3.1293 \(\int \frac{(c+d x)^7}{(a+b x)^{11}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{d^2 (c+d x)^8}{360 (a+b x)^8 (b c-a d)^3}+\frac{d (c+d x)^8}{45 (a+b x)^9 (b c-a d)^2}-\frac{(c+d x)^8}{10 (a+b x)^{10} (b c-a d)} \]

[Out]

-(c + d*x)^8/(10*(b*c - a*d)*(a + b*x)^10) + (d*(c + d*x)^8)/(45*(b*c - a*d)^2*(a + b*x)^9) - (d^2*(c + d*x)^8
)/(360*(b*c - a*d)^3*(a + b*x)^8)

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Rubi [A]  time = 0.0218066, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac{d^2 (c+d x)^8}{360 (a+b x)^8 (b c-a d)^3}+\frac{d (c+d x)^8}{45 (a+b x)^9 (b c-a d)^2}-\frac{(c+d x)^8}{10 (a+b x)^{10} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^11,x]

[Out]

-(c + d*x)^8/(10*(b*c - a*d)*(a + b*x)^10) + (d*(c + d*x)^8)/(45*(b*c - a*d)^2*(a + b*x)^9) - (d^2*(c + d*x)^8
)/(360*(b*c - a*d)^3*(a + b*x)^8)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(c+d x)^7}{(a+b x)^{11}} \, dx &=-\frac{(c+d x)^8}{10 (b c-a d) (a+b x)^{10}}-\frac{d \int \frac{(c+d x)^7}{(a+b x)^{10}} \, dx}{5 (b c-a d)}\\ &=-\frac{(c+d x)^8}{10 (b c-a d) (a+b x)^{10}}+\frac{d (c+d x)^8}{45 (b c-a d)^2 (a+b x)^9}+\frac{d^2 \int \frac{(c+d x)^7}{(a+b x)^9} \, dx}{45 (b c-a d)^2}\\ &=-\frac{(c+d x)^8}{10 (b c-a d) (a+b x)^{10}}+\frac{d (c+d x)^8}{45 (b c-a d)^2 (a+b x)^9}-\frac{d^2 (c+d x)^8}{360 (b c-a d)^3 (a+b x)^8}\\ \end{align*}

Mathematica [B]  time = 0.118284, size = 371, normalized size = 4.17 \[ -\frac{3 a^2 b^5 d^2 \left (150 c^3 d^2 x^2+240 c^2 d^3 x^3+50 c^4 d x+7 c^5+210 c d^4 x^4+84 d^5 x^5\right )+5 a^3 b^4 d^3 \left (54 c^2 d^2 x^2+20 c^3 d x+3 c^4+72 c d^3 x^3+42 d^4 x^4\right )+5 a^4 b^3 d^4 \left (12 c^2 d x+2 c^3+27 c d^2 x^2+24 d^3 x^3\right )+3 a^5 b^2 d^5 \left (2 c^2+10 c d x+15 d^2 x^2\right )+a^6 b d^6 (3 c+10 d x)+a^7 d^7+a b^6 d \left (675 c^4 d^2 x^2+1200 c^3 d^3 x^3+1260 c^2 d^4 x^4+210 c^5 d x+28 c^6+756 c d^5 x^5+210 d^6 x^6\right )+b^7 \left (945 c^5 d^2 x^2+1800 c^4 d^3 x^3+2100 c^3 d^4 x^4+1512 c^2 d^5 x^5+280 c^6 d x+36 c^7+630 c d^6 x^6+120 d^7 x^7\right )}{360 b^8 (a+b x)^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^11,x]

[Out]

-(a^7*d^7 + a^6*b*d^6*(3*c + 10*d*x) + 3*a^5*b^2*d^5*(2*c^2 + 10*c*d*x + 15*d^2*x^2) + 5*a^4*b^3*d^4*(2*c^3 +
12*c^2*d*x + 27*c*d^2*x^2 + 24*d^3*x^3) + 5*a^3*b^4*d^3*(3*c^4 + 20*c^3*d*x + 54*c^2*d^2*x^2 + 72*c*d^3*x^3 +
42*d^4*x^4) + 3*a^2*b^5*d^2*(7*c^5 + 50*c^4*d*x + 150*c^3*d^2*x^2 + 240*c^2*d^3*x^3 + 210*c*d^4*x^4 + 84*d^5*x
^5) + a*b^6*d*(28*c^6 + 210*c^5*d*x + 675*c^4*d^2*x^2 + 1200*c^3*d^3*x^3 + 1260*c^2*d^4*x^4 + 756*c*d^5*x^5 +
210*d^6*x^6) + b^7*(36*c^7 + 280*c^6*d*x + 945*c^5*d^2*x^2 + 1800*c^4*d^3*x^3 + 2100*c^3*d^4*x^4 + 1512*c^2*d^
5*x^5 + 630*c*d^6*x^6 + 120*d^7*x^7))/(360*b^8*(a + b*x)^10)

