∫ 1_______ (d+ex)5∕2(a−cx2)2 dx

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

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

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Rubi [A] time = 0.72, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {741, 829, 827, 1166, 208} \begin {gather*} -\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} \end {gather*}

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 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]

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 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 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 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]

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*}

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Mathematica [C] time = 0.31, size = 327, normalized size = 1.05 \begin {gather*} \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 )} \end {gather*}

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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IntegrateAlgebraic [A] time = 1.37, size = 455, normalized size = 1.46 \begin {gather*} \frac {\left (7 \sqrt {a} c e+2 c^{3/2} d\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{4 a^{3/2} \left (\sqrt {a} e+\sqrt {c} d\right )^3 \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}+\frac {\left (7 \sqrt {a} c e-2 c^{3/2} d\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{4 a^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right )^3 \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}+\frac {-4 a^3 e^7+8 a^2 c d^2 e^5+7 a^2 c e^5 (d+e x)^2+40 a^2 c d e^5 (d+e x)-4 a c^2 d^4 e^3-40 a c^2 d^3 e^3 (d+e x)+110 a c^2 d^2 e^3 (d+e x)^2-57 a c^2 d e^3 (d+e x)^3+3 c^3 d^4 e (d+e x)^2-3 c^3 d^3 e (d+e x)^3}{6 a (d+e x)^{3/2} \left (a e^2-c d^2\right )^3 \left (a e^2-c d^2+2 c d (d+e x)-c (d+e x)^2\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-4*a*c^2*d^4*e^3 + 8*a^2*c*d^2*e^5 - 4*a^3*e^7 - 40*a*c^2*d^3*e^3*(d + e*x) + 40*a^2*c*d*e^5*(d + e*x) + 3*c^

3*d^4*e*(d + e*x)^2 + 110*a*c^2*d^2*e^3*(d + e*x)^2 + 7*a^2*c*e^5*(d + e*x)^2 - 3*c^3*d^3*e*(d + e*x)^3 - 57*a

*c^2*d*e^3*(d + e*x)^3)/(6*a*(-(c*d^2) + a*e^2)^3*(d + e*x)^(3/2)*(-(c*d^2) + a*e^2 + 2*c*d*(d + e*x) - c*(d +

e*x)^2)) + ((2*c^(3/2)*d + 7*Sqrt[a]*c*e)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d

+ Sqrt[a]*e)])/(4*a^(3/2)*(Sqrt[c]*d + Sqrt[a]*e)^3*Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]) + ((-2*c^(3/2)*d

+ 7*Sqrt[a]*c*e)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(4*a^(3/2)

*(Sqrt[c]*d - Sqrt[a]*e)^3*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])

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fricas [B] time = 2.44, size = 8308, normalized size = 26.71

result too large to display

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)

________________________________________________________________________________________

giac [B] time = 1.31, size = 1872, normalized size = 6.02

result too large to display

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]

1/4*((a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - a^4*e^7)^2*(sqrt(a*c)*c*d^3*e + 19*sqrt(a*c)*a*d*e^3

)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(c) - (a*c^5*d^10*e - 37*a^2*c^4*d^8*e^3 + 98*a^3*c^3*d^6*e^5 - 82*a^4*c^2*d

^4*e^7 + 13*a^5*c*d^2*e^9 + 7*a^6*e^11)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a

^3*c*d^2*e^5 - a^4*e^7)*abs(c) - (2*sqrt(a*c)*a*c^8*d^17*e - 27*sqrt(a*c)*a^2*c^7*d^15*e^3 + 113*sqrt(a*c)*a^3

*c^6*d^13*e^5 - 223*sqrt(a*c)*a^4*c^5*d^11*e^7 + 225*sqrt(a*c)*a^5*c^4*d^9*e^9 - 97*sqrt(a*c)*a^6*c^3*d^7*e^11

- 13*sqrt(a*c)*a^7*c^2*d^5*e^13 + 27*sqrt(a*c)*a^8*c*d^3*e^15 - 7*sqrt(a*c)*a^9*d*e^17)*sqrt(-c^2*d + sqrt(a*

c)*c*e)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^4*d^7 - 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 - a^4*c*d*e^6 -

sqrt((a*c^4*d^7 - 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 - a^4*c*d*e^6)^2 - (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)*(a*c^4*d^6 - 3*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 - a^4*c*e^6)))

