\(\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)^2} \, dx\) [13]
Optimal result
Integrand size = 37, antiderivative size = 163 \[
\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)^2} \, dx=\frac {\left (C e^2-B e f+A f^2\right ) \sqrt {1-d^2 x^2}}{f \left (d^2 e^2-f^2\right ) (e+f x)}+\frac {C \arcsin (d x)}{d f^2}-\frac {\left (C d^2 e^3-2 C e f^2-A d^2 e f^2+B f^3\right ) \arctan \left (\frac {f+d^2 e x}{\sqrt {d^2 e^2-f^2} \sqrt {1-d^2 x^2}}\right )}{f^2 \left (d^2 e^2-f^2\right )^{3/2}}
\]
[Out]
C*arcsin(d*x)/d/f^2-(-A*d^2*e*f^2+C*d^2*e^3+B*f^3-2*C*e*f^2)*arctan((d^2*e*x+f)/(d^2*e^2-f^2)^(1/2)/(-d^2*x^2+
1)^(1/2))/f^2/(d^2*e^2-f^2)^(3/2)+(A*f^2-B*e*f+C*e^2)*(-d^2*x^2+1)^(1/2)/f/(d^2*e^2-f^2)/(f*x+e)
Rubi [A] (verified)
Time = 0.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1623, 1665, 858, 222, 739,
210} \[
\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)^2} \, dx=-\frac {\arctan \left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right ) \left (-A d^2 e f^2+B f^3+C d^2 e^3-2 C e f^2\right )}{f^2 \left (d^2 e^2-f^2\right )^{3/2}}+\frac {\sqrt {1-d^2 x^2} \left (A f^2-B e f+C e^2\right )}{f \left (d^2 e^2-f^2\right ) (e+f x)}+\frac {C \arcsin (d x)}{d f^2}
\]
[In]
Int[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^2),x]
[Out]
((C*e^2 - B*e*f + A*f^2)*Sqrt[1 - d^2*x^2])/(f*(d^2*e^2 - f^2)*(e + f*x)) + (C*ArcSin[d*x])/(d*f^2) - ((C*d^2*
e^3 - 2*C*e*f^2 - A*d^2*e*f^2 + B*f^3)*ArcTan[(f + d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/(f^2*(d^
2*e^2 - f^2)^(3/2))
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 222
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]
Rule 739
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]
Rule 858
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && !IGtQ[m, 0]
Rule 1623
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[P
x*(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d,
0] && EqQ[m, n] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
Rule 1665
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]
Rubi steps \begin{align*}
\text {integral}& = \int \frac {A+B x+C x^2}{(e+f x)^2 \sqrt {1-d^2 x^2}} \, dx \\ & = \frac {\left (C e^2-B e f+A f^2\right ) \sqrt {1-d^2 x^2}}{f \left (d^2 e^2-f^2\right ) (e+f x)}+\frac {\int \frac {C e+A d^2 e-B f+C \left (\frac {d^2 e^2}{f}-f\right ) x}{(e+f x) \sqrt {1-d^2 x^2}} \, dx}{d^2 e^2-f^2} \\ & = \frac {\left (C e^2-B e f+A f^2\right ) \sqrt {1-d^2 x^2}}{f \left (d^2 e^2-f^2\right ) (e+f x)}+\frac {C \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{f^2}+\frac {\left (2 C e+A d^2 e-\frac {C d^2 e^3}{f^2}-B f\right ) \int \frac {1}{(e+f x) \sqrt {1-d^2 x^2}} \, dx}{d^2 e^2-f^2} \\ & = \frac {\left (C e^2-B e f+A f^2\right ) \sqrt {1-d^2 x^2}}{f \left (d^2 e^2-f^2\right ) (e+f x)}+\frac {C \sin ^{-1}(d x)}{d f^2}-\frac {\left (2 C e+A d^2 e-\frac {C d^2 e^3}{f^2}-B f\right ) \text {Subst}\left (\int \frac {1}{-d^2 e^2+f^2-x^2} \, dx,x,\frac {f+d^2 e x}{\sqrt {1-d^2 x^2}}\right )}{d^2 e^2-f^2} \\ & = \frac {\left (C e^2-B e f+A f^2\right ) \sqrt {1-d^2 x^2}}{f \left (d^2 e^2-f^2\right ) (e+f x)}+\frac {C \sin ^{-1}(d x)}{d f^2}+\frac {\left (2 C e+A d^2 e-\frac {C d^2 e^3}{f^2}-B f\right ) \tan ^{-1}\left (\frac {f+d^2 e x}{\sqrt {d^2 e^2-f^2} \sqrt {1-d^2 x^2}}\right )}{\left (d^2 e^2-f^2\right )^{3/2}} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.10 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.21
\[
\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)^2} \, dx=\frac {\frac {f \left (C e^2+f (-B e+A f)\right ) \sqrt {1-d^2 x^2}}{(d e-f) (d e+f) (e+f x)}+\frac {2 C \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )}{d}+\frac {2 \sqrt {d^2 e^2-f^2} \left (C d^2 e^3-2 C e f^2-A d^2 e f^2+B f^3\right ) \arctan \left (\frac {\sqrt {d^2 e^2-f^2} x}{e+f x-e \sqrt {1-d^2 x^2}}\right )}{(-d e+f)^2 (d e+f)^2}}{f^2}
\]
[In]
Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^2),x]
[Out]
((f*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[1 - d^2*x^2])/((d*e - f)*(d*e + f)*(e + f*x)) + (2*C*ArcTan[(d*x)/(-1 + Sq
rt[1 - d^2*x^2])])/d + (2*Sqrt[d^2*e^2 - f^2]*(C*d^2*e^3 - 2*C*e*f^2 - A*d^2*e*f^2 + B*f^3)*ArcTan[(Sqrt[d^2*e
^2 - f^2]*x)/(e + f*x - e*Sqrt[1 - d^2*x^2])])/((-(d*e) + f)^2*(d*e + f)^2))/f^2
Maple [C] (verified)
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.66 (sec) , antiderivative size = 899, normalized size of antiderivative = 5.52
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\(\frac {\left (-A \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{3} e \,f^{3} x +C \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{3} e^{3} f x -A \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{3} e^{2} f^{2}+C \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{3} e^{4}+C \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d^{2} e^{2} f^{2} x \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}+A \,\operatorname {csgn}\left (d \right ) d \,f^{4} \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}+B \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d \,f^{4} x -B \,\operatorname {csgn}\left (d \right ) d e \,f^{3} \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}-2 C \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d e \,f^{3} x +C \,\operatorname {csgn}\left (d \right ) d \,e^{2} f^{2} \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}+C \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d^{2} e^{3} f \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}+B \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d e \,f^{3}-2 C \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d \,e^{2} f^{2}-C \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) f^{4} x \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}-C \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) e \,f^{3} \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\right ) \operatorname {csgn}\left (d \right ) \sqrt {-d x +1}\, \sqrt {d x +1}}{\sqrt {-d^{2} x^{2}+1}\, \left (d e -f \right ) d \left (d e +f \right ) \left (f x +e \right ) \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f^{3}}\) |
\(899\) |
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[In]
int((C*x^2+B*x+A)/(f*x+e)^2/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x,method=_RETURNVERBOSE)
[Out]
(-A*csgn(d)*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))*d^3*e*f^3*x+C*csgn(d)*ln
(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))*d^3*e^3*f*x-A*csgn(d)*ln(2*(d^2*e*x+(-
d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))*d^3*e^2*f^2+C*csgn(d)*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/
2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))*d^3*e^4+C*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*d^2*e^2*f^2*x*(-(
d^2*e^2-f^2)/f^2)^(1/2)+A*csgn(d)*d*f^4*(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)+B*csgn(d)*ln(2*(d^2*e*x+
(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))*d*f^4*x-B*csgn(d)*d*e*f^3*(-d^2*x^2+1)^(1/2)*(-(d^
2*e^2-f^2)/f^2)^(1/2)-2*C*csgn(d)*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))*d*
e*f^3*x+C*csgn(d)*d*e^2*f^2*(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)+C*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1
/2))*d^2*e^3*f*(-(d^2*e^2-f^2)/f^2)^(1/2)+B*csgn(d)*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2
)*f+f)/(f*x+e))*d*e*f^3-2*C*csgn(d)*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))*
d*e^2*f^2-C*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*f^4*x*(-(d^2*e^2-f^2)/f^2)^(1/2)-C*arctan(csgn(d)*d*x/(-d^2
*x^2+1)^(1/2))*e*f^3*(-(d^2*e^2-f^2)/f^2)^(1/2))*csgn(d)*(-d*x+1)^(1/2)*(d*x+1)^(1/2)/(-d^2*x^2+1)^(1/2)/(d*e-
f)/d/(d*e+f)/(f*x+e)/(-(d^2*e^2-f^2)/f^2)^(1/2)/f^3
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (155) = 310\).
