3.13.77 \(\int \frac {A+B x}{(d+e x)^{3/2} (a-c x^2)} \, dx\)
Optimal. Leaf size=197 \[ -\frac {2 (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} \sqrt [4]{c} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}} \]
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Rubi [A] time = 0.40, antiderivative size = 197, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used =
{829, 827, 1166, 208} \begin {gather*} -\frac {2 (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}+\frac {\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} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\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} \sqrt [4]{c} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)),x]
[Out]
(-2*(B*d - A*e))/((c*d^2 - a*e^2)*Sqrt[d + e*x]) + ((Sqrt[a]*B - A*Sqrt[c])*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sq
rt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)^(3/2)) + ((Sqrt[a]*B + A*Sqrt[c])*ArcTanh
[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(1/4)*(Sqrt[c]*d + Sqrt[a]*e)^(3/2))
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 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 {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx &=-\frac {2 (B d-A e)}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}+\frac {\int \frac {-A c d+a B e-c (B d-A e) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{-c d^2+a e^2}\\ &=-\frac {2 (B d-A e)}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}-\frac {2 \operatorname {Subst}\left (\int \frac {c d (B d-A e)+e (-A c d+a B e)-c (B d-A e) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{c d^2-a e^2}\\ &=-\frac {2 (B d-A e)}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}+\frac {\left (\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \sqrt {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 {c} d-\sqrt {a} e}+\frac {\left (\left (B+\frac {A \sqrt {c}}{\sqrt {a}}\right ) \sqrt {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 {c} d+\sqrt {a} e}\\ &=-\frac {2 (B d-A e)}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}+\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (B+\frac {A \sqrt {c}}{\sqrt {a}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt [4]{c} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.33, size = 271, normalized size = 1.38 \begin {gather*} \frac {\frac {B \left (\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt [4]{c}}-\frac {(B d-A e) \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 {d+e x} \left (c d^2-a e^2\right )}}{\sqrt {a} e} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)),x]
[Out]
((B*(-(ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]/Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ArcTanh[(c^
(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]/Sqrt[Sqrt[c]*d + Sqrt[a]*e]))/c^(1/4) - ((B*d - 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[c]*d)
+ Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]))/((c*d^2 - a*e^2)*
Sqrt[d + e*x]))/(Sqrt[a]*e)
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IntegrateAlgebraic [A] time = 0.63, size = 265, normalized size = 1.35 \begin {gather*} -\frac {2 (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right ) \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)),x]
[Out]
(-2*(B*d - A*e))/((c*d^2 - a*e^2)*Sqrt[d + e*x]) + ((Sqrt[a]*B + A*Sqrt[c])*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt
[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)*Sqrt[-(Sqrt[c]*(Sqrt[c]*d + S
qrt[a]*e))]) + ((Sqrt[a]*B - A*Sqrt[c])*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - S
qrt[a]*e)])/(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])
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fricas [B] time = 2.76, size = 6448, normalized size = 32.73
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a),x, algorithm="fricas")
[Out]
1/2*((c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3
- 3*(B^2*a^2 + A^2*a*c)*d*e^2 + (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)*sqrt((4*A^2*B^2*c^
4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*
B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c +
A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a
^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(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))*log((2*(A*B^3*a*c^2 - A^3*B*c^3)*d^3 - 3*(B^4*a^2*c - A^4*c^3)*d^
2*e + 6*(A*B^3*a^2*c - A^3*B*a*c^2)*d*e^2 - (B^4*a^3 - A^4*a*c^2)*e^3)*sqrt(e*x + d) + (2*A*B^2*a*c^3*d^5 - (3
*B^3*a^2*c^2 + 7*A^2*B*a*c^3)*d^4*e + 2*(7*A*B^2*a^2*c^2 + 3*A^3*a*c^3)*d^3*e^2 - 4*(B^3*a^3*c + 4*A^2*B*a^2*c