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Maple [B]  time = 0.006, size = 464, normalized size = 5.2 \begin{align*} -{\frac{-{a}^{7}{d}^{7}+7\,{a}^{6}c{d}^{6}b-21\,{a}^{5}{b}^{2}{c}^{2}{d}^{5}+35\,{c}^{3}{d}^{4}{a}^{4}{b}^{3}-35\,{a}^{3}{b}^{4}{c}^{4}{d}^{3}+21\,{a}^{2}{c}^{5}{d}^{2}{b}^{5}-7\,a{c}^{6}d{b}^{6}+{b}^{7}{c}^{7}}{10\,{b}^{8} \left ( bx+a \right ) ^{10}}}+{\frac{21\,{d}^{2} \left ({a}^{5}{d}^{5}-5\,{a}^{4}bc{d}^{4}+10\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-10\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+5\,a{b}^{4}{c}^{4}d-{b}^{5}{c}^{5} \right ) }{8\,{b}^{8} \left ( bx+a \right ) ^{8}}}+{\frac{35\,{d}^{4} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }{6\,{b}^{8} \left ( bx+a \right ) ^{6}}}-{\frac{{d}^{7}}{3\,{b}^{8} \left ( bx+a \right ) ^{3}}}-{\frac{21\,{d}^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{5\,{b}^{8} \left ( bx+a \right ) ^{5}}}+{\frac{7\,{d}^{6} \left ( ad-bc \right ) }{4\,{b}^{8} \left ( bx+a \right ) ^{4}}}-{\frac{7\,d \left ({a}^{6}{d}^{6}-6\,{a}^{5}bc{d}^{5}+15\,{a}^{4}{b}^{2}{c}^{2}{d}^{4}-20\,{a}^{3}{b}^{3}{c}^{3}{d}^{3}+15\,{a}^{2}{b}^{4}{c}^{4}{d}^{2}-6\,a{b}^{5}{c}^{5}d+{b}^{6}{c}^{6} \right ) }{9\,{b}^{8} \left ( bx+a \right ) ^{9}}}-5\,{\frac{{d}^{3} \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) }{{b}^{8} \left ( bx+a \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^11,x)

[Out]

-1/10*(-a^7*d^7+7*a^6*b*c*d^6-21*a^5*b^2*c^2*d^5+35*a^4*b^3*c^3*d^4-35*a^3*b^4*c^4*d^3+21*a^2*b^5*c^5*d^2-7*a*
b^6*c^6*d+b^7*c^7)/b^8/(b*x+a)^10+21/8*d^2*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^
4*c^4*d-b^5*c^5)/b^8/(b*x+a)^8+35/6*d^4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^8/(b*x+a)^6-1/3*d^7/b^
8/(b*x+a)^3-21/5*d^5*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^8/(b*x+a)^5+7/4*d^6*(a*d-b*c)/b^8/(b*x+a)^4-7/9*d*(a^6*d^6-
6*a^5*b*c*d^5+15*a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)/b^8/(b*x+a)^9-5*
d^3*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/b^8/(b*x+a)^7

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Maxima [B]  time = 1.08508, size = 755, normalized size = 8.48 \begin{align*} -\frac{120 \, b^{7} d^{7} x^{7} + 36 \, b^{7} c^{7} + 28 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 15 \, a^{3} b^{4} c^{4} d^{3} + 10 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} + a^{7} d^{7} + 210 \,{\left (3 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 252 \,{\left (6 \, b^{7} c^{2} d^{5} + 3 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 210 \,{\left (10 \, b^{7} c^{3} d^{4} + 6 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 120 \,{\left (15 \, b^{7} c^{4} d^{3} + 10 \, a b^{6} c^{3} d^{4} + 6 \, 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} + 45 \,{\left (21 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 6 \, 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} + 10 \,{\left (28 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 6 \, a^{4} b^{3} c^{2} d^{5} + 3 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{360 \,{\left (b^{18} x^{10} + 10 \, a b^{17} x^{9} + 45 \, a^{2} b^{16} x^{8} + 120 \, a^{3} b^{15} x^{7} + 210 \, a^{4} b^{14} x^{6} + 252 \, a^{5} b^{13} x^{5} + 210 \, a^{6} b^{12} x^{4} + 120 \, a^{7} b^{11} x^{3} + 45 \, a^{8} b^{10} x^{2} + 10 \, a^{9} b^{9} x + a^{10} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^11,x, algorithm="maxima")

