3.2090 \(\int \frac{a+b x}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=206 \[ \frac{315 e^4}{64 \sqrt{d+e x} (b d-a e)^5}+\frac{105 e^3}{64 (a+b x) \sqrt{d+e x} (b d-a e)^4}-\frac{21 e^2}{32 (a+b x)^2 \sqrt{d+e x} (b d-a e)^3}-\frac{315 \sqrt{b} e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 (b d-a e)^{11/2}}+\frac{3 e}{8 (a+b x)^3 \sqrt{d+e x} (b d-a e)^2}-\frac{1}{4 (a+b x)^4 \sqrt{d+e x} (b d-a e)} \]

[Out]

(315*e^4)/(64*(b*d - a*e)^5*Sqrt[d + e*x]) - 1/(4*(b*d - a*e)*(a + b*x)^4*Sqrt[d + e*x]) + (3*e)/(8*(b*d - a*e
)^2*(a + b*x)^3*Sqrt[d + e*x]) - (21*e^2)/(32*(b*d - a*e)^3*(a + b*x)^2*Sqrt[d + e*x]) + (105*e^3)/(64*(b*d -
a*e)^4*(a + b*x)*Sqrt[d + e*x]) - (315*Sqrt[b]*e^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*(b*d
- a*e)^(11/2))

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Rubi [A]  time = 0.12396, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {27, 51, 63, 208} \[ \frac{315 e^4}{64 \sqrt{d+e x} (b d-a e)^5}+\frac{105 e^3}{64 (a+b x) \sqrt{d+e x} (b d-a e)^4}-\frac{21 e^2}{32 (a+b x)^2 \sqrt{d+e x} (b d-a e)^3}-\frac{315 \sqrt{b} e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 (b d-a e)^{11/2}}+\frac{3 e}{8 (a+b x)^3 \sqrt{d+e x} (b d-a e)^2}-\frac{1}{4 (a+b x)^4 \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(315*e^4)/(64*(b*d - a*e)^5*Sqrt[d + e*x]) - 1/(4*(b*d - a*e)*(a + b*x)^4*Sqrt[d + e*x]) + (3*e)/(8*(b*d - a*e
)^2*(a + b*x)^3*Sqrt[d + e*x]) - (21*e^2)/(32*(b*d - a*e)^3*(a + b*x)^2*Sqrt[d + e*x]) + (105*e^3)/(64*(b*d -
a*e)^4*(a + b*x)*Sqrt[d + e*x]) - (315*Sqrt[b]*e^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*(b*d
- a*e)^(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 51

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*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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{a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^5 (d+e x)^{3/2}} \, dx\\ &=-\frac{1}{4 (b d-a e) (a+b x)^4 \sqrt{d+e x}}-\frac{(9 e) \int \frac{1}{(a+b x)^4 (d+e x)^{3/2}} \, dx}{8 (b d-a e)}\\ &=-\frac{1}{4 (b d-a e) (a+b x)^4 \sqrt{d+e x}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt{d+e x}}+\frac{\left (21 e^2\right ) \int \frac{1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{16 (b d-a e)^2}\\ &=-\frac{1}{4 (b d-a e) (a+b x)^4 \sqrt{d+e x}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt{d+e x}}-\frac{21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt{d+e x}}-\frac{\left (105 e^3\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{64 (b d-a e)^3}\\ &=-\frac{1}{4 (b d-a e) (a+b x)^4 \sqrt{d+e x}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt{d+e x}}-\frac{21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt{d+e x}}+\frac{105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt{d+e x}}+\frac{\left (315 e^4\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{128 (b d-a e)^4}\\ &=\frac{315 e^4}{64 (b d-a e)^5 \sqrt{d+e x}}-\frac{1}{4 (b d-a e) (a+b x)^4 \sqrt{d+e x}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt{d+e x}}-\frac{21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt{d+e x}}+\frac{105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt{d+e x}}+\frac{\left (315 b e^4\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{128 (b d-a e)^5}\\ &=\frac{315 e^4}{64 (b d-a e)^5 \sqrt{d+e x}}-\frac{1}{4 (b d-a e) (a+b x)^4 \sqrt{d+e x}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt{d+e x}}-\frac{21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt{d+e x}}+\frac{105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt{d+e x}}+\frac{\left (315 b e^3\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{64 (b d-a e)^5}\\ &=\frac{315 e^4}{64 (b d-a e)^5 \sqrt{d+e x}}-\frac{1}{4 (b d-a e) (a+b x)^4 \sqrt{d+e x}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt{d+e x}}-\frac{21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt{d+e x}}+\frac{105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt{d+e x}}-\frac{315 \sqrt{b} e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 (b d-a e)^{11/2}}\\ \end{align*}

Mathematica [C]  time = 0.023864, size = 50, normalized size = 0.24 \[ -\frac{2 e^4 \, _2F_1\left (-\frac{1}{2},5;\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right )}{\sqrt{d+e x} (a e-b d)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e^4*Hypergeometric2F1[-1/2, 5, 1/2, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^5*Sqrt[d + e*x])

