3.12 \(\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx\)

Optimal. Leaf size=122 \[ \frac {\left (A f^2-B e f+C e^2\right ) \tan ^{-1}\left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right )}{f^2 \sqrt {d^2 e^2-f^2}}-\frac {\sin ^{-1}(d x) (C e-B f)}{d f^2}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f} \]

[Out]

-(-B*f+C*e)*arcsin(d*x)/d/f^2+(A*f^2-B*e*f+C*e^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)^(1/2)-C*(-d^2*x^2+1)^(1/2)/d^2/f

________________________________________________________________________________________

Rubi [A]  time = 0.28, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1609, 1654, 844, 216, 725, 204} \[ \frac {\left (A f^2-B e f+C e^2\right ) \tan ^{-1}\left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right )}{f^2 \sqrt {d^2 e^2-f^2}}-\frac {\sin ^{-1}(d x) (C e-B f)}{d f^2}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)),x]

[Out]

-((C*Sqrt[1 - d^2*x^2])/(d^2*f)) - ((C*e - B*f)*ArcSin[d*x])/(d*f^2) + ((C*e^2 - B*e*f + A*f^2)*ArcTan[(f + d^
2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/(f^2*Sqrt[d^2*e^2 - f^2])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 216

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 725

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 844

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 1609

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 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin {align*} \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx &=\int \frac {A+B x+C x^2}{(e+f x) \sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}-\frac {\int \frac {-A d^2 f^2+d^2 f (C e-B f) x}{(e+f x) \sqrt {1-d^2 x^2}} \, dx}{d^2 f^2}\\ &=-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}-\frac {(C e-B f) \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{f^2}+\frac {\left (C e^2-B e f+A f^2\right ) \int \frac {1}{(e+f x) \sqrt {1-d^2 x^2}} \, dx}{f^2}\\ &=-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}-\frac {(C e-B f) \sin ^{-1}(d x)}{d f^2}-\frac {\left (C e^2-B e f+A f^2\right ) \operatorname {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 )}{f^2}\\ &=-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}-\frac {(C e-B f) \sin ^{-1}(d x)}{d f^2}+\frac {\left (C e^2-B e f+A f^2\right ) \tan ^{-1}\left (\frac {f+d^2 e x}{\sqrt {d^2 e^2-f^2} \sqrt {1-d^2 x^2}}\right )}{f^2 \sqrt {d^2 e^2-f^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 117, normalized size = 0.96 \[ \frac {\frac {\left (f (A f-B e)+C e^2\right ) \tan ^{-1}\left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right )}{\sqrt {d^2 e^2-f^2}}+\frac {\sin ^{-1}(d x) (B f-C e)}{d}-\frac {C f \sqrt {1-d^2 x^2}}{d^2}}{f^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)),x]

[Out]

(-((C*f*Sqrt[1 - d^2*x^2])/d^2) + ((-(C*e) + B*f)*ArcSin[d*x])/d + ((C*e^2 + f*(-(B*e) + A*f))*ArcTan[(f + d^2
*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/Sqrt[d^2*e^2 - f^2])/f^2

