3.12.3 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(b x+c x^2)^3} \, dx\)
Optimal. Leaf size=363 \[ -\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac {d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (7 b^2 e (5 A e+4 B d)-12 b c d (7 A e+2 B d)+48 A c^2 d^2\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (-b^2 c e (A e+8 B d)-12 b c^2 d (A e+2 B d)+48 A c^3 d^2-3 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{5/2}}+\frac {\sqrt {d+e x} \left (b c d^2 \left (-b c (11 A e+6 B d)+12 A c^2 d+2 b^2 B e\right )+x \left (-A b^3 c e^3+b^2 c^2 d e (14 A e+11 B d)-12 b c^3 d^2 (3 A e+B d)+24 A c^4 d^3-3 b^4 B e^3\right )\right )}{4 b^4 c^2 \left (b x+c x^2\right )} \]
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Rubi [A] time = 0.83, antiderivative size = 363, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used =
{818, 826, 1166, 208} \begin {gather*} \frac {\sqrt {d+e x} \left (x \left (b^2 c^2 d e (14 A e+11 B d)-A b^3 c e^3-12 b c^3 d^2 (3 A e+B d)+24 A c^4 d^3-3 b^4 B e^3\right )+b c d^2 \left (-b c (11 A e+6 B d)+12 A c^2 d+2 b^2 B e\right )\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac {(c d-b e)^{3/2} \left (-b^2 c e (A e+8 B d)-12 b c^2 d (A e+2 B d)+48 A c^3 d^2-3 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{5/2}}-\frac {d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (7 b^2 e (5 A e+4 B d)-12 b c d (7 A e+2 B d)+48 A c^2 d^2\right )}{4 b^5}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^3,x]
[Out]
-((d + e*x)^(5/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(2*b^2*c*(b*x + c*x^2)^2) + (Sqrt[d +
e*x]*(b*c*d^2*(12*A*c^2*d + 2*b^2*B*e - b*c*(6*B*d + 11*A*e)) + (24*A*c^4*d^3 - 3*b^4*B*e^3 - A*b^3*c*e^3 - 1
2*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(11*B*d + 14*A*e))*x))/(4*b^4*c^2*(b*x + c*x^2)) - (d^(3/2)*(48*A*c^2*
d^2 + 7*b^2*e*(4*B*d + 5*A*e) - 12*b*c*d*(2*B*d + 7*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + ((c*d - b*
e)^(3/2)*(48*A*c^3*d^2 - 3*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + A*e) - b^2*c*e*(8*B*d + A*e))*ArcTanh[(Sqrt[c]*Sqrt
[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(5/2))
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 818
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 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]
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\int \frac {(d+e x)^{3/2} \left (-\frac {1}{2} d \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )-\frac {1}{2} e \left (2 A c^2 d-3 b^2 B e-b c (B d+A e)\right ) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2 c}\\ &=-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} c^2 d^2 \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right )+\frac {1}{4} e \left (24 A c^4 d^3+3 b^4 B e^3+b^3 c e^2 (2 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+10 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 c^2}\\ &=-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} c^2 d^2 e \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right )-\frac {1}{4} d e \left (24 A c^4 d^3+3 b^4 B e^3+b^3 c e^2 (2 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+10 A e)\right )+\frac {1}{4} e \left (24 A c^4 d^3+3 b^4 B e^3+b^3 c e^2 (2 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+10 A e)\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4 c^2}\\ &=-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac {\left ((c d-b e)^2 \left (48 A c^3 d^2-3 b^3 B e^2-12 b c^2 d (2 B d+A e)-b^2 c e (8 B d+A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 c^2}+\frac {\left (c d^2 \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5}\\ &=-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac {d^{3/2} \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (48 A c^3 d^2-3 b^3 B e^2-12 b c^2 d (2 B d+A e)-b^2 c e (8 B d+A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 4.05, size = 554, normalized size = 1.53 \begin {gather*} \frac {-\frac {210 c (d+e x)^{9/2} \left (b^2 e (5 A e+4 B d)-3 b c d (5 A e+2 B d)+12 A c^2 d^2\right )}{b^2 d (b e-c d)}+\frac {(b+c x) \left ((b+c x) \left (105 c^{9/2} (c d-b e)^2 \left (-2 d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+\frac {2}{15} d \sqrt {d+e x} \left (23 d^2+11 d e x+3 e^2 x^2\right )+\frac {2}{7} (d+e x)^{7/2}\right ) \left (7 b^2 e (5 A e+4 B d)-12 b c d (7 A e+2 B d)+48 A c^2 d^2\right )-2 c^2 d^2 \left (-b^2 c e (A e+8 B d)-12 b c^2 d (A e+2 B d)+48 A c^3 d^2-3 b^3 B e^2\right ) \left (7 (c d-b e) \left (5 (c d-b e) \left (\sqrt {c} \sqrt {d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )\right )+3 c^{5/2} (d+e x)^{5/2}\right )+15 c^{7/2} (d+e x)^{7/2}\right )\right )-210 b c^{11/2} (d+e x)^{9/2} \left (b^3 e^2 (5 A e+4 B d)-11 b^2 c d e (2 A e+B d)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )\right )}{b^4 c^{9/2} d (c d-b e)^2}-\frac {210 (d+e x)^{9/2} (5 A b e-8 A c d+4 b B d)}{b d x}-\frac {420 A (d+e x)^{9/2}}{x^2}}{840 b d (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^3,x]
