3.13.86 \(\int \frac {(A+B x) (d+e x)^{3/2}}{(a-c x^2)^3} \, dx\)
Optimal. Leaf size=350 \[ \frac {3 \left (a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-A \left (-2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{7/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}-\frac {3 \left (a B e \left (\sqrt {a} e+2 \sqrt {c} d\right )-A \left (2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{7/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {\sqrt {d+e x} (a A e-3 x (2 A c d-a B e))}{16 a^2 c \left (a-c x^2\right )}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2} \]
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Rubi [A] time = 0.80, antiderivative size = 350, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used =
{819, 823, 827, 1166, 208} \begin {gather*} \frac {3 \left (a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-A \left (-2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{7/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}-\frac {3 \left (a B e \left (\sqrt {a} e+2 \sqrt {c} d\right )-A \left (2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{7/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {\sqrt {d+e x} (a A e-3 x (2 A c d-a B e))}{16 a^2 c \left (a-c x^2\right )}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^3,x]
[Out]
(Sqrt[d + e*x]*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(4*a*c*(a - c*x^2)^2) - (Sqrt[d + e*x]*(a*A*e - 3*(2*A*c*d
- a*B*e)*x))/(16*a^2*c*(a - c*x^2)) + (3*(a*B*e*(2*Sqrt[c]*d - Sqrt[a]*e) - A*(4*c^(3/2)*d^2 - 2*Sqrt[a]*c*d*
e - a*Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(7/4)*Sqrt[Sqr
t[c]*d - Sqrt[a]*e]) - (3*(a*B*e*(2*Sqrt[c]*d + Sqrt[a]*e) - A*(4*c^(3/2)*d^2 + 2*Sqrt[a]*c*d*e - a*Sqrt[c]*e^
2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(7/4)*Sqrt[Sqrt[c]*d + Sqrt[a]
*e])
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 819
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
Rule 823
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx &=\frac {\sqrt {d+e x} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (-6 A c d^2+3 a B d e+a A e^2\right )-\frac {1}{2} e (5 A c d-3 a B e) x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx}{4 a c}\\ &=\frac {\sqrt {d+e x} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} (a A e-3 (2 A c d-a B e) x)}{16 a^2 c \left (a-c x^2\right )}+\frac {\int \frac {\frac {3}{4} c \left (c d^2-a e^2\right ) \left (4 A c d^2-2 a B d e-a A e^2\right )+\frac {3}{4} c e (2 A c d-a B e) \left (c d^2-a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c^2 \left (c d^2-a e^2\right )}\\ &=\frac {\sqrt {d+e x} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} (a A e-3 (2 A c d-a B e) x)}{16 a^2 c \left (a-c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{4} c d e (2 A c d-a B e) \left (c d^2-a e^2\right )+\frac {3}{4} c e \left (c d^2-a e^2\right ) \left (4 A c d^2-2 a B d e-a A e^2\right )+\frac {3}{4} c e (2 A c d-a B e) \left (c d^2-a e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{4 a^2 c^2 \left (c d^2-a e^2\right )}\\ &=\frac {\sqrt {d+e x} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} (a A e-3 (2 A c d-a B e) x)}{16 a^2 c \left (a-c x^2\right )}+\frac {\left (3 \left (a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-A \left (4 c^{3/2} d^2-2 \sqrt {a} c d e-a \sqrt {c} e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{32 a^{5/2} c}-\frac {\left (3 \left (a B e \left (2 \sqrt {c} d+\sqrt {a} e\right )-A \left (4 c^{3/2} d^2+2 \sqrt {a} c d e-a \sqrt {c} e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{32 a^{5/2} c}\\ &=\frac {\sqrt {d+e x} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} (a A e-3 (2 A c d-a B e) x)}{16 a^2 c \left (a-c x^2\right )}+\frac {3 \left (a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-A \left (4 c^{3/2} d^2-2 \sqrt {a} c d e-a \sqrt {c} e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{7/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}-\frac {3 \left (a B e \left (2 \sqrt {c} d+\sqrt {a} e\right )-A \left (4 c^{3/2} d^2+2 \sqrt {a} c d e-a \sqrt {c} e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{7/4} \sqrt {\sqrt {c} d+\sqrt {a} e}}\\ \end {align*}