Time = 16.95 (sec) , antiderivative size = 1025, normalized size of antiderivative = 6.29
\[
\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)^2} \, dx=\left [\frac {C d^{3} e^{5} f - B d^{3} e^{4} f^{2} + B d e^{2} f^{4} - A d e f^{5} + {\left (A d^{3} - C d\right )} e^{3} f^{3} - {\left (C d^{3} e^{5} + B d e^{2} f^{3} - {\left (A d^{3} + 2 \, C d\right )} e^{3} f^{2} + {\left (C d^{3} e^{4} f + B d e f^{4} - {\left (A d^{3} + 2 \, C d\right )} e^{2} f^{3}\right )} x\right )} \sqrt {-d^{2} e^{2} + f^{2}} \log \left (\frac {d^{2} e f x + f^{2} + \sqrt {-d^{2} e^{2} + f^{2}} {\left (d^{2} e x + f\right )} + {\left (\sqrt {-d^{2} e^{2} + f^{2}} \sqrt {-d x + 1} f - {\left (d^{2} e^{2} - f^{2}\right )} \sqrt {-d x + 1}\right )} \sqrt {d x + 1}}{f x + e}\right ) + {\left (C d^{3} e^{5} f - B d^{3} e^{4} f^{2} + B d e^{2} f^{4} - A d e f^{5} + {\left (A d^{3} - C d\right )} e^{3} f^{3}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + {\left (C d^{3} e^{4} f^{2} - B d^{3} e^{3} f^{3} + B d e f^{5} - A d f^{6} + {\left (A d^{3} - C d\right )} e^{2} f^{4}\right )} x - 2 \, {\left (C d^{4} e^{6} - 2 \, C d^{2} e^{4} f^{2} + C e^{2} f^{4} + {\left (C d^{4} e^{5} f - 2 \, C d^{2} e^{3} f^{3} + C e f^{5}\right )} x\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{d^{5} e^{6} f^{2} - 2 \, d^{3} e^{4} f^{4} + d e^{2} f^{6} + {\left (d^{5} e^{5} f^{3} - 2 \, d^{3} e^{3} f^{5} + d e f^{7}\right )} x}, \frac {C d^{3} e^{5} f - B d^{3} e^{4} f^{2} + B d e^{2} f^{4} - A d e f^{5} + {\left (A d^{3} - C d\right )} e^{3} f^{3} - 2 \, {\left (C d^{3} e^{5} + B d e^{2} f^{3} - {\left (A d^{3} + 2 \, C d\right )} e^{3} f^{2} + {\left (C d^{3} e^{4} f + B d e f^{4} - {\left (A d^{3} + 2 \, C d\right )} e^{2} f^{3}\right )} x\right )} \sqrt {d^{2} e^{2} - f^{2}} \arctan \left (-\frac {\sqrt {d^{2} e^{2} - f^{2}} \sqrt {d x + 1} \sqrt {-d x + 1} e - \sqrt {d^{2} e^{2} - f^{2}} {\left (f x + e\right )}}{{\left (d^{2} e^{2} - f^{2}\right )} x}\right ) + {\left (C d^{3} e^{5} f - B d^{3} e^{4} f^{2} + B d e^{2} f^{4} - A d e f^{5} + {\left (A d^{3} - C d\right )} e^{3} f^{3}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + {\left (C d^{3} e^{4} f^{2} - B d^{3} e^{3} f^{3} + B d e f^{5} - A d f^{6} + {\left (A d^{3} - C d\right )} e^{2} f^{4}\right )} x - 2 \, {\left (C d^{4} e^{6} - 2 \, C d^{2} e^{4} f^{2} + C e^{2} f^{4} + {\left (C d^{4} e^{5} f - 2 \, C d^{2} e^{3} f^{3} + C e f^{5}\right )} x\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{d^{5} e^{6} f^{2} - 2 \, d^{3} e^{4} f^{4} + d e^{2} f^{6} + {\left (d^{5} e^{5} f^{3} - 2 \, d^{3} e^{3} f^{5} + d e f^{7}\right )} x}\right ]
\]
[In]
integrate((C*x^2+B*x+A)/(f*x+e)^2/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="fricas")
[Out]
[(C*d^3*e^5*f - B*d^3*e^4*f^2 + B*d*e^2*f^4 - A*d*e*f^5 + (A*d^3 - C*d)*e^3*f^3 - (C*d^3*e^5 + B*d*e^2*f^3 - (
A*d^3 + 2*C*d)*e^3*f^2 + (C*d^3*e^4*f + B*d*e*f^4 - (A*d^3 + 2*C*d)*e^2*f^3)*x)*sqrt(-d^2*e^2 + f^2)*log((d^2*
e*f*x + f^2 + sqrt(-d^2*e^2 + f^2)*(d^2*e*x + f) + (sqrt(-d^2*e^2 + f^2)*sqrt(-d*x + 1)*f - (d^2*e^2 - f^2)*sq
rt(-d*x + 1))*sqrt(d*x + 1))/(f*x + e)) + (C*d^3*e^5*f - B*d^3*e^4*f^2 + B*d*e^2*f^4 - A*d*e*f^5 + (A*d^3 - C*
d)*e^3*f^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) + (C*d^3*e^4*f^2 - B*d^3*e^3*f^3 + B*d*e*f^5 - A*d*f^6 + (A*d^3 - C*d
)*e^2*f^4)*x - 2*(C*d^4*e^6 - 2*C*d^2*e^4*f^2 + C*e^2*f^4 + (C*d^4*e^5*f - 2*C*d^2*e^3*f^3 + C*e*f^5)*x)*arcta
n((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^5*e^6*f^2 - 2*d^3*e^4*f^4 + d*e^2*f^6 + (d^5*e^5*f^3 - 2*d^3*e
^3*f^5 + d*e*f^7)*x), (C*d^3*e^5*f - B*d^3*e^4*f^2 + B*d*e^2*f^4 - A*d*e*f^5 + (A*d^3 - C*d)*e^3*f^3 - 2*(C*d^
3*e^5 + B*d*e^2*f^3 - (A*d^3 + 2*C*d)*e^3*f^2 + (C*d^3*e^4*f + B*d*e*f^4 - (A*d^3 + 2*C*d)*e^2*f^3)*x)*sqrt(d^
2*e^2 - f^2)*arctan(-(sqrt(d^2*e^2 - f^2)*sqrt(d*x + 1)*sqrt(-d*x + 1)*e - sqrt(d^2*e^2 - f^2)*(f*x + e))/((d^