[Out]
((-210*c*(12*A*c^2*d^2 - 3*b*c*d*(2*B*d + 5*A*e) + b^2*e*(4*B*d + 5*A*e))*(d + e*x)^(9/2))/(b^2*d*(-(c*d) + b*
e)) - (420*A*(d + e*x)^(9/2))/x^2 - (210*(4*b*B*d - 8*A*c*d + 5*A*b*e)*(d + e*x)^(9/2))/(b*d*x) + ((b + c*x)*(
-210*b*c^(11/2)*(-24*A*c^3*d^3 - 11*b^2*c*d*e*(B*d + 2*A*e) + 12*b*c^2*d^2*(B*d + 3*A*e) + b^3*e^2*(4*B*d + 5*
A*e))*(d + e*x)^(9/2) + (b + c*x)*(105*c^(9/2)*(c*d - b*e)^2*(48*A*c^2*d^2 + 7*b^2*e*(4*B*d + 5*A*e) - 12*b*c*
d*(2*B*d + 7*A*e))*((2*(d + e*x)^(7/2))/7 + (2*d*Sqrt[d + e*x]*(23*d^2 + 11*d*e*x + 3*e^2*x^2))/15 - 2*d^(7/2)
*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) - 2*c^2*d^2*(48*A*c^3*d^2 - 3*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + A*e) - b^2*c*e*
(8*B*d + A*e))*(15*c^(7/2)*(d + e*x)^(7/2) + 7*(c*d - b*e)*(3*c^(5/2)*(d + e*x)^(5/2) + 5*(c*d - b*e)*(Sqrt[c]
*Sqrt[d + e*x]*(4*c*d - 3*b*e + c*e*x) - 3*(c*d - b*e)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])
)))))/(b^4*c^(9/2)*d*(c*d - b*e)^2))/(840*b*d*(b + c*x)^2)
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IntegrateAlgebraic [B] time = 2.67, size = 1116, normalized size = 3.07 \begin {gather*} \frac {-24 A c^5 e \sqrt {d+e x} d^6+12 b B c^4 e \sqrt {d+e x} d^6+72 A c^5 e (d+e x)^{3/2} d^5-36 b B c^4 e (d+e x)^{3/2} d^5+72 A b c^4 e^2 \sqrt {d+e x} d^5-29 b^2 B c^3 e^2 \sqrt {d+e x} d^5-72 A c^5 e (d+e x)^{5/2} d^4+36 b B c^4 e (d+e x)^{5/2} d^4-180 A b c^4 e^2 (d+e x)^{3/2} d^4+69 b^2 B c^3 e^2 (d+e x)^{3/2} d^4-73 A b^2 c^3 e^3 \sqrt {d+e x} d^4+19 b^3 B c^2 e^3 \sqrt {d+e x} d^4+24 A c^5 e (d+e x)^{7/2} d^3-12 b B c^4 e (d+e x)^{7/2} d^3+144 A b c^4 e^2 (d+e x)^{5/2} d^3-51 b^2 B c^3 e^2 (d+e x)^{5/2} d^3+148 A b^2 c^3 e^3 (d+e x)^{3/2} d^3-32 b^3 B c^2 e^3 (d+e x)^{3/2} d^3+26 A b^3 c^2 e^4 \sqrt {d+e x} d^3+b^4 B c e^4 \sqrt {d+e x} d^3-36 A b c^4 e^2 (d+e x)^{7/2} d^2+11 b^2 B c^3 e^2 (d+e x)^{7/2} d^2-85 A b^2 c^3 e^3 (d+e x)^{5/2} d^2+11 b^3 B c^2 e^3 (d+e x)^{5/2} d^2-42 A b^3 c^2 e^4 (d+e x)^{3/2} d^2-7 b^4 B c e^4 (d+e x)^{3/2} d^2-3 b^5 B e^5 \sqrt {d+e x} d^2-A b^4 c e^5 \sqrt {d+e x} d^2+10 A b^2 c^3 e^3 (d+e x)^{7/2} d+2 b^3 B c^2 e^3 (d+e x)^{7/2} d+13 A b^3 c^2 e^4 (d+e x)^{5/2} d+11 b^4 B c e^4 (d+e x)^{5/2} d+6 b^5 B e^5 (d+e x)^{3/2} d+2 A b^4 c e^5 (d+e x)^{3/2} d+A b^3 c^2 e^4 (d+e x)^{7/2}-5 b^4 B c e^4 (d+e x)^{7/2}-3 b^5 B e^5 (d+e x)^{5/2}-A b^4 c e^5 (d+e x)^{5/2}}{4 b^4 c^2 e^2 x^2 (c d-b e-c (d+e x))^2}+\frac {\left (-3 B e^4 b^5-A c e^4 b^4-2 B c d e^3 b^4-10 A c^2 d e^3 b^3-11 B c^2 d^2 e^2 b^3+71 A c^3 d^2 e^2 b^2+40 B c^3 d^3 e b^2-24 B c^4 d^4 b-108 A c^4 d^3 e b+48 A c^5 d^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {b e-c d} \sqrt {d+e x}}{c d-b e}\right )}{4 b^5 c^{5/2} \sqrt {b e-c d}}+\frac {\left (-48 A c^2 d^{7/2}+24 b B c d^{7/2}-28 b^2 B e d^{5/2}+84 A b c e d^{5/2}-35 A b^2 e^2 d^{3/2}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^3,x]
[Out]
(12*b*B*c^4*d^6*e*Sqrt[d + e*x] - 24*A*c^5*d^6*e*Sqrt[d + e*x] - 29*b^2*B*c^3*d^5*e^2*Sqrt[d + e*x] + 72*A*b*c
^4*d^5*e^2*Sqrt[d + e*x] + 19*b^3*B*c^2*d^4*e^3*Sqrt[d + e*x] - 73*A*b^2*c^3*d^4*e^3*Sqrt[d + e*x] + b^4*B*c*d
^3*e^4*Sqrt[d + e*x] + 26*A*b^3*c^2*d^3*e^4*Sqrt[d + e*x] - 3*b^5*B*d^2*e^5*Sqrt[d + e*x] - A*b^4*c*d^2*e^5*Sq
rt[d + e*x] - 36*b*B*c^4*d^5*e*(d + e*x)^(3/2) + 72*A*c^5*d^5*e*(d + e*x)^(3/2) + 69*b^2*B*c^3*d^4*e^2*(d + e*
x)^(3/2) - 180*A*b*c^4*d^4*e^2*(d + e*x)^(3/2) - 32*b^3*B*c^2*d^3*e^3*(d + e*x)^(3/2) + 148*A*b^2*c^3*d^3*e^3*
(d + e*x)^(3/2) - 7*b^4*B*c*d^2*e^4*(d + e*x)^(3/2) - 42*A*b^3*c^2*d^2*e^4*(d + e*x)^(3/2) + 6*b^5*B*d*e^5*(d
+ e*x)^(3/2) + 2*A*b^4*c*d*e^5*(d + e*x)^(3/2) + 36*b*B*c^4*d^4*e*(d + e*x)^(5/2) - 72*A*c^5*d^4*e*(d + e*x)^(
5/2) - 51*b^2*B*c^3*d^3*e^2*(d + e*x)^(5/2) + 144*A*b*c^4*d^3*e^2*(d + e*x)^(5/2) + 11*b^3*B*c^2*d^2*e^3*(d +
e*x)^(5/2) - 85*A*b^2*c^3*d^2*e^3*(d + e*x)^(5/2) + 11*b^4*B*c*d*e^4*(d + e*x)^(5/2) + 13*A*b^3*c^2*d*e^4*(d +
e*x)^(5/2) - 3*b^5*B*e^5*(d + e*x)^(5/2) - A*b^4*c*e^5*(d + e*x)^(5/2) - 12*b*B*c^4*d^3*e*(d + e*x)^(7/2) + 2