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Mathematica [A] time = 1.42, size = 550, normalized size = 1.57 \begin {gather*} \frac {\frac {c^2 (d+e x)^{5/2} \left (a^2 e^2 (3 A e+2 B d+B e x)-a c d e (5 A d+4 A e x+3 B d x)+6 A c^2 d^3 x\right )}{2 \left (a-c x^2\right )}+\frac {3 c^{3/4} \left (A \left (a^2 e^4+5 a c d^2 e^2-10 c^2 d^4\right )+a B d e \left (5 c d^2-a e^2\right )\right ) \left (2 \sqrt {a} \sqrt [4]{c} e \sqrt {d+e x}+\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )-\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )\right )}{4 \sqrt {a}}+\frac {\sqrt [4]{c} \left (2 A c d \left (3 c d^2-2 a e^2\right )+a B e \left (a e^2-3 c d^2\right )\right ) \left (2 \sqrt {a} c^{3/4} e \sqrt {d+e x} (7 d+e x)+3 \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )-3 \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )\right )}{4 \sqrt {a}}+\frac {2 a c^2 (d+e x)^{5/2} \left (c d^2-a e^2\right ) (-a A e+a B (d-e x)+A c d x)}{\left (a-c x^2\right )^2}}{8 a^2 c^2 \left (c d^2-a e^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^3,x]
[Out]
((2*a*c^2*(c*d^2 - a*e^2)*(d + e*x)^(5/2)*(-(a*A*e) + A*c*d*x + a*B*(d - e*x)))/(a - c*x^2)^2 + (c^2*(d + e*x)
^(5/2)*(6*A*c^2*d^3*x - a*c*d*e*(5*A*d + 3*B*d*x + 4*A*e*x) + a^2*e^2*(2*B*d + 3*A*e + B*e*x)))/(2*(a - c*x^2)
) + (3*c^(3/4)*(a*B*d*e*(5*c*d^2 - a*e^2) + A*(-10*c^2*d^4 + 5*a*c*d^2*e^2 + a^2*e^4))*(2*Sqrt[a]*c^(1/4)*e*Sq
rt[d + e*x] + (Sqrt[c]*d - Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] - (Sq
rt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]))/(4*Sqrt[a]) + (c^(1/
4)*(2*A*c*d*(3*c*d^2 - 2*a*e^2) + a*B*e*(-3*c*d^2 + a*e^2))*(2*Sqrt[a]*c^(3/4)*e*Sqrt[d + e*x]*(7*d + e*x) + 3
*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] - 3*(Sqrt[c]*d + S
qrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]))/(4*Sqrt[a]))/(8*a^2*c^2*(c*d^2
- a*e^2)^2)
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IntegrateAlgebraic [A] time = 3.17, size = 523, normalized size = 1.49 \begin {gather*} \frac {3 \left (-a^{3/2} B e^2+2 \sqrt {a} A c d e-a A \sqrt {c} e^2-2 a B \sqrt {c} d e+4 A c^{3/2} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{32 a^{5/2} c^{3/2} \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}-\frac {3 \left (a^{3/2} B e^2-2 \sqrt {a} A c d e-a A \sqrt {c} e^2-2 a B \sqrt {c} d e+4 A c^{3/2} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{32 a^{5/2} c^{3/2} \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}+\frac {e \sqrt {d+e x} \left (3 a^2 A e^4+a^2 B e^3 (d+e x)+3 a^2 B d e^3-9 a A c d^2 e^2+8 a A c d e^2 (d+e x)+a A c e^2 (d+e x)^2-3 a B c d^3 e+9 a B c d^2 e (d+e x)-9 a B c d e (d+e x)^2+3 a B c e (d+e x)^3+6 A c^2 d^4-18 A c^2 d^3 (d+e x)+18 A c^2 d^2 (d+e x)^2-6 A c^2 d (d+e x)^3\right )}{16 a^2 c \left (a e^2-c d^2+2 c d (d+e x)-c (d+e x)^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^3,x]
[Out]
(e*Sqrt[d + e*x]*(6*A*c^2*d^4 - 3*a*B*c*d^3*e - 9*a*A*c*d^2*e^2 + 3*a^2*B*d*e^3 + 3*a^2*A*e^4 - 18*A*c^2*d^3*(
d + e*x) + 9*a*B*c*d^2*e*(d + e*x) + 8*a*A*c*d*e^2*(d + e*x) + a^2*B*e^3*(d + e*x) + 18*A*c^2*d^2*(d + e*x)^2
- 9*a*B*c*d*e*(d + e*x)^2 + a*A*c*e^2*(d + e*x)^2 - 6*A*c^2*d*(d + e*x)^3 + 3*a*B*c*e*(d + e*x)^3))/(16*a^2*c*
(-(c*d^2) + a*e^2 + 2*c*d*(d + e*x) - c*(d + e*x)^2)^2) + (3*(4*A*c^(3/2)*d^2 - 2*a*B*Sqrt[c]*d*e + 2*Sqrt[a]*
A*c*d*e - a^(3/2)*B*e^2 - a*A*Sqrt[c]*e^2)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d
+ Sqrt[a]*e)])/(32*a^(5/2)*c^(3/2)*Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]) - (3*(4*A*c^(3/2)*d^2 - 2*a*B*Sqr
t[c]*d*e - 2*Sqrt[a]*A*c*d*e + a^(3/2)*B*e^2 - a*A*Sqrt[c]*e^2)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[
d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(32*a^(5/2)*c^(3/2)*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])
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fricas [B] time = 1.86, size = 4176, normalized size = 11.93