^2)*d^2*e^3 + 2*(4*A*B^2*a^3*c + A^3*a^2*c^2)*d*e^4 - (B^3*a^4 + A^2*B*a^3*c)*e^5 + (A*a*c^5*d^8 - 2*B*a^2*c^4
*d^7*e - 2*A*a^2*c^4*d^6*e^2 + 6*B*a^3*c^3*d^5*e^3 - 6*B*a^4*c^2*d^3*e^5 + 2*A*a^4*c^2*d^2*e^6 + 2*B*a^5*c*d*e
^7 - A*a^5*c*e^8)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2
*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^
4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*
c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e
^10 + a^7*c*e^12)))*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d
*e^2 + (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)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 +
A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3
)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (
B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c
^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(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))) - (c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a
*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 + (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)*
sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4
)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 -
12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c
^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12))
)/(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))*log((2*(A*B^3*a*c^2 - A^3*B*c^3)*d^3 - 3*(B^4*a
^2*c - A^4*c^3)*d^2*e + 6*(A*B^3*a^2*c - A^3*B*a*c^2)*d*e^2 - (B^4*a^3 - A^4*a*c^2)*e^3)*sqrt(e*x + d) - (2*A*
B^2*a*c^3*d^5 - (3*B^3*a^2*c^2 + 7*A^2*B*a*c^3)*d^4*e + 2*(7*A*B^2*a^2*c^2 + 3*A^3*a*c^3)*d^3*e^2 - 4*(B^3*a^3
*c + 4*A^2*B*a^2*c^2)*d^2*e^3 + 2*(4*A*B^2*a^3*c + A^3*a^2*c^2)*d*e^4 - (B^3*a^4 + A^2*B*a^3*c)*e^5 + (A*a*c^5
*d^8 - 2*B*a^2*c^4*d^7*e - 2*A*a^2*c^4*d^6*e^2 + 6*B*a^3*c^3*d^5*e^3 - 6*B*a^4*c^2*d^3*e^5 + 2*A*a^4*c^2*d^2*e
^6 + 2*B*a^5*c*d*e^7 - A*a^5*c*e^8)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^
2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^
2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4
*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8
- 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^
2*a^2 + A^2*a*c)*d*e^2 + (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)*sqrt((4*A^2*B^2*c^4*d^6 -
12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2
*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*
a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*
d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(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))) + (c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt(-(6*A*B*a*c*d^2*e + 2*A*
B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 - (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)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2
*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^
4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*
c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e
^10 + a^7*c*e^12)))/(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))*log((2*(A*B^3*a*c^2 - A^3*B*c
^3)*d^3 - 3*(B^4*a^2*c - A^4*c^3)*d^2*e + 6*(A*B^3*a^2*c - A^3*B*a*c^2)*d*e^2 - (B^4*a^3 - A^4*a*c^2)*e^3)*sqr
t(e*x + d) + (2*A*B^2*a*c^3*d^5 - (3*B^3*a^2*c^2 + 7*A^2*B*a*c^3)*d^4*e + 2*(7*A*B^2*a^2*c^2 + 3*A^3*a*c^3)*d^
3*e^2 - 4*(B^3*a^3*c + 4*A^2*B*a^2*c^2)*d^2*e^3 + 2*(4*A*B^2*a^3*c + A^3*a^2*c^2)*d*e^4 - (B^3*a^4 + A^2*B*a^3
*c)*e^5 - (A*a*c^5*d^8 - 2*B*a^2*c^4*d^7*e - 2*A*a^2*c^4*d^6*e^2 + 6*B*a^3*c^3*d^5*e^3 - 6*B*a^4*c^2*d^3*e^5 +
2*A*a^4*c^2*d^2*e^6 + 2*B*a^5*c*d*e^7 - A*a^5*c*e^8)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d
^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6
*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A
^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 +
15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a*c + A^
2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 - (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)*sqrt((4
*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e
^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*
B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10
*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(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))) - (c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt(-(6*A*
B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 - (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)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^
2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^
2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4
*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8