4*A*c^5*d^3*e*(d + e*x)^(7/2) + 11*b^2*B*c^3*d^2*e^2*(d + e*x)^(7/2) - 36*A*b*c^4*d^2*e^2*(d + e*x)^(7/2) + 2*
b^3*B*c^2*d*e^3*(d + e*x)^(7/2) + 10*A*b^2*c^3*d*e^3*(d + e*x)^(7/2) - 5*b^4*B*c*e^4*(d + e*x)^(7/2) + A*b^3*c
^2*e^4*(d + e*x)^(7/2))/(4*b^4*c^2*e^2*x^2*(c*d - b*e - c*(d + e*x))^2) + ((-24*b*B*c^4*d^4 + 48*A*c^5*d^4 + 4
0*b^2*B*c^3*d^3*e - 108*A*b*c^4*d^3*e - 11*b^3*B*c^2*d^2*e^2 + 71*A*b^2*c^3*d^2*e^2 - 2*b^4*B*c*d*e^3 - 10*A*b
^3*c^2*d*e^3 - 3*b^5*B*e^4 - A*b^4*c*e^4)*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(4*b
^5*c^(5/2)*Sqrt[-(c*d) + b*e]) + ((24*b*B*c*d^(7/2) - 48*A*c^2*d^(7/2) - 28*b^2*B*d^(5/2)*e + 84*A*b*c*d^(5/2)
*e - 35*A*b^2*d^(3/2)*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5)
________________________________________________________________________________________
fricas [B] time = 34.35, size = 3378, normalized size = 9.31
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^3,x, algorithm="fricas")
[Out]
[-1/8*(((24*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 - 15*A*b*c^5)*d^2*e - (5*B*b^3*c^3 + 11*A*b^2*c^4)*d*e^2
- (3*B*b^4*c^2 + A*b^3*c^3)*e^3)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3 - 15*A*b^2*c^4)*d^2*
e - (5*B*b^4*c^2 + 11*A*b^3*c^3)*d*e^2 - (3*B*b^5*c + A*b^4*c^2)*e^3)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3
- 4*(4*B*b^4*c^2 - 15*A*b^3*c^3)*d^2*e - (5*B*b^5*c + 11*A*b^4*c^2)*d*e^2 - (3*B*b^6 + A*b^5*c)*e^3)*x^2)*sqrt
((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - ((35*A*b^2*c^4*
d*e^2 - 24*(B*b*c^5 - 2*A*c^6)*d^3 + 28*(B*b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c^3*d*e^2 - 24*(B*b^2
*c^4 - 2*A*b*c^5)*d^3 + 28*(B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (35*A*b^4*c^2*d*e^2 - 24*(B*b^3*c^3 - 2*A*b^
2*c^4)*d^3 + 28*(B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2
*(2*A*b^4*c^2*d^3 + (12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (11*B*b^3*c^3 - 36*A*b^2*c^4)*d^2*e - 2*(B*b^4*c^2 + 5*A
*b^3*c^3)*d*e^2 + (5*B*b^5*c - A*b^4*c^2)*e^3)*x^3 + (18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (17*B*b^4*c^2 - 55*A*
b^3*c^3)*d^2*e + 4*(B*b^5*c - 4*A*b^4*c^2)*d*e^2 + (3*B*b^6 + A*b^5*c)*e^3)*x^2 + (13*A*b^4*c^2*d^2*e + 4*(B*b
^4*c^2 - 2*A*b^3*c^3)*d^3)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), -1/8*(2*((24*(B*b*c^
5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 - 15*A*b*c^5)*d^2*e - (5*B*b^3*c^3 + 11*A*b^2*c^4)*d*e^2 - (3*B*b^4*c^2 + A*
b^3*c^3)*e^3)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3 - 15*A*b^2*c^4)*d^2*e - (5*B*b^4*c^2 +
11*A*b^3*c^3)*d*e^2 - (3*B*b^5*c + A*b^4*c^2)*e^3)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 4*(4*B*b^4*c^2 -
15*A*b^3*c^3)*d^2*e - (5*B*b^5*c + 11*A*b^4*c^2)*d*e^2 - (3*B*b^6 + A*b^5*c)*e^3)*x^2)*sqrt(-(c*d - b*e)/c)*ar
ctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) - ((35*A*b^2*c^4*d*e^2 - 24*(B*b*c^5 - 2*A*c^6)*d^3 +
28*(B*b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c^3*d*e^2 - 24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 28*(B*b^3*c^3
- 3*A*b^2*c^4)*d^2*e)*x^3 + (35*A*b^4*c^2*d*e^2 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 + 28*(B*b^4*c^2 - 3*A*b^3*
c^3)*d^2*e)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c^2*d^3 + (12*(B*b^2*c^4 -
2*A*b*c^5)*d^3 - (11*B*b^3*c^3 - 36*A*b^2*c^4)*d^2*e - 2*(B*b^4*c^2 + 5*A*b^3*c^3)*d*e^2 + (5*B*b^5*c - A*b^4*
c^2)*e^3)*x^3 + (18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (17*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e + 4*(B*b^5*c - 4*A*b^4
*c^2)*d*e^2 + (3*B*b^6 + A*b^5*c)*e^3)*x^2 + (13*A*b^4*c^2*d^2*e + 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3)*x)*sqrt(e*
x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), 1/8*(2*((35*A*b^2*c^4*d*e^2 - 24*(B*b*c^5 - 2*A*c^6)*d^3
+ 28*(B*b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c^3*d*e^2 - 24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 28*(B*b^3*c
^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (35*A*b^4*c^2*d*e^2 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 + 28*(B*b^4*c^2 - 3*A*b^
3*c^3)*d^2*e)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - ((24*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 -
15*A*b*c^5)*d^2*e - (5*B*b^3*c^3 + 11*A*b^2*c^4)*d*e^2 - (3*B*b^4*c^2 + A*b^3*c^3)*e^3)*x^4 + 2*(24*(B*b^2*c^
4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3 - 15*A*b^2*c^4)*d^2*e - (5*B*b^4*c^2 + 11*A*b^3*c^3)*d*e^2 - (3*B*b^5*c +