2*e^2 - f^2)*x)) + (C*d^3*e^5*f - B*d^3*e^4*f^2 + B*d*e^2*f^4 - A*d*e*f^5 + (A*d^3 - C*d)*e^3*f^3)*sqrt(d*x +
1)*sqrt(-d*x + 1) + (C*d^3*e^4*f^2 - B*d^3*e^3*f^3 + B*d*e*f^5 - A*d*f^6 + (A*d^3 - C*d)*e^2*f^4)*x - 2*(C*d^4
*e^6 - 2*C*d^2*e^4*f^2 + C*e^2*f^4 + (C*d^4*e^5*f - 2*C*d^2*e^3*f^3 + C*e*f^5)*x)*arctan((sqrt(d*x + 1)*sqrt(-
d*x + 1) - 1)/(d*x)))/(d^5*e^6*f^2 - 2*d^3*e^4*f^4 + d*e^2*f^6 + (d^5*e^5*f^3 - 2*d^3*e^3*f^5 + d*e*f^7)*x)]
Sympy [F]
\[
\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)^2} \, dx=\int \frac {A + B x + C x^{2}}{\left (e + f x\right )^{2} \sqrt {- d x + 1} \sqrt {d x + 1}}\, dx
\]
[In]
integrate((C*x**2+B*x+A)/(f*x+e)**2/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
Integral((A + B*x + C*x**2)/((e + f*x)**2*sqrt(-d*x + 1)*sqrt(d*x + 1)), x)
Maxima [F(-2)]
Exception generated. \[
\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)^2} \, dx=\text {Exception raised: ValueError}
\]
[In]
integrate((C*x^2+B*x+A)/(f*x+e)^2/(-d*x+1)^(1/2)/(d*x+1)^(1/2),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(e>0)', see `assume?` for more
details)Is e
Giac [F(-2)]
Exception generated. \[
\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)^2} \, dx=\text {Exception raised: TypeError}
\]
[In]
integrate((C*x^2+B*x+A)/(f*x+e)^2/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Mupad [B] (verification not implemented)
Time = 0.01 (sec) , antiderivative size = 10198, normalized size of antiderivative = 62.56
\[
\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)^2} \, dx=\text {Too large to display}
\]
[In]
int((A + B*x + C*x^2)/((e + f*x)^2*(1 - d*x)^(1/2)*(d*x + 1)^(1/2)),x)
[Out]
(A*d^5*e^5*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)
*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^
3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) +
(d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*2i - A*d^3*e^3*f^2*atan(((f + d*e)^(3/2)*(f - d
*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^
2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)
^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((
d*x + 1)^(1/2) - 1)^2))*2i + (4*A*f^2*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2)
- 1) + (A*d^5*e^5*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e
)^(3/2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2
- (2*d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2)
- 1) + (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((1 - d*x)^(1/2) - 1)^2*4i)/((d*x + 1)^(
1/2) - 1)^2 + (A*d^5*e^5*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(
f - d*e)^(3/2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2)
- 1)^2 - (2*d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1
)^(1/2) - 1) + (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((1 - d*x)^(1/2) - 1)^4*2i)/((d*x
+ 1)^(1/2) - 1)^4 - (4*A*f^2*((1 - d*x)^(1/2) - 1)^3*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^3
- (A*d^3*e^3*f^2*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e
)^(3/2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2
- (2*d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2)
- 1) + (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((1 - d*x)^(1/2) - 1)^2*4i)/((d*x + 1)^(
1/2) - 1)^2 + (A*d^2*e^2*f^3*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/
2)*(f - d*e)^(3/2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1
/2) - 1)^2 - (2*d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x
+ 1)^(1/2) - 1) + (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((1 - d*x)^(1/2) - 1)^3*8i)/(
(d*x + 1)^(1/2) - 1)^3 - (A*d^3*e^3*f^2*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f
+ d*e)^(3/2)*(f - d*e)^(3/2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((
d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2)
