3.1009 \(\int \frac{x^{5/2} (A+B x)}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=347 \[ -\frac{\sqrt{2} \left (-\frac{2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{c^{7/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (\frac{2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{c^{7/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 \sqrt{x} \left (-a B c-A b c+b^2 B\right )}{c^3}-\frac{2 x^{3/2} (b B-A c)}{3 c^2}+\frac{2 B x^{5/2}}{5 c} \]

[Out]

(2*(b^2*B - A*b*c - a*B*c)*Sqrt[x])/c^3 - (2*(b*B - A*c)*x^(3/2))/(3*c^2) + (2*B*x^(5/2))/(5*c) - (Sqrt[2]*(b^
3*B - A*b^2*c - 2*a*b*B*c + a*A*c^2 - (b^4*B - A*b^3*c - 4*a*b^2*B*c + 3*a*A*b*c^2 + 2*a^2*B*c^2)/Sqrt[b^2 - 4
*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(c^(7/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) -
(Sqrt[2]*(b^3*B - A*b^2*c - 2*a*b*B*c + a*A*c^2 + (b^4*B - A*b^3*c - 4*a*b^2*B*c + 3*a*A*b*c^2 + 2*a^2*B*c^2)/
Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(7/2)*Sqrt[b + Sqrt[b^2 -
4*a*c]])

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Rubi [A]  time = 4.54477, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {824, 826, 1166, 205} \[ -\frac{\sqrt{2} \left (-\frac{2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{c^{7/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (\frac{2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{c^{7/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 \sqrt{x} \left (-a B c-A b c+b^2 B\right )}{c^3}-\frac{2 x^{3/2} (b B-A c)}{3 c^2}+\frac{2 B x^{5/2}}{5 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b^2*B - A*b*c - a*B*c)*Sqrt[x])/c^3 - (2*(b*B - A*c)*x^(3/2))/(3*c^2) + (2*B*x^(5/2))/(5*c) - (Sqrt[2]*(b^
3*B - A*b^2*c - 2*a*b*B*c + a*A*c^2 - (b^4*B - A*b^3*c - 4*a*b^2*B*c + 3*a*A*b*c^2 + 2*a^2*B*c^2)/Sqrt[b^2 - 4
*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(c^(7/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) -
(Sqrt[2]*(b^3*B - A*b^2*c - 2*a*b*B*c + a*A*c^2 + (b^4*B - A*b^3*c - 4*a*b^2*B*c + 3*a*A*b*c^2 + 2*a^2*B*c^2)/
Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(7/2)*Sqrt[b + Sqrt[b^2 -
4*a*c]])