A*b^4*c^2)*e^3)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 4*(4*B*b^4*c^2 - 15*A*b^3*c^3)*d^2*e - (5*B*b^5*c +
11*A*b^4*c^2)*d*e^2 - (3*B*b^6 + A*b^5*c)*e^3)*x^2)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x
+ d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - 2*(2*A*b^4*c^2*d^3 + (12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (11*B*b^3*c^3
- 36*A*b^2*c^4)*d^2*e - 2*(B*b^4*c^2 + 5*A*b^3*c^3)*d*e^2 + (5*B*b^5*c - A*b^4*c^2)*e^3)*x^3 + (18*(B*b^3*c^3
- 2*A*b^2*c^4)*d^3 - (17*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e + 4*(B*b^5*c - 4*A*b^4*c^2)*d*e^2 + (3*B*b^6 + A*b^5*
c)*e^3)*x^2 + (13*A*b^4*c^2*d^2*e + 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^
3*x^3 + b^7*c^2*x^2), -1/4*(((24*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 - 15*A*b*c^5)*d^2*e - (5*B*b^3*c^3 +
11*A*b^2*c^4)*d*e^2 - (3*B*b^4*c^2 + A*b^3*c^3)*e^3)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3
- 15*A*b^2*c^4)*d^2*e - (5*B*b^4*c^2 + 11*A*b^3*c^3)*d*e^2 - (3*B*b^5*c + A*b^4*c^2)*e^3)*x^3 + (24*(B*b^3*c^
3 - 2*A*b^2*c^4)*d^3 - 4*(4*B*b^4*c^2 - 15*A*b^3*c^3)*d^2*e - (5*B*b^5*c + 11*A*b^4*c^2)*d*e^2 - (3*B*b^6 + A*
b^5*c)*e^3)*x^2)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) - ((35*A*b^2*c
^4*d*e^2 - 24*(B*b*c^5 - 2*A*c^6)*d^3 + 28*(B*b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c^3*d*e^2 - 24*(B*
b^2*c^4 - 2*A*b*c^5)*d^3 + 28*(B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (35*A*b^4*c^2*d*e^2 - 24*(B*b^3*c^3 - 2*A
*b^2*c^4)*d^3 + 28*(B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (2*A*b^4*
c^2*d^3 + (12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (11*B*b^3*c^3 - 36*A*b^2*c^4)*d^2*e - 2*(B*b^4*c^2 + 5*A*b^3*c^3)*
d*e^2 + (5*B*b^5*c - A*b^4*c^2)*e^3)*x^3 + (18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (17*B*b^4*c^2 - 55*A*b^3*c^3)*d
^2*e + 4*(B*b^5*c - 4*A*b^4*c^2)*d*e^2 + (3*B*b^6 + A*b^5*c)*e^3)*x^2 + (13*A*b^4*c^2*d^2*e + 4*(B*b^4*c^2 - 2
*A*b^3*c^3)*d^3)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2)]
________________________________________________________________________________________
giac [B] time = 0.34, size = 1047, normalized size = 2.88
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^3,x, algorithm="giac")
[Out]
-1/4*(24*B*b*c*d^4 - 48*A*c^2*d^4 - 28*B*b^2*d^3*e + 84*A*b*c*d^3*e - 35*A*b^2*d^2*e^2)*arctan(sqrt(x*e + d)/s
qrt(-d))/(b^5*sqrt(-d)) + 1/4*(24*B*b*c^4*d^4 - 48*A*c^5*d^4 - 40*B*b^2*c^3*d^3*e + 108*A*b*c^4*d^3*e + 11*B*b
^3*c^2*d^2*e^2 - 71*A*b^2*c^3*d^2*e^2 + 2*B*b^4*c*d*e^3 + 10*A*b^3*c^2*d*e^3 + 3*B*b^5*e^4 + A*b^4*c*e^4)*arct
an(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^5*c^2) - 1/4*(12*(x*e + d)^(7/2)*B*b*c^4*d^3*
e - 24*(x*e + d)^(7/2)*A*c^5*d^3*e - 36*(x*e + d)^(5/2)*B*b*c^4*d^4*e + 72*(x*e + d)^(5/2)*A*c^5*d^4*e + 36*(x
*e + d)^(3/2)*B*b*c^4*d^5*e - 72*(x*e + d)^(3/2)*A*c^5*d^5*e - 12*sqrt(x*e + d)*B*b*c^4*d^6*e + 24*sqrt(x*e +
d)*A*c^5*d^6*e - 11*(x*e + d)^(7/2)*B*b^2*c^3*d^2*e^2 + 36*(x*e + d)^(7/2)*A*b*c^4*d^2*e^2 + 51*(x*e + d)^(5/2
)*B*b^2*c^3*d^3*e^2 - 144*(x*e + d)^(5/2)*A*b*c^4*d^3*e^2 - 69*(x*e + d)^(3/2)*B*b^2*c^3*d^4*e^2 + 180*(x*e +
d)^(3/2)*A*b*c^4*d^4*e^2 + 29*sqrt(x*e + d)*B*b^2*c^3*d^5*e^2 - 72*sqrt(x*e + d)*A*b*c^4*d^5*e^2 - 2*(x*e + d)
^(7/2)*B*b^3*c^2*d*e^3 - 10*(x*e + d)^(7/2)*A*b^2*c^3*d*e^3 - 11*(x*e + d)^(5/2)*B*b^3*c^2*d^2*e^3 + 85*(x*e +
d)^(5/2)*A*b^2*c^3*d^2*e^3 + 32*(x*e + d)^(3/2)*B*b^3*c^2*d^3*e^3 - 148*(x*e + d)^(3/2)*A*b^2*c^3*d^3*e^3 - 1
9*sqrt(x*e + d)*B*b^3*c^2*d^4*e^3 + 73*sqrt(x*e + d)*A*b^2*c^3*d^4*e^3 + 5*(x*e + d)^(7/2)*B*b^4*c*e^4 - (x*e
+ d)^(7/2)*A*b^3*c^2*e^4 - 11*(x*e + d)^(5/2)*B*b^4*c*d*e^4 - 13*(x*e + d)^(5/2)*A*b^3*c^2*d*e^4 + 7*(x*e + d)
^(3/2)*B*b^4*c*d^2*e^4 + 42*(x*e + d)^(3/2)*A*b^3*c^2*d^2*e^4 - sqrt(x*e + d)*B*b^4*c*d^3*e^4 - 26*sqrt(x*e +
d)*A*b^3*c^2*d^3*e^4 + 3*(x*e + d)^(5/2)*B*b^5*e^5 + (x*e + d)^(5/2)*A*b^4*c*e^5 - 6*(x*e + d)^(3/2)*B*b^5*d*e
^5 - 2*(x*e + d)^(3/2)*A*b^4*c*d*e^5 + 3*sqrt(x*e + d)*B*b^5*d^2*e^5 + sqrt(x*e + d)*A*b^4*c*d^2*e^5)/(((x*e +
d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2*b^4*c^2)
________________________________________________________________________________________
maple [B] time = 0.12, size = 1218, normalized size = 3.36
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^3,x)
[Out]
21*e*d^(5/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c+15/4*e^3/b/(c*e*x+b*e)^2*(e*x+d)^(1/2)*B*d^2+5/2*e^3/b^2/(