________________________________________________________________________________________

fricas [B]  time = 15.02, size = 493, normalized size = 4.04 \[ \left [-\frac {{\left (C d^{2} e^{2} - B d^{2} e f + A d^{2} f^{2}\right )} \sqrt {-d^{2} e^{2} + f^{2}} \log \left (\frac {d^{2} e f x + f^{2} - \sqrt {-d^{2} e^{2} + f^{2}} {\left (d^{2} e x + f\right )} - {\left (\sqrt {-d^{2} e^{2} + f^{2}} \sqrt {-d x + 1} f + {\left (d^{2} e^{2} - f^{2}\right )} \sqrt {-d x + 1}\right )} \sqrt {d x + 1}}{f x + e}\right ) + {\left (C d^{2} e^{2} f - C f^{3}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} - 2 \, {\left (C d^{3} e^{3} - B d^{3} e^{2} f - C d e f^{2} + B d f^{3}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{d^{4} e^{2} f^{2} - d^{2} f^{4}}, \frac {2 \, {\left (C d^{2} e^{2} - B d^{2} e f + A d^{2} f^{2}\right )} \sqrt {d^{2} e^{2} - f^{2}} \arctan \left (-\frac {\sqrt {d^{2} e^{2} - f^{2}} \sqrt {d x + 1} \sqrt {-d x + 1} e - \sqrt {d^{2} e^{2} - f^{2}} {\left (f x + e\right )}}{{\left (d^{2} e^{2} - f^{2}\right )} x}\right ) - {\left (C d^{2} e^{2} f - C f^{3}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 2 \, {\left (C d^{3} e^{3} - B d^{3} e^{2} f - C d e f^{2} + B d f^{3}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{d^{4} e^{2} f^{2} - d^{2} f^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(f*x+e)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="fricas")

[Out]

[-((C*d^2*e^2 - B*d^2*e*f + A*d^2*f^2)*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)*sqrt(-d*x + 1))*sqrt(d*x + 1))/(f*x + e)) +
(C*d^2*e^2*f - C*f^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 2*(C*d^3*e^3 - B*d^3*e^2*f - C*d*e*f^2 + B*d*f^3)*arctan
((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^4*e^2*f^2 - d^2*f^4), (2*(C*d^2*e^2 - B*d^2*e*f + A*d^2*f^2)*sq
rt(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^2*e^2*f - C*f^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*(C*d^3*e^3 - B*d^3*e^2*f - C*d*
e*f^2 + B*d*f^3)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^4*e^2*f^2 - d^2*f^4)]

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(f*x+e)/(-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,x):;OUTPUT:Unde
f/Unsigned Inf encountered in limit

________________________________________________________________________________________

maple [C]  time = 0.00, size = 373, normalized size = 3.06 \[ \frac {\left (-A \,d^{2} f^{2} \mathrm {csgn}\relax (d ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right )+B \,d^{2} e f \,\mathrm {csgn}\relax (d ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right )-C \,d^{2} e^{2} \mathrm {csgn}\relax (d ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right )+\sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, B d \,f^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )-\sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, C d e f \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )-\sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, C \,f^{2} \mathrm {csgn}\relax (d )\right ) \sqrt {-d x +1}\, \sqrt {d x +1}\, \mathrm {csgn}\relax (d )}{\sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, \sqrt {-d^{2} x^{2}+1}\, d^{2} f^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/(f*x+e)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x)

[Out]

(-A*d^2*f^2*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))+B*d^2*e*f*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))-C*d^2*e^2*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^2*e^2-f^2)/f^2)^(1/2)*B*d*f^2*arctan(1/(-d^2*x^
2+1)^(1/2)*d*x*csgn(d))-(-(d^2*e^2-f^2)/f^2)^(1/2)*C*d*e*f*arctan(1/(-d^2*x^2+1)^(1/2)*d*x*csgn(d))-(-d^2*x^2+
1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*C*f^2*csgn(d))*(-d*x+1)^(1/2)*(d*x+1)^(1/2)/(-(d^2*e^2-f^2)/f^2)^(1/2)/(-d
^2*x^2+1)^(1/2)/d^2/f^3*csgn(d)

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(f*x+e)/(-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(f-d*e>0)', see `assume?` for m
ore details)Is f-d*e positive, negative or zero?