Rule 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + 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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^{5/2} (A+B x)}{a+b x+c x^2} \, dx &=\frac{2 B x^{5/2}}{5 c}+\frac{\int \frac{x^{3/2} (-a B-(b B-A c) x)}{a+b x+c x^2} \, dx}{c}\\ &=-\frac{2 (b B-A c) x^{3/2}}{3 c^2}+\frac{2 B x^{5/2}}{5 c}+\frac{\int \frac{\sqrt{x} \left (a (b B-A c)+\left (b^2 B-A b c-a B c\right ) x\right )}{a+b x+c x^2} \, dx}{c^2}\\ &=\frac{2 \left (b^2 B-A b c-a B c\right ) \sqrt{x}}{c^3}-\frac{2 (b B-A c) x^{3/2}}{3 c^2}+\frac{2 B x^{5/2}}{5 c}+\frac{\int \frac{-a \left (b^2 B-A b c-a B c\right )-\left (b^3 B-A b^2 c-2 a b B c+a A c^2\right ) x}{\sqrt{x} \left (a+b x+c x^2\right )} \, dx}{c^3}\\ &=\frac{2 \left (b^2 B-A b c-a B c\right ) \sqrt{x}}{c^3}-\frac{2 (b B-A c) x^{3/2}}{3 c^2}+\frac{2 B x^{5/2}}{5 c}+\frac{2 \operatorname{Subst}\left (\int \frac{-a \left (b^2 B-A b c-a B c\right )+\left (-b^3 B+A b^2 c+2 a b B c-a A c^2\right ) x^2}{a+b x^2+c x^4} \, dx,x,\sqrt{x}\right )}{c^3}\\ &=\frac{2 \left (b^2 B-A b c-a B c\right ) \sqrt{x}}{c^3}-\frac{2 (b B-A c) x^{3/2}}{3 c^2}+\frac{2 B x^{5/2}}{5 c}-\frac{\left (b^3 B-A b^2 c-2 a b B c+a A c^2-\frac{b^4 B-A b^3 c-4 a b^2 B c+3 a A b c^2+2 a^2 B c^2}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx,x,\sqrt{x}\right )}{c^3}-\frac{\left (b^3 B-A b^2 c-2 a b B c+a A c^2+\frac{b^4 B-A b^3 c-4 a b^2 B c+3 a A b c^2+2 a^2 B c^2}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx,x,\sqrt{x}\right )}{c^3}\\ &=\frac{2 \left (b^2 B-A b c-a B c\right ) \sqrt{x}}{c^3}-\frac{2 (b B-A c) x^{3/2}}{3 c^2}+\frac{2 B x^{5/2}}{5 c}-\frac{\sqrt{2} \left (b^3 B-A b^2 c-2 a b B c+a A c^2-\frac{b^4 B-A b^3 c-4 a b^2 B c+3 a A b c^2+2 a^2 B c^2}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{c^{7/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (b^3 B-A b^2 c-2 a b B c+a A c^2+\frac{b^4 B-A b^3 c-4 a b^2 B c+3 a A b c^2+2 a^2 B c^2}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{c^{7/2} \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [A]  time = 0.913493, size = 480, normalized size = 1.38 \[ \frac{\sqrt{2} B \left (\frac{\left (\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}+2 a b c-b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (-\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}+2 a b c-b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{c^{7/2}}+\frac{\sqrt{2} A \left (\frac{3 a b c-b^3}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{c^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} A \left (\frac{b^3-3 a b c}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{c^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 B \sqrt{x} \left (b^2-a c\right )}{c^3}-\frac{2 A b \sqrt{x}}{c^2}+\frac{2 A x^{3/2}}{3 c}-\frac{2 b B x^{3/2}}{3 c^2}+\frac{2 B x^{5/2}}{5 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*b*Sqrt[x])/c^2 + (2*B*(b^2 - a*c)*Sqrt[x])/c^3 - (2*b*B*x^(3/2))/(3*c^2) + (2*A*x^(3/2))/(3*c) + (2*B*x^
(5/2))/(5*c) + (Sqrt[2]*A*(b^2 - a*c + (-b^3 + 3*a*b*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sq
rt[b - Sqrt[b^2 - 4*a*c]]])/(c^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*A*(b^2 - a*c + (b^3 - 3*a*b*c)/Sq
rt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(5/2)*Sqrt[b + Sqrt[b^2 - 4
*a*c]]) + (Sqrt[2]*B*(((-b^3 + 2*a*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt
[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]] + ((-b^3 + 2*a*b*c - (b^4 - 4*a*b^2*c +
2*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^
2 - 4*a*c]]))/c^(7/2)

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Maple [B]  time = 0.059, size = 1141, 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(x^(5/2)*(B*x+A)/(c*x^2+b*x+a),x)

[Out]

2/5*B*x^(5/2)/c+2/3*A*x^(3/2)/c-2/3/c^2*B*x^(3/2)*b-2/c^2*A*b*x^(1/2)-2*a*B*x^(1/2)/c^2+2/c^3*b^2*B*x^(1/2)+1/
c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*A-1
/c^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*
b^2-3/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^
2)^(1/2))*c)^(1/2))*a*A*b+1/c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c
*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^3-2/c^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1
/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b*B+1/c^3*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh
(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3*B-2/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(
1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a^2*B+4/c^2/(-4*a*c+b^2)^(1/2)*2^(
1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b^2*B-1/
c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(
1/2))*c)^(1/2))*b^4*B-1/c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(
1/2))*c)^(1/2))*a*A+1/c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(
1/2))*c)^(1/2))*A*b^2-3/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)
/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*A*b+1/c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arc
tan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^3+2/c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*a
rctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b*B-1/c^3*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)
*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3*B-2/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b
^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a^2*B+4/c^2/(-4*a*c+b^2)^(1/2)*
2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b^2*B-1/
c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2
))*c)^(1/2))*b^4*B

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (3 \, B c x^{\frac{5}{2}} - 5 \,{\left (B b - A c\right )} x^{\frac{3}{2}}\right )}}{15 \, c^{2}} - \int \frac{{\left (A b c -{\left (b^{2} - a c\right )} B\right )} x^{\frac{3}{2}} -{\left (B a b - A a c\right )} \sqrt{x}}{c^{3} x^{2} + b c^{2} x + a c^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*B*c*x^(5/2) - 5*(B*b - A*c)*x^(3/2))/c^2 - integrate(((A*b*c - (b^2 - a*c)*B)*x^(3/2) - (B*a*b - A*a*c
)*sqrt(x))/(c^3*x^2 + b*c^2*x + a*c^2), x)