- 1))/((d*x + 1)^(1/2) - 1) + (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((1 - d*x)^(1/2) -
1)^4*2i)/((d*x + 1)^(1/2) - 1)^4 + (A*d^4*e^4*f*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2)
- 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) -
1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*
x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((1 - d*x
)^(1/2) - 1)*8i)/((d*x + 1)^(1/2) - 1) - (A*d^2*e^2*f^3*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)
^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^
(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*(
(1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((
1 - d*x)^(1/2) - 1)*8i)/((d*x + 1)^(1/2) - 1) - (A*d^4*e^4*f*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 -
d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 -
d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*
f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2
))*((1 - d*x)^(1/2) - 1)^3*8i)/((d*x + 1)^(1/2) - 1)^3 + (8*A*d*e*f*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f
- d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2)/(d^3*e^4*(f + d*e)^(3/2)*(f - d*e)^(3/2) - d*e^2*f^2*(f + d*e)^(3/2)*(
f - d*e)^(3/2) - (4*e*f^3*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1) + (4*e*
f^3*((1 - d*x)^(1/2) - 1)^3*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^3 + (2*d^3*e^4*((1 - d*x)^(
1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 + (d^3*e^4*((1 - d*x)^(1/2) - 1)^4*(f + d
*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^4 - (2*d*e^2*f^2*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f -
d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 - (4*d^2*e^3*f*((1 - d*x)^(1/2) - 1)^3*(f + d*e)^(3/2)*(f - d*e)^(3/2))/(
(d*x + 1)^(1/2) - 1)^3 - (d*e^2*f^2*((1 - d*x)^(1/2) - 1)^4*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2)
- 1)^4 + (4*d^2*e^3*f*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)) - (B*d^3*e
^3*f*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/(
(d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*
((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2*
e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*2i - (B*f^4*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i
- (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^
3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) +
(2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2
) - 1)^2))*((1 - d*x)^(1/2) - 1)*8i)/((d*x + 1)^(1/2) - 1) + (B*f^4*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i -
(((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3
*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) +
(2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2)
- 1)^2))*((1 - d*x)^(1/2) - 1)^3*8i)/((d*x + 1)^(1/2) - 1)^3 - B*d*e*f^3*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2
)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f
- (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) -
1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)
^(1/2) - 1)^2))*2i - (4*B*f*((1 - d*x)^(1/2) - 1)^3*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^3 +
(4*B*f*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1) - (B*d^2*e^2*f^2*atan(((f
+ d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/((d*x + 1)^(1/
2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*((1 - d*x)^(1
/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2*e^2*f*((1 - d
*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((1 - d*x)^(1/2) - 1)^3*8i)/((d*x + 1)^(1/2) - 1)^3 - (B*d*e*f^3*a
tan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/((d*x +
1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*((1 -
d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2*e^2*f*
((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((1 - d*x)^(1/2) - 1)^2*4i)/((d*x + 1)^(1/2) - 1)^2 - (B*d*
e*f^3*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/