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Maple [B]  time = 0.023, size = 446, normalized size = 2.2 \begin{align*} -2\,{\frac{{e}^{4}}{ \left ( ae-bd \right ) ^{5}\sqrt{ex+d}}}-{\frac{187\,{b}^{4}{e}^{4}}{64\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{643\,{e}^{5}{b}^{3}a}{64\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{643\,{b}^{4}{e}^{4}d}{64\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{765\,{e}^{6}{b}^{2}{a}^{2}}{64\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{765\,{e}^{5}{b}^{3}ad}{32\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{765\,{b}^{4}{e}^{4}{d}^{2}}{64\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{325\,{a}^{3}b{e}^{7}}{64\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{4}}\sqrt{ex+d}}+{\frac{975\,{a}^{2}{b}^{2}d{e}^{6}}{64\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{4}}\sqrt{ex+d}}-{\frac{975\,{e}^{5}{b}^{3}a{d}^{2}}{64\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{4}}\sqrt{ex+d}}+{\frac{325\,{b}^{4}{e}^{4}{d}^{3}}{64\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{4}}\sqrt{ex+d}}-{\frac{315\,{e}^{4}b}{64\, \left ( ae-bd \right ) ^{5}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*e^4/(a*e-b*d)^5/(e*x+d)^(1/2)-187/64*e^4/(a*e-b*d)^5*b^4/(b*e*x+a*e)^4*(e*x+d)^(7/2)-643/64*e^5/(a*e-b*d)^5
*b^3/(b*e*x+a*e)^4*(e*x+d)^(5/2)*a+643/64*e^4/(a*e-b*d)^5*b^4/(b*e*x+a*e)^4*(e*x+d)^(5/2)*d-765/64*e^6/(a*e-b*
d)^5*b^2/(b*e*x+a*e)^4*(e*x+d)^(3/2)*a^2+765/32*e^5/(a*e-b*d)^5*b^3/(b*e*x+a*e)^4*(e*x+d)^(3/2)*a*d-765/64*e^4
/(a*e-b*d)^5*b^4/(b*e*x+a*e)^4*(e*x+d)^(3/2)*d^2-325/64*e^7/(a*e-b*d)^5*b/(b*e*x+a*e)^4*(e*x+d)^(1/2)*a^3+975/
64*e^6/(a*e-b*d)^5*b^2/(b*e*x+a*e)^4*(e*x+d)^(1/2)*d*a^2-975/64*e^5/(a*e-b*d)^5*b^3/(b*e*x+a*e)^4*(e*x+d)^(1/2
)*a*d^2+325/64*e^4/(a*e-b*d)^5*b^4/(b*e*x+a*e)^4*(e*x+d)^(1/2)*d^3-315/64*e^4/(a*e-b*d)^5*b/((a*e-b*d)*b)^(1/2
)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.33735, size = 3549, normalized size = 17.23 \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)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/128*(315*(b^4*e^5*x^5 + a^4*d*e^4 + (b^4*d*e^4 + 4*a*b^3*e^5)*x^4 + 2*(2*a*b^3*d*e^4 + 3*a^2*b^2*e^5)*x^3
+ 2*(3*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*x^2 + (4*a^3*b*d*e^4 + a^4*e^5)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d
- a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) - 2*(315*b^4*e^4*x^4 - 16*b^4*d^4 + 88*a*b
^3*d^3*e - 210*a^2*b^2*d^2*e^2 + 325*a^3*b*d*e^3 + 128*a^4*e^4 + 105*(b^4*d*e^3 + 11*a*b^3*e^4)*x^3 - 21*(2*b^
4*d^2*e^2 - 19*a*b^3*d*e^3 - 73*a^2*b^2*e^4)*x^2 + 3*(8*b^4*d^3*e - 52*a*b^3*d^2*e^2 + 185*a^2*b^2*d*e^3 + 279
*a^3*b*e^4)*x)*sqrt(e*x + d))/(a^4*b^5*d^6 - 5*a^5*b^4*d^5*e + 10*a^6*b^3*d^4*e^2 - 10*a^7*b^2*d^3*e^3 + 5*a^8
*b*d^2*e^4 - a^9*d*e^5 + (b^9*d^5*e - 5*a*b^8*d^4*e^2 + 10*a^2*b^7*d^3*e^3 - 10*a^3*b^6*d^2*e^4 + 5*a^4*b^5*d*
e^5 - a^5*b^4*e^6)*x^5 + (b^9*d^6 - a*b^8*d^5*e - 10*a^2*b^7*d^4*e^2 + 30*a^3*b^6*d^3*e^3 - 35*a^4*b^5*d^2*e^4
+ 19*a^5*b^4*d*e^5 - 4*a^6*b^3*e^6)*x^4 + 2*(2*a*b^8*d^6 - 7*a^2*b^7*d^5*e + 5*a^3*b^6*d^4*e^2 + 10*a^4*b^5*d
^3*e^3 - 20*a^5*b^4*d^2*e^4 + 13*a^6*b^3*d*e^5 - 3*a^7*b^2*e^6)*x^3 + 2*(3*a^2*b^7*d^6 - 13*a^3*b^6*d^5*e + 20
*a^4*b^5*d^4*e^2 - 10*a^5*b^4*d^3*e^3 - 5*a^6*b^3*d^2*e^4 + 7*a^7*b^2*d*e^5 - 2*a^8*b*e^6)*x^2 + (4*a^3*b^6*d^