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Fricas [B]  time = 39.4457, size = 15695, normalized size = 45.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/30*(15*sqrt(2)*c^3*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*
b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c + (b^2*c^7 - 4*a*c^8)*s
qrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b
+ 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B
^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 +
28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 +
(37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^
15)))/(b^2*c^7 - 4*a*c^8))*log(sqrt(2)*(B^3*b^10 + 4*(A^2*B*a^4 + A^3*a^3*b)*c^6 - (4*B^3*a^5 + 28*A*B^2*a^4*b
+ 41*A^2*B*a^3*b^2 + 13*A^3*a^2*b^3)*c^5 + (29*B^3*a^4*b^2 + 87*A*B^2*a^3*b^3 + 58*A^2*B*a^2*b^4 + 7*A^3*a*b^
5)*c^4 - (51*B^3*a^3*b^4 + 80*A*B^2*a^2*b^5 + 24*A^2*B*a*b^6 + A^3*b^7)*c^3 + (35*B^3*a^2*b^6 + 27*A*B^2*a*b^7
+ 3*A^2*B*b^8)*c^2 - (10*B^3*a*b^8 + 3*A*B^2*b^9)*c - (B*b^5*c^7 - 8*A*a^2*c^10 + 6*(2*B*a^2*b + A*a*b^2)*c^9
- (7*B*a*b^3 + A*b^4)*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7
+ (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2
+ 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^
3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^
2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^
3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*
b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c + (b^2*c^
7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 1
2*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*
b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2
*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3
*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2
*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8)) + 4*(B^4*a^3*b^6 - A*B^3*a^2*b^7 + A^4*a^4*c^5 - (7*A^3*B*a^4*b + 3*A
^4*a^3*b^2)*c^4 - (B^4*a^6 + 5*A*B^3*a^5*b - 9*A^2*B^2*a^4*b^2 - 11*A^3*B*a^3*b^3 - A^4*a^2*b^4)*c^3 + (6*B^4*
a^5*b^2 + 2*A*B^3*a^4*b^3 - 12*A^2*B^2*a^3*b^4 - 3*A^3*B*a^2*b^5)*c^2 - (5*B^4*a^4*b^4 - 3*A*B^3*a^3*b^5 - 3*A
^2*B^2*a^2*b^6)*c)*sqrt(x)) - 15*sqrt(2)*c^3*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 1
8*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c
+ (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B
^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*
A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5
+ 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b
^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^1
1)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8))*log(-sqrt(2)*(B^3*b^10 + 4*(A^2*B*a^4 + A^3*a^3*b)*c^6 - (4
*B^3*a^5 + 28*A*B^2*a^4*b + 41*A^2*B*a^3*b^2 + 13*A^3*a^2*b^3)*c^5 + (29*B^3*a^4*b^2 + 87*A*B^2*a^3*b^3 + 58*A
^2*B*a^2*b^4 + 7*A^3*a*b^5)*c^4 - (51*B^3*a^3*b^4 + 80*A*B^2*a^2*b^5 + 24*A^2*B*a*b^6 + A^3*b^7)*c^3 + (35*B^3
*a^2*b^6 + 27*A*B^2*a*b^7 + 3*A^2*B*b^8)*c^2 - (10*B^3*a*b^8 + 3*A*B^2*b^9)*c - (B*b^5*c^7 - 8*A*a^2*c^10 + 6*
(2*B*a^2*b + A*a*b^2)*c^9 - (7*B*a*b^3 + A*b^4)*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a
^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4
)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B
^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*
A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 -
2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (
7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5
+ 2*A*B*b^6)*c + (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^
3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^
4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 16
0*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 +
24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^1
0 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8)) + 4*(B^4*a^3*b^6 - A*B^3*a^2*b^7 + A^4*a^4*c
^5 - (7*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^4 - (B^4*a^6 + 5*A*B^3*a^5*b - 9*A^2*B^2*a^4*b^2 - 11*A^3*B*a^3*b^3 - A
^4*a^2*b^4)*c^3 + (6*B^4*a^5*b^2 + 2*A*B^3*a^4*b^3 - 12*A^2*B^2*a^3*b^4 - 3*A^3*B*a^2*b^5)*c^2 - (5*B^4*a^4*b^
4 - 3*A*B^3*a^3*b^5 - 3*A^2*B^2*a^2*b^6)*c)*sqrt(x)) + 15*sqrt(2)*c^3*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*
b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7