(b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d+11/4*e^2/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*
x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^2+6/b^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*
B*d^4*c^2-12/b^5*c^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^4+1/4*e^4/(c*e*x+b*e)
^2/c*(e*x+d)^(1/2)*B*d-3/4*e^5*b/(c*e*x+b*e)^2/c^2*(e*x+d)^(1/2)*B+1/2*e^3/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d+1
5/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(1/2)*A*d+1/e*d^4/b^3/x^2*(e*x+d)^(1/2)*B+1/4*e^4/b/c/((b*e-c*d)*c)^(1/2)*arct
an((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A-1/e*d^3/b^3/x^2*(e*x+d)^(3/2)*B+11/4*d^3/b^3/x^2*(e*x+d)^(1/2)*A-7*e
*d^(5/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*B+1/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A-13/4*d^2/b^3/x^2*(e*x+d)
^(3/2)*A+6*d^(7/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*B*c+3/4*e^4/c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)
/((b*e-c*d)*c)^(1/2)*c)*B-1/4*e^5/(c*e*x+b*e)^2/c*(e*x+d)^(1/2)*A-5/4*e^4/(c*e*x+b*e)^2/c*(e*x+d)^(3/2)*B-35/4
*e^2*d^(3/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*A-12*d^(7/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c^2+11/4*e^2
/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(3/2)*B*d^2-2*e/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d^3*c^2-3*e/b^4/(c*e*x+b*e)^2*c
^3*(e*x+d)^(1/2)*A*d^4-39/4*e^3/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(1/2)*A*d^2+37/4*e^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(1/
2)*A*d^3*c^2+3*e/b^4/(c*e*x+b*e)^2*c^3*(e*x+d)^(3/2)*A*d^3-21/4*e^2/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(1/2)*B*d^3+2*
e/b^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)*B*d^4*c^2-71/4*e^2/b^3*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)
*c)^(1/2)*c)*A*d^2+27*e/b^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^3*c^2+1/2*e^3/
b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d-10*e/b^3*c/((b*e-c*d)*c)^(1/2)*arctan(
(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^3-3/e*d^4/b^4/x^2*c*(e*x+d)^(1/2)*A-23/4*e^2/b^3/(c*e*x+b*e)^2*(e*x+d
)^(3/2)*A*c^2*d^2+3/e*d^3/b^4/x^2*c*(e*x+d)^(3/2)*A+5/2*e^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*c*d
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?
________________________________________________________________________________________
mupad [B] time = 6.40, size = 11072, normalized size = 30.50
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^3,x)
[Out]
atan(((((64*A*b^13*c^4*d*e^6 + 192*B*b^14*c^3*d*e^6 - 1536*A*b^10*c^7*d^4*e^3 + 3072*A*b^11*c^6*d^3*e^4 - 1600
*A*b^12*c^5*d^2*e^5 + 768*B*b^11*c^6*d^4*e^3 - 1088*B*b^12*c^5*d^3*e^4 + 128*B*b^13*c^4*d^2*e^5)/(64*b^12*c^3)
- ((64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(d + e*x)^(1/2)*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e
^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 2
1*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*
b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*
b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^
3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2
304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3
- 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*
B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*
e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^
5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2) - (
(d + e*x)^(1/2)*(9*B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 4608*A^2*c^10*d^8*e^2 + 28896*A^2*b^2*c^8*d^6*e^4 - 2217
6*A^2*b^3*c^7*d^5*e^5 + 8330*A^2*b^4*c^6*d^4*e^6 - 1204*A^2*b^5*c^5*d^3*e^7 - 42*A^2*b^6*c^4*d^2*e^8 + 1152*B^
2*b^2*c^8*d^8*e^2 - 3264*B^2*b^3*c^7*d^7*e^3 + 2912*B^2*b^4*c^6*d^6*e^4 - 784*B^2*b^5*c^5*d^5*e^5 + 105*B^2*b^
6*c^4*d^4*e^6 - 196*B^2*b^7*c^3*d^3*e^7 + 70*B^2*b^8*c^2*d^2*e^8 + 12*B^2*b^9*c*d*e^9 - 18432*A^2*b*c^9*d^7*e^
3 + 20*A^2*b^7*c^3*d*e^9 + 6*A*B*b^9*c*e^10 - 4608*A*B*b*c^9*d^8*e^2 + 64*A*B*b^8*c^2*d*e^9 + 15744*A*B*b^2*c^
8*d^7*e^3 - 19488*A*B*b^3*c^7*d^6*e^4 + 10304*A*B*b^4*c^6*d^5*e^5 - 2170*A*B*b^5*c^5*d^4*e^6 + 504*A*B*b^6*c^4
*d^3*e^7 - 364*A*B*b^7*c^3*d^2*e^8))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*
B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5
*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6
*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7
- 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A
*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2)*1i - (((64*A*b^13*c^4*d*e^6 + 192*B*b^14*c^