/(a*c^4*d^6 - 3*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 - a^4*c*e^6)))/((a^2*c^8*d^14 - 7*a^3*c^7*d^12*e^2 + 21*a^

4*c^6*d^10*e^4 - 35*a^5*c^5*d^8*e^6 + 35*a^6*c^4*d^6*e^8 - 21*a^7*c^3*d^4*e^10 + 7*a^8*c^2*d^2*e^12 - a^9*c*e^

14)*abs(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - a^4*e^7)) - 1/4*((a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3

+ 3*a^3*c*d^2*e^5 - a^4*e^7)^2*(c^2*d^3*e + 19*a*c*d*e^3)*abs(c) + (sqrt(a*c)*c^5*d^10*e - 37*sqrt(a*c)*a*c^4*

d^8*e^3 + 98*sqrt(a*c)*a^2*c^3*d^6*e^5 - 82*sqrt(a*c)*a^3*c^2*d^4*e^7 + 13*sqrt(a*c)*a^4*c*d^2*e^9 + 7*sqrt(a*

c)*a^5*e^11)*abs(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - a^4*e^7)*abs(c) - (2*a*c^9*d^17*e - 27*a^

2*c^8*d^15*e^3 + 113*a^3*c^7*d^13*e^5 - 223*a^4*c^6*d^11*e^7 + 225*a^5*c^5*d^9*e^9 - 97*a^6*c^4*d^7*e^11 - 13*

a^7*c^3*d^5*e^13 + 27*a^8*c^2*d^3*e^15 - 7*a^9*c*d*e^17)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^4*d^7 - 3*a^2

*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 - a^4*c*d*e^6 + sqrt((a*c^4*d^7 - 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 - a^4

*c*d*e^6)^2 - (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)*(a*c^4*d^6 - 3*a

^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 - a^4*c*e^6)))/(a*c^4*d^6 - 3*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 - a^4*c*e

^6)))/((a^2*c^6*d^12*e - sqrt(a*c)*a*c^6*d^13 + 6*sqrt(a*c)*a^2*c^5*d^11*e^2 - 6*a^3*c^5*d^10*e^3 - 15*sqrt(a*

c)*a^3*c^4*d^9*e^4 + 15*a^4*c^4*d^8*e^5 + 20*sqrt(a*c)*a^4*c^3*d^7*e^6 - 20*a^5*c^3*d^6*e^7 - 15*sqrt(a*c)*a^5

*c^2*d^5*e^8 + 15*a^6*c^2*d^4*e^9 + 6*sqrt(a*c)*a^6*c*d^3*e^10 - 6*a^7*c*d^2*e^11 - sqrt(a*c)*a^7*d*e^12 + a^8

*e^13)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - a^4*e^7)) - 1/2*((

x*e + d)^(3/2)*c^3*d^3*e - sqrt(x*e + d)*c^3*d^4*e + 3*(x*e + d)^(3/2)*a*c^2*d*e^3 - 6*sqrt(x*e + d)*a*c^2*d^2

*e^3 - sqrt(x*e + d)*a^2*c*e^5)/((a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*((x*e + d)^2*c -

2*(x*e + d)*c*d + c*d^2 - a*e^2)) - 2/3*(12*(x*e + d)*c*d*e^3 + c*d^2*e^3 - a*e^5)/((c^3*d^6 - 3*a*c^2*d^4*e^2

+ 3*a^2*c*d^2*e^4 - a^3*e^6)*(x*e + d)^(3/2))

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maple [B] time = 0.10, size = 1021, normalized size = 3.28 \begin {gather*} \frac {7 a \,c^{2} e^{5} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {7 a \,c^{2} e^{5} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {c^{4} d^{4} e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {c^{4} d^{4} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}+\frac {15 c^{3} d^{2} e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {15 c^{3} d^{2} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\sqrt {e x +d}\, a c \,e^{5}}{2 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c \,e^{2} x^{2}-a \,e^{2}\right )}-\frac {\sqrt {e x +d}\, c^{3} d^{4} e}{2 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c \,e^{2} x^{2}-a \,e^{2}\right ) a}-\frac {c^{3} d^{3} e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}+\frac {c^{3} d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {3 \sqrt {e x +d}\, c^{2} d^{2} e^{3}}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c \,e^{2} x^{2}-a \,e^{2}\right )}-\frac {19 c^{2} d \,e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {19 c^{2} d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (e x +d \right )^{\frac {3}{2}} c^{3} d^{3} e}{2 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c \,e^{2} x^{2}-a \,e^{2}\right ) a}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c^{2} d \,e^{3}}{2 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c \,e^{2} x^{2}-a \,e^{2}\right )}+\frac {8 c d \,e^{3}}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {e x +d}}-\frac {2 e^{3}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}} \end {gather*}

