3.1452 \(\int \frac{A+B x}{(d+e x)^{5/2} (a-c x^2)} \, dx\)

Optimal. Leaf size=243 \[ -\frac{2 (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{2 \left (a B e^2-2 A c d e+B c d^2\right )}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}+\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}} \]

[Out]

(-2*(B*d - A*e))/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)) - (2*(B*c*d^2 - 2*A*c*d*e + a*B*e^2))/((c*d^2 - a*e^2)^2*
Sqrt[d + e*x]) + ((Sqrt[a]*B - A*Sqrt[c])*c^(1/4)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]
)/(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)) + ((Sqrt[a]*B + A*Sqrt[c])*c^(1/4)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/S
qrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)^(5/2))

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Rubi [A]  time = 0.53565, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {829, 827, 1166, 208} \[ -\frac{2 (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{2 \left (a B e^2-2 A c d e+B c d^2\right )}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}+\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(B*d - A*e))/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)) - (2*(B*c*d^2 - 2*A*c*d*e + a*B*e^2))/((c*d^2 - a*e^2)^2*
Sqrt[d + e*x]) + ((Sqrt[a]*B - A*Sqrt[c])*c^(1/4)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]
)/(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)) + ((Sqrt[a]*B + A*Sqrt[c])*c^(1/4)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/S
qrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)^(5/2))

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

Mathematica [C]  time = 0.159375, size = 266, normalized size = 1.09 \[ \frac{3 B (d+e x) \left (\left (\sqrt{a} e+\sqrt{c} d\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right )+\left (\sqrt{a} e-\sqrt{c} d\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right )\right )-(B d-A e) \left (\left (\sqrt{a} e+\sqrt{c} d\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right )+\left (\sqrt{a} e-\sqrt{c} d\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right )\right )}{3 \sqrt{a} (d+e x)^{3/2} \left (c d^2 e-a e^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((B*d - 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[c]*d) + Sqrt[a]*e)*Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]
*e)])) + 3*B*(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 + S
qrt[a]*e)]))/(3*Sqrt[a]*(c*d^2*e - a*e^3)*(d + e*x)^(3/2))

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Maple [B]  time = 0.03, size = 973, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2/(a*e^2-c*d^2)^2/(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))*a*A*e^3+c^3/(a*e^2-c*d^2)^2/(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))*A*d^2*e-2*c^2/(a*e^2-c*d^2)^2/(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))*a*B*d*e^2-2*c^2/(a*e^2-c*d^2)^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))*A*d*e+c/(a*e^2-c*d^2)^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))*B*a*e^2+c^2/(a*e^2-c*d^2)^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))*B*d^2+c^2/(a*e^2-c*d^2)^2/
(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))*a*A*
e^3+c^3/(a*e^2-c*d^2)^2/(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))*A*d^2*e-2*c^2/(a*e^2-c*d^2)^2/(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))*a*B*d*e^2+2*c^2/(a*e^2-c*d^2)^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))*A*d*e-c/(a*e^2-c*d^2)^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))*B*a*e^2-c^2/(a*e^2-c*d^2)^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))*B*d^2-2/3/(a*e^2-c*d^2)/(e*x+d)^(3/2)*A*
e+2/3/(a*e^2-c*d^2)/(e*x+d)^(3/2)*B*d+4/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)*A*c*d*e-2/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)*
a*B*e^2-2/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)*B*c*d^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 107.143, size = 22700, normalized size = 93.42 \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)^(5/2)/(-c*x^2+a),x, algorithm="fricas")

[Out]