________________________________________________________________________________________

mupad [B]  time = 0.01, size = 5803, normalized size = 47.57 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2)/((e + f*x)*(1 - d*x)^(1/2)*(d*x + 1)^(1/2)),x)

[Out]

(4*C*e*atan((37748736*C^5*d^4*e^10*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2) - 1)*(37748736*C^5*d^4*e^10 + 6710
8864*C^5*e^6*f^4 - 100663296*C^5*d^2*e^8*f^2)) + (67108864*C^5*e^6*f^4*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2
) - 1)*(37748736*C^5*d^4*e^10 + 67108864*C^5*e^6*f^4 - 100663296*C^5*d^2*e^8*f^2)) - (100663296*C^5*d^2*e^8*f^
2*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2) - 1)*(37748736*C^5*d^4*e^10 + 67108864*C^5*e^6*f^4 - 100663296*C^5*
d^2*e^8*f^2))))/(d*f^2) - (4*B*atan((67108864*B^5*e*f^4*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2) - 1)*(6710886
4*B^5*e*f^4 + 37748736*B^5*d^4*e^5 - 100663296*B^5*d^2*e^3*f^2)) + (37748736*B^5*d^4*e^5*((1 - d*x)^(1/2) - 1)
)/(((d*x + 1)^(1/2) - 1)*(67108864*B^5*e*f^4 + 37748736*B^5*d^4*e^5 - 100663296*B^5*d^2*e^3*f^2)) - (100663296
*B^5*d^2*e^3*f^2*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2) - 1)*(67108864*B^5*e*f^4 + 37748736*B^5*d^4*e^5 - 10
0663296*B^5*d^2*e^3*f^2))))/(d*f) - (8*C*((1 - d*x)^(1/2) - 1)^2)/(f*((d*x + 1)^(1/2) - 1)^2*(d^2 + (2*d^2*((1
- d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (d^2*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4)) - (A*a
tan((f^2*1i - d^2*e^2*1i - (f^2*((1 - d*x)^(1/2) - 1)^2*1i)/((d*x + 1)^(1/2) - 1)^2 + (d^2*e^2*((1 - d*x)^(1/2
) - 1)^2*1i)/((d*x + 1)^(1/2) - 1)^2)/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2) - (f*((1 - d*x)^(1/2) - 1)^2*(f + d*e
)^(1/2)*(f - d*e)^(1/2))/((d*x + 1)^(1/2) - 1)^2 + (2*d*e*((1 - d*x)^(1/2) - 1)*(f + d*e)^(1/2)*(f - d*e)^(1/2
))/((d*x + 1)^(1/2) - 1)))*2i)/((f + d*e)^(1/2)*(f - d*e)^(1/2)) - (C*e^2*atan(((C*e^2*((4096*(32*C^3*e^5*f^3
+ 24*C^3*d^2*e^7*f))/(d*f^4) - (4096*((1 - d*x)^(1/2) - 1)^2*(32*C^3*e^5*f^3 - 96*C^3*d^2*e^7*f))/(d*f^4*((d*x
+ 1)^(1/2) - 1)^2) + (458752*C^3*e^6*((1 - d*x)^(1/2) - 1))/(f^2*((d*x + 1)^(1/2) - 1)) + (C*e^2*((4096*(16*C
^2*e^3*f^6 + 9*C^2*d^4*e^7*f^2))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(8*C^2*e^4*f^3 + 3*C^2*d^2*e^6*f))/(f^