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)^3,x, algorithm="fricas")
[Out]
-1/64*(3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*A^2*c^3*d^5 - 16*A*B*a*c^2*d^4*e + 16*A*B*a^2*c*d^2*e^
3 - 2*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5*A^2*a*c^2)*d^3*e^2 - (3*B^2*a^3 - 5*A^2*a^2*c)*d*e^4 + (a^5*c^4*d^2 - a^6
*c^3*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*
e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))/(a^5*c^4*d^2 - a^6*c^3*e^2))*log(-27*(32*A^3*B*c^4*d^5
*e^4 - 16*(3*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^5 + 8*(3*A*B^3*a^2*c^2 - A^3*B*a*c^3)*d^3*e^6 - 4*(B^4*a^3*c - 6*A
^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^7 - 2*(5*A*B^3*a^3*c + 3*A^3*B*a^2*c^2)*d*e^8 + (B^4*a^4 - A^4*a^2*c^2)*e^
9)*sqrt(e*x + d) + 27*(4*A^2*B*a^3*c^4*d^3*e^5 - 2*(2*A*B^2*a^4*c^3 + A^3*a^3*c^4)*d^2*e^6 + (B^3*a^5*c^2 - A^
2*B*a^4*c^3)*d*e^7 + (A*B^2*a^5*c^2 + A^3*a^4*c^3)*e^8 + (4*A*a^5*c^8*d^5 - 2*B*a^6*c^7*d^4*e - 7*A*a^6*c^7*d^
3*e^2 + 3*B*a^7*c^6*d^2*e^3 + 3*A*a^7*c^6*d*e^4 - B*a^8*c^5*e^5)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c +
A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))
*sqrt((16*A^2*c^3*d^5 - 16*A*B*a*c^2*d^4*e + 16*A*B*a^2*c*d^2*e^3 - 2*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5*A^2*a*c^2
)*d^3*e^2 - (3*B^2*a^3 - 5*A^2*a^2*c)*d*e^4 + (a^5*c^4*d^2 - a^6*c^3*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B
^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c
^7*e^4)))/(a^5*c^4*d^2 - a^6*c^3*e^2))) - 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*A^2*c^3*d^5 - 16*A*
B*a*c^2*d^4*e + 16*A*B*a^2*c*d^2*e^3 - 2*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5*A^2*a*c^2)*d^3*e^2 - (3*B^2*a^3 - 5*A^
2*a^2*c)*d*e^4 + (a^5*c^4*d^2 - a^6*c^3*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (
B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))/(a^5*c^4*d^2 - a^6*
c^3*e^2))*log(-27*(32*A^3*B*c^4*d^5*e^4 - 16*(3*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^5 + 8*(3*A*B^3*a^2*c^2 - A^3*B*
a*c^3)*d^3*e^6 - 4*(B^4*a^3*c - 6*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^7 - 2*(5*A*B^3*a^3*c + 3*A^3*B*a^2*c^2)
*d*e^8 + (B^4*a^4 - A^4*a^2*c^2)*e^9)*sqrt(e*x + d) - 27*(4*A^2*B*a^3*c^4*d^3*e^5 - 2*(2*A*B^2*a^4*c^3 + A^3*a
^3*c^4)*d^2*e^6 + (B^3*a^5*c^2 - A^2*B*a^4*c^3)*d*e^7 + (A*B^2*a^5*c^2 + A^3*a^4*c^3)*e^8 + (4*A*a^5*c^8*d^5 -
2*B*a^6*c^7*d^4*e - 7*A*a^6*c^7*d^3*e^2 + 3*B*a^7*c^6*d^2*e^3 + 3*A*a^7*c^6*d*e^4 - B*a^8*c^5*e^5)*sqrt((4*A^
2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 -
2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))*sqrt((16*A^2*c^3*d^5 - 16*A*B*a*c^2*d^4*e + 16*A*B*a^2*c*d^2*e^3 - 2*A*B*a
^3*e^5 + 4*(B^2*a^2*c - 5*A^2*a*c^2)*d^3*e^2 - (3*B^2*a^3 - 5*A^2*a^2*c)*d*e^4 + (a^5*c^4*d^2 - a^6*c^3*e^2)*s
qrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*
c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))/(a^5*c^4*d^2 - a^6*c^3*e^2))) + 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 +
a^4*c)*sqrt((16*A^2*c^3*d^5 - 16*A*B*a*c^2*d^4*e + 16*A*B*a^2*c*d^2*e^3 - 2*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5*A^2
*a*c^2)*d^3*e^2 - (3*B^2*a^3 - 5*A^2*a^2*c)*d*e^4 - (a^5*c^4*d^2 - a^6*c^3*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^8 -
4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 +
a^7*c^7*e^4)))/(a^5*c^4*d^2 - a^6*c^3*e^2))*log(-27*(32*A^3*B*c^4*d^5*e^4 - 16*(3*A^2*B^2*a*c^3 + A^4*c^4)*d^
4*e^5 + 8*(3*A*B^3*a^2*c^2 - A^3*B*a*c^3)*d^3*e^6 - 4*(B^4*a^3*c - 6*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^7 -
2*(5*A*B^3*a^3*c + 3*A^3*B*a^2*c^2)*d*e^8 + (B^4*a^4 - A^4*a^2*c^2)*e^9)*sqrt(e*x + d) + 27*(4*A^2*B*a^3*c^4*d
^3*e^5 - 2*(2*A*B^2*a^4*c^3 + A^3*a^3*c^4)*d^2*e^6 + (B^3*a^5*c^2 - A^2*B*a^4*c^3)*d*e^7 + (A*B^2*a^5*c^2 + A^
3*a^4*c^3)*e^8 - (4*A*a^5*c^8*d^5 - 2*B*a^6*c^7*d^4*e - 7*A*a^6*c^7*d^3*e^2 + 3*B*a^7*c^6*d^2*e^3 + 3*A*a^7*c^
6*d*e^4 - B*a^8*c^5*e^5)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*
a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))*sqrt((16*A^2*c^3*d^5 - 16*A*B*a*c^2*d^4