3*d*e^6 - 1536*A*b^10*c^7*d^4*e^3 + 3072*A*b^11*c^6*d^3*e^4 - 1600*A*b^12*c^5*d^2*e^5 + 768*B*b^11*c^6*d^4*e^3
- 1088*B*b^12*c^5*d^3*e^4 + 128*B*b^13*c^4*d^2*e^5)/(64*b^12*c^3) + ((64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(
d + e*x)^(1/2)*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7
*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*
e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^
6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*
b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^
3*d^2*e^5)/(64*b^10*c^5))^(1/2))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*
b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4
*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e +
21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6
720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b
^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2) + ((d + e*x)^(1/2)*(9*B^2*b^10*e^10 + A^2*b^8*c
^2*e^10 + 4608*A^2*c^10*d^8*e^2 + 28896*A^2*b^2*c^8*d^6*e^4 - 22176*A^2*b^3*c^7*d^5*e^5 + 8330*A^2*b^4*c^6*d^4
*e^6 - 1204*A^2*b^5*c^5*d^3*e^7 - 42*A^2*b^6*c^4*d^2*e^8 + 1152*B^2*b^2*c^8*d^8*e^2 - 3264*B^2*b^3*c^7*d^7*e^3
+ 2912*B^2*b^4*c^6*d^6*e^4 - 784*B^2*b^5*c^5*d^5*e^5 + 105*B^2*b^6*c^4*d^4*e^6 - 196*B^2*b^7*c^3*d^3*e^7 + 70
*B^2*b^8*c^2*d^2*e^8 + 12*B^2*b^9*c*d*e^9 - 18432*A^2*b*c^9*d^7*e^3 + 20*A^2*b^7*c^3*d*e^9 + 6*A*B*b^9*c*e^10
- 4608*A*B*b*c^9*d^8*e^2 + 64*A*B*b^8*c^2*d*e^9 + 15744*A*B*b^2*c^8*d^7*e^3 - 19488*A*B*b^3*c^7*d^6*e^4 + 1030
4*A*B*b^4*c^6*d^5*e^5 - 2170*A*B*b^5*c^5*d^4*e^6 + 504*A*B*b^6*c^4*d^3*e^7 - 364*A*B*b^7*c^3*d^2*e^8))/(8*b^8*
c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2
+ 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105
*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e
^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d
*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5
)/(64*b^10*c^5))^(1/2)*1i)/((((64*A*b^13*c^4*d*e^6 + 192*B*b^14*c^3*d*e^6 - 1536*A*b^10*c^7*d^4*e^3 + 3072*A*b
^11*c^6*d^3*e^4 - 1600*A*b^12*c^5*d^2*e^5 + 768*B*b^11*c^6*d^4*e^3 - 1088*B*b^12*c^5*d^3*e^4 + 128*B*b^13*c^4*
d^2*e^5)/(64*b^12*c^3) - ((64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(d + e*x)^(1/2)*(-(9*B^2*b^9*e^7 - 2304*A^2*c
^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A
^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c
^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*
A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1
960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2))/(8*b^8*c^3)
)*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 58
80*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2
*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 +
1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6
+ 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(6
4*b^10*c^5))^(1/2) - ((d + e*x)^(1/2)*(9*B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 4608*A^2*c^10*d^8*e^2 + 28896*A^2*
b^2*c^8*d^6*e^4 - 22176*A^2*b^3*c^7*d^5*e^5 + 8330*A^2*b^4*c^6*d^4*e^6 - 1204*A^2*b^5*c^5*d^3*e^7 - 42*A^2*b^6
*c^4*d^2*e^8 + 1152*B^2*b^2*c^8*d^8*e^2 - 3264*B^2*b^3*c^7*d^7*e^3 + 2912*B^2*b^4*c^6*d^6*e^4 - 784*B^2*b^5*c^
5*d^5*e^5 + 105*B^2*b^6*c^4*d^4*e^6 - 196*B^2*b^7*c^3*d^3*e^7 + 70*B^2*b^8*c^2*d^2*e^8 + 12*B^2*b^9*c*d*e^9 -
18432*A^2*b*c^9*d^7*e^3 + 20*A^2*b^7*c^3*d*e^9 + 6*A*B*b^9*c*e^10 - 4608*A*B*b*c^9*d^8*e^2 + 64*A*B*b^8*c^2*d*
e^9 + 15744*A*B*b^2*c^8*d^7*e^3 - 19488*A*B*b^3*c^7*d^6*e^4 + 10304*A*B*b^4*c^6*d^5*e^5 - 2170*A*B*b^5*c^5*d^4
*e^6 + 504*A*B*b^6*c^4*d^3*e^7 - 364*A*B*b^7*c^3*d^2*e^8))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 +
A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^
5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^
5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8
*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b
^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2) + (((64*A*b^13*c^4*d*
e^6 + 192*B*b^14*c^3*d*e^6 - 1536*A*b^10*c^7*d^4*e^3 + 3072*A*b^11*c^6*d^3*e^4 - 1600*A*b^12*c^5*d^2*e^5 + 768
*B*b^11*c^6*d^4*e^3 - 1088*B*b^12*c^5*d^3*e^4 + 128*B*b^13*c^4*d^2*e^5)/(64*b^12*c^3) + ((64*b^11*c^5*e^3 - 12
8*b^10*c^6*d*e^2)*(d + e*x)^(1/2)*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7
- 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 7
84*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*
c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*