2*((d*x + 1)^(1/2) - 1)) + (4096*((1 - d*x)^(1/2) - 1)^2*(128*C^2*d^2*e^5*f^4 - 144*C^2*e^3*f^6 + 9*C^2*d^4*e^
7*f^2))/(d*f^4*((d*x + 1)^(1/2) - 1)^2) - (C*e^2*((4096*(24*C*d^2*e^3*f^7 - 30*C*d^4*e^5*f^5))/(d*f^4) + (1638
4*((1 - d*x)^(1/2) - 1)*(20*C*e^2*f^6 - 22*C*d^2*e^4*f^4))/(f^2*((d*x + 1)^(1/2) - 1)) + (4096*(96*C*d^2*e^3*f
^7 - 90*C*d^4*e^5*f^5)*((1 - d*x)^(1/2) - 1)^2)/(d*f^4*((d*x + 1)^(1/2) - 1)^2) + (C*e^2*((4096*(7*d^4*e^3*f^8
- 9*d^6*e^5*f^6))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(5*d^2*e^2*f^7 - 6*d^4*e^4*f^5))/(f^2*((d*x + 1)^(1/
2) - 1)) + (4096*((1 - d*x)^(1/2) - 1)^2*(11*d^4*e^3*f^8 - 9*d^6*e^5*f^6))/(d*f^4*((d*x + 1)^(1/2) - 1)^2)))/(
f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^
(1/2)))*1i)/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2)) + (C*e^2*((4096*(32*C^3*e^5*f^3 + 24*C^3*d^2*e^7*f))/(d*f^4)
- (4096*((1 - d*x)^(1/2) - 1)^2*(32*C^3*e^5*f^3 - 96*C^3*d^2*e^7*f))/(d*f^4*((d*x + 1)^(1/2) - 1)^2) + (45875
2*C^3*e^6*((1 - d*x)^(1/2) - 1))/(f^2*((d*x + 1)^(1/2) - 1)) - (C*e^2*((4096*(16*C^2*e^3*f^6 + 9*C^2*d^4*e^7*f
^2))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(8*C^2*e^4*f^3 + 3*C^2*d^2*e^6*f))/(f^2*((d*x + 1)^(1/2) - 1)) + (
4096*((1 - d*x)^(1/2) - 1)^2*(128*C^2*d^2*e^5*f^4 - 144*C^2*e^3*f^6 + 9*C^2*d^4*e^7*f^2))/(d*f^4*((d*x + 1)^(1
/2) - 1)^2) + (C*e^2*((4096*(24*C*d^2*e^3*f^7 - 30*C*d^4*e^5*f^5))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(20*
C*e^2*f^6 - 22*C*d^2*e^4*f^4))/(f^2*((d*x + 1)^(1/2) - 1)) + (4096*(96*C*d^2*e^3*f^7 - 90*C*d^4*e^5*f^5)*((1 -
d*x)^(1/2) - 1)^2)/(d*f^4*((d*x + 1)^(1/2) - 1)^2) - (C*e^2*((4096*(7*d^4*e^3*f^8 - 9*d^6*e^5*f^6))/(d*f^4) +
(16384*((1 - d*x)^(1/2) - 1)*(5*d^2*e^2*f^7 - 6*d^4*e^4*f^5))/(f^2*((d*x + 1)^(1/2) - 1)) + (4096*((1 - d*x)^
(1/2) - 1)^2*(11*d^4*e^3*f^8 - 9*d^6*e^5*f^6))/(d*f^4*((d*x + 1)^(1/2) - 1)^2)))/(f^2*(f + d*e)^(1/2)*(f - d*e
)^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2)))*1i)/(f^2*(f + d*e)^(
1/2)*(f - d*e)^(1/2)))/((131072*C^4*e^7)/(d*f^4) + (C*e^2*((4096*(32*C^3*e^5*f^3 + 24*C^3*d^2*e^7*f))/(d*f^4)
- (4096*((1 - d*x)^(1/2) - 1)^2*(32*C^3*e^5*f^3 - 96*C^3*d^2*e^7*f))/(d*f^4*((d*x + 1)^(1/2) - 1)^2) + (458752
*C^3*e^6*((1 - d*x)^(1/2) - 1))/(f^2*((d*x + 1)^(1/2) - 1)) + (C*e^2*((4096*(16*C^2*e^3*f^6 + 9*C^2*d^4*e^7*f^
