### 3.630 $$\int \frac{1}{(d+e x)^{5/2} (a-c x^2)^2} \, dx$$

Optimal. Leaf size=311 $-\frac{c^{3/4} \left (2 \sqrt{c} d-7 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^{7/2}}+\frac{c^{3/4} \left (7 \sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} \left (\sqrt{a} e+\sqrt{c} d\right )^{7/2}}-\frac{a e-c d x}{2 a \left (a-c x^2\right ) (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{e \left (7 a e^2+3 c d^2\right )}{6 a (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}-\frac{c d e \left (19 a e^2+c d^2\right )}{2 a \sqrt{d+e x} \left (c d^2-a e^2\right )^3}$

[Out]

-(e*(3*c*d^2 + 7*a*e^2))/(6*a*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - (c*d*e*(c*d^2 + 19*a*e^2))/(2*a*(c*d^2 - a*
e^2)^3*Sqrt[d + e*x]) - (a*e - c*d*x)/(2*a*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*(a - c*x^2)) - (c^(3/4)*(2*Sqrt[c]*
d - 7*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*(Sqrt[c]*d - Sqrt[a]
*e)^(7/2)) + (c^(3/4)*(2*Sqrt[c]*d + 7*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]
)/(4*a^(3/2)*(Sqrt[c]*d + Sqrt[a]*e)^(7/2))

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Rubi [A]  time = 0.722576, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {741, 829, 827, 1166, 208} $-\frac{c^{3/4} \left (2 \sqrt{c} d-7 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^{7/2}}+\frac{c^{3/4} \left (7 \sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} \left (\sqrt{a} e+\sqrt{c} d\right )^{7/2}}-\frac{a e-c d x}{2 a \left (a-c x^2\right ) (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{e \left (7 a e^2+3 c d^2\right )}{6 a (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}-\frac{c d e \left (19 a e^2+c d^2\right )}{2 a \sqrt{d+e x} \left (c d^2-a e^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^(5/2)*(a - c*x^2)^2),x]

[Out]

-(e*(3*c*d^2 + 7*a*e^2))/(6*a*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - (c*d*e*(c*d^2 + 19*a*e^2))/(2*a*(c*d^2 - a*
e^2)^3*Sqrt[d + e*x]) - (a*e - c*d*x)/(2*a*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*(a - c*x^2)) - (c^(3/4)*(2*Sqrt[c]*
d - 7*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*(Sqrt[c]*d - Sqrt[a]
*e)^(7/2)) + (c^(3/4)*(2*Sqrt[c]*d + 7*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]
)/(4*a^(3/2)*(Sqrt[c]*d + Sqrt[a]*e)^(7/2))

Rule 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 829

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((e*f - d*g)*(d
+ e*x)^(m + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d*f + a*
e*g - c*(e*f - d*g)*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&
FractionQ[m] && LtQ[m, -1]

