3.1287 \(\int \frac{(c+d x)^7}{(a+b x)^5} \, dx\)

Optimal. Leaf size=187 \[ \frac{7 d^6 (a+b x)^2 (b c-a d)}{2 b^8}+\frac{21 d^5 x (b c-a d)^2}{b^7}-\frac{35 d^3 (b c-a d)^4}{b^8 (a+b x)}-\frac{21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}+\frac{35 d^4 (b c-a d)^3 \log (a+b x)}{b^8}-\frac{7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac{(b c-a d)^7}{4 b^8 (a+b x)^4}+\frac{d^7 (a+b x)^3}{3 b^8} \]

[Out]

(21*d^5*(b*c - a*d)^2*x)/b^7 - (b*c - a*d)^7/(4*b^8*(a + b*x)^4) - (7*d*(b*c - a*d)^6)/(3*b^8*(a + b*x)^3) - (
21*d^2*(b*c - a*d)^5)/(2*b^8*(a + b*x)^2) - (35*d^3*(b*c - a*d)^4)/(b^8*(a + b*x)) + (7*d^6*(b*c - a*d)*(a + b
*x)^2)/(2*b^8) + (d^7*(a + b*x)^3)/(3*b^8) + (35*d^4*(b*c - a*d)^3*Log[a + b*x])/b^8

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Rubi [A]  time = 0.197603, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{7 d^6 (a+b x)^2 (b c-a d)}{2 b^8}+\frac{21 d^5 x (b c-a d)^2}{b^7}-\frac{35 d^3 (b c-a d)^4}{b^8 (a+b x)}-\frac{21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}+\frac{35 d^4 (b c-a d)^3 \log (a+b x)}{b^8}-\frac{7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac{(b c-a d)^7}{4 b^8 (a+b x)^4}+\frac{d^7 (a+b x)^3}{3 b^8} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^5,x]

[Out]

(21*d^5*(b*c - a*d)^2*x)/b^7 - (b*c - a*d)^7/(4*b^8*(a + b*x)^4) - (7*d*(b*c - a*d)^6)/(3*b^8*(a + b*x)^3) - (
21*d^2*(b*c - a*d)^5)/(2*b^8*(a + b*x)^2) - (35*d^3*(b*c - a*d)^4)/(b^8*(a + b*x)) + (7*d^6*(b*c - a*d)*(a + b
*x)^2)/(2*b^8) + (d^7*(a + b*x)^3)/(3*b^8) + (35*d^4*(b*c - a*d)^3*Log[a + b*x])/b^8

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(c+d x)^7}{(a+b x)^5} \, dx &=\int \left (\frac{21 d^5 (b c-a d)^2}{b^7}+\frac{(b c-a d)^7}{b^7 (a+b x)^5}+\frac{7 d (b c-a d)^6}{b^7 (a+b x)^4}+\frac{21 d^2 (b c-a d)^5}{b^7 (a+b x)^3}+\frac{35 d^3 (b c-a d)^4}{b^7 (a+b x)^2}+\frac{35 d^4 (b c-a d)^3}{b^7 (a+b x)}+\frac{7 d^6 (b c-a d) (a+b x)}{b^7}+\frac{d^7 (a+b x)^2}{b^7}\right ) \, dx\\ &=\frac{21 d^5 (b c-a d)^2 x}{b^7}-\frac{(b c-a d)^7}{4 b^8 (a+b x)^4}-\frac{7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac{21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}-\frac{35 d^3 (b c-a d)^4}{b^8 (a+b x)}+\frac{7 d^6 (b c-a d) (a+b x)^2}{2 b^8}+\frac{d^7 (a+b x)^3}{3 b^8}+\frac{35 d^4 (b c-a d)^3 \log (a+b x)}{b^8}\\ \end{align*}

Mathematica [A]  time = 0.113926, size = 173, normalized size = 0.93 \[ \frac{12 b d^5 x \left (15 a^2 d^2-35 a b c d+21 b^2 c^2\right )+6 b^2 d^6 x^2 (7 b c-5 a d)-\frac{420 d^3 (b c-a d)^4}{a+b x}+\frac{126 d^2 (a d-b c)^5}{(a+b x)^2}+420 d^4 (b c-a d)^3 \log (a+b x)-\frac{28 d (b c-a d)^6}{(a+b x)^3}-\frac{3 (b c-a d)^7}{(a+b x)^4}+4 b^3 d^7 x^3}{12 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^5,x]