- 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(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))*log((2*(A*B
^3*a*c^2 - A^3*B*c^3)*d^3 - 3*(B^4*a^2*c - A^4*c^3)*d^2*e + 6*(A*B^3*a^2*c - A^3*B*a*c^2)*d*e^2 - (B^4*a^3 - A
^4*a*c^2)*e^3)*sqrt(e*x + d) - (2*A*B^2*a*c^3*d^5 - (3*B^3*a^2*c^2 + 7*A^2*B*a*c^3)*d^4*e + 2*(7*A*B^2*a^2*c^2
+ 3*A^3*a*c^3)*d^3*e^2 - 4*(B^3*a^3*c + 4*A^2*B*a^2*c^2)*d^2*e^3 + 2*(4*A*B^2*a^3*c + A^3*a^2*c^2)*d*e^4 - (B
^3*a^4 + A^2*B*a^3*c)*e^5 - (A*a*c^5*d^8 - 2*B*a^2*c^4*d^7*e - 2*A*a^2*c^4*d^6*e^2 + 6*B*a^3*c^3*d^5*e^3 - 6*B
*a^4*c^2*d^3*e^5 + 2*A*a^4*c^2*d^2*e^6 + 2*B*a^5*c*d*e^7 - A*a^5*c*e^8)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*
c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*
a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^
5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*
a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e
^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 - (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)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3
+ 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3
)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^1
2 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a
^7*c*e^12)))/(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))) - 4*(B*d - A*e)*sqrt(e*x + d))/(c*d
^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)
________________________________________________________________________________________
giac [B] time = 0.70, size = 920, normalized size = 4.67 \begin {gather*} -\frac {2 \, {\left (B d - A e\right )}}{{\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d}} + \frac {{\left ({\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} B a d {\left | c \right |} - {\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} A a {\left | c \right |} e + 2 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} A {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (a c^{2} d^{4} - a^{3} e^{4}\right )} B {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (\sqrt {a c} c^{3} d^{6} e - 2 \, \sqrt {a c} a c^{2} d^{4} e^{3} + \sqrt {a c} a^{2} c d^{2} e^{5}\right )} A {\left | c \right |} + {\left (\sqrt {a c} a c^{2} d^{5} e^{2} - 2 \, \sqrt {a c} a^{2} c d^{3} e^{4} + \sqrt {a c} a^{3} d e^{6}\right )} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{2} d^{3} - a c d e^{2} + \sqrt {{\left (c^{2} d^{3} - a c d e^{2}\right )}^{2} - {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} - a c e^{2}\right )}}}{c^{2} d^{2} - a c e^{2}}}}\right )}{{\left (a c^{3} d^{5} - \sqrt {a c} a c^{2} d^{4} e - 2 \, a^{2} c^{2} d^{3} e^{2} + 2 \, \sqrt {a c} a^{2} c d^{2} e^{3} + a^{3} c d e^{4} - \sqrt {a c} a^{3} e^{5}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | c d^{2} e - a e^{3} \right |}} - \frac {{\left ({\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} B a d {\left | c \right |} - {\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} A a {\left | c \right |} e - 2 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} A {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} + {\left (a c^{2} d^{4} - a^{3} e^{4}\right )} B {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (\sqrt {a c} c^{3} d^{6} e - 2 \, \sqrt {a c} a c^{2} d^{4} e^{3} + \sqrt {a c} a^{2} c d^{2} e^{5}\right )} A {\left | c \right |} + {\left (\sqrt {a c} a c^{2} d^{5} e^{2} - 2 \, \sqrt {a c} a^{2} c d^{3} e^{4} + \sqrt {a c} a^{3} d e^{6}\right )} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{2} d^{3} - a c d e^{2} - \sqrt {{\left (c^{2} d^{3} - a c d e^{2}\right )}^{2} - {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} - a c e^{2}\right )}}}{c^{2} d^{2} - a c e^{2}}}}\right )}{{\left (a c^{3} d^{5} + \sqrt {a c} a c^{2} d^{4} e - 2 \, a^{2} c^{2} d^{3} e^{2} - 2 \, \sqrt {a c} a^{2} c d^{2} e^{3} + a^{3} c d e^{4} + \sqrt {a c} a^{3} e^{5}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | c d^{2} e - a e^{3} \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a),x, algorithm="giac")
[Out]
-2*(B*d - A*e)/((c*d^2 - a*e^2)*sqrt(x*e + d)) + ((c*d^2*e - a*e^3)^2*sqrt(a*c)*B*a*d*abs(c) - (c*d^2*e - a*e^
3)^2*sqrt(a*c)*A*a*abs(c)*e + 2*(a*c^2*d^3*e - a^2*c*d*e^3)*A*abs(c*d^2*e - a*e^3)*abs(c) - (a*c^2*d^4 - a^3*e
^4)*B*abs(c*d^2*e - a*e^3)*abs(c) - (sqrt(a*c)*c^3*d^6*e - 2*sqrt(a*c)*a*c^2*d^4*e^3 + sqrt(a*c)*a^2*c*d^2*e^5
)*A*abs(c) + (sqrt(a*c)*a*c^2*d^5*e^2 - 2*sqrt(a*c)*a^2*c*d^3*e^4 + sqrt(a*c)*a^3*d*e^6)*B*abs(c))*arctan(sqrt
(x*e + d)/sqrt(-(c^2*d^3 - a*c*d*e^2 + sqrt((c^2*d^3 - a*c*d*e^2)^2 - (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(c^2
