3.292 \(\int \frac{x^5}{(a+b x)^2 (c+d x)^3} \, dx\)

Optimal. Leaf size=173 \[ -\frac{c^3 \left (10 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^4}+\frac{a^5}{b^3 (a+b x) (b c-a d)^3}+\frac{a^4 (5 b c-2 a d) \log (a+b x)}{b^3 (b c-a d)^4}-\frac{c^4 (3 b c-5 a d)}{d^4 (c+d x) (b c-a d)^3}+\frac{c^5}{2 d^4 (c+d x)^2 (b c-a d)^2}+\frac{x}{b^2 d^3} \]

[Out]

x/(b^2*d^3) + a^5/(b^3*(b*c - a*d)^3*(a + b*x)) + c^5/(2*d^4*(b*c - a*d)^2*(c + d*x)^2) - (c^4*(3*b*c - 5*a*d)
)/(d^4*(b*c - a*d)^3*(c + d*x)) + (a^4*(5*b*c - 2*a*d)*Log[a + b*x])/(b^3*(b*c - a*d)^4) - (c^3*(3*b^2*c^2 - 1
0*a*b*c*d + 10*a^2*d^2)*Log[c + d*x])/(d^4*(b*c - a*d)^4)

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Rubi [A]  time = 0.204877, antiderivative size = 173, 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} \[ -\frac{c^3 \left (10 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^4}+\frac{a^5}{b^3 (a+b x) (b c-a d)^3}+\frac{a^4 (5 b c-2 a d) \log (a+b x)}{b^3 (b c-a d)^4}-\frac{c^4 (3 b c-5 a d)}{d^4 (c+d x) (b c-a d)^3}+\frac{c^5}{2 d^4 (c+d x)^2 (b c-a d)^2}+\frac{x}{b^2 d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x)^2*(c + d*x)^3),x]

[Out]

x/(b^2*d^3) + a^5/(b^3*(b*c - a*d)^3*(a + b*x)) + c^5/(2*d^4*(b*c - a*d)^2*(c + d*x)^2) - (c^4*(3*b*c - 5*a*d)
)/(d^4*(b*c - a*d)^3*(c + d*x)) + (a^4*(5*b*c - 2*a*d)*Log[a + b*x])/(b^3*(b*c - a*d)^4) - (c^3*(3*b^2*c^2 - 1
0*a*b*c*d + 10*a^2*d^2)*Log[c + d*x])/(d^4*(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^5}{(a+b x)^2 (c+d x)^3} \, dx &=\int \left (\frac{1}{b^2 d^3}-\frac{a^5}{b^2 (b c-a d)^3 (a+b x)^2}-\frac{a^4 (-5 b c+2 a d)}{b^2 (b c-a d)^4 (a+b x)}-\frac{c^5}{d^3 (-b c+a d)^2 (c+d x)^3}-\frac{c^4 (3 b c-5 a d)}{d^3 (-b c+a d)^3 (c+d x)^2}-\frac{c^3 \left (3 b^2 c^2-10 a b c d+10 a^2 d^2\right )}{d^3 (-b c+a d)^4 (c+d x)}\right ) \, dx\\ &=\frac{x}{b^2 d^3}+\frac{a^5}{b^3 (b c-a d)^3 (a+b x)}+\frac{c^5}{2 d^4 (b c-a d)^2 (c+d x)^2}-\frac{c^4 (3 b c-5 a d)}{d^4 (b c-a d)^3 (c+d x)}+\frac{a^4 (5 b c-2 a d) \log (a+b x)}{b^3 (b c-a d)^4}-\frac{c^3 \left (3 b^2 c^2-10 a b c d+10 a^2 d^2\right ) \log (c+d x)}{d^4 (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 0.225364, size = 172, normalized size = 0.99 \[ -\frac{c^3 \left (10 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^4}+\frac{a^5}{b^3 (a+b x) (b c-a d)^3}+\frac{a^4 (5 b c-2 a d) \log (a+b x)}{b^3 (b c-a d)^4}+\frac{c^4 (3 b c-5 a d)}{d^4 (c+d x) (a d-b c)^3}+\frac{c^5}{2 d^4 (c+d x)^2 (b c-a d)^2}+\frac{x}{b^2 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/((a + b*x)^2*(c + d*x)^3),x]

[Out]

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

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Maple [A]  time = 0.015, size = 242, normalized size = 1.4 \begin{align*}{\frac{x}{{b}^{2}{d}^{3}}}-5\,{\frac{{c}^{4}a}{{d}^{3} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}+3\,{\frac{{c}^{5}b}{ \left ( ad-bc \right ) ^{3}{d}^{4} \left ( dx+c \right ) }}+{\frac{{c}^{5}}{2\,{d}^{4} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}-10\,{\frac{{c}^{3}\ln \left ( dx+c \right ){a}^{2}}{{d}^{2} \left ( ad-bc \right ) ^{4}}}+10\,{\frac{{c}^{4}\ln \left ( dx+c \right ) ab}{{d}^{3} \left ( ad-bc \right ) ^{4}}}-3\,{\frac{{c}^{5}\ln \left ( dx+c \right ){b}^{2}}{{d}^{4} \left ( ad-bc \right ) ^{4}}}-{\frac{{a}^{5}}{{b}^{3} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }}-2\,{\frac{{a}^{5}\ln \left ( bx+a \right ) d}{{b}^{3} \left ( ad-bc \right ) ^{4}}}+5\,{\frac{{a}^{4}\ln \left ( bx+a \right ) c}{{b}^{2} \left ( ad-bc \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x+a)^2/(d*x+c)^3,x)