[Out]

(12*b*d^5*(21*b^2*c^2 - 35*a*b*c*d + 15*a^2*d^2)*x + 6*b^2*d^6*(7*b*c - 5*a*d)*x^2 + 4*b^3*d^7*x^3 - (3*(b*c -
a*d)^7)/(a + b*x)^4 - (28*d*(b*c - a*d)^6)/(a + b*x)^3 + (126*d^2*(-(b*c) + a*d)^5)/(a + b*x)^2 - (420*d^3*(b
*c - a*d)^4)/(a + b*x) + 420*d^4*(b*c - a*d)^3*Log[a + b*x])/(12*b^8)

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Maple [B]  time = 0.011, size = 641, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^5,x)

[Out]

-1/4/b/(b*x+a)^4*c^7+1/3*d^7/b^5*x^3-35*d^6/b^6*a*c*x-210/b^6*d^5/(b*x+a)*a^2*c^2-5/2*d^7/b^6*x^2*a+7/2*d^6/b^
5*x^2*c+15*d^7/b^7*a^2*x+21*d^5/b^5*c^2*x-35/b^8*d^7/(b*x+a)*a^4-35/b^4*d^3/(b*x+a)*c^4+21/2/b^8*d^7/(b*x+a)^2
*a^5-21/2/b^3*d^2/(b*x+a)^2*c^5-7/3/b^8*d^7/(b*x+a)^3*a^6-7/3/b^2*d/(b*x+a)^3*c^6-35/b^8*d^7*ln(b*x+a)*a^3+35/
b^5*d^4*ln(b*x+a)*c^3+1/4/b^8/(b*x+a)^4*a^7*d^7+14/b^3*d^2/(b*x+a)^3*a*c^5+35/4/b^4/(b*x+a)^4*a^3*c^4*d^3-21/4
/b^3/(b*x+a)^4*a^2*c^5*d^2+7/4/b^2/(b*x+a)^4*a*c^6*d+105/b^7*d^6*ln(b*x+a)*a^2*c-105/b^6*d^5*ln(b*x+a)*a*c^2-7
/4/b^7/(b*x+a)^4*a^6*c*d^6+21/4/b^6/(b*x+a)^4*a^5*c^2*d^5-35/4/b^5/(b*x+a)^4*a^4*c^3*d^4+140/b^5*d^4/(b*x+a)*a
*c^3-105/2/b^7*d^6/(b*x+a)^2*a^4*c+105/b^6*d^5/(b*x+a)^2*a^3*c^2-105/b^5*d^4/(b*x+a)^2*a^2*c^3+140/b^7*d^6/(b*
x+a)*a^3*c+105/2/b^4*d^3/(b*x+a)^2*a*c^4+14/b^7*d^6/(b*x+a)^3*a^5*c-35/b^6*d^5/(b*x+a)^3*a^4*c^2+140/3/b^5*d^4
/(b*x+a)^3*a^3*c^3-35/b^4*d^3/(b*x+a)^3*a^2*c^4

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Maxima [B]  time = 1.11407, size = 667, normalized size = 3.57 \begin{align*} -\frac{3 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} - 875 \, a^{4} b^{3} c^{3} d^{4} + 1617 \, a^{5} b^{2} c^{2} d^{5} - 1197 \, a^{6} b c d^{6} + 319 \, a^{7} d^{7} + 420 \,{\left (b^{7} c^{4} d^{3} - 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 126 \,{\left (b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} + 50 \, a^{3} b^{4} c^{2} d^{5} - 35 \, a^{4} b^{3} c d^{6} + 9 \, a^{5} b^{2} d^{7}\right )} x^{2} + 28 \,{\left (b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} - 110 \, a^{3} b^{4} c^{3} d^{4} + 195 \, a^{4} b^{3} c^{2} d^{5} - 141 \, a^{5} b^{2} c d^{6} + 37 \, a^{6} b d^{7}\right )} x}{12 \,{\left (b^{12} x^{4} + 4 \, a b^{11} x^{3} + 6 \, a^{2} b^{10} x^{2} + 4 \, a^{3} b^{9} x + a^{4} b^{8}\right )}} + \frac{2 \, b^{2} d^{7} x^{3} + 3 \,{\left (7 \, b^{2} c d^{6} - 5 \, a b d^{7}\right )} x^{2} + 6 \,{\left (21 \, b^{2} c^{2} d^{5} - 35 \, a b c d^{6} + 15 \, a^{2} d^{7}\right )} x}{6 \, b^{7}} + \frac{35 \,{\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^5,x, algorithm="maxima")