*d^2 - a*c*e^2)))/(c^2*d^2 - a*c*e^2)))/((a*c^3*d^5 - sqrt(a*c)*a*c^2*d^4*e - 2*a^2*c^2*d^3*e^2 + 2*sqrt(a*c)*
a^2*c*d^2*e^3 + a^3*c*d*e^4 - sqrt(a*c)*a^3*e^5)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(c*d^2*e - a*e^3)) - ((c*d^2*
e - a*e^3)^2*sqrt(a*c)*B*a*d*abs(c) - (c*d^2*e - a*e^3)^2*sqrt(a*c)*A*a*abs(c)*e - 2*(a*c^2*d^3*e - a^2*c*d*e^
3)*A*abs(c*d^2*e - a*e^3)*abs(c) + (a*c^2*d^4 - a^3*e^4)*B*abs(c*d^2*e - a*e^3)*abs(c) - (sqrt(a*c)*c^3*d^6*e
- 2*sqrt(a*c)*a*c^2*d^4*e^3 + sqrt(a*c)*a^2*c*d^2*e^5)*A*abs(c) + (sqrt(a*c)*a*c^2*d^5*e^2 - 2*sqrt(a*c)*a^2*c
*d^3*e^4 + sqrt(a*c)*a^3*d*e^6)*B*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(c^2*d^3 - a*c*d*e^2 - sqrt((c^2*d^3 - a*
c*d*e^2)^2 - (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(c^2*d^2 - a*c*e^2)))/(c^2*d^2 - a*c*e^2)))/((a*c^3*d^5 + sqr
t(a*c)*a*c^2*d^4*e - 2*a^2*c^2*d^3*e^2 - 2*sqrt(a*c)*a^2*c*d^2*e^3 + a^3*c*d*e^4 + sqrt(a*c)*a^3*e^5)*sqrt(-c^
2*d + sqrt(a*c)*c*e)*abs(c*d^2*e - a*e^3))
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maple [B] time = 0.08, size = 588, normalized size = 2.98 \begin {gather*} -\frac {A \,c^{2} d e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {A \,c^{2} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {B a c \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {B a c \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {A c e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {A c e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {B c d \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {B c d \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {2 A e}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {e x +d}}+\frac {2 B d}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {e x +d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a),x)
[Out]
-2/(a*e^2-c*d^2)/(e*x+d)^(1/2)*A*e+2/(a*e^2-c*d^2)/(e*x+d)^(1/2)*B*d-1/(a*e^2-c*d^2)*c^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*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d*e+1/(a*e^2-c*d^2)*c/(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)*B*a*e^2
+1/(a*e^2-c*d^2)*c/((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)*A*
e-1/(a*e^2-c*d^2)*c/((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)*B
*d-1/(a*e^2-c*d^2)*c^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*d+(a*c*e^2)^
(1/2))*c)^(1/2)*c)*A*d*e+1/(a*e^2-c*d^2)*c/(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)*B*a*e^2-1/(a*e^2-c*d^2)*c/((-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)*A*e+1/(a*e^2-c*d^2)*c/((-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)*B*d
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {B x + A}{{\left (c x^{2} - a\right )} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a),x, algorithm="maxima")
[Out]
-integrate((B*x + A)/((c*x^2 - a)*(e*x + d)^(3/2)), x)
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mupad [B] time = 5.67, size = 10288, normalized size = 52.22
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)/((a - c*x^2)*(d + e*x)^(3/2)),x)
[Out]
atan((((-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^
3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2)
- 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4
*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c
)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*
a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c
)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4))
)^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e
^8 + 320*a^5*c^5*d^3*e^10) - 32*B*a^6*c^3*e^12 + 64*A*a*c^8*d^9*e^3 + 64*A*a^5*c^4*d*e^11 - 32*B*a*c^8*d^10*e^
2 - 256*A*a^2*c^7*d^7*e^5 + 384*A*a^3*c^6*d^5*e^7 - 256*A*a^4*c^5*d^3*e^9 + 96*B*a^2*c^7*d^8*e^4 - 64*B*a^3*c^
6*d^6*e^6 - 64*B*a^4*c^5*d^4*e^8 + 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*B^2*a^5*
c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*
A^2*a*c^7*d^6*e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e
^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*
(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*
c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))
/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*1i + ((-(B^2*a^2*c^2*d^3 + B^2*a
^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(
1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c
*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*
c^2*d^2*e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c
^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 +
3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)
^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a
^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 32*B