*e + 16*A*B*a^2*c*d^2*e^3 - 2*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5*A^2*a*c^2)*d^3*e^2 - (3*B^2*a^3 - 5*A^2*a^2*c)*d*
e^4 - (a^5*c^4*d^2 - a^6*c^3*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2
*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))/(a^5*c^4*d^2 - a^6*c^3*e^2)))
- 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*A^2*c^3*d^5 - 16*A*B*a*c^2*d^4*e + 16*A*B*a^2*c*d^2*e^3 - 2
*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5*A^2*a*c^2)*d^3*e^2 - (3*B^2*a^3 - 5*A^2*a^2*c)*d*e^4 - (a^5*c^4*d^2 - a^6*c^3*
e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)
/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))/(a^5*c^4*d^2 - a^6*c^3*e^2))*log(-27*(32*A^3*B*c^4*d^5*e^4
- 16*(3*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^5 + 8*(3*A*B^3*a^2*c^2 - A^3*B*a*c^3)*d^3*e^6 - 4*(B^4*a^3*c - 6*A^2*B^
2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^7 - 2*(5*A*B^3*a^3*c + 3*A^3*B*a^2*c^2)*d*e^8 + (B^4*a^4 - A^4*a^2*c^2)*e^9)*sq
rt(e*x + d) - 27*(4*A^2*B*a^3*c^4*d^3*e^5 - 2*(2*A*B^2*a^4*c^3 + A^3*a^3*c^4)*d^2*e^6 + (B^3*a^5*c^2 - A^2*B*a
^4*c^3)*d*e^7 + (A*B^2*a^5*c^2 + A^3*a^4*c^3)*e^8 - (4*A*a^5*c^8*d^5 - 2*B*a^6*c^7*d^4*e - 7*A*a^6*c^7*d^3*e^2
+ 3*B*a^7*c^6*d^2*e^3 + 3*A*a^7*c^6*d*e^4 - B*a^8*c^5*e^5)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B
*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))*sqrt
((16*A^2*c^3*d^5 - 16*A*B*a*c^2*d^4*e + 16*A*B*a^2*c*d^2*e^3 - 2*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5*A^2*a*c^2)*d^3
*e^2 - (3*B^2*a^3 - 5*A^2*a^2*c)*d*e^4 - (a^5*c^4*d^2 - a^6*c^3*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*
c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^
4)))/(a^5*c^4*d^2 - a^6*c^3*e^2))) - 4*(A*a*c*e*x^2 + 4*B*a^2*d + 3*A*a^2*e - 3*(2*A*c^2*d - B*a*c*e)*x^3 + (1
0*A*a*c*d + B*a^2*e)*x)*sqrt(e*x + d))/(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)
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giac [B] time = 0.74, size = 754, normalized size = 2.15 \begin {gather*} \frac {3 \, {\left (2 \, \sqrt {a c} B a c^{3} d^{2} e - B a^{2} c^{2} d {\left | c \right |} e^{2} + 2 \, \sqrt {a c} A a c^{3} d e^{2} - \sqrt {a c} B a^{2} c^{2} e^{3} + {\left (2 \, a c^{3} d^{2} e - a^{2} c^{2} e^{3}\right )} A {\left | c \right |} - {\left (4 \, \sqrt {a c} c^{4} d^{3} - \sqrt {a c} a c^{3} d e^{2}\right )} A\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d + \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{4} d - \sqrt {a c} a^{3} c^{3} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e}} - \frac {3 \, {\left (2 \, \sqrt {a c} B a c^{3} d^{2} e + B a^{2} c^{2} d {\left | c \right |} e^{2} + 2 \, \sqrt {a c} A a c^{3} d e^{2} - \sqrt {a c} B a^{2} c^{2} e^{3} - {\left (2 \, a c^{3} d^{2} e - a^{2} c^{2} e^{3}\right )} A {\left | c \right |} - {\left (4 \, \sqrt {a c} c^{4} d^{3} - \sqrt {a c} a c^{3} d e^{2}\right )} A\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d - \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{4} d + \sqrt {a c} a^{3} c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e}} - \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} A c^{2} d e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{2} d^{2} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{2} d^{3} e - 6 \, \sqrt {x e + d} A c^{2} d^{4} e - 3 \, {\left (x e + d\right )}^{\frac {7}{2}} B a c e^{2} + 9 \, {\left (x e + d\right )}^{\frac {5}{2}} B a c d e^{2} - 9 \, {\left (x e + d\right )}^{\frac {3}{2}} B a c d^{2} e^{2} + 3 \, \sqrt {x e + d} B a c d^{3} e^{2} - {\left (x e + d\right )}^{\frac {5}{2}} A a c e^{3} - 8 \, {\left (x e + d\right )}^{\frac {3}{2}} A a c d e^{3} + 9 \, \sqrt {x e + d} A a c d^{2} e^{3} - {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} e^{4} - 3 \, \sqrt {x e + d} B a^{2} d e^{4} - 3 \, \sqrt {x e + d} A a^{2} e^{5}}{16 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )}^{2} a^{2} c} \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)^3,x, algorithm="giac")