[Out]

-1/12*(3*b^7*c^7 + 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4*d^3 - 875*a^4*b^3*c^3*d^4 + 1617*a^5*b
^2*c^2*d^5 - 1197*a^6*b*c*d^6 + 319*a^7*d^7 + 420*(b^7*c^4*d^3 - 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 - 4*a^3*b
^4*c*d^6 + a^4*b^3*d^7)*x^3 + 126*(b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 - 30*a^2*b^5*c^3*d^4 + 50*a^3*b^4*c^2*d^5 - 3
5*a^4*b^3*c*d^6 + 9*a^5*b^2*d^7)*x^2 + 28*(b^7*c^6*d + 3*a*b^6*c^5*d^2 + 15*a^2*b^5*c^4*d^3 - 110*a^3*b^4*c^3*
d^4 + 195*a^4*b^3*c^2*d^5 - 141*a^5*b^2*c*d^6 + 37*a^6*b*d^7)*x)/(b^12*x^4 + 4*a*b^11*x^3 + 6*a^2*b^10*x^2 + 4
*a^3*b^9*x + a^4*b^8) + 1/6*(2*b^2*d^7*x^3 + 3*(7*b^2*c*d^6 - 5*a*b*d^7)*x^2 + 6*(21*b^2*c^2*d^5 - 35*a*b*c*d^
6 + 15*a^2*d^7)*x)/b^7 + 35*(b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7)*log(b*x + a)/b^8

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Fricas [B]  time = 2.32775, size = 1551, normalized size = 8.29 \begin{align*} \frac{4 \, b^{7} d^{7} x^{7} - 3 \, b^{7} c^{7} - 7 \, a b^{6} c^{6} d - 21 \, a^{2} b^{5} c^{5} d^{2} - 105 \, a^{3} b^{4} c^{4} d^{3} + 875 \, a^{4} b^{3} c^{3} d^{4} - 1617 \, a^{5} b^{2} c^{2} d^{5} + 1197 \, a^{6} b c d^{6} - 319 \, a^{7} d^{7} + 14 \,{\left (3 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 84 \,{\left (3 \, b^{7} c^{2} d^{5} - 3 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 4 \,{\left (252 \, a b^{6} c^{2} d^{5} - 357 \, a^{2} b^{5} c d^{6} + 139 \, a^{3} b^{4} d^{7}\right )} x^{4} - 4 \,{\left (105 \, b^{7} c^{4} d^{3} - 420 \, a b^{6} c^{3} d^{4} + 252 \, a^{2} b^{5} c^{2} d^{5} + 168 \, a^{3} b^{4} c d^{6} - 136 \, a^{4} b^{3} d^{7}\right )} x^{3} - 6 \,{\left (21 \, b^{7} c^{5} d^{2} + 105 \, a b^{6} c^{4} d^{3} - 630 \, a^{2} b^{5} c^{3} d^{4} + 882 \, a^{3} b^{4} c^{2} d^{5} - 462 \, a^{4} b^{3} c d^{6} + 74 \, a^{5} b^{2} d^{7}\right )} x^{2} - 4 \,{\left (7 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 105 \, a^{2} b^{5} c^{4} d^{3} - 770 \, a^{3} b^{4} c^{3} d^{4} + 1302 \, a^{4} b^{3} c^{2} d^{5} - 882 \, a^{5} b^{2} c d^{6} + 214 \, a^{6} b d^{7}\right )} x + 420 \,{\left (a^{4} b^{3} c^{3} d^{4} - 3 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} - a^{7} d^{7} +{\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 4 \,{\left (a b^{6} c^{3} d^{4} - 3 \, a^{2} b^{5} c^{2} d^{5} + 3 \, a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{3} + 6 \,{\left (a^{2} b^{5} c^{3} d^{4} - 3 \, a^{3} b^{4} c^{2} d^{5} + 3 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 4 \,{\left (a^{3} b^{4} c^{3} d^{4} - 3 \, a^{4} b^{3} c^{2} d^{5} + 3 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{12 \,{\left (b^{12} x^{4} + 4 \, a b^{11} x^{3} + 6 \, a^{2} b^{10} x^{2} + 4 \, a^{3} b^{9} x + a^{4} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^5,x, algorithm="fricas")