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

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{1}{(d+e x)^{5/2} \left (a-c x^2\right )^2} \, dx &=-\frac{a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}+\frac{\int \frac{\frac{1}{2} \left (2 c d^2-7 a e^2\right )+\frac{5}{2} c d e x}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx}{2 a \left (c d^2-a e^2\right )}\\ &=-\frac{e \left (3 c d^2+7 a e^2\right )}{6 a \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac{a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}-\frac{\int \frac{-c d \left (c d^2-6 a e^2\right )-\frac{1}{2} c e \left (3 c d^2+7 a e^2\right ) x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx}{2 a \left (c d^2-a e^2\right )^2}\\ &=-\frac{e \left (3 c d^2+7 a e^2\right )}{6 a \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac{c d e \left (c d^2+19 a e^2\right )}{2 a \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}-\frac{a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}+\frac{\int \frac{\frac{1}{2} c \left (2 c^2 d^4-15 a c d^2 e^2-7 a^2 e^4\right )+\frac{1}{2} c^2 d e \left (c d^2+19 a e^2\right ) x}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx}{2 a \left (c d^2-a e^2\right )^3}\\ &=-\frac{e \left (3 c d^2+7 a e^2\right )}{6 a \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac{c d e \left (c d^2+19 a e^2\right )}{2 a \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}-\frac{a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2} c^2 d^2 e \left (c d^2+19 a e^2\right )+\frac{1}{2} c e \left (2 c^2 d^4-15 a c d^2 e^2-7 a^2 e^4\right )+\frac{1}{2} c^2 d e \left (c d^2+19 a e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt{d+e x}\right )}{a \left (c d^2-a e^2\right )^3}\\ &=-\frac{e \left (3 c d^2+7 a e^2\right )}{6 a \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac{c d e \left (c d^2+19 a e^2\right )}{2 a \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}-\frac{a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}-\frac{\left (c^{3/2} \left (2 \sqrt{c} d-7 \sqrt{a} e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c d-\sqrt{a} \sqrt{c} e-c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^3}+\frac{\left (c^{3/2} \left (2 \sqrt{c} d+7 \sqrt{a} e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c d+\sqrt{a} \sqrt{c} e-c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 a^{3/2} \left (\sqrt{c} d+\sqrt{a} e\right )^3}\\ &=-\frac{e \left (3 c d^2+7 a e^2\right )}{6 a \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac{c d e \left (c d^2+19 a e^2\right )}{2 a \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}-\frac{a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}-\frac{c^{3/4} \left (2 \sqrt{c} d-7 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^{7/2}}+\frac{c^{3/4} \left (2 \sqrt{c} d+7 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{a} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d+\sqrt{a} e\right )^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.302566, size = 327, normalized size = 1.05 $\frac{\frac{\left (7 a e^2+3 c d^2\right ) \left (\left (\sqrt{a} e+\sqrt{c} d\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right )+\left (\sqrt{a} e-\sqrt{c} d\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right )\right )}{\sqrt{a} \left (c d^2-a e^2\right )}+\frac{15 c d (d+e x) \left (\left (\sqrt{a} e+\sqrt{c} d\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right )+\left (\sqrt{a} e-\sqrt{c} d\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right )\right )}{\sqrt{a} \left (a e^2-c d^2\right )}+\frac{6 a e-6 c d x}{a-c x^2}}{12 a (d+e x)^{3/2} \left (a e^2-c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^(5/2)*(a - c*x^2)^2),x]

[Out]

((6*a*e - 6*c*d*x)/(a - c*x^2) + ((3*c*d^2 + 7*a*e^2)*((Sqrt[c]*d + Sqrt[a]*e)*Hypergeometric2F1[-3/2, 1, -1/2
, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)] + (-(Sqrt[c]*d) + Sqrt[a]*e)*Hypergeometric2F1[-3/2, 1, -1/2, (
Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]))/(Sqrt[a]*(c*d^2 - a*e^2)) + (15*c*d*(d + e*x)*((Sqrt[c]*d + Sqrt
[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)] + (-(Sqrt[c]*d) + Sqrt[a]*
e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]))/(Sqrt[a]*(-(c*d^2) + a*e^2))
)/(12*a*(-(c*d^2) + a*e^2)*(d + e*x)^(3/2))

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Maple [B]  time = 0.24, size = 1021, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(-c*x^2+a)^2,x)

[Out]

-2/3*e^3/(a*e^2-c*d^2)^2/(e*x+d)^(3/2)+8*e^3/(a*e^2-c*d^2)^3*c*d/(e*x+d)^(1/2)+3/2*e^3/(a*e^2-c*d^2)^3*c^2/(c*
e^2*x^2-a*e^2)*d*(e*x+d)^(3/2)+1/2*e/(a*e^2-c*d^2)^3*c^3/(c*e^2*x^2-a*e^2)*d^3/a*(e*x+d)^(3/2)-1/2*e^5/(a*e^2-
c*d^2)^3*c/(c*e^2*x^2-a*e^2)*a*(e*x+d)^(1/2)-3*e^3/(a*e^2-c*d^2)^3*c^2/(c*e^2*x^2-a*e^2)*(e*x+d)^(1/2)*d^2-1/2
*e/(a*e^2-c*d^2)^3*c^3/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(1/2)*d^4+7/4*e^5/(a*e^2-c*d^2)^3*c^2*a/(a*c*e^2)^(1/2)/((-
c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))+15/4*e^3/(a*e^2-c*d^2)^
3*c^3/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)
)*d^2-1/2*e/(a*e^2-c*d^2)^3*c^4/a/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c
*d+(a*c*e^2)^(1/2))*c)^(1/2))*d^4+19/4*e^3/(a*e^2-c*d^2)^3*c^2/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)
^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*d+1/4*e/(a*e^2-c*d^2)^3*c^3/a/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arct
an((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*d^3+7/4*e^5/(a*e^2-c*d^2)^3*c^2*a/(a*c*e^2)^(1/2)/((c*d+(
a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+15/4*e^3/(a*e^2-c*d^2)^3*c^3
/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*d^2-
1/2*e/(a*e^2-c*d^2)^3*c^4/a/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c
*e^2)^(1/2))*c)^(1/2))*d^4-19/4*e^3/(a*e^2-c*d^2)^3*c^2/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*
c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*d-1/4*e/(a*e^2-c*d^2)^3*c^3/a/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+
d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*d^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} - a\right )}^{2}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(-c*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 - a)^2*(e*x + d)^(5/2)), x)