[Out]
3/32*(2*sqrt(a*c)*B*a*c^3*d^2*e - B*a^2*c^2*d*abs(c)*e^2 + 2*sqrt(a*c)*A*a*c^3*d*e^2 - sqrt(a*c)*B*a^2*c^2*e^3
+ (2*a*c^3*d^2*e - a^2*c^2*e^3)*A*abs(c) - (4*sqrt(a*c)*c^4*d^3 - sqrt(a*c)*a*c^3*d*e^2)*A)*arctan(sqrt(x*e +
d)/sqrt(-(a^2*c^2*d + sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 - a^3*c*e^2)*a^2*c^2))/(a^2*c^2)))/((a^3*c^4*d - sqrt(a
*c)*a^3*c^3*e)*sqrt(-c^2*d - sqrt(a*c)*c*e)) - 3/32*(2*sqrt(a*c)*B*a*c^3*d^2*e + B*a^2*c^2*d*abs(c)*e^2 + 2*sq
rt(a*c)*A*a*c^3*d*e^2 - sqrt(a*c)*B*a^2*c^2*e^3 - (2*a*c^3*d^2*e - a^2*c^2*e^3)*A*abs(c) - (4*sqrt(a*c)*c^4*d^
3 - sqrt(a*c)*a*c^3*d*e^2)*A)*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^2*d - sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 - a^3*c*
e^2)*a^2*c^2))/(a^2*c^2)))/((a^3*c^4*d + sqrt(a*c)*a^3*c^3*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)) - 1/16*(6*(x*e + d
)^(7/2)*A*c^2*d*e - 18*(x*e + d)^(5/2)*A*c^2*d^2*e + 18*(x*e + d)^(3/2)*A*c^2*d^3*e - 6*sqrt(x*e + d)*A*c^2*d^
4*e - 3*(x*e + d)^(7/2)*B*a*c*e^2 + 9*(x*e + d)^(5/2)*B*a*c*d*e^2 - 9*(x*e + d)^(3/2)*B*a*c*d^2*e^2 + 3*sqrt(x
*e + d)*B*a*c*d^3*e^2 - (x*e + d)^(5/2)*A*a*c*e^3 - 8*(x*e + d)^(3/2)*A*a*c*d*e^3 + 9*sqrt(x*e + d)*A*a*c*d^2*
e^3 - (x*e + d)^(3/2)*B*a^2*e^4 - 3*sqrt(x*e + d)*B*a^2*d*e^4 - 3*sqrt(x*e + d)*A*a^2*e^5)/(((x*e + d)^2*c - 2
*(x*e + d)*c*d + c*d^2 - a*e^2)^2*a^2*c)
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maple [B] time = 0.09, size = 1060, normalized size = 3.03
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^3,x)
[Out]
-3/8*e/(c*e^2*x^2-a*e^2)^2/a^2*(e*x+d)^(7/2)*A*c*d+3/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(7/2)*B+1/16*e^3/(c*
e^2*x^2-a*e^2)^2/a*(e*x+d)^(5/2)*A+9/8*e/(c*e^2*x^2-a*e^2)^2/a^2*(e*x+d)^(5/2)*A*c*d^2-9/16*e^2/(c*e^2*x^2-a*e
^2)^2/a*(e*x+d)^(5/2)*B*d+1/2*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(3/2)*A*d-9/8*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e*x
+d)^(3/2)*A*d^3+1/16*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(3/2)*B+9/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(3/2)*B*
d^2+3/16*e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)*A-9/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(1/2)*A*d^2+3/8*e/(c
*e^2*x^2-a*e^2)^2/a^2*c*(e*x+d)^(1/2)*A*d^4+3/16*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)*B*d-3/16*e^2/(c*e^2*x
^2-a*e^2)^2/a*(e*x+d)^(1/2)*B*d^3-3/32*e^3/a/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(
1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A+3/8*e/a^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)*A*d^2-3/16*e^2/a/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1
/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B*d+3/16*e/a^2/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*ar
ctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d-3/32*e^2/a/c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan
h((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B-3/32*e^3/a/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/
2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A+3/8*e/a^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)*A*d^2-3/16*e^2/a/(a*c*e^2)^(1/2)/((-c*d+
(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B*d-3/16*e/a^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+3/32*e^2/a/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
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} - a\right )}^{3}}\,{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)^3,x, algorithm="maxima")
[Out]
-integrate((B*x + A)*(e*x + d)^(3/2)/(c*x^2 - a)^3, x)
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mupad [B] time = 8.08, size = 7239, normalized size = 20.68
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^3,x)
[Out]
((3*(B*a*e^2 - 2*A*c*d*e)*(d + e*x)^(7/2))/(16*a^2) + ((d + e*x)^(5/2)*(A*a*e^3 - 9*B*a*d*e^2 + 18*A*c*d^2*e))
/(16*a^2) + (3*(d + e*x)^(1/2)*(A*a^2*e^5 + B*a^2*d*e^4 + 2*A*c^2*d^4*e - 3*A*a*c*d^2*e^3 - B*a*c*d^3*e^2))/(1
6*a^2*c) + ((d + e*x)^(3/2)*(B*a^2*e^4 - 18*A*c^2*d^3*e + 8*A*a*c*d*e^3 + 9*B*a*c*d^2*e^2))/(16*a^2*c))/(c^2*(