2))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(8*C^2*e^4*f^3 + 3*C^2*d^2*e^6*f))/(f^2*((d*x + 1)^(1/2) - 1)) + (4
096*((1 - d*x)^(1/2) - 1)^2*(128*C^2*d^2*e^5*f^4 - 144*C^2*e^3*f^6 + 9*C^2*d^4*e^7*f^2))/(d*f^4*((d*x + 1)^(1/
2) - 1)^2) - (C*e^2*((4096*(24*C*d^2*e^3*f^7 - 30*C*d^4*e^5*f^5))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(20*C
*e^2*f^6 - 22*C*d^2*e^4*f^4))/(f^2*((d*x + 1)^(1/2) - 1)) + (4096*(96*C*d^2*e^3*f^7 - 90*C*d^4*e^5*f^5)*((1 -
d*x)^(1/2) - 1)^2)/(d*f^4*((d*x + 1)^(1/2) - 1)^2) + (C*e^2*((4096*(7*d^4*e^3*f^8 - 9*d^6*e^5*f^6))/(d*f^4) +
(16384*((1 - d*x)^(1/2) - 1)*(5*d^2*e^2*f^7 - 6*d^4*e^4*f^5))/(f^2*((d*x + 1)^(1/2) - 1)) + (4096*((1 - d*x)^(
1/2) - 1)^2*(11*d^4*e^3*f^8 - 9*d^6*e^5*f^6))/(d*f^4*((d*x + 1)^(1/2) - 1)^2)))/(f^2*(f + d*e)^(1/2)*(f - d*e)
^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f^2*(f + d*e)^(1/2)
*(f - d*e)^(1/2)) - (C*e^2*((4096*(32*C^3*e^5*f^3 + 24*C^3*d^2*e^7*f))/(d*f^4) - (4096*((1 - d*x)^(1/2) - 1)^2
*(32*C^3*e^5*f^3 - 96*C^3*d^2*e^7*f))/(d*f^4*((d*x + 1)^(1/2) - 1)^2) + (458752*C^3*e^6*((1 - d*x)^(1/2) - 1))
/(f^2*((d*x + 1)^(1/2) - 1)) - (C*e^2*((4096*(16*C^2*e^3*f^6 + 9*C^2*d^4*e^7*f^2))/(d*f^4) + (16384*((1 - d*x)
^(1/2) - 1)*(8*C^2*e^4*f^3 + 3*C^2*d^2*e^6*f))/(f^2*((d*x + 1)^(1/2) - 1)) + (4096*((1 - d*x)^(1/2) - 1)^2*(12
8*C^2*d^2*e^5*f^4 - 144*C^2*e^3*f^6 + 9*C^2*d^4*e^7*f^2))/(d*f^4*((d*x + 1)^(1/2) - 1)^2) + (C*e^2*((4096*(24*
C*d^2*e^3*f^7 - 30*C*d^4*e^5*f^5))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(20*C*e^2*f^6 - 22*C*d^2*e^4*f^4))/(
f^2*((d*x + 1)^(1/2) - 1)) + (4096*(96*C*d^2*e^3*f^7 - 90*C*d^4*e^5*f^5)*((1 - d*x)^(1/2) - 1)^2)/(d*f^4*((d*x
+ 1)^(1/2) - 1)^2) - (C*e^2*((4096*(7*d^4*e^3*f^8 - 9*d^6*e^5*f^6))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(5
*d^2*e^2*f^7 - 6*d^4*e^4*f^5))/(f^2*((d*x + 1)^(1/2) - 1)) + (4096*((1 - d*x)^(1/2) - 1)^2*(11*d^4*e^3*f^8 - 9
*d^6*e^5*f^6))/(d*f^4*((d*x + 1)^(1/2) - 1)^2)))/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f^2*(f + d*e)^(1/2)*
(f - d*e)^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2)) + (917504*C^4
*e^7*((1 - d*x)^(1/2) - 1)^2)/(d*f^4*((d*x + 1)^(1/2) - 1)^2)))*2i)/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2)) + (B
*e*atan(((B*e*((4096*(24*B^3*d^2*e^4 + 32*B^3*e^2*f^2))/d + (4096*((1 - d*x)^(1/2) - 1)^2*(96*B^3*d^2*e^4 - 32