c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e
^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7
*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e
^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 806
4*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 +
6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*
d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2) + ((d + e*x)^(1/2)*(9*B^2*b^
10*e^10 + A^2*b^8*c^2*e^10 + 4608*A^2*c^10*d^8*e^2 + 28896*A^2*b^2*c^8*d^6*e^4 - 22176*A^2*b^3*c^7*d^5*e^5 + 8
330*A^2*b^4*c^6*d^4*e^6 - 1204*A^2*b^5*c^5*d^3*e^7 - 42*A^2*b^6*c^4*d^2*e^8 + 1152*B^2*b^2*c^8*d^8*e^2 - 3264*
B^2*b^3*c^7*d^7*e^3 + 2912*B^2*b^4*c^6*d^6*e^4 - 784*B^2*b^5*c^5*d^5*e^5 + 105*B^2*b^6*c^4*d^4*e^6 - 196*B^2*b
^7*c^3*d^3*e^7 + 70*B^2*b^8*c^2*d^2*e^8 + 12*B^2*b^9*c*d*e^9 - 18432*A^2*b*c^9*d^7*e^3 + 20*A^2*b^7*c^3*d*e^9
+ 6*A*B*b^9*c*e^10 - 4608*A*B*b*c^9*d^8*e^2 + 64*A*B*b^8*c^2*d*e^9 + 15744*A*B*b^2*c^8*d^7*e^3 - 19488*A*B*b^3
*c^7*d^6*e^4 + 10304*A*B*b^4*c^6*d^5*e^5 - 2170*A*B*b^5*c^5*d^4*e^6 + 504*A*B*b^6*c^4*d^3*e^7 - 364*A*B*b^7*c^
3*d^2*e^8))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A
^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^
4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 +
21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e
+ 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*
A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2) - (252*B^3*b^11*d^3*e^11 - 55296*A^3*c^11*d^11*e^3 - 694656*A^3*b^2*
c^9*d^9*e^5 + 844992*A^3*b^3*c^8*d^8*e^6 - 579096*A^3*b^4*c^7*d^7*e^7 + 212436*A^3*b^5*c^6*d^6*e^8 - 31282*A^3
*b^6*c^5*d^5*e^9 - 1877*A^3*b^7*c^4*d^4*e^10 + 616*A^3*b^8*c^3*d^3*e^11 + 35*A^3*b^9*c^2*d^2*e^12 + 6912*B^3*b
^3*c^8*d^11*e^3 - 25920*B^3*b^4*c^7*d^10*e^4 + 33408*B^3*b^5*c^6*d^9*e^5 - 16808*B^3*b^6*c^5*d^8*e^6 + 5180*B^
3*b^7*c^4*d^7*e^7 - 4816*B^3*b^8*c^3*d^6*e^8 + 1672*B^3*b^9*c^2*d^5*e^9 + 315*A*B^2*b^11*d^2*e^12 + 304128*A^3
*b*c^10*d^10*e^4 + 120*B^3*b^10*c*d^4*e^10 - 41472*A*B^2*b^2*c^9*d^11*e^3 + 179712*A*B^2*b^3*c^8*d^10*e^4 - 29
3184*A*B^2*b^4*c^7*d^9*e^5 + 220464*A*B^2*b^5*c^6*d^8*e^6 - 83076*A*B^2*b^6*c^5*d^7*e^7 + 31899*A*B^2*b^7*c^4*
d^6*e^8 - 18012*A*B^2*b^8*c^3*d^5*e^9 + 3522*A*B^2*b^9*c^2*d^4*e^10 - 407808*A^2*B*b^2*c^9*d^10*e^4 + 800064*A
^2*B*b^3*c^8*d^9*e^5 - 790704*A^2*B*b^4*c^7*d^8*e^6 + 413028*A^2*B*b^5*c^6*d^7*e^7 - 122283*A^2*B*b^6*c^5*d^6*
e^8 + 36402*A^2*B*b^7*c^4*d^5*e^9 - 13617*A^2*B*b^8*c^3*d^4*e^10 + 1764*A^2*B*b^9*c^2*d^3*e^11 - 168*A*B^2*b^1
0*c*d^3*e^11 + 82944*A^2*B*b*c^10*d^11*e^3 + 210*A^2*B*b^10*c*d^2*e^12)/(32*b^12*c^3)))*(-(9*B^2*b^9*e^7 - 230
4*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 -
1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^
2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e
+ 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*
e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/(64*b^10*c^5))^(1/2)*2i -
log((d^2*e^3*(b*e - c*d)^2*(35*A^3*b^7*c^2*e^7 - 55296*A^3*c^9*d^7 + 6912*B^3*b^3*c^6*d^7 + 315*A*B^2*b^9*e^7
+ 252*B^3*b^9*d*e^6 - 252288*A^3*b^2*c^7*d^5*e^2 + 146880*A^3*b^3*c^6*d^4*e^3 - 33048*A^3*b^4*c^5*d^3*e^4 - 54
0*A^3*b^5*c^4*d^2*e^5 + 2304*B^3*b^5*c^4*d^5*e^2 - 104*B^3*b^6*c^3*d^4*e^3 + 2668*B^3*b^7*c^2*d^3*e^4 + 82944*
A^2*B*b*c^8*d^7 + 210*A^2*B*b^8*c*e^7 + 193536*A^3*b*c^8*d^6*e - 41472*A*B^2*b^2*c^7*d^7 + 686*A^3*b^6*c^3*d*e
^6 - 12096*B^3*b^4*c^5*d^6*e + 624*B^3*b^8*c*d^2*e^5 - 58176*A*B^2*b^4*c^5*d^5*e^2 + 7344*A*B^2*b^5*c^4*d^4*e^
3 - 10212*A*B^2*b^6*c^3*d^3*e^4 + 4131*A*B^2*b^7*c^2*d^2*e^5 + 233280*A^2*B*b^3*c^6*d^5*e^2 - 82224*A^2*B*b^4*
c^5*d^4*e^3 + 15300*A^2*B*b^5*c^4*d^3*e^4 - 9459*A^2*B*b^6*c^3*d^2*e^5 + 462*A*B^2*b^8*c*d*e^6 + 96768*A*B^2*b
^3*c^6*d^6*e - 241920*A^2*B*b^2*c^7*d^6*e + 2184*A^2*B*b^7*c^2*d*e^6))/(64*b^12*c^3) - (((((d*e^3*(b*e - c*d)*
(24*A*c^3*d^2 + 3*B*b^3*e^2 + A*b^2*c*e^2 - 12*B*b*c^2*d^2 - 24*A*b*c^2*d*e + 5*B*b^2*c*d*e))/b^2 - b^2*c^2*e^
2*(b*e - 2*c*d)*(d + e*x)^(1/2)*((d^3*(35*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 28*B*b^2*d*e - 84*A*b*c*d*
e)^2)/b^10)^(1/2))*((d^3*(35*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 28*B*b^2*d*e - 84*A*b*c*d*e)^2)/b^10)^(
1/2))/8 - ((d + e*x)^(1/2)*(9*B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 4608*A^2*c^10*d^8*e^2 + 28896*A^2*b^2*c^8*d^6
*e^4 - 22176*A^2*b^3*c^7*d^5*e^5 + 8330*A^2*b^4*c^6*d^4*e^6 - 1204*A^2*b^5*c^5*d^3*e^7 - 42*A^2*b^6*c^4*d^2*e^
8 + 1152*B^2*b^2*c^8*d^8*e^2 - 3264*B^2*b^3*c^7*d^7*e^3 + 2912*B^2*b^4*c^6*d^6*e^4 - 784*B^2*b^5*c^5*d^5*e^5 +