d + e*x)^4 + a^2*e^4 + c^2*d^4 + (6*c^2*d^2 - 2*a*c*e^2)*(d + e*x)^2 - (4*c^2*d^3 - 4*a*c*d*e^2)*(d + e*x) - 4
*c^2*d*(d + e*x)^3 - 2*a*c*d^2*e^2) + atan(((((3*(4096*A*a^6*c^4*e^5 + 4096*B*a^6*c^4*d*e^4 - 8192*A*a^5*c^5*d
^2*e^3))/(4096*a^6*c^2) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(B^2*a*e^5*(a^15*c^7)^(1/2) - 16*A^2*a^5*c^7*d^5
+ A^2*c*e^5*(a^15*c^7)^(1/2) + 20*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7*c^5*d^3*e^2 + 2*A*B*a^8*c^4*e^5 - 5*A^2*a^7
*c^5*d*e^4 + 3*B^2*a^8*c^4*d*e^4 + 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) - 16*A*B*a^7*c^5*d^2*
e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2))*(-(9*(B^2*a*e^5*(a^15*c^7)^(1/2) - 16*A^2*a^5*c^7*d^5 + A^2
*c*e^5*(a^15*c^7)^(1/2) + 20*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7*c^5*d^3*e^2 + 2*A*B*a^8*c^4*e^5 - 5*A^2*a^7*c^5*d
*e^4 + 3*B^2*a^8*c^4*d*e^4 + 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) - 16*A*B*a^7*c^5*d^2*e^3))/
(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 144*A^2*c^3*d^4*e^2 + 9*A^2*a^
2*c*e^6 - 36*A^2*a*c^2*d^2*e^4 + 36*B^2*a^2*c*d^2*e^4 - 144*A*B*a*c^2*d^3*e^3))/(64*a^4))*(-(9*(B^2*a*e^5*(a^1
5*c^7)^(1/2) - 16*A^2*a^5*c^7*d^5 + A^2*c*e^5*(a^15*c^7)^(1/2) + 20*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7*c^5*d^3*e^
2 + 2*A*B*a^8*c^4*e^5 - 5*A^2*a^7*c^5*d*e^4 + 3*B^2*a^8*c^4*d*e^4 + 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15
*c^7)^(1/2) - 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2)*1i - (((3*(4096*A*a^6*c^4*e
^5 + 4096*B*a^6*c^4*d*e^4 - 8192*A*a^5*c^5*d^2*e^3))/(4096*a^6*c^2) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(B^2
*a*e^5*(a^15*c^7)^(1/2) - 16*A^2*a^5*c^7*d^5 + A^2*c*e^5*(a^15*c^7)^(1/2) + 20*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7
*c^5*d^3*e^2 + 2*A*B*a^8*c^4*e^5 - 5*A^2*a^7*c^5*d*e^4 + 3*B^2*a^8*c^4*d*e^4 + 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*
d*e^4*(a^15*c^7)^(1/2) - 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2))*(-(9*(B^2*a*e^5
*(a^15*c^7)^(1/2) - 16*A^2*a^5*c^7*d^5 + A^2*c*e^5*(a^15*c^7)^(1/2) + 20*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7*c^5*d
^3*e^2 + 2*A*B*a^8*c^4*e^5 - 5*A^2*a^7*c^5*d*e^4 + 3*B^2*a^8*c^4*d*e^4 + 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*
(a^15*c^7)^(1/2) - 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2) - ((d + e*x)^(1/2)*(9*
B^2*a^3*e^6 + 144*A^2*c^3*d^4*e^2 + 9*A^2*a^2*c*e^6 - 36*A^2*a*c^2*d^2*e^4 + 36*B^2*a^2*c*d^2*e^4 - 144*A*B*a*
c^2*d^3*e^3))/(64*a^4))*(-(9*(B^2*a*e^5*(a^15*c^7)^(1/2) - 16*A^2*a^5*c^7*d^5 + A^2*c*e^5*(a^15*c^7)^(1/2) + 2
0*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7*c^5*d^3*e^2 + 2*A*B*a^8*c^4*e^5 - 5*A^2*a^7*c^5*d*e^4 + 3*B^2*a^8*c^4*d*e^4
+ 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) - 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*
c^7*e^2)))^(1/2)*1i)/((((3*(4096*A*a^6*c^4*e^5 + 4096*B*a^6*c^4*d*e^4 - 8192*A*a^5*c^5*d^2*e^3))/(4096*a^6*c^2
) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(B^2*a*e^5*(a^15*c^7)^(1/2) - 16*A^2*a^5*c^7*d^5 + A^2*c*e^5*(a^15*c^7
)^(1/2) + 20*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7*c^5*d^3*e^2 + 2*A*B*a^8*c^4*e^5 - 5*A^2*a^7*c^5*d*e^4 + 3*B^2*a^8
*c^4*d*e^4 + 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) - 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*
d^2 - a^11*c^7*e^2)))^(1/2))*(-(9*(B^2*a*e^5*(a^15*c^7)^(1/2) - 16*A^2*a^5*c^7*d^5 + A^2*c*e^5*(a^15*c^7)^(1/2
) + 20*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7*c^5*d^3*e^2 + 2*A*B*a^8*c^4*e^5 - 5*A^2*a^7*c^5*d*e^4 + 3*B^2*a^8*c^4*d
*e^4 + 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) - 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 -
a^11*c^7*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 144*A^2*c^3*d^4*e^2 + 9*A^2*a^2*c*e^6 - 36*A^2*a*c^2
*d^2*e^4 + 36*B^2*a^2*c*d^2*e^4 - 144*A*B*a*c^2*d^3*e^3))/(64*a^4))*(-(9*(B^2*a*e^5*(a^15*c^7)^(1/2) - 16*A^2*
a^5*c^7*d^5 + A^2*c*e^5*(a^15*c^7)^(1/2) + 20*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7*c^5*d^3*e^2 + 2*A*B*a^8*c^4*e^5