*B^3*e^2*f^2))/(d*((d*x + 1)^(1/2) - 1)^2) + (458752*B^3*e^3*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) +
(B*e*((4096*(16*B^2*e*f^4 + 9*B^2*d^4*e^5))/d + (((1 - d*x)^(1/2) - 1)*(131072*B^2*e^2*f^3 + 49152*B^2*d^2*e^4
*f))/((d*x + 1)^(1/2) - 1) + (4096*((1 - d*x)^(1/2) - 1)^2*(9*B^2*d^4*e^5 - 144*B^2*e*f^4 + 128*B^2*d^2*e^3*f^
2))/(d*((d*x + 1)^(1/2) - 1)^2) - (B*e*((4096*(24*B*d^2*e^2*f^4 - 30*B*d^4*e^4*f^2))/d + ((327680*B*e*f^5 - 36
0448*B*d^2*e^3*f^3)*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (4096*(96*B*d^2*e^2*f^4 - 90*B*d^4*e^4*f^2)
*((1 - d*x)^(1/2) - 1)^2)/(d*((d*x + 1)^(1/2) - 1)^2) + (B*e*((4096*(7*d^4*e^3*f^4 - 9*d^6*e^5*f^2))/d + (((1
- d*x)^(1/2) - 1)*(81920*d^2*e^2*f^5 - 98304*d^4*e^4*f^3))/((d*x + 1)^(1/2) - 1) + (4096*((1 - d*x)^(1/2) - 1)
^2*(11*d^4*e^3*f^4 - 9*d^6*e^5*f^2))/(d*((d*x + 1)^(1/2) - 1)^2)))/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f*(f
+ d*e)^(1/2)*(f - d*e)^(1/2))))/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2)))*1i)/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2))
+ (B*e*((4096*(24*B^3*d^2*e^4 + 32*B^3*e^2*f^2))/d + (4096*((1 - d*x)^(1/2) - 1)^2*(96*B^3*d^2*e^4 - 32*B^3*e^
2*f^2))/(d*((d*x + 1)^(1/2) - 1)^2) + (458752*B^3*e^3*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) - (B*e*((
4096*(16*B^2*e*f^4 + 9*B^2*d^4*e^5))/d + (((1 - d*x)^(1/2) - 1)*(131072*B^2*e^2*f^3 + 49152*B^2*d^2*e^4*f))/((
d*x + 1)^(1/2) - 1) + (4096*((1 - d*x)^(1/2) - 1)^2*(9*B^2*d^4*e^5 - 144*B^2*e*f^4 + 128*B^2*d^2*e^3*f^2))/(d*
((d*x + 1)^(1/2) - 1)^2) + (B*e*((4096*(24*B*d^2*e^2*f^4 - 30*B*d^4*e^4*f^2))/d + ((327680*B*e*f^5 - 360448*B*
d^2*e^3*f^3)*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (4096*(96*B*d^2*e^2*f^4 - 90*B*d^4*e^4*f^2)*((1 -
d*x)^(1/2) - 1)^2)/(d*((d*x + 1)^(1/2) - 1)^2) - (B*e*((4096*(7*d^4*e^3*f^4 - 9*d^6*e^5*f^2))/d + (((1 - d*x)^
(1/2) - 1)*(81920*d^2*e^2*f^5 - 98304*d^4*e^4*f^3))/((d*x + 1)^(1/2) - 1) + (4096*((1 - d*x)^(1/2) - 1)^2*(11*
d^4*e^3*f^4 - 9*d^6*e^5*f^2))/(d*((d*x + 1)^(1/2) - 1)^2)))/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f*(f + d*e)
^(1/2)*(f - d*e)^(1/2))))/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2)))*1i)/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2)))/((1310
72*B^4*e^3)/d + (917504*B^4*e^3*((1 - d*x)^(1/2) - 1)^2)/(d*((d*x + 1)^(1/2) - 1)^2) + (B*e*((4096*(24*B^3*d^2
