3.1789 \(\int \frac{(a+b x)^3}{(c+d x) (e+f x)^{9/2}} \, dx\)

Optimal. Leaf size=260 \[ -\frac{2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )}{3 f^3 (e+f x)^{3/2} (d e-c f)^3}+\frac{2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{5 f^3 (e+f x)^{5/2} (d e-c f)^2}-\frac{2 (b e-a f)^3}{7 f^3 (e+f x)^{7/2} (d e-c f)}-\frac{2 (b c-a d)^3}{\sqrt{e+f x} (d e-c f)^4}+\frac{2 \sqrt{d} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{9/2}} \]

[Out]

(-2*(b*e - a*f)^3)/(7*f^3*(d*e - c*f)*(e + f*x)^(7/2)) + (2*(b*e - a*f)^2*(2*b*d*e - 3*b*c*f + a*d*f))/(5*f^3*
(d*e - c*f)^2*(e + f*x)^(5/2)) - (2*(b*e - a*f)*(a^2*d^2*f^2 + a*b*d*f*(d*e - 3*c*f) + b^2*(d^2*e^2 - 3*c*d*e*
f + 3*c^2*f^2)))/(3*f^3*(d*e - c*f)^3*(e + f*x)^(3/2)) - (2*(b*c - a*d)^3)/((d*e - c*f)^4*Sqrt[e + f*x]) + (2*
Sqrt[d]*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(d*e - c*f)^(9/2)

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Rubi [A]  time = 0.357535, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {87, 63, 208} \[ -\frac{2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )}{3 f^3 (e+f x)^{3/2} (d e-c f)^3}+\frac{2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{5 f^3 (e+f x)^{5/2} (d e-c f)^2}-\frac{2 (b e-a f)^3}{7 f^3 (e+f x)^{7/2} (d e-c f)}-\frac{2 (b c-a d)^3}{\sqrt{e+f x} (d e-c f)^4}+\frac{2 \sqrt{d} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/((c + d*x)*(e + f*x)^(9/2)),x]

[Out]

(-2*(b*e - a*f)^3)/(7*f^3*(d*e - c*f)*(e + f*x)^(7/2)) + (2*(b*e - a*f)^2*(2*b*d*e - 3*b*c*f + a*d*f))/(5*f^3*
(d*e - c*f)^2*(e + f*x)^(5/2)) - (2*(b*e - a*f)*(a^2*d^2*f^2 + a*b*d*f*(d*e - 3*c*f) + b^2*(d^2*e^2 - 3*c*d*e*
f + 3*c^2*f^2)))/(3*f^3*(d*e - c*f)^3*(e + f*x)^(3/2)) - (2*(b*c - a*d)^3)/((d*e - c*f)^4*Sqrt[e + f*x]) + (2*
Sqrt[d]*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(d*e - c*f)^(9/2)

Rule 87

Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], ((c + d*x)^n*(e + f*x)^IntegerPart[p])/(a + b*x), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{(a+b x)^3}{(c+d x) (e+f x)^{9/2}} \, dx &=\int \left (\frac{(-b e+a f)^3}{f^2 (-d e+c f) (e+f x)^{9/2}}+\frac{(-b e+a f)^2 (-2 b d e+3 b c f-a d f)}{f^2 (-d e+c f)^2 (e+f x)^{7/2}}+\frac{-3 a b^2 c^2 f^3+3 a^2 b c d f^3-a^3 d^2 f^3+b^3 e \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )}{f^2 (d e-c f)^3 (e+f x)^{5/2}}+\frac{(b c-a d)^3 f}{(-d e+c f)^4 (e+f x)^{3/2}}+\frac{d (-b c+a d)^3}{(d e-c f)^4 (c+d x) \sqrt{e+f x}}\right ) \, dx\\ &=-\frac{2 (b e-a f)^3}{7 f^3 (d e-c f) (e+f x)^{7/2}}+\frac{2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{5 f^3 (d e-c f)^2 (e+f x)^{5/2}}-\frac{2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right )}{3 f^3 (d e-c f)^3 (e+f x)^{3/2}}-\frac{2 (b c-a d)^3}{(d e-c f)^4 \sqrt{e+f x}}-\frac{\left (d (b c-a d)^3\right ) \int \frac{1}{(c+d x) \sqrt{e+f x}} \, dx}{(d e-c f)^4}\\ &=-\frac{2 (b e-a f)^3}{7 f^3 (d e-c f) (e+f x)^{7/2}}+\frac{2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{5 f^3 (d e-c f)^2 (e+f x)^{5/2}}-\frac{2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right )}{3 f^3 (d e-c f)^3 (e+f x)^{3/2}}-\frac{2 (b c-a d)^3}{(d e-c f)^4 \sqrt{e+f x}}-\frac{\left (2 d (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{c-\frac{d e}{f}+\frac{d x^2}{f}} \, dx,x,\sqrt{e+f x}\right )}{f (d e-c f)^4}\\ &=-\frac{2 (b e-a f)^3}{7 f^3 (d e-c f) (e+f x)^{7/2}}+\frac{2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{5 f^3 (d e-c f)^2 (e+f x)^{5/2}}-\frac{2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right )}{3 f^3 (d e-c f)^3 (e+f x)^{3/2}}-\frac{2 (b c-a d)^3}{(d e-c f)^4 \sqrt{e+f x}}+\frac{2 \sqrt{d} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.144532, size = 166, normalized size = 0.64 \[ \frac{2 \left (-\frac{15 b \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{f^3}+\frac{21 b^2 d (e+f x) (-3 a d f+b c f+2 b d e)}{f^3}+\frac{15 (b c-a d)^3 \, _2F_1\left (-\frac{7}{2},1;-\frac{5}{2};\frac{d (e+f x)}{d e-c f}\right )}{c f-d e}-\frac{35 b^3 d^2 (e+f x)^2}{f^3}\right )}{105 d^3 (e+f x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/((c + d*x)*(e + f*x)^(9/2)),x]