6 - 19*a^4*b^5*d^5*e + 35*a^5*b^4*d^4*e^2 - 30*a^6*b^3*d^3*e^3 + 10*a^7*b^2*d^2*e^4 + a^8*b*d*e^5 - a^9*e^6)*x
), -1/64*(315*(b^4*e^5*x^5 + a^4*d*e^4 + (b^4*d*e^4 + 4*a*b^3*e^5)*x^4 + 2*(2*a*b^3*d*e^4 + 3*a^2*b^2*e^5)*x^3
+ 2*(3*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*x^2 + (4*a^3*b*d*e^4 + a^4*e^5)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*
e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (315*b^4*e^4*x^4 - 16*b^4*d^4 + 88*a*b^3*d^3*e - 210*a^
2*b^2*d^2*e^2 + 325*a^3*b*d*e^3 + 128*a^4*e^4 + 105*(b^4*d*e^3 + 11*a*b^3*e^4)*x^3 - 21*(2*b^4*d^2*e^2 - 19*a*
b^3*d*e^3 - 73*a^2*b^2*e^4)*x^2 + 3*(8*b^4*d^3*e - 52*a*b^3*d^2*e^2 + 185*a^2*b^2*d*e^3 + 279*a^3*b*e^4)*x)*sq
rt(e*x + d))/(a^4*b^5*d^6 - 5*a^5*b^4*d^5*e + 10*a^6*b^3*d^4*e^2 - 10*a^7*b^2*d^3*e^3 + 5*a^8*b*d^2*e^4 - a^9*
d*e^5 + (b^9*d^5*e - 5*a*b^8*d^4*e^2 + 10*a^2*b^7*d^3*e^3 - 10*a^3*b^6*d^2*e^4 + 5*a^4*b^5*d*e^5 - a^5*b^4*e^6
)*x^5 + (b^9*d^6 - a*b^8*d^5*e - 10*a^2*b^7*d^4*e^2 + 30*a^3*b^6*d^3*e^3 - 35*a^4*b^5*d^2*e^4 + 19*a^5*b^4*d*e
^5 - 4*a^6*b^3*e^6)*x^4 + 2*(2*a*b^8*d^6 - 7*a^2*b^7*d^5*e + 5*a^3*b^6*d^4*e^2 + 10*a^4*b^5*d^3*e^3 - 20*a^5*b
^4*d^2*e^4 + 13*a^6*b^3*d*e^5 - 3*a^7*b^2*e^6)*x^3 + 2*(3*a^2*b^7*d^6 - 13*a^3*b^6*d^5*e + 20*a^4*b^5*d^4*e^2
- 10*a^5*b^4*d^3*e^3 - 5*a^6*b^3*d^2*e^4 + 7*a^7*b^2*d*e^5 - 2*a^8*b*e^6)*x^2 + (4*a^3*b^6*d^6 - 19*a^4*b^5*d^
5*e + 35*a^5*b^4*d^4*e^2 - 30*a^6*b^3*d^3*e^3 + 10*a^7*b^2*d^2*e^4 + a^8*b*d*e^5 - a^9*e^6)*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)/(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.22642, size = 594, normalized size = 2.88 \begin{align*} \frac{315 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{4}}{64 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt{-b^{2} d + a b e}} + \frac{2 \, e^{4}}{{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt{x e + d}} + \frac{187 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} e^{4} - 643 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d e^{4} + 765 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{2} e^{4} - 325 \, \sqrt{x e + d} b^{4} d^{3} e^{4} + 643 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} e^{5} - 1530 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d e^{5} + 975 \, \sqrt{x e + d} a b^{3} d^{2} e^{5} + 765 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} e^{6} - 975 \, \sqrt{x e + d} a^{2} b^{2} d e^{6} + 325 \, \sqrt{x e + d} a^{3} b e^{7}}{64 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

315/64*b*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*
a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(-b^2*d + a*b*e)) + 2*e^4/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^
3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(x*e + d)) + 1/64*(187*(x*e + d)^(7/2)*b^4*e^4 -
643*(x*e + d)^(5/2)*b^4*d*e^4 + 765*(x*e + d)^(3/2)*b^4*d^2*e^4 - 325*sqrt(x*e + d)*b^4*d^3*e^4 + 643*(x*e +
d)^(5/2)*a*b^3*e^5 - 1530*(x*e + d)^(3/2)*a*b^3*d*e^5 + 975*sqrt(x*e + d)*a*b^3*d^2*e^5 + 765*(x*e + d)^(3/2)*
a^2*b^2*e^6 - 975*sqrt(x*e + d)*a^2*b^2*d*e^6 + 325*sqrt(x*e + d)*a^3*b*e^7)/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^
2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*((x*e + d)*b - b*d + a*e)^4)