3.279 \(\int \frac{(c+d x)^3}{x^6 (a+b x)^2} \, dx\)

Optimal. Leaf size=199 \[ -\frac{b^2 (b c-a d)^3}{a^6 (a+b x)}-\frac{3 b^2 \log (x) (b c-a d)^2 (2 b c-a d)}{a^7}+\frac{3 b^2 (b c-a d)^2 (2 b c-a d) \log (a+b x)}{a^7}+\frac{c^2 (2 b c-3 a d)}{4 a^3 x^4}-\frac{c (b c-a d)^2}{a^4 x^3}+\frac{(b c-a d)^2 (4 b c-a d)}{2 a^5 x^2}-\frac{b (5 b c-2 a d) (b c-a d)^2}{a^6 x}-\frac{c^3}{5 a^2 x^5} \]

[Out]

-c^3/(5*a^2*x^5) + (c^2*(2*b*c - 3*a*d))/(4*a^3*x^4) - (c*(b*c - a*d)^2)/(a^4*x^3) + ((b*c - a*d)^2*(4*b*c - a
*d))/(2*a^5*x^2) - (b*(5*b*c - 2*a*d)*(b*c - a*d)^2)/(a^6*x) - (b^2*(b*c - a*d)^3)/(a^6*(a + b*x)) - (3*b^2*(b
*c - a*d)^2*(2*b*c - a*d)*Log[x])/a^7 + (3*b^2*(b*c - a*d)^2*(2*b*c - a*d)*Log[a + b*x])/a^7

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Rubi [A]  time = 0.1816, antiderivative size = 199, normalized size of antiderivative = 1., 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} \[ -\frac{b^2 (b c-a d)^3}{a^6 (a+b x)}-\frac{3 b^2 \log (x) (b c-a d)^2 (2 b c-a d)}{a^7}+\frac{3 b^2 (b c-a d)^2 (2 b c-a d) \log (a+b x)}{a^7}+\frac{c^2 (2 b c-3 a d)}{4 a^3 x^4}-\frac{c (b c-a d)^2}{a^4 x^3}+\frac{(b c-a d)^2 (4 b c-a d)}{2 a^5 x^2}-\frac{b (5 b c-2 a d) (b c-a d)^2}{a^6 x}-\frac{c^3}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^6*(a + b*x)^2),x]

[Out]

-c^3/(5*a^2*x^5) + (c^2*(2*b*c - 3*a*d))/(4*a^3*x^4) - (c*(b*c - a*d)^2)/(a^4*x^3) + ((b*c - a*d)^2*(4*b*c - a
*d))/(2*a^5*x^2) - (b*(5*b*c - 2*a*d)*(b*c - a*d)^2)/(a^6*x) - (b^2*(b*c - a*d)^3)/(a^6*(a + b*x)) - (3*b^2*(b
*c - a*d)^2*(2*b*c - a*d)*Log[x])/a^7 + (3*b^2*(b*c - a*d)^2*(2*b*c - a*d)*Log[a + b*x])/a^7

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^6 (a+b x)^2} \, dx &=\int \left (\frac{c^3}{a^2 x^6}+\frac{c^2 (-2 b c+3 a d)}{a^3 x^5}+\frac{3 c (-b c+a d)^2}{a^4 x^4}+\frac{(-4 b c+a d) (-b c+a d)^2}{a^5 x^3}-\frac{b (-b c+a d)^2 (-5 b c+2 a d)}{a^6 x^2}+\frac{3 b^2 (-2 b c+a d) (-b c+a d)^2}{a^7 x}-\frac{b^3 (-b c+a d)^3}{a^6 (a+b x)^2}-\frac{3 b^3 (-2 b c+a d) (-b c+a d)^2}{a^7 (a+b x)}\right ) \, dx\\ &=-\frac{c^3}{5 a^2 x^5}+\frac{c^2 (2 b c-3 a d)}{4 a^3 x^4}-\frac{c (b c-a d)^2}{a^4 x^3}+\frac{(b c-a d)^2 (4 b c-a d)}{2 a^5 x^2}-\frac{b (5 b c-2 a d) (b c-a d)^2}{a^6 x}-\frac{b^2 (b c-a d)^3}{a^6 (a+b x)}-\frac{3 b^2 (b c-a d)^2 (2 b c-a d) \log (x)}{a^7}+\frac{3 b^2 (b c-a d)^2 (2 b c-a d) \log (a+b x)}{a^7}\\ \end{align*}

