3.1645 \(\int \frac{(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=162 \[ \frac{9 e (d+e x)^{5/2} (b d-a e)}{5 b^3}+\frac{3 e (d+e x)^{3/2} (b d-a e)^2}{b^4}+\frac{9 e \sqrt{d+e x} (b d-a e)^3}{b^5}-\frac{9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{9 e (d+e x)^{7/2}}{7 b^2} \]

[Out]

(9*e*(b*d - a*e)^3*Sqrt[d + e*x])/b^5 + (3*e*(b*d - a*e)^2*(d + e*x)^(3/2))/b^4 + (9*e*(b*d - a*e)*(d + e*x)^(
5/2))/(5*b^3) + (9*e*(d + e*x)^(7/2))/(7*b^2) - (d + e*x)^(9/2)/(b*(a + b*x)) - (9*e*(b*d - a*e)^(7/2)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)

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Rubi [A]  time = 0.182752, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {27, 47, 50, 63, 208} \[ \frac{9 e (d+e x)^{5/2} (b d-a e)}{5 b^3}+\frac{3 e (d+e x)^{3/2} (b d-a e)^2}{b^4}+\frac{9 e \sqrt{d+e x} (b d-a e)^3}{b^5}-\frac{9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{9 e (d+e x)^{7/2}}{7 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*e*(b*d - a*e)^3*Sqrt[d + e*x])/b^5 + (3*e*(b*d - a*e)^2*(d + e*x)^(3/2))/b^4 + (9*e*(b*d - a*e)*(d + e*x)^(
5/2))/(5*b^3) + (9*e*(d + e*x)^(7/2))/(7*b^2) - (d + e*x)^(9/2)/(b*(a + b*x)) - (9*e*(b*d - a*e)^(7/2)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 47

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

Rule 50

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

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{(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^{9/2}}{(a+b x)^2} \, dx\\ &=-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{(9 e) \int \frac{(d+e x)^{7/2}}{a+b x} \, dx}{2 b}\\ &=\frac{9 e (d+e x)^{7/2}}{7 b^2}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{(9 e (b d-a e)) \int \frac{(d+e x)^{5/2}}{a+b x} \, dx}{2 b^2}\\ &=\frac{9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac{9 e (d+e x)^{7/2}}{7 b^2}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{\left (9 e (b d-a e)^2\right ) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{2 b^3}\\ &=\frac{3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac{9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac{9 e (d+e x)^{7/2}}{7 b^2}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{\left (9 e (b d-a e)^3\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{2 b^4}\\ &=\frac{9 e (b d-a e)^3 \sqrt{d+e x}}{b^5}+\frac{3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac{9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac{9 e (d+e x)^{7/2}}{7 b^2}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{\left (9 e (b d-a e)^4\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{2 b^5}\\ &=\frac{9 e (b d-a e)^3 \sqrt{d+e x}}{b^5}+\frac{3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac{9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac{9 e (d+e x)^{7/2}}{7 b^2}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{\left (9 (b d-a e)^4\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{b^5}\\ &=\frac{9 e (b d-a e)^3 \sqrt{d+e x}}{b^5}+\frac{3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac{9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac{9 e (d+e x)^{7/2}}{7 b^2}-\frac{(d+e x)^{9/2}}{b (a+b x)}-\frac{9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}\\ \end{align*}

Mathematica [C]  time = 0.024523, size = 50, normalized size = 0.31 \[ \frac{2 e (d+e x)^{11/2} \, _2F_1\left (2,\frac{11}{2};\frac{13}{2};-\frac{b (d+e x)}{a e-b d}\right )}{11 (a e-b d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*(d + e*x)^(11/2)*Hypergeometric2F1[2, 11/2, 13/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(11*(-(b*d) + a*e)^2)