*B^2*a*b^5 + 2*A*B*b^6)*c - (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b
+ 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6
- 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^
4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3
*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5
*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8))*log(sqrt(2)*(B^3*b^10 + 4*(A^2*B*a
^4 + A^3*a^3*b)*c^6 - (4*B^3*a^5 + 28*A*B^2*a^4*b + 41*A^2*B*a^3*b^2 + 13*A^3*a^2*b^3)*c^5 + (29*B^3*a^4*b^2 +
87*A*B^2*a^3*b^3 + 58*A^2*B*a^2*b^4 + 7*A^3*a*b^5)*c^4 - (51*B^3*a^3*b^4 + 80*A*B^2*a^2*b^5 + 24*A^2*B*a*b^6
+ A^3*b^7)*c^3 + (35*B^3*a^2*b^6 + 27*A*B^2*a*b^7 + 3*A^2*B*b^8)*c^2 - (10*B^3*a*b^8 + 3*A*B^2*b^9)*c + (B*b^5
*c^7 - 8*A*a^2*c^10 + 6*(2*B*a^2*b + A*a*b^2)*c^9 - (7*B*a*b^3 + A*b^4)*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*
(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*
a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 +
3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 -
2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9
+ 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))*sqrt(-(B^2*b^7 + (4*A*B*a^
3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2
*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c - (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 +
6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^
4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^
5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*
b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^
10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8)) + 4*(B^4*a^3*b^6 - A
*B^3*a^2*b^7 + A^4*a^4*c^5 - (7*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^4 - (B^4*a^6 + 5*A*B^3*a^5*b - 9*A^2*B^2*a^4*b^
2 - 11*A^3*B*a^3*b^3 - A^4*a^2*b^4)*c^3 + (6*B^4*a^5*b^2 + 2*A*B^3*a^4*b^3 - 12*A^2*B^2*a^3*b^4 - 3*A^3*B*a^2*
b^5)*c^2 - (5*B^4*a^4*b^4 - 3*A*B^3*a^3*b^5 - 3*A^2*B^2*a^2*b^6)*c)*sqrt(x)) - 15*sqrt(2)*c^3*sqrt(-(B^2*b^7 +
(4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a
*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c - (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*
B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b
^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4
*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(3
1*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*
A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8))*log(-sqrt(
2)*(B^3*b^10 + 4*(A^2*B*a^4 + A^3*a^3*b)*c^6 - (4*B^3*a^5 + 28*A*B^2*a^4*b + 41*A^2*B*a^3*b^2 + 13*A^3*a^2*b^3
)*c^5 + (29*B^3*a^4*b^2 + 87*A*B^2*a^3*b^3 + 58*A^2*B*a^2*b^4 + 7*A^3*a*b^5)*c^4 - (51*B^3*a^3*b^4 + 80*A*B^2*
a^2*b^5 + 24*A^2*B*a*b^6 + A^3*b^7)*c^3 + (35*B^3*a^2*b^6 + 27*A*B^2*a*b^7 + 3*A^2*B*b^8)*c^2 - (10*B^3*a*b^8
+ 3*A*B^2*b^9)*c + (B*b^5*c^7 - 8*A*a^2*c^10 + 6*(2*B*a^2*b + A*a*b^2)*c^9 - (7*B*a*b^3 + A*b^4)*c^8)*sqrt((B^
4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^
2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*
b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3
*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4
*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))*s
qrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2
*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c - (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^
4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 +
52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*
a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*
b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A
*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c
^8)) + 4*(B^4*a^3*b^6 - A*B^3*a^2*b^7 + A^4*a^4*c^5 - (7*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^4 - (B^4*a^6 + 5*A*B^3
*a^5*b - 9*A^2*B^2*a^4*b^2 - 11*A^3*B*a^3*b^3 - A^4*a^2*b^4)*c^3 + (6*B^4*a^5*b^2 + 2*A*B^3*a^4*b^3 - 12*A^2*B
^2*a^3*b^4 - 3*A^3*B*a^2*b^5)*c^2 - (5*B^4*a^4*b^4 - 3*A*B^3*a^3*b^5 - 3*A^2*B^2*a^2*b^6)*c)*sqrt(x)) + 4*(3*B
*c^2*x^2 + 15*B*b^2 - 15*(B*a + A*b)*c - 5*(B*b*c - A*c^2)*x)*sqrt(x))/c^3

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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(x**(5/2)*(B*x+A)/(c*x**2+b*x+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(x^(5/2)*(B*x+A)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

Timed out