Mathematica [A]  time = 0.0955193, size = 189, normalized size = 0.95 \[ -\frac{\frac{5 a^4 c^2 (3 a d-2 b c)}{x^4}+\frac{20 a^3 c (b c-a d)^2}{x^3}+\frac{10 a^2 (b c-a d)^2 (a d-4 b c)}{x^2}+\frac{4 a^5 c^3}{x^5}-\frac{20 a b^2 (a d-b c)^3}{a+b x}+60 b^2 \log (x) (b c-a d)^2 (2 b c-a d)-60 b^2 (b c-a d)^2 (2 b c-a d) \log (a+b x)-\frac{20 a b (b c-a d)^2 (2 a d-5 b c)}{x}}{20 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(x^6*(a + b*x)^2),x]

[Out]

-((4*a^5*c^3)/x^5 + (5*a^4*c^2*(-2*b*c + 3*a*d))/x^4 + (20*a^3*c*(b*c - a*d)^2)/x^3 + (10*a^2*(b*c - a*d)^2*(-
4*b*c + a*d))/x^2 - (20*a*b*(b*c - a*d)^2*(-5*b*c + 2*a*d))/x - (20*a*b^2*(-(b*c) + a*d)^3)/(a + b*x) + 60*b^2
*(b*c - a*d)^2*(2*b*c - a*d)*Log[x] - 60*b^2*(b*c - a*d)^2*(2*b*c - a*d)*Log[a + b*x])/(20*a^7)

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Maple [A]  time = 0.013, size = 382, normalized size = 1.9 \begin{align*} 3\,{\frac{c{d}^{2}b}{{a}^{3}{x}^{2}}}-{\frac{9\,{c}^{2}d{b}^{2}}{2\,{a}^{4}{x}^{2}}}-{\frac{{c}^{3}}{5\,{a}^{2}{x}^{5}}}-{\frac{{d}^{3}}{2\,{a}^{2}{x}^{2}}}+2\,{\frac{{c}^{2}db}{{a}^{3}{x}^{3}}}-9\,{\frac{c{d}^{2}{b}^{2}}{{a}^{4}x}}+12\,{\frac{{c}^{2}d{b}^{3}}{{a}^{5}x}}-12\,{\frac{{b}^{3}\ln \left ( x \right ) c{d}^{2}}{{a}^{5}}}+15\,{\frac{{b}^{4}\ln \left ( x \right ){c}^{2}d}{{a}^{6}}}-3\,{\frac{c{d}^{2}{b}^{3}}{{a}^{4} \left ( bx+a \right ) }}+3\,{\frac{{c}^{2}d{b}^{4}}{{a}^{5} \left ( bx+a \right ) }}+12\,{\frac{{b}^{3}\ln \left ( bx+a \right ) c{d}^{2}}{{a}^{5}}}-15\,{\frac{{b}^{4}\ln \left ( bx+a \right ){c}^{2}d}{{a}^{6}}}+2\,{\frac{{d}^{3}b}{{a}^{3}x}}-5\,{\frac{{c}^{3}{b}^{4}}{{a}^{6}x}}+3\,{\frac{{b}^{2}\ln \left ( x \right ){d}^{3}}{{a}^{4}}}-6\,{\frac{{b}^{5}\ln \left ( x \right ){c}^{3}}{{a}^{7}}}-{\frac{c{d}^{2}}{{a}^{2}{x}^{3}}}-{\frac{{c}^{3}{b}^{2}}{{a}^{4}{x}^{3}}}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ){d}^{3}}{{a}^{4}}}+6\,{\frac{{b}^{5}\ln \left ( bx+a \right ){c}^{3}}{{a}^{7}}}+{\frac{{d}^{3}{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}-{\frac{{c}^{3}{b}^{5}}{{a}^{6} \left ( bx+a \right ) }}+2\,{\frac{{b}^{3}{c}^{3}}{{a}^{5}{x}^{2}}}-{\frac{3\,{c}^{2}d}{4\,{a}^{2}{x}^{4}}}+{\frac{{c}^{3}b}{2\,{a}^{3}{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/x^6/(b*x+a)^2,x)

[Out]