[Out]

(2*((-15*b*(3*a^2*d^2*f^2 - 3*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + c*d*e*f + c^2*f^2)))/f^3 + (21*b^2*d*(2*b*d
*e + b*c*f - 3*a*d*f)*(e + f*x))/f^3 - (35*b^3*d^2*(e + f*x)^2)/f^3 + (15*(b*c - a*d)^3*Hypergeometric2F1[-7/2
, 1, -5/2, (d*(e + f*x))/(d*e - c*f)])/(-(d*e) + c*f)))/(105*d^3*(e + f*x)^(7/2))

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Maple [B]  time = 0.023, size = 756, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/(d*x+c)/(f*x+e)^(9/2),x)

[Out]

-2/7/(c*f-d*e)/(f*x+e)^(7/2)*a^3+6/7/f/(c*f-d*e)/(f*x+e)^(7/2)*a^2*b*e-6/7/f^2/(c*f-d*e)/(f*x+e)^(7/2)*a*b^2*e
^2+2/7/f^3/(c*f-d*e)/(f*x+e)^(7/2)*b^3*e^3+2/5/(c*f-d*e)^2/(f*x+e)^(5/2)*a^3*d-6/5/(c*f-d*e)^2/(f*x+e)^(5/2)*a
^2*b*c+12/5/f/(c*f-d*e)^2/(f*x+e)^(5/2)*a*b^2*c*e-6/5/f^2/(c*f-d*e)^2/(f*x+e)^(5/2)*a*b^2*d*e^2-6/5/f^2/(c*f-d
*e)^2/(f*x+e)^(5/2)*b^3*c*e^2+4/5/f^3/(c*f-d*e)^2/(f*x+e)^(5/2)*b^3*d*e^3-2/3/(c*f-d*e)^3/(f*x+e)^(3/2)*a^3*d^
2+2/(c*f-d*e)^3/(f*x+e)^(3/2)*a^2*b*c*d-2/(c*f-d*e)^3/(f*x+e)^(3/2)*a*b^2*c^2+2/f/(c*f-d*e)^3/(f*x+e)^(3/2)*b^
3*c^2*e-2/f^2/(c*f-d*e)^3/(f*x+e)^(3/2)*b^3*c*d*e^2+2/3/f^3/(c*f-d*e)^3/(f*x+e)^(3/2)*b^3*d^2*e^3+2/(c*f-d*e)^
4/(f*x+e)^(1/2)*a^3*d^3-6/(c*f-d*e)^4/(f*x+e)^(1/2)*a^2*c*b*d^2+6/(c*f-d*e)^4/(f*x+e)^(1/2)*a*b^2*c^2*d-2/(c*f
-d*e)^4/(f*x+e)^(1/2)*b^3*c^3+2*d^4/(c*f-d*e)^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2)
)*a^3-6*d^3/(c*f-d*e)^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a^2*c*b+6*d^2/(c*f-d*e
)^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a*b^2*c^2-2*d/(c*f-d*e)^4/((c*f-d*e)*d)^(1
/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b^3*c^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)/(f*x+e)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.64295, size = 4632, normalized size = 17.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)/(f*x+e)^(9/2),x, algorithm="fricas")

[Out]