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Fricas [B]  time = 16.4105, size = 18263, normalized size = 58.72 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(-c*x^2+a)^2,x, algorithm="fricas")

[Out]

1/24*(3*(a^2*c^3*d^8 - 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 - a^5*d^2*e^6 - (a*c^4*d^6*e^2 - 3*a^2*c^3*d^4*e^4
+ 3*a^3*c^2*d^2*e^6 - a^4*c*e^8)*x^4 - 2*(a*c^4*d^7*e - 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 - a^4*c*d*e^7)*x
^3 - (a*c^4*d^8 - 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 - 4*a^4*c*d^2*e^6 + a^5*e^8)*x^2 + 2*(a^2*c^3*d^7*e -
3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 - a^5*d*e^7)*x)*sqrt((4*c^6*d^9 - 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 +
1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*
c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14)*sqrt((11025*c^9*d^12*e^
6 - 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 + 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 + 820
26*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 - 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^
6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 - 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^1
4*e^14 + 3003*a^11*c^6*d^12*e^16 - 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 +
91*a^15*c^2*d^4*e^24 - 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*
e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14))*log((420*
c^6*d^8*e^3 - 8421*a*c^5*d^6*e^5 + 36783*a^2*c^4*d^4*e^7 + 40817*a^3*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)*sqrt(e*x
+ d) + (105*a^2*c^6*d^10*e^4 - 4389*a^3*c^5*d^8*e^6 + 26274*a^4*c^4*d^6*e^8 + 34142*a^5*c^3*d^4*e^10 + 7525*a
^6*c^2*d^2*e^12 + 343*a^7*c*e^14 + 2*(a^3*c^9*d^19 - 15*a^4*c^8*d^17*e^2 + 64*a^5*c^7*d^15*e^4 - 112*a^6*c^6*d
^13*e^6 + 42*a^7*c^5*d^11*e^8 + 154*a^8*c^4*d^9*e^10 - 280*a^9*c^3*d^7*e^12 + 216*a^10*c^2*d^5*e^14 - 83*a^11*
c*d^3*e^16 + 13*a^12*d*e^18)*sqrt((11025*c^9*d^12*e^6 - 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 + 1360
716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 -
14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 - 2002*a^8*c^9*d^
18*e^10 + 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 - 2002*a^12*c^5*d^10*e^18
+ 1001*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 - 14*a^16*c*d^2*e^26 + a^17*e^28)))*s
qrt((4*c^6*d^9 - 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 + 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3*c^7*
d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^1
0 + 7*a^9*c*d^2*e^12 - a^10*e^14)*sqrt((11025*c^9*d^12*e^6 - 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 +
1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^
28 - 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 - 2002*a^8*c
^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 - 2002*a^12*c^5*d^10
*e^18 + 1001*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 - 14*a^16*c*d^2*e^26 + a^17*e^28
)))/(a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^
8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14))) - 3*(a^2*c^3*d^8 - 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 - a^5*
d^2*e^6 - (a*c^4*d^6*e^2 - 3*a^2*c^3*d^4*e^4 + 3*a^3*c^2*d^2*e^6 - a^4*c*e^8)*x^4 - 2*(a*c^4*d^7*e - 3*a^2*c^3
*d^5*e^3 + 3*a^3*c^2*d^3*e^5 - a^4*c*d*e^7)*x^3 - (a*c^4*d^8 - 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 - 4*a^4*c
*d^2*e^6 + a^5*e^8)*x^2 + 2*(a^2*c^3*d^7*e - 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 - a^5*d*e^7)*x)*sqrt((4*c^6*d
^9 - 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 + 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3*c^7*d^14 - 7*a^4
*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*
d^2*e^12 - a^10*e^14)*sqrt((11025*c^9*d^12*e^6 - 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 + 1360716*a^3
*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 - 14*a^4*
c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 - 2002*a^8*c^9*d^18*e^10
+ 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 - 2002*a^12*c^5*d^10*e^18 + 1001
*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 - 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7
*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^