3/a^3/x^2*c*b*d^2-9/2/a^4/x^2*b^2*c^2*d-1/5*c^3/a^2/x^5-1/2/a^2/x^2*d^3+2*c^2/a^3/x^3*b*d-9*b^2/a^4/x*c*d^2+12
*b^3/a^5/x*c^2*d-12*b^3/a^5*ln(x)*c*d^2+15*b^4/a^6*ln(x)*c^2*d-3*b^3/a^4/(b*x+a)*c*d^2+3*b^4/a^5/(b*x+a)*c^2*d
+12*b^3/a^5*ln(b*x+a)*c*d^2-15*b^4/a^6*ln(b*x+a)*c^2*d+2*b/a^3/x*d^3-5*b^4/a^6/x*c^3+3*b^2/a^4*ln(x)*d^3-6*b^5
/a^7*ln(x)*c^3-c/a^2/x^3*d^2-c^3/a^4/x^3*b^2-3*b^2/a^4*ln(b*x+a)*d^3+6*b^5/a^7*ln(b*x+a)*c^3+b^2/a^3/(b*x+a)*d
^3-b^5/a^6/(b*x+a)*c^3+2/a^5/x^2*b^3*c^3-3/4*c^2/a^2/x^4*d+1/2*c^3/a^3/x^4*b

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Maxima [A]  time = 2.44701, size = 448, normalized size = 2.25 \begin{align*} -\frac{4 \, a^{5} c^{3} + 60 \,{\left (2 \, b^{5} c^{3} - 5 \, a b^{4} c^{2} d + 4 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 30 \,{\left (2 \, a b^{4} c^{3} - 5 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} - 10 \,{\left (2 \, a^{2} b^{3} c^{3} - 5 \, a^{3} b^{2} c^{2} d + 4 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 5 \,{\left (2 \, a^{3} b^{2} c^{3} - 5 \, a^{4} b c^{2} d + 4 \, a^{5} c d^{2}\right )} x^{2} - 3 \,{\left (2 \, a^{4} b c^{3} - 5 \, a^{5} c^{2} d\right )} x}{20 \,{\left (a^{6} b x^{6} + a^{7} x^{5}\right )}} + \frac{3 \,{\left (2 \, b^{5} c^{3} - 5 \, a b^{4} c^{2} d + 4 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (b x + a\right )}{a^{7}} - \frac{3 \,{\left (2 \, b^{5} c^{3} - 5 \, a b^{4} c^{2} d + 4 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (x\right )}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^6/(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/20*(4*a^5*c^3 + 60*(2*b^5*c^3 - 5*a*b^4*c^2*d + 4*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^5 + 30*(2*a*b^4*c^3 - 5*a^
2*b^3*c^2*d + 4*a^3*b^2*c*d^2 - a^4*b*d^3)*x^4 - 10*(2*a^2*b^3*c^3 - 5*a^3*b^2*c^2*d + 4*a^4*b*c*d^2 - a^5*d^3
)*x^3 + 5*(2*a^3*b^2*c^3 - 5*a^4*b*c^2*d + 4*a^5*c*d^2)*x^2 - 3*(2*a^4*b*c^3 - 5*a^5*c^2*d)*x)/(a^6*b*x^6 + a^
7*x^5) + 3*(2*b^5*c^3 - 5*a*b^4*c^2*d + 4*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(b*x + a)/a^7 - 3*(2*b^5*c^3 - 5*a*b
^4*c^2*d + 4*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(x)/a^7

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Fricas [B]  time = 2.41576, size = 872, normalized size = 4.38 \begin{align*} -\frac{4 \, a^{6} c^{3} + 60 \,{\left (2 \, a b^{5} c^{3} - 5 \, a^{2} b^{4} c^{2} d + 4 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{5} + 30 \,{\left (2 \, a^{2} b^{4} c^{3} - 5 \, a^{3} b^{3} c^{2} d + 4 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x^{4} - 10 \,{\left (2 \, a^{3} b^{3} c^{3} - 5 \, a^{4} b^{2} c^{2} d + 4 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} x^{3} + 5 \,{\left (2 \, a^{4} b^{2} c^{3} - 5 \, a^{5} b c^{2} d + 4 \, a^{6} c d^{2}\right )} x^{2} - 3 \,{\left (2 \, a^{5} b c^{3} - 5 \, a^{6} c^{2} d\right )} x - 60 \,{\left ({\left (2 \, b^{6} c^{3} - 5 \, a b^{5} c^{2} d + 4 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{6} +{\left (2 \, a b^{5} c^{3} - 5 \, a^{2} b^{4} c^{2} d + 4 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{5}\right )} \log \left (b x + a\right ) + 60 \,{\left ({\left (2 \, b^{6} c^{3} - 5 \, a b^{5} c^{2} d + 4 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{6} +{\left (2 \, a b^{5} c^{3} - 5 \, a^{2} b^{4} c^{2} d + 4 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{5}\right )} \log \left (x\right )}{20 \,{\left (a^{7} b x^{6} + a^{8} x^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^6/(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/20*(4*a^6*c^3 + 60*(2*a*b^5*c^3 - 5*a^2*b^4*c^2*d + 4*a^3*b^3*c*d^2 - a^4*b^2*d^3)*x^5 + 30*(2*a^2*b^4*c^3
- 5*a^3*b^3*c^2*d + 4*a^4*b^2*c*d^2 - a^5*b*d^3)*x^4 - 10*(2*a^3*b^3*c^3 - 5*a^4*b^2*c^2*d + 4*a^5*b*c*d^2 - a
^6*d^3)*x^3 + 5*(2*a^4*b^2*c^3 - 5*a^5*b*c^2*d + 4*a^6*c*d^2)*x^2 - 3*(2*a^5*b*c^3 - 5*a^6*c^2*d)*x - 60*((2*b
^6*c^3 - 5*a*b^5*c^2*d + 4*a^2*b^4*c*d^2 - a^3*b^3*d^3)*x^6 + (2*a*b^5*c^3 - 5*a^2*b^4*c^2*d + 4*a^3*b^3*c*d^2
- a^4*b^2*d^3)*x^5)*log(b*x + a) + 60*((2*b^6*c^3 - 5*a*b^5*c^2*d + 4*a^2*b^4*c*d^2 - a^3*b^3*d^3)*x^6 + (2*a
*b^5*c^3 - 5*a^2*b^4*c^2*d + 4*a^3*b^3*c*d^2 - a^4*b^2*d^3)*x^5)*log(x))/(a^7*b*x^6 + a^8*x^5)