[-1/105*(105*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^7*x^4 + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2
*b*c*d^2 - a^3*d^3)*e*f^6*x^3 + 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^2*f^5*x^2 + 4*(b^3*c^3
- 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^3*f^4*x + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^
4*f^3)*sqrt(d/(d*e - c*f))*log((d*f*x + 2*d*e - c*f - 2*(d*e - c*f)*sqrt(f*x + e)*sqrt(d/(d*e - c*f)))/(d*x +
c)) + 2*(8*b^3*d^3*e^6 + 15*a^3*c^3*f^6 + 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^6*x^3 - 2*
(19*b^3*c*d^2 - 9*a*b^2*d^3)*e^5*f + 3*(29*b^3*c^2*d - 39*a*b^2*c*d^2 + 15*a^2*b*d^3)*e^4*f^2 + 4*(12*b^3*c^3
- 60*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 44*a^3*d^3)*e^3*f^3 + 2*(12*a*b^2*c^3 - 48*a^2*b*c^2*d + 61*a^3*c*d^2)*e^2
*f^4 + 6*(3*a^2*b*c^3 - 11*a^3*c^2*d)*e*f^5 + 35*(b^3*d^3*e^4*f^2 - 4*b^3*c*d^2*e^3*f^3 + 6*b^3*c^2*d*e^2*f^4
+ 2*(3*b^3*c^3 - 15*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 5*a^3*d^3)*e*f^5 + (3*a*b^2*c^3 - 3*a^2*b*c^2*d + a^3*c*d^2
)*f^6)*x^2 + 7*(4*b^3*d^3*e^5*f - (19*b^3*c*d^2 - 9*a*b^2*d^3)*e^4*f^2 + 36*(b^3*c^2*d - a*b^2*c*d^2)*e^3*f^3
+ 2*(12*b^3*c^3 - 60*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 29*a^3*d^3)*e^2*f^4 + 4*(3*a*b^2*c^3 - 12*a^2*b*c^2*d + 4*
a^3*c*d^2)*e*f^5 + 3*(3*a^2*b*c^3 - a^3*c^2*d)*f^6)*x)*sqrt(f*x + e))/(d^4*e^8*f^3 - 4*c*d^3*e^7*f^4 + 6*c^2*d
^2*e^6*f^5 - 4*c^3*d*e^5*f^6 + c^4*e^4*f^7 + (d^4*e^4*f^7 - 4*c*d^3*e^3*f^8 + 6*c^2*d^2*e^2*f^9 - 4*c^3*d*e*f^
10 + c^4*f^11)*x^4 + 4*(d^4*e^5*f^6 - 4*c*d^3*e^4*f^7 + 6*c^2*d^2*e^3*f^8 - 4*c^3*d*e^2*f^9 + c^4*e*f^10)*x^3
+ 6*(d^4*e^6*f^5 - 4*c*d^3*e^5*f^6 + 6*c^2*d^2*e^4*f^7 - 4*c^3*d*e^3*f^8 + c^4*e^2*f^9)*x^2 + 4*(d^4*e^7*f^4 -
4*c*d^3*e^6*f^5 + 6*c^2*d^2*e^5*f^6 - 4*c^3*d*e^4*f^7 + c^4*e^3*f^8)*x), 2/105*(105*((b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*f^7*x^4 + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e*f^6*x^3 + 6*(b^3
*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^2*f^5*x^2 + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3
*d^3)*e^3*f^4*x + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^4*f^3)*sqrt(-d/(d*e - c*f))*arctan(-(d
*e - c*f)*sqrt(f*x + e)*sqrt(-d/(d*e - c*f))/(d*f*x + d*e)) - (8*b^3*d^3*e^6 + 15*a^3*c^3*f^6 + 105*(b^3*c^3 -
3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^6*x^3 - 2*(19*b^3*c*d^2 - 9*a*b^2*d^3)*e^5*f + 3*(29*b^3*c^2*d - 3
9*a*b^2*c*d^2 + 15*a^2*b*d^3)*e^4*f^2 + 4*(12*b^3*c^3 - 60*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 44*a^3*d^3)*e^3*f^3
+ 2*(12*a*b^2*c^3 - 48*a^2*b*c^2*d + 61*a^3*c*d^2)*e^2*f^4 + 6*(3*a^2*b*c^3 - 11*a^3*c^2*d)*e*f^5 + 35*(b^3*d^
3*e^4*f^2 - 4*b^3*c*d^2*e^3*f^3 + 6*b^3*c^2*d*e^2*f^4 + 2*(3*b^3*c^3 - 15*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 5*a^3
*d^3)*e*f^5 + (3*a*b^2*c^3 - 3*a^2*b*c^2*d + a^3*c*d^2)*f^6)*x^2 + 7*(4*b^3*d^3*e^5*f - (19*b^3*c*d^2 - 9*a*b^
2*d^3)*e^4*f^2 + 36*(b^3*c^2*d - a*b^2*c*d^2)*e^3*f^3 + 2*(12*b^3*c^3 - 60*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 29*a
^3*d^3)*e^2*f^4 + 4*(3*a*b^2*c^3 - 12*a^2*b*c^2*d + 4*a^3*c*d^2)*e*f^5 + 3*(3*a^2*b*c^3 - a^3*c^2*d)*f^6)*x)*s
qrt(f*x + e))/(d^4*e^8*f^3 - 4*c*d^3*e^7*f^4 + 6*c^2*d^2*e^6*f^5 - 4*c^3*d*e^5*f^6 + c^4*e^4*f^7 + (d^4*e^4*f^
7 - 4*c*d^3*e^3*f^8 + 6*c^2*d^2*e^2*f^9 - 4*c^3*d*e*f^10 + c^4*f^11)*x^4 + 4*(d^4*e^5*f^6 - 4*c*d^3*e^4*f^7 +
6*c^2*d^2*e^3*f^8 - 4*c^3*d*e^2*f^9 + c^4*e*f^10)*x^3 + 6*(d^4*e^6*f^5 - 4*c*d^3*e^5*f^6 + 6*c^2*d^2*e^4*f^7 -
4*c^3*d*e^3*f^8 + c^4*e^2*f^9)*x^2 + 4*(d^4*e^7*f^4 - 4*c*d^3*e^6*f^5 + 6*c^2*d^2*e^5*f^6 - 4*c^3*d*e^4*f^7 +
c^4*e^3*f^8)*x)]