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]

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)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1

/2)*c)+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*

d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*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*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*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*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*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*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*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)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)+1

5/4*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*d+(a*c*

e^2)^(1/2))*c)^(1/2)*c)*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)*arcta

n((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*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*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*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)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^3-2/3*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)

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (c x^{2} - a\right )}^{2} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}

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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mupad [B] time = 4.84, size = 12290, normalized size = 39.52

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*e^3)/(3*(a*e^2 - c*d^2)) - (20*c*d*e^3*(d + e*x))/(3*(a*e^2 - c*d^2)^2) - (c*e*(d + e*x)^2*(7*a^2*e^4 +

3*c^2*d^4 + 110*a*c*d^2*e^2))/(6*a*(a*e^2 - c*d^2)^3) + (c^2*d*e*(19*a*e^2 + c*d^2)*(d + e*x)^3)/(2*a*(a*e^2 -

c*d^2)^3))/((a*e^2 - c*d^2)*(d + e*x)^(3/2) - c*(d + e*x)^(7/2) + 2*c*d*(d + e*x)^(5/2)) - atan((((d + e*x)^(

1/2)*(1568*a^16*c^5*e^28 - 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 - 29120*a^5*c^16*d^22*e^6 + 128128*a

^6*c^15*d^20*e^8 - 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 + 282240*a^9*c^12*d^14*e^14 - 1059840

*a^10*c^11*d^12*e^16 + 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 + 456512*a^13*c^8*d^6*e^22 - 10

0480*a^14*c^7*d^4*e^24 + 4160*a^15*c^6*d^2*e^26) - (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2

*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 8

19*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c

*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*

d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 - 63*a^4*

c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5

*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^

7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2

)*(2048*a^21*c^4*d*e^32 - 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 - 215040*a^8*c^17*d^27*e^6 + 931840

*a^9*c^16*d^25*e^8 - 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 - 10250240*a^12*c^13*d^19*e^14

+ 13178880*a^13*c^12*d^17*e^16 - 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 - 6150144*a^16*c^

9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 - 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 - 30720*a^20*c^5

*d^3*e^30) - 1792*a^19*c^4*e^31 + 256*a^5*c^18*d^28*e^3 - 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 -

609280*a^8*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 - 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*

e^15 - 6825984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 - 2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c

^8*d^8*e^23 - 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 + 12800*a^18*c^5*d^2*e^29))*(-(4*a^3*c^6*d^9

- 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3

*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2)

)/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^

8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*1i + ((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 - 128*a^

3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 - 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 - 282560*a^7*c^1

4*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 + 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 + 1403904*a^

11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 + 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 + 4160*a^1

5*c^6*d^2*e^26) - (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189

*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2)

- 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 -

21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2

)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4

+ 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*

e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*

e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 - 2048*a^

6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 - 215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 - 2795520*a^10

*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 - 10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 -

13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 - 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^

9*e^24 - 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 - 30720*a^20*c^5*d^3*e^30) + 1792*a^19*c^4*e^31 -

256*a^5*c^18*d^28*e^3 + 11776*a^6*c^17*d^26*e^5 - 119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15*d^22*e^9 - 19233

28*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 - 6243072*a^11*c^12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^1

7 - 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 - 1044736*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6

*e^25 + 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 31

5*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^

(1/2) - 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 -

7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*

a^11*c^2*d^4*e^10)))^(1/2)*1i)/(((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 - 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d

^24*e^4 - 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 - 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^1

6*e^12 + 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 + 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*

c^9*d^8*e^20 + 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 + 4160*a^15*c^6*d^2*e^26) - (-(4*a^3*c^6*d^

9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d

^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/

2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*

d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(

a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 + 105*c^3

*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^

14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^1

0*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 - 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*

d^29*e^4 - 215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 - 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^

14*d^21*e^12 - 10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 - 13178880*a^14*c^11*d^15*e^18 + 10

250240*a^15*c^10*d^13*e^20 - 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 - 931840*a^18*c^7*d^7*e^26

+ 215040*a^19*c^6*d^5*e^28 - 30720*a^20*c^5*d^3*e^30) - 1792*a^19*c^4*e^31 + 256*a^5*c^18*d^28*e^3 - 11776*a^