10 + 7*a^9*c*d^2*e^12 - a^10*e^14))*log((420*c^6*d^8*e^3 - 8421*a*c^5*d^6*e^5 + 36783*a^2*c^4*d^4*e^7 + 40817*
a^3*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)*sqrt(e*x + d) - (105*a^2*c^6*d^10*e^4 - 4389*a^3*c^5*d^8*e^6 + 26274*a^4*
c^4*d^6*e^8 + 34142*a^5*c^3*d^4*e^10 + 7525*a^6*c^2*d^2*e^12 + 343*a^7*c*e^14 + 2*(a^3*c^9*d^19 - 15*a^4*c^8*d
^17*e^2 + 64*a^5*c^7*d^15*e^4 - 112*a^6*c^6*d^13*e^6 + 42*a^7*c^5*d^11*e^8 + 154*a^8*c^4*d^9*e^10 - 280*a^9*c^
3*d^7*e^12 + 216*a^10*c^2*d^5*e^14 - 83*a^11*c*d^3*e^16 + 13*a^12*d*e^18)*sqrt((11025*c^9*d^12*e^6 - 171990*a*
c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 + 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^
2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 - 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e
^6 + 1001*a^7*c^10*d^20*e^8 - 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003
*a^11*c^6*d^12*e^16 - 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d
^4*e^24 - 14*a^16*c*d^2*e^26 + a^17*e^28)))*sqrt((4*c^6*d^9 - 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 + 1155*a^
3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*
e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14)*sqrt((11025*c^9*d^12*e^6 - 1719
90*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 + 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c
^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 - 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d
^22*e^6 + 1001*a^7*c^10*d^20*e^8 - 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 +
3003*a^11*c^6*d^12*e^16 - 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*
c^2*d^4*e^24 - 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35
*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14))) + 3*(a^2*c^3*d^8
- 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 - a^5*d^2*e^6 - (a*c^4*d^6*e^2 - 3*a^2*c^3*d^4*e^4 + 3*a^3*c^2*d^2*e^6
- a^4*c*e^8)*x^4 - 2*(a*c^4*d^7*e - 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 - a^4*c*d*e^7)*x^3 - (a*c^4*d^8 - 4*
a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 - 4*a^4*c*d^2*e^6 + a^5*e^8)*x^2 + 2*(a^2*c^3*d^7*e - 3*a^3*c^2*d^5*e^3 +
3*a^4*c*d^3*e^5 - a^5*d*e^7)*x)*sqrt((4*c^6*d^9 - 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 + 1155*a^3*c^3*d^3*e^
6 + 315*a^4*c^2*d*e^8 - (a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7
*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14)*sqrt((11025*c^9*d^12*e^6 - 171990*a*c^8*d^1
0*e^8 + 494991*a^2*c^7*d^8*e^10 + 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^2*e^16
+ 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 - 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e^6 + 10
01*a^7*c^10*d^20*e^8 - 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c
^6*d^12*e^16 - 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24
- 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8
*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14))*log((420*c^6*d^8*e^3 - 8421*a
*c^5*d^6*e^5 + 36783*a^2*c^4*d^4*e^7 + 40817*a^3*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)*sqrt(e*x + d) + (105*a^2*c^6
*d^10*e^4 - 4389*a^3*c^5*d^8*e^6 + 26274*a^4*c^4*d^6*e^8 + 34142*a^5*c^3*d^4*e^10 + 7525*a^6*c^2*d^2*e^12 + 34
3*a^7*c*e^14 - 2*(a^3*c^9*d^19 - 15*a^4*c^8*d^17*e^2 + 64*a^5*c^7*d^15*e^4 - 112*a^6*c^6*d^13*e^6 + 42*a^7*c^5
*d^11*e^8 + 154*a^8*c^4*d^9*e^10 - 280*a^9*c^3*d^7*e^12 + 216*a^10*c^2*d^5*e^14 - 83*a^11*c*d^3*e^16 + 13*a^12
*d*e^18)*sqrt((11025*c^9*d^12*e^6 - 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 + 1360716*a^3*c^6*d^6*e^12
+ 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 - 14*a^4*c^13*d^26*e^2
+ 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 - 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c
^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 - 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8
*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 - 14*a^16*c*d^2*e^26 + a^17*e^28)))*sqrt((4*c^6*d^9 - 63*
a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 + 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 - (a^3*c^7*d^14 - 7*a^4*c^6*d^1
2*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12
- a^10*e^14)*sqrt((11025*c^9*d^12*e^6 - 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 + 1360716*a^3*c^6*d^6