[Out]

1/12*(4*b^7*d^7*x^7 - 3*b^7*c^7 - 7*a*b^6*c^6*d - 21*a^2*b^5*c^5*d^2 - 105*a^3*b^4*c^4*d^3 + 875*a^4*b^3*c^3*d
^4 - 1617*a^5*b^2*c^2*d^5 + 1197*a^6*b*c*d^6 - 319*a^7*d^7 + 14*(3*b^7*c*d^6 - a*b^6*d^7)*x^6 + 84*(3*b^7*c^2*
d^5 - 3*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 4*(252*a*b^6*c^2*d^5 - 357*a^2*b^5*c*d^6 + 139*a^3*b^4*d^7)*x^4 - 4*(
105*b^7*c^4*d^3 - 420*a*b^6*c^3*d^4 + 252*a^2*b^5*c^2*d^5 + 168*a^3*b^4*c*d^6 - 136*a^4*b^3*d^7)*x^3 - 6*(21*b
^7*c^5*d^2 + 105*a*b^6*c^4*d^3 - 630*a^2*b^5*c^3*d^4 + 882*a^3*b^4*c^2*d^5 - 462*a^4*b^3*c*d^6 + 74*a^5*b^2*d^
7)*x^2 - 4*(7*b^7*c^6*d + 21*a*b^6*c^5*d^2 + 105*a^2*b^5*c^4*d^3 - 770*a^3*b^4*c^3*d^4 + 1302*a^4*b^3*c^2*d^5
- 882*a^5*b^2*c*d^6 + 214*a^6*b*d^7)*x + 420*(a^4*b^3*c^3*d^4 - 3*a^5*b^2*c^2*d^5 + 3*a^6*b*c*d^6 - a^7*d^7 +
(b^7*c^3*d^4 - 3*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6 - a^3*b^4*d^7)*x^4 + 4*(a*b^6*c^3*d^4 - 3*a^2*b^5*c^2*d^5 + 3
*a^3*b^4*c*d^6 - a^4*b^3*d^7)*x^3 + 6*(a^2*b^5*c^3*d^4 - 3*a^3*b^4*c^2*d^5 + 3*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^
2 + 4*(a^3*b^4*c^3*d^4 - 3*a^4*b^3*c^2*d^5 + 3*a^5*b^2*c*d^6 - a^6*b*d^7)*x)*log(b*x + a))/(b^12*x^4 + 4*a*b^1
1*x^3 + 6*a^2*b^10*x^2 + 4*a^3*b^9*x + a^4*b^8)

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Sympy [B]  time = 26.0549, size = 495, normalized size = 2.65 \begin{align*} - \frac{319 a^{7} d^{7} - 1197 a^{6} b c d^{6} + 1617 a^{5} b^{2} c^{2} d^{5} - 875 a^{4} b^{3} c^{3} d^{4} + 105 a^{3} b^{4} c^{4} d^{3} + 21 a^{2} b^{5} c^{5} d^{2} + 7 a b^{6} c^{6} d + 3 b^{7} c^{7} + x^{3} \left (420 a^{4} b^{3} d^{7} - 1680 a^{3} b^{4} c d^{6} + 2520 a^{2} b^{5} c^{2} d^{5} - 1680 a b^{6} c^{3} d^{4} + 420 b^{7} c^{4} d^{3}\right ) + x^{2} \left (1134 a^{5} b^{2} d^{7} - 4410 a^{4} b^{3} c d^{6} + 6300 a^{3} b^{4} c^{2} d^{5} - 3780 a^{2} b^{5} c^{3} d^{4} + 630 a b^{6} c^{4} d^{3} + 126 b^{7} c^{5} d^{2}\right ) + x \left (1036 a^{6} b d^{7} - 3948 a^{5} b^{2} c d^{6} + 5460 a^{4} b^{3} c^{2} d^{5} - 3080 a^{3} b^{4} c^{3} d^{4} + 420 a^{2} b^{5} c^{4} d^{3} + 84 a b^{6} c^{5} d^{2} + 28 b^{7} c^{6} d\right )}{12 a^{4} b^{8} + 48 a^{3} b^{9} x + 72 a^{2} b^{10} x^{2} + 48 a b^{11} x^{3} + 12 b^{12} x^{4}} + \frac{d^{7} x^{3}}{3 b^{5}} - \frac{x^{2} \left (5 a d^{7} - 7 b c d^{6}\right )}{2 b^{6}} + \frac{x \left (15 a^{2} d^{7} - 35 a b c d^{6} + 21 b^{2} c^{2} d^{5}\right )}{b^{7}} - \frac{35 d^{4} \left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**7/(b*x+a)**5,x)