[Out]

-1/360*(120*b^7*d^7*x^7 + 36*b^7*c^7 + 28*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 15*a^3*b^4*c^4*d^3 + 10*a^4*b^3*c
^3*d^4 + 6*a^5*b^2*c^2*d^5 + 3*a^6*b*c*d^6 + a^7*d^7 + 210*(3*b^7*c*d^6 + a*b^6*d^7)*x^6 + 252*(6*b^7*c^2*d^5
+ 3*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 210*(10*b^7*c^3*d^4 + 6*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^
4 + 120*(15*b^7*c^4*d^3 + 10*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 + 3*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 45*(21*b
^7*c^5*d^2 + 15*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 + 6*a^3*b^4*c^2*d^5 + 3*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 +
10*(28*b^7*c^6*d + 21*a*b^6*c^5*d^2 + 15*a^2*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^4 + 6*a^4*b^3*c^2*d^5 + 3*a^5*b^2*
c*d^6 + a^6*b*d^7)*x)/(b^18*x^10 + 10*a*b^17*x^9 + 45*a^2*b^16*x^8 + 120*a^3*b^15*x^7 + 210*a^4*b^14*x^6 + 252
*a^5*b^13*x^5 + 210*a^6*b^12*x^4 + 120*a^7*b^11*x^3 + 45*a^8*b^10*x^2 + 10*a^9*b^9*x + a^10*b^8)

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Fricas [B]  time = 2.03056, size = 1172, normalized size = 13.17 \begin{align*} -\frac{120 \, b^{7} d^{7} x^{7} + 36 \, b^{7} c^{7} + 28 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 15 \, a^{3} b^{4} c^{4} d^{3} + 10 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} + a^{7} d^{7} + 210 \,{\left (3 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 252 \,{\left (6 \, b^{7} c^{2} d^{5} + 3 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 210 \,{\left (10 \, b^{7} c^{3} d^{4} + 6 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 120 \,{\left (15 \, b^{7} c^{4} d^{3} + 10 \, a b^{6} c^{3} d^{4} + 6 \, 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} + 45 \,{\left (21 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 6 \, 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} + 10 \,{\left (28 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 6 \, a^{4} b^{3} c^{2} d^{5} + 3 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{360 \,{\left (b^{18} x^{10} + 10 \, a b^{17} x^{9} + 45 \, a^{2} b^{16} x^{8} + 120 \, a^{3} b^{15} x^{7} + 210 \, a^{4} b^{14} x^{6} + 252 \, a^{5} b^{13} x^{5} + 210 \, a^{6} b^{12} x^{4} + 120 \, a^{7} b^{11} x^{3} + 45 \, a^{8} b^{10} x^{2} + 10 \, a^{9} b^{9} x + a^{10} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^11,x, algorithm="fricas")

[Out]

-1/360*(120*b^7*d^7*x^7 + 36*b^7*c^7 + 28*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 15*a^3*b^4*c^4*d^3 + 10*a^4*b^3*c
^3*d^4 + 6*a^5*b^2*c^2*d^5 + 3*a^6*b*c*d^6 + a^7*d^7 + 210*(3*b^7*c*d^6 + a*b^6*d^7)*x^6 + 252*(6*b^7*c^2*d^5
+ 3*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 210*(10*b^7*c^3*d^4 + 6*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^
4 + 120*(15*b^7*c^4*d^3 + 10*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 + 3*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 45*(21*b
^7*c^5*d^2 + 15*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 + 6*a^3*b^4*c^2*d^5 + 3*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 +
10*(28*b^7*c^6*d + 21*a*b^6*c^5*d^2 + 15*a^2*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^4 + 6*a^4*b^3*c^2*d^5 + 3*a^5*b^2*
c*d^6 + a^6*b*d^7)*x)/(b^18*x^10 + 10*a*b^17*x^9 + 45*a^2*b^16*x^8 + 120*a^3*b^15*x^7 + 210*a^4*b^14*x^6 + 252
*a^5*b^13*x^5 + 210*a^6*b^12*x^4 + 120*a^7*b^11*x^3 + 45*a^8*b^10*x^2 + 10*a^9*b^9*x + a^10*b^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**7/(b*x+a)**11,x)