*e^12 + 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 - 14*a^4*c^13*d^2
6*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 - 2002*a^8*c^9*d^18*e^10 + 3003*
a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 - 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^
4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 - 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 -
7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a
^9*c*d^2*e^12 - a^10*e^14))) - 3*(a^2*c^3*d^8 - 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 - a^5*d^2*e^6 - (a*c^4*d^6
*e^2 - 3*a^2*c^3*d^4*e^4 + 3*a^3*c^2*d^2*e^6 - a^4*c*e^8)*x^4 - 2*(a*c^4*d^7*e - 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2
*d^3*e^5 - a^4*c*d*e^7)*x^3 - (a*c^4*d^8 - 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 - 4*a^4*c*d^2*e^6 + a^5*e^8)*
x^2 + 2*(a^2*c^3*d^7*e - 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 - a^5*d*e^7)*x)*sqrt((4*c^6*d^9 - 63*a*c^5*d^7*e^
2 + 189*a^2*c^4*d^5*e^4 + 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 - (a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a
^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14
)*sqrt((11025*c^9*d^12*e^6 - 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 + 1360716*a^3*c^6*d^6*e^12 + 7808
31*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 - 14*a^4*c^13*d^26*e^2 + 91*a
^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 - 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16
*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 - 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 -
364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 - 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 - 7*a^4*c^6*d^
12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^1
2 - a^10*e^14))*log((420*c^6*d^8*e^3 - 8421*a*c^5*d^6*e^5 + 36783*a^2*c^4*d^4*e^7 + 40817*a^3*c^3*d^2*e^9 + 24
01*a^4*c^2*e^11)*sqrt(e*x + d) - (105*a^2*c^6*d^10*e^4 - 4389*a^3*c^5*d^8*e^6 + 26274*a^4*c^4*d^6*e^8 + 34142*
a^5*c^3*d^4*e^10 + 7525*a^6*c^2*d^2*e^12 + 343*a^7*c*e^14 - 2*(a^3*c^9*d^19 - 15*a^4*c^8*d^17*e^2 + 64*a^5*c^7
*d^15*e^4 - 112*a^6*c^6*d^13*e^6 + 42*a^7*c^5*d^11*e^8 + 154*a^8*c^4*d^9*e^10 - 280*a^9*c^3*d^7*e^12 + 216*a^1
0*c^2*d^5*e^14 - 83*a^11*c*d^3*e^16 + 13*a^12*d*e^18)*sqrt((11025*c^9*d^12*e^6 - 171990*a*c^8*d^10*e^8 + 49499
1*a^2*c^7*d^8*e^10 + 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^
3*e^18)/(a^3*c^14*d^28 - 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d
^20*e^8 - 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16
- 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 - 14*a^16*c*
d^2*e^26 + a^17*e^28)))*sqrt((4*c^6*d^9 - 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 + 1155*a^3*c^3*d^3*e^6 + 315*
a^4*c^2*d*e^8 - (a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6
*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14)*sqrt((11025*c^9*d^12*e^6 - 171990*a*c^8*d^10*e^8 +
494991*a^2*c^7*d^8*e^10 + 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^2*e^16 + 2401*a
^6*c^3*e^18)/(a^3*c^14*d^28 - 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e^6 + 1001*a^7*c
^10*d^20*e^8 - 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*
e^16 - 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 - 14*a^
16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 3
5*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14))) - 4*(9*a*c^2*d^4*e + 55*a^2*c*d^2*e^
3 - 4*a^3*e^5 - 3*(c^3*d^3*e^2 + 19*a*c^2*d*e^4)*x^3 - (6*c^3*d^4*e + 61*a*c^2*d^2*e^3 - 7*a^2*c*e^5)*x^2 - 3*
(c^3*d^5 - 3*a*c^2*d^3*e^2 - 18*a^2*c*d*e^4)*x)*sqrt(e*x + d))/(a^2*c^3*d^8 - 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*
e^4 - a^5*d^2*e^6 - (a*c^4*d^6*e^2 - 3*a^2*c^3*d^4*e^4 + 3*a^3*c^2*d^2*e^6 - a^4*c*e^8)*x^4 - 2*(a*c^4*d^7*e -
3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 - a^4*c*d*e^7)*x^3 - (a*c^4*d^8 - 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4
- 4*a^4*c*d^2*e^6 + a^5*e^8)*x^2 + 2*(a^2*c^3*d^7*e - 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 - a^5*d*e^7)*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(1/(e*x+d)**(5/2)/(-c*x**2+a)**2,x)

[Out]

Timed out

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Giac [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(1/(e*x+d)^(5/2)/(-c*x^2+a)^2,x, algorithm="giac")

[Out]

Timed out