[Out]

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

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Maxima [B]  time = 1.72402, size = 711, normalized size = 4.11 \begin{align*} \frac{{\left (5 \, a^{4} b c - 2 \, a^{5} d\right )} \log \left (b x + a\right )}{b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}} - \frac{{\left (3 \, b^{2} c^{5} - 10 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{4} - 4 \, a b^{3} c^{3} d^{5} + 6 \, a^{2} b^{2} c^{2} d^{6} - 4 \, a^{3} b c d^{7} + a^{4} d^{8}} - \frac{5 \, a b^{4} c^{6} - 9 \, a^{2} b^{3} c^{5} d - 2 \, a^{5} c^{2} d^{4} + 2 \,{\left (3 \, b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} - a^{5} d^{6}\right )} x^{2} +{\left (5 \, b^{5} c^{6} - 3 \, a b^{4} c^{5} d - 10 \, a^{2} b^{3} c^{4} d^{2} - 4 \, a^{5} c d^{5}\right )} x}{2 \,{\left (a b^{6} c^{5} d^{4} - 3 \, a^{2} b^{5} c^{4} d^{5} + 3 \, a^{3} b^{4} c^{3} d^{6} - a^{4} b^{3} c^{2} d^{7} +{\left (b^{7} c^{3} d^{6} - 3 \, a b^{6} c^{2} d^{7} + 3 \, a^{2} b^{5} c d^{8} - a^{3} b^{4} d^{9}\right )} x^{3} +{\left (2 \, b^{7} c^{4} d^{5} - 5 \, a b^{6} c^{3} d^{6} + 3 \, a^{2} b^{5} c^{2} d^{7} + a^{3} b^{4} c d^{8} - a^{4} b^{3} d^{9}\right )} x^{2} +{\left (b^{7} c^{5} d^{4} - a b^{6} c^{4} d^{5} - 3 \, a^{2} b^{5} c^{3} d^{6} + 5 \, a^{3} b^{4} c^{2} d^{7} - 2 \, a^{4} b^{3} c d^{8}\right )} x\right )}} + \frac{x}{b^{2} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(d*x+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 3.7459, size = 1949, normalized size = 11.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/2*(5*a*b^5*c^7 - 14*a^2*b^4*c^6*d + 9*a^3*b^3*c^5*d^2 - 2*a^5*b*c^3*d^4 + 2*a^6*c^2*d^5 - 2*(b^6*c^4*d^3 -
4*a*b^5*c^3*d^4 + 6*a^2*b^4*c^2*d^5 - 4*a^3*b^3*c*d^6 + a^4*b^2*d^7)*x^4 - 2*(2*b^6*c^5*d^2 - 7*a*b^5*c^4*d^3
+ 8*a^2*b^4*c^3*d^4 - 2*a^3*b^3*c^2*d^5 - 2*a^4*b^2*c*d^6 + a^5*b*d^7)*x^3 + 2*(2*b^6*c^6*d - 6*a*b^5*c^5*d^2
+ 7*a^2*b^4*c^4*d^3 - 8*a^3*b^3*c^3*d^4 + 7*a^4*b^2*c^2*d^5 - 3*a^5*b*c*d^6 + a^6*d^7)*x^2 + (5*b^6*c^7 - 10*a
*b^5*c^6*d + a^2*b^4*c^5*d^2 - 2*a^3*b^3*c^4*d^3 + 8*a^4*b^2*c^3*d^4 - 6*a^5*b*c^2*d^5 + 4*a^6*c*d^6)*x - 2*(5
*a^5*b*c^3*d^4 - 2*a^6*c^2*d^5 + (5*a^4*b^2*c*d^6 - 2*a^5*b*d^7)*x^3 + (10*a^4*b^2*c^2*d^5 + a^5*b*c*d^6 - 2*a
^6*d^7)*x^2 + (5*a^4*b^2*c^3*d^4 + 8*a^5*b*c^2*d^5 - 4*a^6*c*d^6)*x)*log(b*x + a) + 2*(3*a*b^5*c^7 - 10*a^2*b^
4*c^6*d + 10*a^3*b^3*c^5*d^2 + (3*b^6*c^5*d^2 - 10*a*b^5*c^4*d^3 + 10*a^2*b^4*c^3*d^4)*x^3 + (6*b^6*c^6*d - 17
*a*b^5*c^5*d^2 + 10*a^2*b^4*c^4*d^3 + 10*a^3*b^3*c^3*d^4)*x^2 + (3*b^6*c^7 - 4*a*b^5*c^6*d - 10*a^2*b^4*c^5*d^
2 + 20*a^3*b^3*c^4*d^3)*x)*log(d*x + c))/(a*b^7*c^6*d^4 - 4*a^2*b^6*c^5*d^5 + 6*a^3*b^5*c^4*d^6 - 4*a^4*b^4*c^
3*d^7 + a^5*b^3*c^2*d^8 + (b^8*c^4*d^6 - 4*a*b^7*c^3*d^7 + 6*a^2*b^6*c^2*d^8 - 4*a^3*b^5*c*d^9 + a^4*b^4*d^10)
*x^3 + (2*b^8*c^5*d^5 - 7*a*b^7*c^4*d^6 + 8*a^2*b^6*c^3*d^7 - 2*a^3*b^5*c^2*d^8 - 2*a^4*b^4*c*d^9 + a^5*b^3*d^
10)*x^2 + (b^8*c^6*d^4 - 2*a*b^7*c^5*d^5 - 2*a^2*b^6*c^4*d^6 + 8*a^3*b^5*c^3*d^7 - 7*a^4*b^4*c^2*d^8 + 2*a^5*b
^3*c*d^9)*x)