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Maple [B]  time = 0.213, size = 539, normalized size = 3.3 \begin{align*}{\frac{2\,e}{7\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{4\,a{e}^{2}}{5\,{b}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{4\,de}{5\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+2\,{\frac{ \left ( ex+d \right ) ^{3/2}{a}^{2}{e}^{3}}{{b}^{4}}}-4\,{\frac{ \left ( ex+d \right ) ^{3/2}ad{e}^{2}}{{b}^{3}}}+2\,{\frac{e \left ( ex+d \right ) ^{3/2}{d}^{2}}{{b}^{2}}}-8\,{\frac{{a}^{3}{e}^{4}\sqrt{ex+d}}{{b}^{5}}}+24\,{\frac{{a}^{2}d{e}^{3}\sqrt{ex+d}}{{b}^{4}}}-24\,{\frac{a{d}^{2}{e}^{2}\sqrt{ex+d}}{{b}^{3}}}+8\,{\frac{e{d}^{3}\sqrt{ex+d}}{{b}^{2}}}-{\frac{{a}^{4}{e}^{5}}{{b}^{5} \left ( bxe+ae \right ) }\sqrt{ex+d}}+4\,{\frac{\sqrt{ex+d}{a}^{3}d{e}^{4}}{{b}^{4} \left ( bxe+ae \right ) }}-6\,{\frac{\sqrt{ex+d}{d}^{2}{e}^{3}{a}^{2}}{{b}^{3} \left ( bxe+ae \right ) }}+4\,{\frac{\sqrt{ex+d}a{d}^{3}{e}^{2}}{{b}^{2} \left ( bxe+ae \right ) }}-{\frac{e{d}^{4}}{b \left ( bxe+ae \right ) }\sqrt{ex+d}}+9\,{\frac{{a}^{4}{e}^{5}}{{b}^{5}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-36\,{\frac{{a}^{3}d{e}^{4}}{{b}^{4}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+54\,{\frac{{d}^{2}{e}^{3}{a}^{2}}{{b}^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-36\,{\frac{a{d}^{3}{e}^{2}}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+9\,{\frac{e{d}^{4}}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*e*(e*x+d)^(7/2)/b^2-4/5/b^3*(e*x+d)^(5/2)*a*e^2+4/5*e/b^2*(e*x+d)^(5/2)*d+2/b^4*(e*x+d)^(3/2)*a^2*e^3-4/b^
3*(e*x+d)^(3/2)*a*d*e^2+2*e/b^2*(e*x+d)^(3/2)*d^2-8/b^5*a^3*e^4*(e*x+d)^(1/2)+24/b^4*a^2*d*e^3*(e*x+d)^(1/2)-2
4/b^3*a*d^2*e^2*(e*x+d)^(1/2)+8*e/b^2*d^3*(e*x+d)^(1/2)-1/b^5*(e*x+d)^(1/2)/(b*e*x+a*e)*a^4*e^5+4/b^4*(e*x+d)^
(1/2)/(b*e*x+a*e)*a^3*d*e^4-6/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*d^2*e^3*a^2+4/b^2*(e*x+d)^(1/2)/(b*e*x+a*e)*a*d^3*
e^2-e/b*(e*x+d)^(1/2)/(b*e*x+a*e)*d^4+9/b^5/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^
4*e^5-36/b^4/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^3*d*e^4+54/b^3/((a*e-b*d)*b)^(1
/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*d^2*e^3*a^2-36/b^2/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/
((a*e-b*d)*b)^(1/2))*a*d^3*e^2+9*e/b/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*d^4