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Sympy [B]  time = 2.81629, size = 530, normalized size = 2.66 \begin{align*} \frac{- 4 a^{5} c^{3} + x^{5} \left (60 a^{3} b^{2} d^{3} - 240 a^{2} b^{3} c d^{2} + 300 a b^{4} c^{2} d - 120 b^{5} c^{3}\right ) + x^{4} \left (30 a^{4} b d^{3} - 120 a^{3} b^{2} c d^{2} + 150 a^{2} b^{3} c^{2} d - 60 a b^{4} c^{3}\right ) + x^{3} \left (- 10 a^{5} d^{3} + 40 a^{4} b c d^{2} - 50 a^{3} b^{2} c^{2} d + 20 a^{2} b^{3} c^{3}\right ) + x^{2} \left (- 20 a^{5} c d^{2} + 25 a^{4} b c^{2} d - 10 a^{3} b^{2} c^{3}\right ) + x \left (- 15 a^{5} c^{2} d + 6 a^{4} b c^{3}\right )}{20 a^{7} x^{5} + 20 a^{6} b x^{6}} + \frac{3 b^{2} \left (a d - 2 b c\right ) \left (a d - b c\right )^{2} \log{\left (x + \frac{3 a^{4} b^{2} d^{3} - 12 a^{3} b^{3} c d^{2} + 15 a^{2} b^{4} c^{2} d - 6 a b^{5} c^{3} - 3 a b^{2} \left (a d - 2 b c\right ) \left (a d - b c\right )^{2}}{6 a^{3} b^{3} d^{3} - 24 a^{2} b^{4} c d^{2} + 30 a b^{5} c^{2} d - 12 b^{6} c^{3}} \right )}}{a^{7}} - \frac{3 b^{2} \left (a d - 2 b c\right ) \left (a d - b c\right )^{2} \log{\left (x + \frac{3 a^{4} b^{2} d^{3} - 12 a^{3} b^{3} c d^{2} + 15 a^{2} b^{4} c^{2} d - 6 a b^{5} c^{3} + 3 a b^{2} \left (a d - 2 b c\right ) \left (a d - b c\right )^{2}}{6 a^{3} b^{3} d^{3} - 24 a^{2} b^{4} c d^{2} + 30 a b^{5} c^{2} d - 12 b^{6} c^{3}} \right )}}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/x**6/(b*x+a)**2,x)

[Out]