1/6*(3*(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2
*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt(-(10*A*B*a*c^3*d^4*e + 20*A*B*a^2*c^2*d^2*e^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3
+ A^2*c^4)*d^5 - 10*(B^2*a^2*c^2 + A^2*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c + A^2*a^2*c^2)*d*e^4 + (a*c^5*d^10 - 5*a
^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((4*A^2*B^2*c^7*d^1
0 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3
*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a
^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B
^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a
^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*
d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7
*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5
*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))*log((2*(A*B^3*a*c^4
- A^3*B*c^5)*d^5 - 5*(B^4*a^2*c^3 - A^4*c^5)*d^4*e + 20*(A*B^3*a^2*c^3 - A^3*B*a*c^4)*d^3*e^2 - 10*(B^4*a^3*c^
2 - A^4*a*c^4)*d^2*e^3 + 10*(A*B^3*a^3*c^2 - A^3*B*a^2*c^3)*d*e^4 - (B^4*a^4*c - A^4*a^2*c^3)*e^5)*sqrt(e*x +
d) + (2*A*B^2*a*c^5*d^8 - (5*B^3*a^2*c^4 + 11*A^2*B*a*c^5)*d^7*e + (41*A*B^2*a^2*c^4 + 15*A^3*a*c^5)*d^6*e^2 -
(25*B^3*a^3*c^3 + 87*A^2*B*a^2*c^4)*d^5*e^3 + 35*(3*A*B^2*a^3*c^3 + A^3*a^2*c^4)*d^4*e^4 - (31*B^3*a^4*c^2 +
81*A^2*B*a^3*c^3)*d^3*e^5 + (43*A*B^2*a^4*c^2 + 13*A^3*a^3*c^3)*d^2*e^6 - (3*B^3*a^5*c + 13*A^2*B*a^4*c^2)*d*e
^7 + (A*B^2*a^5*c + A^3*a^4*c^2)*e^8 + (A*a*c^7*d^13 - 3*B*a^2*c^6*d^12*e - 2*A*a^2*c^6*d^11*e^2 + 14*B*a^3*c^
5*d^10*e^3 - 5*A*a^3*c^5*d^9*e^4 - 25*B*a^4*c^4*d^8*e^5 + 20*A*a^4*c^4*d^7*e^6 + 20*B*a^5*c^3*d^6*e^7 - 25*A*a
^5*c^3*d^5*e^8 - 5*B*a^6*c^2*d^4*e^9 + 14*A*a^6*c^2*d^3*e^10 - 2*B*a^7*c*d^2*e^11 - 3*A*a^7*c*d*e^12 + B*a^8*e
^13)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A
^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a
*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4
*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^
3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10
)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252
*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 +
a^11*e^20)))*sqrt(-(10*A*B*a*c^3*d^4*e + 20*A*B*a^2*c^2*d^2*e^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3 + A^2*c^4)*d^5
- 10*(B^2*a^2*c^2 + A^2*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c + A^2*a^2*c^2)*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 +
10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*
c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B
*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a
^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3
*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a
^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^
3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 -
120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2
+ 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))) - 3*(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*
d^2*e^4 + (c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt(-(10
*A*B*a*c^3*d^4*e + 20*A*B*a^2*c^2*d^2*e^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3 + A^2*c^4)*d^5 - 10*(B^2*a^2*c^2 + A^
2*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c + A^2*a^2*c^2)*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 -
10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e
+ 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(
5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(
11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 +
20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a
^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a
^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 +
45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4
- 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))*log((2*(A*B^3*a*c^4 - A^3*B*c^5)*d^5 - 5*(B^4*a^2*c^3 - A^
4*c^5)*d^4*e + 20*(A*B^3*a^2*c^3 - A^3*B*a*c^4)*d^3*e^2 - 10*(B^4*a^3*c^2 - A^4*a*c^4)*d^2*e^3 + 10*(A*B^3*a^3
*c^2 - A^3*B*a^2*c^3)*d*e^4 - (B^4*a^4*c - A^4*a^2*c^3)*e^5)*sqrt(e*x + d) - (2*A*B^2*a*c^5*d^8 - (5*B^3*a^2*c