*e^4 + 32*B^3*e^2*f^2))/d + (4096*((1 - d*x)^(1/2) - 1)^2*(96*B^3*d^2*e^4 - 32*B^3*e^2*f^2))/(d*((d*x + 1)^(1/
2) - 1)^2) + (458752*B^3*e^3*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (B*e*((4096*(16*B^2*e*f^4 + 9*B^
2*d^4*e^5))/d + (((1 - d*x)^(1/2) - 1)*(131072*B^2*e^2*f^3 + 49152*B^2*d^2*e^4*f))/((d*x + 1)^(1/2) - 1) + (40
96*((1 - d*x)^(1/2) - 1)^2*(9*B^2*d^4*e^5 - 144*B^2*e*f^4 + 128*B^2*d^2*e^3*f^2))/(d*((d*x + 1)^(1/2) - 1)^2)
- (B*e*((4096*(24*B*d^2*e^2*f^4 - 30*B*d^4*e^4*f^2))/d + ((327680*B*e*f^5 - 360448*B*d^2*e^3*f^3)*((1 - d*x)^(
1/2) - 1))/((d*x + 1)^(1/2) - 1) + (4096*(96*B*d^2*e^2*f^4 - 90*B*d^4*e^4*f^2)*((1 - d*x)^(1/2) - 1)^2)/(d*((d
*x + 1)^(1/2) - 1)^2) + (B*e*((4096*(7*d^4*e^3*f^4 - 9*d^6*e^5*f^2))/d + (((1 - d*x)^(1/2) - 1)*(81920*d^2*e^2
*f^5 - 98304*d^4*e^4*f^3))/((d*x + 1)^(1/2) - 1) + (4096*((1 - d*x)^(1/2) - 1)^2*(11*d^4*e^3*f^4 - 9*d^6*e^5*f
^2))/(d*((d*x + 1)^(1/2) - 1)^2)))/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2))))
/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2)) - (B*e*((4096*(24*B^3*d^2*e^4 + 32*
B^3*e^2*f^2))/d + (4096*((1 - d*x)^(1/2) - 1)^2*(96*B^3*d^2*e^4 - 32*B^3*e^2*f^2))/(d*((d*x + 1)^(1/2) - 1)^2)
+ (458752*B^3*e^3*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) - (B*e*((4096*(16*B^2*e*f^4 + 9*B^2*d^4*e^5)
)/d + (((1 - d*x)^(1/2) - 1)*(131072*B^2*e^2*f^3 + 49152*B^2*d^2*e^4*f))/((d*x + 1)^(1/2) - 1) + (4096*((1 - d
*x)^(1/2) - 1)^2*(9*B^2*d^4*e^5 - 144*B^2*e*f^4 + 128*B^2*d^2*e^3*f^2))/(d*((d*x + 1)^(1/2) - 1)^2) + (B*e*((4
096*(24*B*d^2*e^2*f^4 - 30*B*d^4*e^4*f^2))/d + ((327680*B*e*f^5 - 360448*B*d^2*e^3*f^3)*((1 - d*x)^(1/2) - 1))
/((d*x + 1)^(1/2) - 1) + (4096*(96*B*d^2*e^2*f^4 - 90*B*d^4*e^4*f^2)*((1 - d*x)^(1/2) - 1)^2)/(d*((d*x + 1)^(1
/2) - 1)^2) - (B*e*((4096*(7*d^4*e^3*f^4 - 9*d^6*e^5*f^2))/d + (((1 - d*x)^(1/2) - 1)*(81920*d^2*e^2*f^5 - 983
04*d^4*e^4*f^3))/((d*x + 1)^(1/2) - 1) + (4096*((1 - d*x)^(1/2) - 1)^2*(11*d^4*e^3*f^4 - 9*d^6*e^5*f^2))/(d*((
d*x + 1)^(1/2) - 1)^2)))/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f*(f + d
*e)^(1/2)*(f - d*e)^(1/2))))/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2))))*2i)/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)/(f*x+e)/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________