((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3
*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2
*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((1 - d*x)^(1/2) - 1)^4*2i)/((d*x + 1)^(1/2) - 1)^4
+ (8*B*d*e*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 + (B*d^2*e^2*f^2*a
tan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/((d*x +
1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*((1 -
d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2*e^2*f*
((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((1 - d*x)^(1/2) - 1)*8i)/((d*x + 1)^(1/2) - 1) + (B*d^3*e^
3*f*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2)*1i)/((
d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*d^3*e^3*(
(1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (d^2*e
^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((1 - d*x)^(1/2) - 1)^2*4i)/((d*x + 1)^(1/2) - 1)^2 +
(B*d^3*e^3*f*atan(((f + d*e)^(3/2)*(f - d*e)^(3/2)*1i - (((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/
2)*1i)/((d*x + 1)^(1/2) - 1)^2)/(f^3 - d^2*e^2*f - (f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (2*
d^3*e^3*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (2*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1)
+ (d^2*e^2*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2))*((1 - d*x)^(1/2) - 1)^4*2i)/((d*x + 1)^(1/2)
- 1)^4)/(d^3*e^3*(f + d*e)^(3/2)*(f - d*e)^(3/2) + (4*f^3*((1 - d*x)^(1/2) - 1)^3*(f + d*e)^(3/2)*(f - d*e)^(3
/2))/((d*x + 1)^(1/2) - 1)^3 - d*e*f^2*(f + d*e)^(3/2)*(f - d*e)^(3/2) - (4*f^3*((1 - d*x)^(1/2) - 1)*(f + d*e
)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1) + (2*d^3*e^3*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^
(3/2))/((d*x + 1)^(1/2) - 1)^2 + (d^3*e^3*((1 - d*x)^(1/2) - 1)^4*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^
(1/2) - 1)^4 - (4*d^2*e^2*f*((1 - d*x)^(1/2) - 1)^3*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^3 +
(4*d^2*e^2*f*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1) - (2*d*e*f^2*((1 -
d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 - (d*e*f^2*((1 - d*x)^(1/2) - 1)^4*
(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^4) - ((4*C*d*e*((1 - d*x)^(1/2) - 1))/((f^2 - d^2*e^2)*
((d*x + 1)^(1/2) - 1)) - (4*C*d*e*((1 - d*x)^(1/2) - 1)^3)/((f^2 - d^2*e^2)*((d*x + 1)^(1/2) - 1)^3) + (8*C*d^
2*e^2*((1 - d*x)^(1/2) - 1)^2)/(f*(f^2 - d^2*e^2)*((d*x + 1)^(1/2) - 1)^2))/(d^2*e + (4*d*f*((1 - d*x)^(1/2) -
1))/((d*x + 1)^(1/2) - 1) - (4*d*f*((1 - d*x)^(1/2) - 1)^3)/((d*x + 1)^(1/2) - 1)^3 + (2*d^2*e*((1 - d*x)^(1/
2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (d^2*e*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4) + (4*C*atan(((((
(1 - d*x)^(1/2) - 1)*((2097152*(288*e^3*f^11 - 6*d^10*e^13*f - 912*d^2*e^5*f^9 + 1048*d^4*e^7*f^7 - 532*d^6*e^
9*f^5 + 112*d^8*e^11*f^3))/(d*f^2*(d*f^13 - 4*d^3*e^2*f^11 + 6*d^5*e^4*f^9 - 4*d^7*e^6*f^7 + d^9*e^8*f^5)) - (
33554432*(20*d^2*e*f^21 - 103*d^4*e^3*f^19 + 215*d^6*e^5*f^17 - 230*d^8*e^7*f^15 + 130*d^10*e^9*f^13 - 35*d^12
*e^11*f^11 + 3*d^14*e^13*f^9))/(d^5*f^10*(d*f^13 - 4*d^3*e^2*f^11 + 6*d^5*e^4*f^9 - 4*d^7*e^6*f^7 + d^9*e^8*f^
5)) + (8388608*(72*e*f^17 - 452*d^2*e^3*f^15 + 1024*d^4*e^5*f^13 - 1106*d^6*e^7*f^11 + 597*d^8*e^9*f^9 - 144*d
^10*e^11*f^7 + 9*d^12*e^13*f^5))/(d^3*f^6*(d*f^13 - 4*d^3*e^2*f^11 + 6*d^5*e^4*f^9 - 4*d^7*e^6*f^7 + d^9*e^8*f
^5))))/((d*x + 1)^(1/2) - 1) - (33554432*(7*d^2*e^2*f^19 - 35*d^4*e^4*f^17 + 70*d^6*e^6*f^15 - 70*d^8*e^8*f^13
+ 35*d^10*e^10*f^11 - 7*d^12*e^12*f^9))/(d^5*f^10*(f^12 - 4*d^2*e^2*f^10 + 6*d^4*e^4*f^8 - 4*d^6*e^6*f^6 + d^
8*e^8*f^4)) + (2097152*(112*e^4*f^9 + 28*d^8*e^12*f - 336*d^2*e^6*f^7 + 364*d^4*e^8*f^5 - 168*d^6*e^10*f^3))/(
d*f^2*(f^12 - 4*d^2*e^2*f^10 + 6*d^4*e^4*f^8 - 4*d^6*e^6*f^6 + d^8*e^8*f^4)) + (8388608*(28*e^2*f^15 - 168*d^2