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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((b*x+a)**3/(d*x+c)/(f*x+e)**(9/2),x)

[Out]

Timed out

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Giac [B]  time = 1.69782, size = 1308, normalized size = 5.03 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)/(f*x+e)^(9/2),x, algorithm="giac")

[Out]

-2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/((c^4*f
^4 - 4*c^3*d*f^3*e + 6*c^2*d^2*f^2*e^2 - 4*c*d^3*f*e^3 + d^4*e^4)*sqrt(c*d*f - d^2*e)) - 2/105*(105*(f*x + e)^
3*b^3*c^3*f^3 - 315*(f*x + e)^3*a*b^2*c^2*d*f^3 + 315*(f*x + e)^3*a^2*b*c*d^2*f^3 - 105*(f*x + e)^3*a^3*d^3*f^
3 + 105*(f*x + e)^2*a*b^2*c^3*f^4 - 105*(f*x + e)^2*a^2*b*c^2*d*f^4 + 35*(f*x + e)^2*a^3*c*d^2*f^4 + 63*(f*x +
e)*a^2*b*c^3*f^5 - 21*(f*x + e)*a^3*c^2*d*f^5 + 15*a^3*c^3*f^6 - 105*(f*x + e)^2*b^3*c^3*f^3*e - 105*(f*x + e
)^2*a*b^2*c^2*d*f^3*e + 105*(f*x + e)^2*a^2*b*c*d^2*f^3*e - 35*(f*x + e)^2*a^3*d^3*f^3*e - 126*(f*x + e)*a*b^2
*c^3*f^4*e - 126*(f*x + e)*a^2*b*c^2*d*f^4*e + 42*(f*x + e)*a^3*c*d^2*f^4*e - 45*a^2*b*c^3*f^5*e - 45*a^3*c^2*
d*f^5*e + 210*(f*x + e)^2*b^3*c^2*d*f^2*e^2 + 63*(f*x + e)*b^3*c^3*f^3*e^2 + 315*(f*x + e)*a*b^2*c^2*d*f^3*e^2
+ 63*(f*x + e)*a^2*b*c*d^2*f^3*e^2 - 21*(f*x + e)*a^3*d^3*f^3*e^2 + 45*a*b^2*c^3*f^4*e^2 + 135*a^2*b*c^2*d*f^
4*e^2 + 45*a^3*c*d^2*f^4*e^2 - 140*(f*x + e)^2*b^3*c*d^2*f*e^3 - 168*(f*x + e)*b^3*c^2*d*f^2*e^3 - 252*(f*x +
e)*a*b^2*c*d^2*f^2*e^3 - 15*b^3*c^3*f^3*e^3 - 135*a*b^2*c^2*d*f^3*e^3 - 135*a^2*b*c*d^2*f^3*e^3 - 15*a^3*d^3*f
^3*e^3 + 35*(f*x + e)^2*b^3*d^3*e^4 + 147*(f*x + e)*b^3*c*d^2*f*e^4 + 63*(f*x + e)*a*b^2*d^3*f*e^4 + 45*b^3*c^
2*d*f^2*e^4 + 135*a*b^2*c*d^2*f^2*e^4 + 45*a^2*b*d^3*f^2*e^4 - 42*(f*x + e)*b^3*d^3*e^5 - 45*b^3*c*d^2*f*e^5 -
45*a*b^2*d^3*f*e^5 + 15*b^3*d^3*e^6)/((c^4*f^7 - 4*c^3*d*f^6*e + 6*c^2*d^2*f^5*e^2 - 4*c*d^3*f^4*e^3 + d^4*f^
3*e^4)*(f*x + e)^(7/2))