6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 - 609280*a^8*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 - 4116992*a

^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*e^15 - 6825984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 -

2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 - 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 +

12800*a^18*c^5*d^2*e^29))*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*

e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(a^9*c^

3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^

12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2) - ((d

+ e*x)^(1/2)*(1568*a^16*c^5*e^28 - 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 - 29120*a^5*c^16*d^22*e^6 +

128128*a^6*c^15*d^20*e^8 - 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 + 282240*a^9*c^12*d^14*e^14 -

1059840*a^10*c^11*d^12*e^16 + 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 + 456512*a^13*c^8*d^6*e

^22 - 100480*a^14*c^7*d^4*e^24 + 4160*a^15*c^6*d^2*e^26) - (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315

*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(

1/2) - 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 -

7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a

^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 -

63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2

*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^1

2 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10

)))^(1/2)*(2048*a^21*c^4*d*e^32 - 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 - 215040*a^8*c^17*d^27*e^6

+ 931840*a^9*c^16*d^25*e^8 - 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 - 10250240*a^12*c^13*d^

19*e^14 + 13178880*a^13*c^12*d^17*e^16 - 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 - 6150144

*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 - 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 - 30720*

a^20*c^5*d^3*e^30) + 1792*a^19*c^4*e^31 - 256*a^5*c^18*d^28*e^3 + 11776*a^6*c^17*d^26*e^5 - 119552*a^7*c^16*d^

24*e^7 + 609280*a^8*c^15*d^22*e^9 - 1923328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 - 6243072*a^11*c^

12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^17 - 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 - 104473

6*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6*e^25 + 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(4*a^3*

c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6

*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^

3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^

9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2) + 7448*a^13*c^6*d*e^25 + 32*a^2*c^17*d^23*

e^3 - 72*a^3*c^16*d^21*e^5 - 8240*a^4*c^15*d^19*e^7 + 72120*a^5*c^14*d^17*e^9 - 282240*a^6*c^13*d^15*e^11 + 64

8816*a^7*c^12*d^13*e^13 - 962976*a^8*c^11*d^11*e^15 + 955440*a^9*c^10*d^9*e^17 - 633120*a^10*c^9*d^7*e^19 + 27

0040*a^11*c^8*d^5*e^21 - 67248*a^12*c^7*d^3*e^23))*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2

*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 8

19*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c

*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*

d^4*e^10)))^(1/2)*2i - atan((((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 - 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24

*e^4 - 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 - 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e

^12 + 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 + 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9

*d^8*e^20 + 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 + 4160*a^15*c^6*d^2*e^26) - (-(49*a^3*e^9*(a^9

*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*

e^6 - 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))

/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8

*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(49*a^3*e^9*(a^9*c^3)^(1/2) + 4*

a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^

6*e^3*(a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14

- a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c

^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 - 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^2

9*e^4 - 215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 - 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*

d^21*e^12 - 10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 - 13178880*a^14*c^11*d^15*e^18 + 10250

240*a^15*c^10*d^13*e^20 - 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 - 931840*a^18*c^7*d^7*e^26 +

215040*a^19*c^6*d^5*e^28 - 30720*a^20*c^5*d^3*e^30) - 1792*a^19*c^4*e^31 + 256*a^5*c^18*d^28*e^3 - 11776*a^6*c

^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 - 609280*a^8*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 - 4116992*a^10

*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*e^15 - 6825984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 - 29

45536*a^14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 - 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 + 12

800*a^18*c^5*d^2*e^29))*(-(49*a^3*e^9*(a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2

+ 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(a^9*c^3)^

(1/2) + 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*

e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*1i + ((d

+ e*x)^(1/2)*(1568*a^16*c^5*e^28 - 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 - 29120*a^5*c^16*d^22*e^6 +

128128*a^6*c^15*d^20*e^8 - 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 + 282240*a^9*c^12*d^14*e^14 -

1059840*a^10*c^11*d^12*e^16 + 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 + 456512*a^13*c^8*d^6*e

^22 - 100480*a^14*c^7*d^4*e^24 + 4160*a^15*c^6*d^2*e^26) - (-(49*a^3*e^9*(a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315

*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(a^9*c^3)^(

1/2) + 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 -

7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a

^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(49*a^3*e^9*(a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 -

63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) + 819*a*c^2

*d^4*e^5*(a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^1