*a^6*c^3*e^12 - 64*A*a*c^8*d^9*e^3 - 64*A*a^5*c^4*d*e^11 + 32*B*a*c^8*d^10*e^2 + 256*A*a^2*c^7*d^7*e^5 - 384*A
*a^3*c^6*d^5*e^7 + 256*A*a^4*c^5*d^3*e^9 - 96*B*a^2*c^7*d^8*e^4 + 64*B*a^3*c^6*d^6*e^6 + 64*B*a^4*c^5*d^4*e^8
- 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 3
2*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^6*e^4 - 16*B^2*a*c^7*
d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-
(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2
+ A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a
^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*
a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*1i)/(((-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*
d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*
B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*
d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*((d + e*x)^(
1/2)*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*
c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) -
6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d
^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e
^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) - 32*B*a^6*c^3*e^12 + 64*A*a*c^8*d^9*e^
3 + 64*A*a^5*c^4*d*e^11 - 32*B*a*c^8*d^10*e^2 - 256*A*a^2*c^7*d^7*e^5 + 384*A*a^3*c^6*d^5*e^7 - 256*A*a^4*c^5*
d^3*e^9 + 96*B*a^2*c^7*d^8*e^4 - 64*B*a^3*c^6*d^6*e^6 - 64*B*a^4*c^5*d^4*e^8 + 96*B*a^5*c^4*d^2*e^10) + (d + e
*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^
2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^6*e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 -
64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a
^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*
A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a
^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e
^4)))^(1/2) - ((-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) +
3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c
)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 -
a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^
3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2)
+ 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*
e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d
^2*e^4)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c
^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 32*B*a^6*c^3*e^12 - 64*A*a*c^8*d^9*e^3 - 64*A*a^5*c^4*d*e^11 + 32*B*a*c^8
*d^10*e^2 + 256*A*a^2*c^7*d^7*e^5 - 384*A*a^3*c^6*d^5*e^7 + 256*A*a^4*c^5*d^3*e^9 - 96*B*a^2*c^7*d^8*e^4 + 64*
B*a^3*c^6*d^6*e^6 + 64*B*a^4*c^5*d^4*e^8 - 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*
B^2*a^5*c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e
^8 + 32*A^2*a*c^7*d^6*e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c
^6*d^5*e^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*
c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3
+ 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c
)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2) - 16*A^3*a^3*c^4*e^9 + 1
6*A^3*c^7*d^6*e^3 + 48*A^3*a^2*c^5*d^2*e^7 - 48*B^3*a^2*c^5*d^5*e^4 + 48*B^3*a^3*c^4*d^3*e^6 + 16*A*B^2*a^4*c^
3*e^9 - 16*A^2*B*c^7*d^7*e^2 - 48*A^3*a*c^6*d^4*e^5 + 16*B^3*a*c^6*d^7*e^2 - 16*B^3*a^4*c^3*d*e^8 + 48*A*B^2*a
^2*c^5*d^4*e^5 - 48*A*B^2*a^3*c^4*d^2*e^7 - 48*A^2*B*a^2*c^5*d^3*e^6 - 16*A*B^2*a*c^6*d^6*e^3 + 48*A^2*B*a*c^6
*d^5*e^4 + 16*A^2*B*a^3*c^4*d*e^8))*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2*A*B*c^2
*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3
*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3*c)^(
1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*2i + atan((((-(B^2*a^2*c^2*
d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^
3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e
- 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e
^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^
3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*
a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*
e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*c^9*d^11
*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e
^10) - 32*B*a^6*c^3*e^12 + 64*A*a*c^8*d^9*e^3 + 64*A*a^5*c^4*d*e^11 - 32*B*a*c^8*d^10*e^2 - 256*A*a^2*c^7*d^7*
e^5 + 384*A*a^3*c^6*d^5*e^7 - 256*A*a^4*c^5*d^3*e^9 + 96*B*a^2*c^7*d^8*e^4 - 64*B*a^3*c^6*d^6*e^6 - 64*B*a^4*c
^5*d^4*e^8 + 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*A^2*c^8*
d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^6*e^4 - 16
*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a^3*c^5*d
^3*e^7))*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*
a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2
) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c