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.0296, size = 1450, normalized size = 8.95 \begin{align*} \left [-\frac{315 \,{\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (10 \, b^{4} e^{4} x^{4} - 35 \, b^{4} d^{4} + 528 \, a b^{3} d^{3} e - 1218 \, a^{2} b^{2} d^{2} e^{2} + 1050 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 2 \,{\left (29 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (26 \, b^{4} d^{2} e^{2} - 23 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (194 \, b^{4} d^{3} e - 426 \, a b^{3} d^{2} e^{2} + 357 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{70 \,{\left (b^{6} x + a b^{5}\right )}}, -\frac{315 \,{\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (-\frac{\sqrt{e x + d} b \sqrt{-\frac{b d - a e}{b}}}{b d - a e}\right ) -{\left (10 \, b^{4} e^{4} x^{4} - 35 \, b^{4} d^{4} + 528 \, a b^{3} d^{3} e - 1218 \, a^{2} b^{2} d^{2} e^{2} + 1050 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 2 \,{\left (29 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (26 \, b^{4} d^{2} e^{2} - 23 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (194 \, b^{4} d^{3} e - 426 \, a b^{3} d^{2} e^{2} + 357 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (b^{6} x + a b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/70*(315*(a*b^3*d^3*e - 3*a^2*b^2*d^2*e^2 + 3*a^3*b*d*e^3 - a^4*e^4 + (b^4*d^3*e - 3*a*b^3*d^2*e^2 + 3*a^2*
b^2*d*e^3 - a^3*b*e^4)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b)
)/(b*x + a)) - 2*(10*b^4*e^4*x^4 - 35*b^4*d^4 + 528*a*b^3*d^3*e - 1218*a^2*b^2*d^2*e^2 + 1050*a^3*b*d*e^3 - 31
5*a^4*e^4 + 2*(29*b^4*d*e^3 - 9*a*b^3*e^4)*x^3 + 6*(26*b^4*d^2*e^2 - 23*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 + 2*(
194*b^4*d^3*e - 426*a*b^3*d^2*e^2 + 357*a^2*b^2*d*e^3 - 105*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^6*x + a*b^5), -1/3
5*(315*(a*b^3*d^3*e - 3*a^2*b^2*d^2*e^2 + 3*a^3*b*d*e^3 - a^4*e^4 + (b^4*d^3*e - 3*a*b^3*d^2*e^2 + 3*a^2*b^2*d
*e^3 - a^3*b*e^4)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (10*b^4*
e^4*x^4 - 35*b^4*d^4 + 528*a*b^3*d^3*e - 1218*a^2*b^2*d^2*e^2 + 1050*a^3*b*d*e^3 - 315*a^4*e^4 + 2*(29*b^4*d*e
^3 - 9*a*b^3*e^4)*x^3 + 6*(26*b^4*d^2*e^2 - 23*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 + 2*(194*b^4*d^3*e - 426*a*b^3
*d^2*e^2 + 357*a^2*b^2*d*e^3 - 105*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^6*x + a*b^5)]

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

[Out]

Timed out

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Giac [B]  time = 1.26353, size = 522, normalized size = 3.22 \begin{align*} \frac{9 \,{\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{5}} - \frac{\sqrt{x e + d} b^{4} d^{4} e - 4 \, \sqrt{x e + d} a b^{3} d^{3} e^{2} + 6 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{3} - 4 \, \sqrt{x e + d} a^{3} b d e^{4} + \sqrt{x e + d} a^{4} e^{5}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{5}} + \frac{2 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{12} e + 14 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{12} d e + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{12} d^{2} e + 140 \, \sqrt{x e + d} b^{12} d^{3} e - 14 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{11} e^{2} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{11} d e^{2} - 420 \, \sqrt{x e + d} a b^{11} d^{2} e^{2} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{10} e^{3} + 420 \, \sqrt{x e + d} a^{2} b^{10} d e^{3} - 140 \, \sqrt{x e + d} a^{3} b^{9} e^{4}\right )}}{35 \, b^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9*(b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2
*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5) - (sqrt(x*e + d)*b^4*d^4*e - 4*sqrt(x*e + d)*a*b^3*d^3*e^2 + 6*sqrt(x*
e + d)*a^2*b^2*d^2*e^3 - 4*sqrt(x*e + d)*a^3*b*d*e^4 + sqrt(x*e + d)*a^4*e^5)/(((x*e + d)*b - b*d + a*e)*b^5)
+ 2/35*(5*(x*e + d)^(7/2)*b^12*e + 14*(x*e + d)^(5/2)*b^12*d*e + 35*(x*e + d)^(3/2)*b^12*d^2*e + 140*sqrt(x*e
+ d)*b^12*d^3*e - 14*(x*e + d)^(5/2)*a*b^11*e^2 - 70*(x*e + d)^(3/2)*a*b^11*d*e^2 - 420*sqrt(x*e + d)*a*b^11*d
^2*e^2 + 35*(x*e + d)^(3/2)*a^2*b^10*e^3 + 420*sqrt(x*e + d)*a^2*b^10*d*e^3 - 140*sqrt(x*e + d)*a^3*b^9*e^4)/b
^14