105*B^2*b^6*c^4*d^4*e^6 - 196*B^2*b^7*c^3*d^3*e^7 + 70*B^2*b^8*c^2*d^2*e^8 + 12*B^2*b^9*c*d*e^9 - 18432*A^2*b
*c^9*d^7*e^3 + 20*A^2*b^7*c^3*d*e^9 + 6*A*B*b^9*c*e^10 - 4608*A*B*b*c^9*d^8*e^2 + 64*A*B*b^8*c^2*d*e^9 + 15744
*A*B*b^2*c^8*d^7*e^3 - 19488*A*B*b^3*c^7*d^6*e^4 + 10304*A*B*b^4*c^6*d^5*e^5 - 2170*A*B*b^5*c^5*d^4*e^6 + 504*
A*B*b^6*c^4*d^3*e^7 - 364*A*B*b^7*c^3*d^2*e^8))/(8*b^8*c^3))*((d^3*(35*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2
+ 28*B*b^2*d*e - 84*A*b*c*d*e)^2)/b^10)^(1/2))/8)*((36*A^2*c^4*d^7 + 9*B^2*b^2*c^2*d^7 + (1225*A^2*b^4*d^3*e^
4)/64 + (49*B^2*b^4*d^5*e^2)/4 + (651*A^2*b^2*c^2*d^5*e^2)/4 + (245*A*B*b^4*d^4*e^3)/8 - 126*A^2*b*c^3*d^6*e -
21*B^2*b^3*c*d^6*e - (735*A^2*b^3*c*d^4*e^3)/8 - 36*A*B*b*c^3*d^7 + 105*A*B*b^2*c^2*d^6*e - (399*A*B*b^3*c*d^
5*e^2)/4)/b^10)^(1/2) + log((d^2*e^3*(b*e - c*d)^2*(35*A^3*b^7*c^2*e^7 - 55296*A^3*c^9*d^7 + 6912*B^3*b^3*c^6*
d^7 + 315*A*B^2*b^9*e^7 + 252*B^3*b^9*d*e^6 - 252288*A^3*b^2*c^7*d^5*e^2 + 146880*A^3*b^3*c^6*d^4*e^3 - 33048*
A^3*b^4*c^5*d^3*e^4 - 540*A^3*b^5*c^4*d^2*e^5 + 2304*B^3*b^5*c^4*d^5*e^2 - 104*B^3*b^6*c^3*d^4*e^3 + 2668*B^3*
b^7*c^2*d^3*e^4 + 82944*A^2*B*b*c^8*d^7 + 210*A^2*B*b^8*c*e^7 + 193536*A^3*b*c^8*d^6*e - 41472*A*B^2*b^2*c^7*d
^7 + 686*A^3*b^6*c^3*d*e^6 - 12096*B^3*b^4*c^5*d^6*e + 624*B^3*b^8*c*d^2*e^5 - 58176*A*B^2*b^4*c^5*d^5*e^2 + 7
344*A*B^2*b^5*c^4*d^4*e^3 - 10212*A*B^2*b^6*c^3*d^3*e^4 + 4131*A*B^2*b^7*c^2*d^2*e^5 + 233280*A^2*B*b^3*c^6*d^
5*e^2 - 82224*A^2*B*b^4*c^5*d^4*e^3 + 15300*A^2*B*b^5*c^4*d^3*e^4 - 9459*A^2*B*b^6*c^3*d^2*e^5 + 462*A*B^2*b^8
*c*d*e^6 + 96768*A*B^2*b^3*c^6*d^6*e - 241920*A^2*B*b^2*c^7*d^6*e + 2184*A^2*B*b^7*c^2*d*e^6))/(64*b^12*c^3) -
(((((d*e^3*(b*e - c*d)*(24*A*c^3*d^2 + 3*B*b^3*e^2 + A*b^2*c*e^2 - 12*B*b*c^2*d^2 - 24*A*b*c^2*d*e + 5*B*b^2*
c*d*e))/b^2 + b^2*c^2*e^2*(b*e - 2*c*d)*(d + e*x)^(1/2)*((d^3*(35*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 28
*B*b^2*d*e - 84*A*b*c*d*e)^2)/b^10)^(1/2))*((d^3*(35*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 28*B*b^2*d*e -
84*A*b*c*d*e)^2)/b^10)^(1/2))/8 + ((d + e*x)^(1/2)*(9*B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 4608*A^2*c^10*d^8*e^2
+ 28896*A^2*b^2*c^8*d^6*e^4 - 22176*A^2*b^3*c^7*d^5*e^5 + 8330*A^2*b^4*c^6*d^4*e^6 - 1204*A^2*b^5*c^5*d^3*e^7
- 42*A^2*b^6*c^4*d^2*e^8 + 1152*B^2*b^2*c^8*d^8*e^2 - 3264*B^2*b^3*c^7*d^7*e^3 + 2912*B^2*b^4*c^6*d^6*e^4 - 7
84*B^2*b^5*c^5*d^5*e^5 + 105*B^2*b^6*c^4*d^4*e^6 - 196*B^2*b^7*c^3*d^3*e^7 + 70*B^2*b^8*c^2*d^2*e^8 + 12*B^2*b
^9*c*d*e^9 - 18432*A^2*b*c^9*d^7*e^3 + 20*A^2*b^7*c^3*d*e^9 + 6*A*B*b^9*c*e^10 - 4608*A*B*b*c^9*d^8*e^2 + 64*A
*B*b^8*c^2*d*e^9 + 15744*A*B*b^2*c^8*d^7*e^3 - 19488*A*B*b^3*c^7*d^6*e^4 + 10304*A*B*b^4*c^6*d^5*e^5 - 2170*A*
B*b^5*c^5*d^4*e^6 + 504*A*B*b^6*c^4*d^3*e^7 - 364*A*B*b^7*c^3*d^2*e^8))/(8*b^8*c^3))*((d^3*(35*A*b^2*e^2 + 48*
A*c^2*d^2 - 24*B*b*c*d^2 + 28*B*b^2*d*e - 84*A*b*c*d*e)^2)/b^10)^(1/2))/8)*((2304*A^2*c^4*d^7 + 576*B^2*b^2*c^
2*d^7 + 1225*A^2*b^4*d^3*e^4 + 784*B^2*b^4*d^5*e^2 + 10416*A^2*b^2*c^2*d^5*e^2 + 1960*A*B*b^4*d^4*e^3 - 8064*A
^2*b*c^3*d^6*e - 1344*B^2*b^3*c*d^6*e - 5880*A^2*b^3*c*d^4*e^3 - 2304*A*B*b*c^3*d^7 + 6720*A*B*b^2*c^2*d^6*e -
6384*A*B*b^3*c*d^5*e^2)/(64*b^10))^(1/2) + (((d + e*x)^(7/2)*(A*b^3*c*e^4 - 5*B*b^4*e^4 + 24*A*c^4*d^3*e - 36
*A*b*c^3*d^2*e^2 + 10*A*b^2*c^2*d*e^3 + 11*B*b^2*c^2*d^2*e^2 - 12*B*b*c^3*d^3*e + 2*B*b^3*c*d*e^3))/(4*b^4*c)
- ((d + e*x)^(5/2)*(3*B*b^5*e^5 + A*b^4*c*e^5 + 72*A*c^5*d^4*e - 144*A*b*c^4*d^3*e^2 - 13*A*b^3*c^2*d*e^4 + 85
*A*b^2*c^3*d^2*e^3 + 51*B*b^2*c^3*d^3*e^2 - 11*B*b^3*c^2*d^2*e^3 - 36*B*b*c^4*d^4*e - 11*B*b^4*c*d*e^4))/(4*b^
4*c^2) - ((d + e*x)^(1/2)*(24*A*c^5*d^6*e + 3*B*b^5*d^2*e^5 - 72*A*b*c^4*d^5*e^2 + A*b^4*c*d^2*e^5 - B*b^4*c*d
^3*e^4 + 73*A*b^2*c^3*d^4*e^3 - 26*A*b^3*c^2*d^3*e^4 + 29*B*b^2*c^3*d^5*e^2 - 19*B*b^3*c^2*d^4*e^3 - 12*B*b*c^
4*d^6*e))/(4*b^4*c^2) + ((d + e*x)^(3/2)*(72*A*c^5*d^5*e + 6*B*b^5*d*e^5 - 180*A*b*c^4*d^4*e^2 - 7*B*b^4*c*d^2
*e^4 + 148*A*b^2*c^3*d^3*e^3 - 42*A*b^3*c^2*d^2*e^4 + 69*B*b^2*c^3*d^4*e^2 - 32*B*b^3*c^2*d^3*e^3 + 2*A*b^4*c*
d*e^5 - 36*B*b*c^4*d^5*e))/(4*b^4*c^2))/(c^2*(d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) -
(4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b
*c*d^3*e)
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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)**(7/2)/(c*x**2+b*x)**3,x)
[Out]
Timed out
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