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Sympy [B]  time = 9.74337, size = 1161, normalized size = 6.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x+a)**2/(d*x+c)**3,x)

[Out]

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

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Giac [B]  time = 1.20767, size = 670, normalized size = 3.87 \begin{align*} \frac{a^{5} b^{4}}{{\left (b^{10} c^{3} - 3 \, a b^{9} c^{2} d + 3 \, a^{2} b^{8} c d^{2} - a^{3} b^{7} d^{3}\right )}{\left (b x + a\right )}} - \frac{{\left (3 \, b^{3} c^{5} - 10 \, a b^{2} c^{4} d + 10 \, a^{2} b c^{3} d^{2}\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{4} - 4 \, a b^{4} c^{3} d^{5} + 6 \, a^{2} b^{3} c^{2} d^{6} - 4 \, a^{3} b^{2} c d^{7} + a^{4} b d^{8}} + \frac{{\left (3 \, b c + 2 \, a d\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3} d^{4}} + \frac{{\left (2 \, b^{4} c^{4} d^{3} - 8 \, a b^{3} c^{3} d^{4} + 12 \, a^{2} b^{2} c^{2} d^{5} - 8 \, a^{3} b c d^{6} + 2 \, a^{4} d^{7} + \frac{9 \, b^{6} c^{5} d^{2} - 30 \, a b^{5} c^{4} d^{3} + 40 \, a^{2} b^{4} c^{3} d^{4} - 40 \, a^{3} b^{3} c^{2} d^{5} + 20 \, a^{4} b^{2} c d^{6} - 4 \, a^{5} b d^{7}}{{\left (b x + a\right )} b} + \frac{2 \,{\left (3 \, b^{8} c^{6} d - 13 \, a b^{7} c^{5} d^{2} + 20 \, a^{2} b^{6} c^{4} d^{3} - 20 \, a^{3} b^{5} c^{3} d^{4} + 15 \, a^{4} b^{4} c^{2} d^{5} - 6 \, a^{5} b^{3} c d^{6} + a^{6} b^{2} d^{7}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )}{\left (b x + a\right )}}{2 \,{\left (b c - a d\right )}^{4} b^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(d*x+c)^3,x, algorithm="giac")

[Out]

a^5*b^4/((b^10*c^3 - 3*a*b^9*c^2*d + 3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*(b*x + a)) - (3*b^3*c^5 - 10*a*b^2*c^4*d +
10*a^2*b*c^3*d^2)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^5*c^4*d^4 - 4*a*b^4*c^3*d^5 + 6*a^2*b^3*c^2*
d^6 - 4*a^3*b^2*c*d^7 + a^4*b*d^8) + (3*b*c + 2*a*d)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/(b^3*d^4) + 1/2*(2
*b^4*c^4*d^3 - 8*a*b^3*c^3*d^4 + 12*a^2*b^2*c^2*d^5 - 8*a^3*b*c*d^6 + 2*a^4*d^7 + (9*b^6*c^5*d^2 - 30*a*b^5*c^
4*d^3 + 40*a^2*b^4*c^3*d^4 - 40*a^3*b^3*c^2*d^5 + 20*a^4*b^2*c*d^6 - 4*a^5*b*d^7)/((b*x + a)*b) + 2*(3*b^8*c^6
*d - 13*a*b^7*c^5*d^2 + 20*a^2*b^6*c^4*d^3 - 20*a^3*b^5*c^3*d^4 + 15*a^4*b^4*c^2*d^5 - 6*a^5*b^3*c*d^6 + a^6*b
^2*d^7)/((b*x + a)^2*b^2))*(b*x + a)/((b*c - a*d)^4*b^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^4)