*e^4*f^13 + 364*d^4*e^6*f^11 - 371*d^6*e^8*f^9 + 182*d^8*e^10*f^7 - 35*d^10*e^12*f^5))/(d^3*f^6*(f^12 - 4*d^2*
e^2*f^10 + 6*d^4*e^4*f^8 - 4*d^6*e^6*f^6 + d^8*e^8*f^4)))*(d^4*f^14 - 4*d^6*e^2*f^12 + 6*d^8*e^4*f^10 - 4*d^10
*e^6*f^8 + d^12*e^8*f^6))/(67108864*e*f^12 + 37748736*d^12*e^13 - 268435456*d^2*e^3*f^10 + 536870912*d^4*e^5*f
^8 - 637534208*d^6*e^7*f^6 + 469762048*d^8*e^9*f^4 - 201326592*d^10*e^11*f^2)))/(d*f^2) + (log(16*f^15 - 9*d^1
4*e^14*f - (16*f^15*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - 92*d^2*e^2*f^13 + 236*d^4*e^4*f^11 - 35
2*d^6*e^6*f^9 + 329*d^8*e^8*f^7 - 191*d^10*e^10*f^5 + 63*d^12*e^12*f^3 + 16*f^6*(f + d*e)^(9/2)*(f - d*e)^(9/2
) + 12*d^6*e^6*(f + d*e)^(9/2)*(f - d*e)^(9/2) + 15*d^12*e^12*(f + d*e)^(3/2)*(f - d*e)^(3/2) - (6*d^15*e^15*(
(1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (16*d*e*f^14*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (92*
d^2*e^2*f^13*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (236*d^4*e^4*f^11*((1 - d*x)^(1/2) - 1)^2)/((d
*x + 1)^(1/2) - 1)^2 + (352*d^6*e^6*f^9*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (329*d^8*e^8*f^7*((
1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (191*d^10*e^10*f^5*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2)
- 1)^2 - (63*d^12*e^12*f^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (16*f^6*((1 - d*x)^(1/2) - 1)^2*
(f + d*e)^(9/2)*(f - d*e)^(9/2))/((d*x + 1)^(1/2) - 1)^2 - 24*d^2*e^2*f^10*(f + d*e)^(3/2)*(f - d*e)^(3/2) + 1
20*d^4*e^4*f^8*(f + d*e)^(3/2)*(f - d*e)^(3/2) - 228*d^6*e^6*f^6*(f + d*e)^(3/2)*(f - d*e)^(3/2) + 4*d^2*e^2*f
^4*(f + d*e)^(9/2)*(f - d*e)^(9/2) + 207*d^8*e^8*f^4*(f + d*e)^(3/2)*(f - d*e)^(3/2) - 28*d^4*e^4*f^2*(f + d*e
)^(9/2)*(f - d*e)^(9/2) - 90*d^10*e^10*f^2*(f + d*e)^(3/2)*(f - d*e)^(3/2) - (88*d^3*e^3*f^12*((1 - d*x)^(1/2)
- 1))/((d*x + 1)^(1/2) - 1) + (216*d^5*e^5*f^10*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) - (308*d^7*e^7*f
^8*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (274*d^9*e^9*f^6*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1
) - (150*d^11*e^11*f^4*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (46*d^13*e^13*f^2*((1 - d*x)^(1/2) - 1))
/((d*x + 1)^(1/2) - 1) + (9*d^14*e^14*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (48*d^6*e^6*((1 - d
*x)^(1/2) - 1)^2*(f + d*e)^(9/2)*(f - d*e)^(9/2))/((d*x + 1)^(1/2) - 1)^2 + (45*d^12*e^12*((1 - d*x)^(1/2) - 1
)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 + (376*d^3*e^3*f^9*((1 - d*x)^(1/2) - 1)*(f + d*e
)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1) - (688*d^5*e^5*f^7*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d
*e)^(3/2))/((d*x + 1)^(1/2) - 1) + (612*d^7*e^7*f^5*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d
*x + 1)^(1/2) - 1) - (152*d^3*e^3*f^3*((1 - d*x)^(1/2) - 1)*(f + d*e)^(9/2)*(f - d*e)^(9/2))/((d*x + 1)^(1/2)
- 1) - (264*d^9*e^9*f^3*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1) - (80*d*e
*f^11*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1) + (96*d*e*f^5*((1 - d*x)^(1
/2) - 1)*(f + d*e)^(9/2)*(f - d*e)^(9/2))/((d*x + 1)^(1/2) - 1) - (136*d^2*e^2*f^10*((1 - d*x)^(1/2) - 1)^2*(f
+ d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 + (560*d^4*e^4*f^8*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3
/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 - (912*d^6*e^6*f^6*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d
*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 + (156*d^2*e^2*f^4*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(9/2)*(f - d*e)^(9/2))
/((d*x + 1)^(1/2) - 1)^2 + (733*d^8*e^8*f^4*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1
)^(1/2) - 1)^2 - (172*d^4*e^4*f^2*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(9/2)*(f - d*e)^(9/2))/((d*x + 1)^(1/2) -
1)^2 - (290*d^10*e^10*f^2*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 + (