^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*1i + ((-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) +
A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d
*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) +
6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(
(d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) +
3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*
c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6
- a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^
2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 32*B*a^6*c^3*e^12 - 64*A*a
*c^8*d^9*e^3 - 64*A*a^5*c^4*d*e^11 + 32*B*a*c^8*d^10*e^2 + 256*A*a^2*c^7*d^7*e^5 - 384*A*a^3*c^6*d^5*e^7 + 256
*A*a^4*c^5*d^3*e^9 - 96*B*a^2*c^7*d^8*e^4 + 64*B*a^3*c^6*d^6*e^6 + 64*B*a^4*c^5*d^4*e^8 - 96*B*a^5*c^4*d^2*e^1
0) + (d + e*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8
+ 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^6*e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7
*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d^3 - B^2
*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)
^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a
*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^
4*c^2*d^2*e^4)))^(1/2)*1i)/(((-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a
^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^
2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(
4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^
3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*
(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e -
3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2
- 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e
^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) - 32*B*a^6*c^3*e^12 + 64*A*a*c^8*d^9*e^3 + 64*A*a^5*c^4*d*e^1
1 - 32*B*a*c^8*d^10*e^2 - 256*A*a^2*c^7*d^7*e^5 + 384*A*a^3*c^6*d^5*e^7 - 256*A*a^4*c^5*d^3*e^9 + 96*B*a^2*c^7
*d^8*e^4 - 64*B*a^3*c^6*d^6*e^6 - 64*B*a^4*c^5*d^4*e^8 + 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2*a^4*
c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2
*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^6*e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 -
192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^
3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*
A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*
c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2) - ((-(B^2*
a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^
2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^
2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c
^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*
a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2
- 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*
B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*
c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c
^5*d^3*e^10) + 32*B*a^6*c^3*e^12 - 64*A*a*c^8*d^9*e^3 - 64*A*a^5*c^4*d*e^11 + 32*B*a*c^8*d^10*e^2 + 256*A*a^2*
c^7*d^7*e^5 - 384*A*a^3*c^6*d^5*e^7 + 256*A*a^4*c^5*d^3*e^9 - 96*B*a^2*c^7*d^8*e^4 + 64*B*a^3*c^6*d^6*e^6 + 64
*B*a^4*c^5*d^4*e^8 - 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2*a^4*c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*
A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^6*
e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a
^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2)
+ 3*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3
*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6
- a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2) - 16*A^3*a^3*c^4*e^9 + 16*A^3*c^7*d^6*e^3 + 48
*A^3*a^2*c^5*d^2*e^7 - 48*B^3*a^2*c^5*d^5*e^4 + 48*B^3*a^3*c^4*d^3*e^6 + 16*A*B^2*a^4*c^3*e^9 - 16*A^2*B*c^7*d
^7*e^2 - 48*A^3*a*c^6*d^4*e^5 + 16*B^3*a*c^6*d^7*e^2 - 16*B^3*a^4*c^3*d*e^8 + 48*A*B^2*a^2*c^5*d^4*e^5 - 48*A*
B^2*a^3*c^4*d^2*e^7 - 48*A^2*B*a^2*c^5*d^3*e^6 - 16*A*B^2*a*c^6*d^6*e^3 + 48*A^2*B*a*c^6*d^5*e^4 + 16*A^2*B*a^
3*c^4*d*e^8))*(-(B^2*a^2*c^2*d^3 - B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 + 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3
*B^2*a^3*c*d*e^2 - A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 - 3*A^2*c^2*d^2*e*(a^3*c)
^(1/2) - 6*A*B*a^2*c^2*d^2*e - 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) + 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 -
a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*2i - (2*(A*e - B*d))/((a*e^2 - c*d^2)*(d + e*x)^(
1/2))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)**(3/2)/(-c*x**2+a),x)
[Out]
Timed out
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