- 5*A^2*a^7*c^5*d*e^4 + 3*B^2*a^8*c^4*d*e^4 + 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) - 16*A*B*a
^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2) + (((3*(4096*A*a^6*c^4*e^5 + 4096*B*a^6*c^4*d*e^4
- 8192*A*a^5*c^5*d^2*e^3))/(4096*a^6*c^2) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(B^2*a*e^5*(a^15*c^7)^(1/2) -
16*A^2*a^5*c^7*d^5 + A^2*c*e^5*(a^15*c^7)^(1/2) + 20*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7*c^5*d^3*e^2 + 2*A*B*a^8*
c^4*e^5 - 5*A^2*a^7*c^5*d*e^4 + 3*B^2*a^8*c^4*d*e^4 + 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) -
16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2))*(-(9*(B^2*a*e^5*(a^15*c^7)^(1/2) - 16*A^
2*a^5*c^7*d^5 + A^2*c*e^5*(a^15*c^7)^(1/2) + 20*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7*c^5*d^3*e^2 + 2*A*B*a^8*c^4*e^
5 - 5*A^2*a^7*c^5*d*e^4 + 3*B^2*a^8*c^4*d*e^4 + 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) - 16*A*B
*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2) - ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 144*A^2*c^3
*d^4*e^2 + 9*A^2*a^2*c*e^6 - 36*A^2*a*c^2*d^2*e^4 + 36*B^2*a^2*c*d^2*e^4 - 144*A*B*a*c^2*d^3*e^3))/(64*a^4))*(
-(9*(B^2*a*e^5*(a^15*c^7)^(1/2) - 16*A^2*a^5*c^7*d^5 + A^2*c*e^5*(a^15*c^7)^(1/2) + 20*A^2*a^6*c^6*d^3*e^2 - 4
*B^2*a^7*c^5*d^3*e^2 + 2*A*B*a^8*c^4*e^5 - 5*A^2*a^7*c^5*d*e^4 + 3*B^2*a^8*c^4*d*e^4 + 16*A*B*a^6*c^6*d^4*e -
2*A*B*c*d*e^4*(a^15*c^7)^(1/2) - 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2) + (3*(9*
B^3*a^4*e^8 + 288*A^3*c^4*d^5*e^3 - 9*A^2*B*a^3*c*e^8 - 216*A^3*a*c^3*d^3*e^5 + 18*A^3*a^2*c^2*d*e^7 - 36*B^3*
a^3*c*d^2*e^6 + 216*A*B^2*a^2*c^2*d^3*e^5 + 252*A^2*B*a^2*c^2*d^2*e^6 - 90*A*B^2*a^3*c*d*e^7 - 432*A^2*B*a*c^3
*d^4*e^4))/(2048*a^6*c^2)))*(-(9*(B^2*a*e^5*(a^15*c^7)^(1/2) - 16*A^2*a^5*c^7*d^5 + A^2*c*e^5*(a^15*c^7)^(1/2)
+ 20*A^2*a^6*c^6*d^3*e^2 - 4*B^2*a^7*c^5*d^3*e^2 + 2*A*B*a^8*c^4*e^5 - 5*A^2*a^7*c^5*d*e^4 + 3*B^2*a^8*c^4*d*
e^4 + 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) - 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a
^11*c^7*e^2)))^(1/2)*2i + atan(((((3*(4096*A*a^6*c^4*e^5 + 4096*B*a^6*c^4*d*e^4 - 8192*A*a^5*c^5*d^2*e^3))/(40
96*a^6*c^2) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((9*(16*A^2*a^5*c^7*d^5 + B^2*a*e^5*(a^15*c^7)^(1/2) + A^2*c*e^5*
(a^15*c^7)^(1/2) - 20*A^2*a^6*c^6*d^3*e^2 + 4*B^2*a^7*c^5*d^3*e^2 - 2*A*B*a^8*c^4*e^5 + 5*A^2*a^7*c^5*d*e^4 -
3*B^2*a^8*c^4*d*e^4 - 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) + 16*A*B*a^7*c^5*d^2*e^3))/(4096*(
a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2))*((9*(16*A^2*a^5*c^7*d^5 + B^2*a*e^5*(a^15*c^7)^(1/2) + A^2*c*e^5*(a^15*c
^7)^(1/2) - 20*A^2*a^6*c^6*d^3*e^2 + 4*B^2*a^7*c^5*d^3*e^2 - 2*A*B*a^8*c^4*e^5 + 5*A^2*a^7*c^5*d*e^4 - 3*B^2*a
^8*c^4*d*e^4 - 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) + 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^
8*d^2 - a^11*c^7*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 144*A^2*c^3*d^4*e^2 + 9*A^2*a^2*c*e^6 - 36*A
^2*a*c^2*d^2*e^4 + 36*B^2*a^2*c*d^2*e^4 - 144*A*B*a*c^2*d^3*e^3))/(64*a^4))*((9*(16*A^2*a^5*c^7*d^5 + B^2*a*e^
5*(a^15*c^7)^(1/2) + A^2*c*e^5*(a^15*c^7)^(1/2) - 20*A^2*a^6*c^6*d^3*e^2 + 4*B^2*a^7*c^5*d^3*e^2 - 2*A*B*a^8*c
^4*e^5 + 5*A^2*a^7*c^5*d*e^4 - 3*B^2*a^8*c^4*d*e^4 - 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) + 1
6*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2)*1i - (((3*(4096*A*a^6*c^4*e^5 + 4096*B*a^6
*c^4*d*e^4 - 8192*A*a^5*c^5*d^2*e^3))/(4096*a^6*c^2) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((9*(16*A^2*a^5*c^7*d^5
+ B^2*a*e^5*(a^15*c^7)^(1/2) + A^2*c*e^5*(a^15*c^7)^(1/2) - 20*A^2*a^6*c^6*d^3*e^2 + 4*B^2*a^7*c^5*d^3*e^2 - 2
*A*B*a^8*c^4*e^5 + 5*A^2*a^7*c^5*d*e^4 - 3*B^2*a^8*c^4*d*e^4 - 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)
^(1/2) + 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2))*((9*(16*A^2*a^5*c^7*d^5 + B^2*a
*e^5*(a^15*c^7)^(1/2) + A^2*c*e^5*(a^15*c^7)^(1/2) - 20*A^2*a^6*c^6*d^3*e^2 + 4*B^2*a^7*c^5*d^3*e^2 - 2*A*B*a^