^4 + 11*A^2*B*a*c^5)*d^7*e + (41*A*B^2*a^2*c^4 + 15*A^3*a*c^5)*d^6*e^2 - (25*B^3*a^3*c^3 + 87*A^2*B*a^2*c^4)*d
^5*e^3 + 35*(3*A*B^2*a^3*c^3 + A^3*a^2*c^4)*d^4*e^4 - (31*B^3*a^4*c^2 + 81*A^2*B*a^3*c^3)*d^3*e^5 + (43*A*B^2*
a^4*c^2 + 13*A^3*a^3*c^3)*d^2*e^6 - (3*B^3*a^5*c + 13*A^2*B*a^4*c^2)*d*e^7 + (A*B^2*a^5*c + A^3*a^4*c^2)*e^8 +
(A*a*c^7*d^13 - 3*B*a^2*c^6*d^12*e - 2*A*a^2*c^6*d^11*e^2 + 14*B*a^3*c^5*d^10*e^3 - 5*A*a^3*c^5*d^9*e^4 - 25*
B*a^4*c^4*d^8*e^5 + 20*A*a^4*c^4*d^7*e^6 + 20*B*a^5*c^3*d^6*e^7 - 25*A*a^5*c^3*d^5*e^8 - 5*B*a^6*c^2*d^4*e^9 +
14*A*a^6*c^2*d^3*e^10 - 2*B*a^7*c*d^2*e^11 - 3*A*a^7*c*d*e^12 + B*a^8*e^13)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*
B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 +
A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A
^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3
+ A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A
^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 +
45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e
^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt(-(10*A*B*a*c^3*d^4*e
+ 20*A*B*a^2*c^2*d^2*e^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3 + A^2*c^4)*d^5 - 10*(B^2*a^2*c^2 + A^2*a*c^3)*d^3*e^2
- 5*(B^2*a^3*c + A^2*a^2*c^2)*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e
^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*
c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 +
32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 +
62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2
+ 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2
*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6
+ 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*
e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4
*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))) + 3*(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 - 2*a*c*d^2*e^4
+ a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt(-(10*A*B*a*c^3*d^4*e + 20*A*B*a^2*c^2*d^2*
e^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3 + A^2*c^4)*d^5 - 10*(B^2*a^2*c^2 + A^2*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c + A^2*
a^2*c^2)*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 -
a^6*e^10)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6
+ 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5
*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 +
11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 +
A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3
)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8
- 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e
^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8
- a^6*e^10))*log((2*(A*B^3*a*c^4 - A^3*B*c^5)*d^5 - 5*(B^4*a^2*c^3 - A^4*c^5)*d^4*e + 20*(A*B^3*a^2*c^3 - A^3
*B*a*c^4)*d^3*e^2 - 10*(B^4*a^3*c^2 - A^4*a*c^4)*d^2*e^3 + 10*(A*B^3*a^3*c^2 - A^3*B*a^2*c^3)*d*e^4 - (B^4*a^4
*c - A^4*a^2*c^3)*e^5)*sqrt(e*x + d) + (2*A*B^2*a*c^5*d^8 - (5*B^3*a^2*c^4 + 11*A^2*B*a*c^5)*d^7*e + (41*A*B^2
*a^2*c^4 + 15*A^3*a*c^5)*d^6*e^2 - (25*B^3*a^3*c^3 + 87*A^2*B*a^2*c^4)*d^5*e^3 + 35*(3*A*B^2*a^3*c^3 + A^3*a^2
*c^4)*d^4*e^4 - (31*B^3*a^4*c^2 + 81*A^2*B*a^3*c^3)*d^3*e^5 + (43*A*B^2*a^4*c^2 + 13*A^3*a^3*c^3)*d^2*e^6 - (3
*B^3*a^5*c + 13*A^2*B*a^4*c^2)*d*e^7 + (A*B^2*a^5*c + A^3*a^4*c^2)*e^8 - (A*a*c^7*d^13 - 3*B*a^2*c^6*d^12*e -
2*A*a^2*c^6*d^11*e^2 + 14*B*a^3*c^5*d^10*e^3 - 5*A*a^3*c^5*d^9*e^4 - 25*B*a^4*c^4*d^8*e^5 + 20*A*a^4*c^4*d^7*e
^6 + 20*B*a^5*c^3*d^6*e^7 - 25*A*a^5*c^3*d^5*e^8 - 5*B*a^6*c^2*d^4*e^9 + 14*A*a^6*c^2*d^3*e^10 - 2*B*a^7*c*d^2
*e^11 - 3*A*a^7*c*d*e^12 + B*a^8*e^13)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^
4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*
c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4
*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^
5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A
^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^1
4*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^