[Out]

Timed out

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Giac [B]  time = 1.05768, size = 670, normalized size = 7.53 \begin{align*} -\frac{120 \, b^{7} d^{7} x^{7} + 630 \, b^{7} c d^{6} x^{6} + 210 \, a b^{6} d^{7} x^{6} + 1512 \, b^{7} c^{2} d^{5} x^{5} + 756 \, a b^{6} c d^{6} x^{5} + 252 \, a^{2} b^{5} d^{7} x^{5} + 2100 \, b^{7} c^{3} d^{4} x^{4} + 1260 \, a b^{6} c^{2} d^{5} x^{4} + 630 \, a^{2} b^{5} c d^{6} x^{4} + 210 \, a^{3} b^{4} d^{7} x^{4} + 1800 \, b^{7} c^{4} d^{3} x^{3} + 1200 \, a b^{6} c^{3} d^{4} x^{3} + 720 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 360 \, a^{3} b^{4} c d^{6} x^{3} + 120 \, a^{4} b^{3} d^{7} x^{3} + 945 \, b^{7} c^{5} d^{2} x^{2} + 675 \, a b^{6} c^{4} d^{3} x^{2} + 450 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 270 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 135 \, a^{4} b^{3} c d^{6} x^{2} + 45 \, a^{5} b^{2} d^{7} x^{2} + 280 \, b^{7} c^{6} d x + 210 \, a b^{6} c^{5} d^{2} x + 150 \, a^{2} b^{5} c^{4} d^{3} x + 100 \, a^{3} b^{4} c^{3} d^{4} x + 60 \, a^{4} b^{3} c^{2} d^{5} x + 30 \, a^{5} b^{2} c d^{6} x + 10 \, a^{6} b d^{7} x + 36 \, b^{7} c^{7} + 28 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 15 \, a^{3} b^{4} c^{4} d^{3} + 10 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} + a^{7} d^{7}}{360 \,{\left (b x + a\right )}^{10} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^11,x, algorithm="giac")

[Out]

-1/360*(120*b^7*d^7*x^7 + 630*b^7*c*d^6*x^6 + 210*a*b^6*d^7*x^6 + 1512*b^7*c^2*d^5*x^5 + 756*a*b^6*c*d^6*x^5 +
252*a^2*b^5*d^7*x^5 + 2100*b^7*c^3*d^4*x^4 + 1260*a*b^6*c^2*d^5*x^4 + 630*a^2*b^5*c*d^6*x^4 + 210*a^3*b^4*d^7
*x^4 + 1800*b^7*c^4*d^3*x^3 + 1200*a*b^6*c^3*d^4*x^3 + 720*a^2*b^5*c^2*d^5*x^3 + 360*a^3*b^4*c*d^6*x^3 + 120*a
^4*b^3*d^7*x^3 + 945*b^7*c^5*d^2*x^2 + 675*a*b^6*c^4*d^3*x^2 + 450*a^2*b^5*c^3*d^4*x^2 + 270*a^3*b^4*c^2*d^5*x
^2 + 135*a^4*b^3*c*d^6*x^2 + 45*a^5*b^2*d^7*x^2 + 280*b^7*c^6*d*x + 210*a*b^6*c^5*d^2*x + 150*a^2*b^5*c^4*d^3*
x + 100*a^3*b^4*c^3*d^4*x + 60*a^4*b^3*c^2*d^5*x + 30*a^5*b^2*c*d^6*x + 10*a^6*b*d^7*x + 36*b^7*c^7 + 28*a*b^6
*c^6*d + 21*a^2*b^5*c^5*d^2 + 15*a^3*b^4*c^4*d^3 + 10*a^4*b^3*c^3*d^4 + 6*a^5*b^2*c^2*d^5 + 3*a^6*b*c*d^6 + a^
7*d^7)/((b*x + a)^10*b^8)