56*d^5*e^5*f*((1 - d*x)^(1/2) - 1)*(f + d*e)^(9/2)*(f - d*e)^(9/2))/((d*x + 1)^(1/2) - 1) + (44*d^11*e^11*f*((
1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1))*(C*d^2*e^3 - 2*C*e*f^2))/(f^2*(f +
d*e)^(3/2)*(f - d*e)^(3/2)) + (C*e*log(9*d^14*e^14*f - 16*f^15 + (16*f^15*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)
^(1/2) - 1)^2 + 92*d^2*e^2*f^13 - 236*d^4*e^4*f^11 + 352*d^6*e^6*f^9 - 329*d^8*e^8*f^7 + 191*d^10*e^10*f^5 - 6
3*d^12*e^12*f^3 + 16*f^6*(f + d*e)^(9/2)*(f - d*e)^(9/2) + 12*d^6*e^6*(f + d*e)^(9/2)*(f - d*e)^(9/2) + 15*d^1
2*e^12*(f + d*e)^(3/2)*(f - d*e)^(3/2) + (6*d^15*e^15*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) - (16*d*e*f
^14*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) - (92*d^2*e^2*f^13*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2)
- 1)^2 + (236*d^4*e^4*f^11*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (352*d^6*e^6*f^9*((1 - d*x)^(1/2
) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (329*d^8*e^8*f^7*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 - (191*d
^10*e^10*f^5*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (63*d^12*e^12*f^3*((1 - d*x)^(1/2) - 1)^2)/((d
*x + 1)^(1/2) - 1)^2 - (16*f^6*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(9/2)*(f - d*e)^(9/2))/((d*x + 1)^(1/2) - 1)^
2 - 24*d^2*e^2*f^10*(f + d*e)^(3/2)*(f - d*e)^(3/2) + 120*d^4*e^4*f^8*(f + d*e)^(3/2)*(f - d*e)^(3/2) - 228*d^
6*e^6*f^6*(f + d*e)^(3/2)*(f - d*e)^(3/2) + 4*d^2*e^2*f^4*(f + d*e)^(9/2)*(f - d*e)^(9/2) + 207*d^8*e^8*f^4*(f
+ d*e)^(3/2)*(f - d*e)^(3/2) - 28*d^4*e^4*f^2*(f + d*e)^(9/2)*(f - d*e)^(9/2) - 90*d^10*e^10*f^2*(f + d*e)^(3
/2)*(f - d*e)^(3/2) + (88*d^3*e^3*f^12*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) - (216*d^5*e^5*f^10*((1 -
d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (308*d^7*e^7*f^8*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) - (274*
d^9*e^9*f^6*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (150*d^11*e^11*f^4*((1 - d*x)^(1/2) - 1))/((d*x + 1
)^(1/2) - 1) - (46*d^13*e^13*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) - (9*d^14*e^14*f*((1 - d*x)^(1/2
) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (48*d^6*e^6*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(9/2)*(f - d*e)^(9/2))/((d*x
+ 1)^(1/2) - 1)^2 + (45*d^12*e^12*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) -
1)^2 + (376*d^3*e^3*f^9*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1) - (688*d
^5*e^5*f^7*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1) + (612*d^7*e^7*f^5*((1
- d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1) - (152*d^3*e^3*f^3*((1 - d*x)^(1/2)
- 1)*(f + d*e)^(9/2)*(f - d*e)^(9/2))/((d*x + 1)^(1/2) - 1) - (264*d^9*e^9*f^3*((1 - d*x)^(1/2) - 1)*(f + d*e)
^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1) - (80*d*e*f^11*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(
3/2))/((d*x + 1)^(1/2) - 1) + (96*d*e*f^5*((1 - d*x)^(1/2) - 1)*(f + d*e)^(9/2)*(f - d*e)^(9/2))/((d*x + 1)^(1
/2) - 1) - (136*d^2*e^2*f^10*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2
+ (560*d^4*e^4*f^8*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 - (912*d^6
*e^6*f^6*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 + (156*d^2*e^2*f^4*(
(1 - d*x)^(1/2) - 1)^2*(f + d*e)^(9/2)*(f - d*e)^(9/2))/((d*x + 1)^(1/2) - 1)^2 + (733*d^8*e^8*f^4*((1 - d*x)^
(1/2) - 1)^2*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 - (172*d^4*e^4*f^2*((1 - d*x)^(1/2) - 1)
^2*(f + d*e)^(9/2)*(f - d*e)^(9/2))/((d*x + 1)^(1/2) - 1)^2 - (290*d^10*e^10*f^2*((1 - d*x)^(1/2) - 1)^2*(f +
d*e)^(3/2)*(f - d*e)^(3/2))/((d*x + 1)^(1/2) - 1)^2 + (56*d^5*e^5*f*((1 - d*x)^(1/2) - 1)*(f + d*e)^(9/2)*(f -
d*e)^(9/2))/((d*x + 1)^(1/2) - 1) + (44*d^11*e^11*f*((1 - d*x)^(1/2) - 1)*(f + d*e)^(3/2)*(f - d*e)^(3/2))/((
d*x + 1)^(1/2) - 1))*(2*f^2 - d^2*e^2))/(f^2*(f + d*e)^(3/2)*(f - d*e)^(3/2))