c^{2} d - 10 a b^{3} c^{3} + a \left (a d - b c\right ) \left (a^{2} d^{2} - 8 a b c d + 10 b^{2} c^{2}\right )}{2 a^{3} b d^{3} - 18 a^{2} b^{2} c d^{2} + 36 a b^{3} c^{2} d - 20 b^{4} c^{3}} \right )}}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/x**4/(b*x+a)**3,x)

[Out]

(-2*a**4*c**3 + x**4*(6*a**3*b*d**3 - 54*a**2*b**2*c*d**2 + 108*a*b**3*c**2*d - 60*b**4*c**3) + x**3*(9*a**4*d
**3 - 81*a**3*b*c*d**2 + 162*a**2*b**2*c**2*d - 90*a*b**3*c**3) + x**2*(-18*a**4*c*d**2 + 36*a**3*b*c**2*d - 2
0*a**2*b**2*c**3) + x*(-9*a**4*c**2*d + 5*a**3*b*c**3))/(6*a**7*x**3 + 12*a**6*b*x**4 + 6*a**5*b**2*x**5) + (a
*d - b*c)*(a**2*d**2 - 8*a*b*c*d + 10*b**2*c**2)*log(x + (a**4*d**3 - 9*a**3*b*c*d**2 + 18*a**2*b**2*c**2*d -
10*a*b**3*c**3 - a*(a*d - b*c)*(a**2*d**2 - 8*a*b*c*d + 10*b**2*c**2))/(2*a**3*b*d**3 - 18*a**2*b**2*c*d**2 +
36*a*b**3*c**2*d - 20*b**4*c**3))/a**6 - (a*d - b*c)*(a**2*d**2 - 8*a*b*c*d + 10*b**2*c**2)*log(x + (a**4*d**3
- 9*a**3*b*c*d**2 + 18*a**2*b**2*c**2*d - 10*a*b**3*c**3 + a*(a*d - b*c)*(a**2*d**2 - 8*a*b*c*d + 10*b**2*c**
2))/(2*a**3*b*d**3 - 18*a**2*b**2*c*d**2 + 36*a*b**3*c**2*d - 20*b**4*c**3))/a**6

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