8*c^4*e^5 + 5*A^2*a^7*c^5*d*e^4 - 3*B^2*a^8*c^4*d*e^4 - 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2)
+ 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2) - ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 144
*A^2*c^3*d^4*e^2 + 9*A^2*a^2*c*e^6 - 36*A^2*a*c^2*d^2*e^4 + 36*B^2*a^2*c*d^2*e^4 - 144*A*B*a*c^2*d^3*e^3))/(64
*a^4))*((9*(16*A^2*a^5*c^7*d^5 + B^2*a*e^5*(a^15*c^7)^(1/2) + A^2*c*e^5*(a^15*c^7)^(1/2) - 20*A^2*a^6*c^6*d^3*
e^2 + 4*B^2*a^7*c^5*d^3*e^2 - 2*A*B*a^8*c^4*e^5 + 5*A^2*a^7*c^5*d*e^4 - 3*B^2*a^8*c^4*d*e^4 - 16*A*B*a^6*c^6*d
^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) + 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2)*1
i)/((((3*(4096*A*a^6*c^4*e^5 + 4096*B*a^6*c^4*d*e^4 - 8192*A*a^5*c^5*d^2*e^3))/(4096*a^6*c^2) - 64*a*c^4*d*e^2
*(d + e*x)^(1/2)*((9*(16*A^2*a^5*c^7*d^5 + B^2*a*e^5*(a^15*c^7)^(1/2) + A^2*c*e^5*(a^15*c^7)^(1/2) - 20*A^2*a^
6*c^6*d^3*e^2 + 4*B^2*a^7*c^5*d^3*e^2 - 2*A*B*a^8*c^4*e^5 + 5*A^2*a^7*c^5*d*e^4 - 3*B^2*a^8*c^4*d*e^4 - 16*A*B
*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) + 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)
))^(1/2))*((9*(16*A^2*a^5*c^7*d^5 + B^2*a*e^5*(a^15*c^7)^(1/2) + A^2*c*e^5*(a^15*c^7)^(1/2) - 20*A^2*a^6*c^6*d
^3*e^2 + 4*B^2*a^7*c^5*d^3*e^2 - 2*A*B*a^8*c^4*e^5 + 5*A^2*a^7*c^5*d*e^4 - 3*B^2*a^8*c^4*d*e^4 - 16*A*B*a^6*c^
6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) + 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2
) + ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 144*A^2*c^3*d^4*e^2 + 9*A^2*a^2*c*e^6 - 36*A^2*a*c^2*d^2*e^4 + 36*B^2*a^
2*c*d^2*e^4 - 144*A*B*a*c^2*d^3*e^3))/(64*a^4))*((9*(16*A^2*a^5*c^7*d^5 + B^2*a*e^5*(a^15*c^7)^(1/2) + A^2*c*e
^5*(a^15*c^7)^(1/2) - 20*A^2*a^6*c^6*d^3*e^2 + 4*B^2*a^7*c^5*d^3*e^2 - 2*A*B*a^8*c^4*e^5 + 5*A^2*a^7*c^5*d*e^4
- 3*B^2*a^8*c^4*d*e^4 - 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) + 16*A*B*a^7*c^5*d^2*e^3))/(409
6*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2) + (((3*(4096*A*a^6*c^4*e^5 + 4096*B*a^6*c^4*d*e^4 - 8192*A*a^5*c^5*d^2
*e^3))/(4096*a^6*c^2) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((9*(16*A^2*a^5*c^7*d^5 + B^2*a*e^5*(a^15*c^7)^(1/2) +
A^2*c*e^5*(a^15*c^7)^(1/2) - 20*A^2*a^6*c^6*d^3*e^2 + 4*B^2*a^7*c^5*d^3*e^2 - 2*A*B*a^8*c^4*e^5 + 5*A^2*a^7*c^
5*d*e^4 - 3*B^2*a^8*c^4*d*e^4 - 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) + 16*A*B*a^7*c^5*d^2*e^3
))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2))*((9*(16*A^2*a^5*c^7*d^5 + B^2*a*e^5*(a^15*c^7)^(1/2) + A^2*c*e
^5*(a^15*c^7)^(1/2) - 20*A^2*a^6*c^6*d^3*e^2 + 4*B^2*a^7*c^5*d^3*e^2 - 2*A*B*a^8*c^4*e^5 + 5*A^2*a^7*c^5*d*e^4
- 3*B^2*a^8*c^4*d*e^4 - 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) + 16*A*B*a^7*c^5*d^2*e^3))/(409
6*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2) - ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 144*A^2*c^3*d^4*e^2 + 9*A^2*a^2*c*
e^6 - 36*A^2*a*c^2*d^2*e^4 + 36*B^2*a^2*c*d^2*e^4 - 144*A*B*a*c^2*d^3*e^3))/(64*a^4))*((9*(16*A^2*a^5*c^7*d^5
+ B^2*a*e^5*(a^15*c^7)^(1/2) + A^2*c*e^5*(a^15*c^7)^(1/2) - 20*A^2*a^6*c^6*d^3*e^2 + 4*B^2*a^7*c^5*d^3*e^2 - 2
*A*B*a^8*c^4*e^5 + 5*A^2*a^7*c^5*d*e^4 - 3*B^2*a^8*c^4*d*e^4 - 16*A*B*a^6*c^6*d^4*e - 2*A*B*c*d*e^4*(a^15*c^7)
^(1/2) + 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2) + (3*(9*B^3*a^4*e^8 + 288*A^3*c^
4*d^5*e^3 - 9*A^2*B*a^3*c*e^8 - 216*A^3*a*c^3*d^3*e^5 + 18*A^3*a^2*c^2*d*e^7 - 36*B^3*a^3*c*d^2*e^6 + 216*A*B^
2*a^2*c^2*d^3*e^5 + 252*A^2*B*a^2*c^2*d^2*e^6 - 90*A*B^2*a^3*c*d*e^7 - 432*A^2*B*a*c^3*d^4*e^4))/(2048*a^6*c^2
)))*((9*(16*A^2*a^5*c^7*d^5 + B^2*a*e^5*(a^15*c^7)^(1/2) + A^2*c*e^5*(a^15*c^7)^(1/2) - 20*A^2*a^6*c^6*d^3*e^2
+ 4*B^2*a^7*c^5*d^3*e^2 - 2*A*B*a^8*c^4*e^5 + 5*A^2*a^7*c^5*d*e^4 - 3*B^2*a^8*c^4*d*e^4 - 16*A*B*a^6*c^6*d^4*
e - 2*A*B*c*d*e^4*(a^15*c^7)^(1/2) + 16*A*B*a^7*c^5*d^2*e^3))/(4096*(a^10*c^8*d^2 - a^11*c^7*e^2)))^(1/2)*2i
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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)**3,x)
[Out]
Timed out
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