(-4*a**5*c**3 + x**5*(60*a**3*b**2*d**3 - 240*a**2*b**3*c*d**2 + 300*a*b**4*c**2*d - 120*b**5*c**3) + x**4*(30
*a**4*b*d**3 - 120*a**3*b**2*c*d**2 + 150*a**2*b**3*c**2*d - 60*a*b**4*c**3) + x**3*(-10*a**5*d**3 + 40*a**4*b
*c*d**2 - 50*a**3*b**2*c**2*d + 20*a**2*b**3*c**3) + x**2*(-20*a**5*c*d**2 + 25*a**4*b*c**2*d - 10*a**3*b**2*c
**3) + x*(-15*a**5*c**2*d + 6*a**4*b*c**3))/(20*a**7*x**5 + 20*a**6*b*x**6) + 3*b**2*(a*d - 2*b*c)*(a*d - b*c)
**2*log(x + (3*a**4*b**2*d**3 - 12*a**3*b**3*c*d**2 + 15*a**2*b**4*c**2*d - 6*a*b**5*c**3 - 3*a*b**2*(a*d - 2*
b*c)*(a*d - b*c)**2)/(6*a**3*b**3*d**3 - 24*a**2*b**4*c*d**2 + 30*a*b**5*c**2*d - 12*b**6*c**3))/a**7 - 3*b**2
*(a*d - 2*b*c)*(a*d - b*c)**2*log(x + (3*a**4*b**2*d**3 - 12*a**3*b**3*c*d**2 + 15*a**2*b**4*c**2*d - 6*a*b**5
*c**3 + 3*a*b**2*(a*d - 2*b*c)*(a*d - b*c)**2)/(6*a**3*b**3*d**3 - 24*a**2*b**4*c*d**2 + 30*a*b**5*c**2*d - 12
*b**6*c**3))/a**7

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Giac [B]  time = 1.18006, size = 589, normalized size = 2.96 \begin{align*} -\frac{3 \,{\left (2 \, b^{6} c^{3} - 5 \, a b^{5} c^{2} d + 4 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{7} b} - \frac{\frac{b^{11} c^{3}}{b x + a} - \frac{3 \, a b^{10} c^{2} d}{b x + a} + \frac{3 \, a^{2} b^{9} c d^{2}}{b x + a} - \frac{a^{3} b^{8} d^{3}}{b x + a}}{a^{6} b^{6}} + \frac{174 \, b^{5} c^{3} - 385 \, a b^{4} c^{2} d + 260 \, a^{2} b^{3} c d^{2} - 50 \, a^{3} b^{2} d^{3} - \frac{5 \,{\left (154 \, a b^{6} c^{3} - 337 \, a^{2} b^{5} c^{2} d + 224 \, a^{3} b^{4} c d^{2} - 42 \, a^{4} b^{3} d^{3}\right )}}{{\left (b x + a\right )} b} + \frac{10 \,{\left (130 \, a^{2} b^{7} c^{3} - 280 \, a^{3} b^{6} c^{2} d + 182 \, a^{4} b^{5} c d^{2} - 33 \, a^{5} b^{4} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac{10 \,{\left (100 \, a^{3} b^{8} c^{3} - 210 \, a^{4} b^{7} c^{2} d + 132 \, a^{5} b^{6} c d^{2} - 23 \, a^{6} b^{5} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac{60 \,{\left (5 \, a^{4} b^{9} c^{3} - 10 \, a^{5} b^{8} c^{2} d + 6 \, a^{6} b^{7} c d^{2} - a^{7} b^{6} d^{3}\right )}}{{\left (b x + a\right )}^{4} b^{4}}}{20 \, a^{7}{\left (\frac{a}{b x + a} - 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^6/(b*x+a)^2,x, algorithm="giac")

[Out]

-3*(2*b^6*c^3 - 5*a*b^5*c^2*d + 4*a^2*b^4*c*d^2 - a^3*b^3*d^3)*log(abs(-a/(b*x + a) + 1))/(a^7*b) - (b^11*c^3/
(b*x + a) - 3*a*b^10*c^2*d/(b*x + a) + 3*a^2*b^9*c*d^2/(b*x + a) - a^3*b^8*d^3/(b*x + a))/(a^6*b^6) + 1/20*(17
4*b^5*c^3 - 385*a*b^4*c^2*d + 260*a^2*b^3*c*d^2 - 50*a^3*b^2*d^3 - 5*(154*a*b^6*c^3 - 337*a^2*b^5*c^2*d + 224*
a^3*b^4*c*d^2 - 42*a^4*b^3*d^3)/((b*x + a)*b) + 10*(130*a^2*b^7*c^3 - 280*a^3*b^6*c^2*d + 182*a^4*b^5*c*d^2 -
33*a^5*b^4*d^3)/((b*x + a)^2*b^2) - 10*(100*a^3*b^8*c^3 - 210*a^4*b^7*c^2*d + 132*a^5*b^6*c*d^2 - 23*a^6*b^5*d
^3)/((b*x + a)^3*b^3) + 60*(5*a^4*b^9*c^3 - 10*a^5*b^8*c^2*d + 6*a^6*b^7*c*d^2 - a^7*b^6*d^3)/((b*x + a)^4*b^4
))/(a^7*(a/(b*x + a) - 1)^5)