[Out]

-(319*a**7*d**7 - 1197*a**6*b*c*d**6 + 1617*a**5*b**2*c**2*d**5 - 875*a**4*b**3*c**3*d**4 + 105*a**3*b**4*c**4
*d**3 + 21*a**2*b**5*c**5*d**2 + 7*a*b**6*c**6*d + 3*b**7*c**7 + x**3*(420*a**4*b**3*d**7 - 1680*a**3*b**4*c*d
**6 + 2520*a**2*b**5*c**2*d**5 - 1680*a*b**6*c**3*d**4 + 420*b**7*c**4*d**3) + x**2*(1134*a**5*b**2*d**7 - 441
0*a**4*b**3*c*d**6 + 6300*a**3*b**4*c**2*d**5 - 3780*a**2*b**5*c**3*d**4 + 630*a*b**6*c**4*d**3 + 126*b**7*c**
5*d**2) + x*(1036*a**6*b*d**7 - 3948*a**5*b**2*c*d**6 + 5460*a**4*b**3*c**2*d**5 - 3080*a**3*b**4*c**3*d**4 +
420*a**2*b**5*c**4*d**3 + 84*a*b**6*c**5*d**2 + 28*b**7*c**6*d))/(12*a**4*b**8 + 48*a**3*b**9*x + 72*a**2*b**1
0*x**2 + 48*a*b**11*x**3 + 12*b**12*x**4) + d**7*x**3/(3*b**5) - x**2*(5*a*d**7 - 7*b*c*d**6)/(2*b**6) + x*(15
*a**2*d**7 - 35*a*b*c*d**6 + 21*b**2*c**2*d**5)/b**7 - 35*d**4*(a*d - b*c)**3*log(a + b*x)/b**8

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Giac [B]  time = 1.08144, size = 891, normalized size = 4.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^5,x, algorithm="giac")

[Out]

1/6*(2*d^7 + 21*(b^2*c*d^6 - a*b*d^7)/((b*x + a)*b) + 126*(b^4*c^2*d^5 - 2*a*b^3*c*d^6 + a^2*b^2*d^7)/((b*x +
a)^2*b^2))*(b*x + a)^3/b^8 - 35*(b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7)*log(abs(b*x + a)/((b
*x + a)^2*abs(b)))/b^8 - 1/12*(3*b^43*c^7/(b*x + a)^4 + 28*b^42*c^6*d/(b*x + a)^3 - 21*a*b^42*c^6*d/(b*x + a)^
4 + 126*b^41*c^5*d^2/(b*x + a)^2 - 168*a*b^41*c^5*d^2/(b*x + a)^3 + 63*a^2*b^41*c^5*d^2/(b*x + a)^4 + 420*b^40
*c^4*d^3/(b*x + a) - 630*a*b^40*c^4*d^3/(b*x + a)^2 + 420*a^2*b^40*c^4*d^3/(b*x + a)^3 - 105*a^3*b^40*c^4*d^3/
(b*x + a)^4 - 1680*a*b^39*c^3*d^4/(b*x + a) + 1260*a^2*b^39*c^3*d^4/(b*x + a)^2 - 560*a^3*b^39*c^3*d^4/(b*x +
a)^3 + 105*a^4*b^39*c^3*d^4/(b*x + a)^4 + 2520*a^2*b^38*c^2*d^5/(b*x + a) - 1260*a^3*b^38*c^2*d^5/(b*x + a)^2
+ 420*a^4*b^38*c^2*d^5/(b*x + a)^3 - 63*a^5*b^38*c^2*d^5/(b*x + a)^4 - 1680*a^3*b^37*c*d^6/(b*x + a) + 630*a^4
*b^37*c*d^6/(b*x + a)^2 - 168*a^5*b^37*c*d^6/(b*x + a)^3 + 21*a^6*b^37*c*d^6/(b*x + a)^4 + 420*a^4*b^36*d^7/(b
*x + a) - 126*a^5*b^36*d^7/(b*x + a)^2 + 28*a^6*b^36*d^7/(b*x + a)^3 - 3*a^7*b^36*d^7/(b*x + a)^4)/b^44