2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt(-(10*A*B*a*c^3*d^4*e + 20*A*B*a^2*c^2*d^2*e^3 + 2*A*B*a^3*
c*e^5 - (B^2*a*c^3 + A^2*c^4)*d^5 - 10*(B^2*a^2*c^2 + A^2*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c + A^2*a^2*c^2)*d*e^4 -
(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt(
(4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^
8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*
e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*
d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2
*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*
d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d
^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)
))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))) -
3*(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2*a*c
*d^3*e^3 + a^2*d*e^5)*x)*sqrt(-(10*A*B*a*c^3*d^4*e + 20*A*B*a^2*c^2*d^2*e^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3 + A
^2*c^4)*d^5 - 10*(B^2*a^2*c^2 + A^2*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c + A^2*a^2*c^2)*d*e^4 - (a*c^5*d^10 - 5*a^2*c
^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((4*A^2*B^2*c^7*d^10 -
20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2
*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c
^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a
^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c
^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18
*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4
*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2
*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))*log((2*(A*B^3*a*c^4 - A^
3*B*c^5)*d^5 - 5*(B^4*a^2*c^3 - A^4*c^5)*d^4*e + 20*(A*B^3*a^2*c^3 - A^3*B*a*c^4)*d^3*e^2 - 10*(B^4*a^3*c^2 -
A^4*a*c^4)*d^2*e^3 + 10*(A*B^3*a^3*c^2 - A^3*B*a^2*c^3)*d*e^4 - (B^4*a^4*c - A^4*a^2*c^3)*e^5)*sqrt(e*x + d) -
(2*A*B^2*a*c^5*d^8 - (5*B^3*a^2*c^4 + 11*A^2*B*a*c^5)*d^7*e + (41*A*B^2*a^2*c^4 + 15*A^3*a*c^5)*d^6*e^2 - (25
*B^3*a^3*c^3 + 87*A^2*B*a^2*c^4)*d^5*e^3 + 35*(3*A*B^2*a^3*c^3 + A^3*a^2*c^4)*d^4*e^4 - (31*B^3*a^4*c^2 + 81*A
^2*B*a^3*c^3)*d^3*e^5 + (43*A*B^2*a^4*c^2 + 13*A^3*a^3*c^3)*d^2*e^6 - (3*B^3*a^5*c + 13*A^2*B*a^4*c^2)*d*e^7 +
(A*B^2*a^5*c + A^3*a^4*c^2)*e^8 - (A*a*c^7*d^13 - 3*B*a^2*c^6*d^12*e - 2*A*a^2*c^6*d^11*e^2 + 14*B*a^3*c^5*d^
10*e^3 - 5*A*a^3*c^5*d^9*e^4 - 25*B*a^4*c^4*d^8*e^5 + 20*A*a^4*c^4*d^7*e^6 + 20*B*a^5*c^3*d^6*e^7 - 25*A*a^5*c
^3*d^5*e^8 - 5*B*a^6*c^2*d^4*e^9 + 14*A*a^6*c^2*d^3*e^10 - 2*B*a^7*c*d^2*e^11 - 3*A*a^7*c*d*e^12 + B*a^8*e^13)
*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c
^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6
)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2
*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^
4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a
*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6
*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11
*e^20)))*sqrt(-(10*A*B*a*c^3*d^4*e + 20*A*B*a^2*c^2*d^2*e^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3 + A^2*c^4)*d^5 - 10
*(B^2*a^2*c^2 + A^2*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c + A^2*a^2*c^2)*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*
a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6
+ A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c
^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c
^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a
^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c
^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^
8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*
a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 1
0*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))) - 4*(4*B*c*d^3 - 7*A*c*d^2*e + 2*B*a*d*
e^2 + A*a*e^3 + 3*(B*c*d^2*e - 2*A*c*d*e^2 + B*a*e^3)*x)*sqrt(e*x + d))/(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4
+ (c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2*a*c*d^3*e^3 + a^2*d*e^5)*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)**(5/2)/(-c*x**2+a),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((B*x+A)/(e*x+d)^(5/2)/(-c*x^2+a),x, algorithm="giac")

[Out]

Timed out