3.225 \(\int \frac{\left (d+e x^2\right )^{3/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx\)
Optimal. Leaf size=108 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{c \sqrt{e}}-\frac{\sqrt{2 c d-b e} \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{c \sqrt{e} \sqrt{c d-b e}} \]
[Out]
ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/(c*Sqrt[e]) - (Sqrt[2*c*d - b*e]*ArcTanh[(S
qrt[e]*Sqrt[2*c*d - b*e]*x)/(Sqrt[c*d - b*e]*Sqrt[d + e*x^2])])/(c*Sqrt[e]*Sqrt[
c*d - b*e])
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Rubi [A] time = 0.257815, antiderivative size = 108, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146
\[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{c \sqrt{e}}-\frac{\sqrt{2 c d-b e} \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{c \sqrt{e} \sqrt{c d-b e}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^2)^(3/2)/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/(c*Sqrt[e]) - (Sqrt[2*c*d - b*e]*ArcTanh[(S
qrt[e]*Sqrt[2*c*d - b*e]*x)/(Sqrt[c*d - b*e]*Sqrt[d + e*x^2])])/(c*Sqrt[e]*Sqrt[
c*d - b*e])
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Rubi in Sympy [A] time = 66.5288, size = 94, normalized size = 0.87 \[ - \frac{\sqrt{b e - 2 c d} \operatorname{atanh}{\left (\frac{\sqrt{e} x \sqrt{b e - 2 c d}}{\sqrt{d + e x^{2}} \sqrt{b e - c d}} \right )}}{c \sqrt{e} \sqrt{b e - c d}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{c \sqrt{e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d)**(3/2)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)
[Out]
-sqrt(b*e - 2*c*d)*atanh(sqrt(e)*x*sqrt(b*e - 2*c*d)/(sqrt(d + e*x**2)*sqrt(b*e
- c*d)))/(c*sqrt(e)*sqrt(b*e - c*d)) + atanh(sqrt(e)*x/sqrt(d + e*x**2))/(c*sqrt
(e))
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Mathematica [A] time = 0.0925788, size = 103, normalized size = 0.95 \[ -\frac{\frac{\sqrt{b e-2 c d} \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{b e-2 c d}}{\sqrt{d+e x^2} \sqrt{b e-c d}}\right )}{\sqrt{b e-c d}}-\log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{c \sqrt{e}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^2)^(3/2)/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
-(((Sqrt[-2*c*d + b*e]*ArcTanh[(Sqrt[e]*Sqrt[-2*c*d + b*e]*x)/(Sqrt[-(c*d) + b*e
]*Sqrt[d + e*x^2])])/Sqrt[-(c*d) + b*e] - Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(c
*Sqrt[e]))
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Maple [B] time = 0.031, size = 4332, normalized size = 40.1 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)^(3/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)
[Out]
1/6*e*c/(-d*e)^(1/2)/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(
b*e-c*d)*c*e)^(1/2))*((x-1/e*(-d*e)^(1/2))^2*e+2*(-d*e)^(1/2)*(x-1/e*(-d*e)^(1/2
)))^(3/2)+1/4*e*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e
-c*d)*c*e)^(1/2))*((x-1/e*(-d*e)^(1/2))^2*e+2*(-d*e)^(1/2)*(x-1/e*(-d*e)^(1/2)))
^(1/2)*x+1/4*e^(1/2)*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(
-(b*e-c*d)*c*e)^(1/2))*d*ln(((x-1/e*(-d*e)^(1/2))*e+(-d*e)^(1/2))/e^(1/2)+((x-1/
e*(-d*e)^(1/2))^2*e+2*(-d*e)^(1/2)*(x-1/e*(-d*e)^(1/2)))^(1/2))-1/6*e*c/(-d*e)^(
1/2)/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1
/2))*((x+1/e*(-d*e)^(1/2))^2*e-2*(-d*e)^(1/2)*(x+1/e*(-d*e)^(1/2)))^(3/2)+1/4*e*
c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2)
)*((x+1/e*(-d*e)^(1/2))^2*e-2*(-d*e)^(1/2)*(x+1/e*(-d*e)^(1/2)))^(1/2)*x+1/4*e^(
1/2)*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^
(1/2))*d*ln(((x+1/e*(-d*e)^(1/2))*e-(-d*e)^(1/2))/e^(1/2)+((x+1/e*(-d*e)^(1/2))^
2*e-2*(-d*e)^(1/2)*(x+1/e*(-d*e)^(1/2)))^(1/2))-1/6*e*c^2/((-d*e)^(1/2)*c+(-(b*e
-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)
*((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c
*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(3/2)-1/4*e*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/
2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))*((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+
2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2)*x
-5/4*e^(1/2)*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*
d)*c*e)^(1/2))*ln(((-(b*e-c*d)*c*e)^(1/2)/c+(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)*e)/e^
(1/2)+((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c
*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))*d+1/2*e^2*c/((-d*e)^(1/2)*c+(-(b*e-c*d
)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)*((x
-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^
(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2)*b-e*c^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/
((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)*((x-(-(b*e-c*d)*c
*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*
e-2*c*d)/c)^(1/2)*d+1/2*e^(3/2)/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^
(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))*ln(((-(b*e-c*d)*c*e)^(1/2)/c+(x-(-(b*e-c*d)*c*e)
^(1/2)/c/e)*e)/e^(1/2)+((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1
/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))*b+1/2*e^3/((-d*e)^(1/
2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)
*c*e)^(1/2)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c+2*(-(b*e-c*d)*c*e)^(1/2)
/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x-(-(b*e-c*d)*c*e)^
(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*
c*d)/c)^(1/2))/(x-(-(b*e-c*d)*c*e)^(1/2)/c/e))*b^2-2*e^2*c/((-d*e)^(1/2)*c+(-(b*
e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2
)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b
*e-c*d)*c*e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)
^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1
/2))/(x-(-(b*e-c*d)*c*e)^(1/2)/c/e))*b*d+2*e*c^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e
)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(-(b*e-2
*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*
e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(
b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x-(-
(b*e-c*d)*c*e)^(1/2)/c/e))*d^2+1/6*e*c^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))
/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)*((x+(-(b*e-c*d)*
c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b
*e-2*c*d)/c)^(3/2)-1/4*e*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)
*c-(-(b*e-c*d)*c*e)^(1/2))*((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e
)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2)*x-5/4*e^(1/2)*c/((
-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))*ln
((-(-(b*e-c*d)*c*e)^(1/2)/c+(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)*e)/e^(1/2)+((x+(-(b*e
-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c
/e)-(b*e-2*c*d)/c)^(1/2))*d-1/2*e^2*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((
-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)*((x+(-(b*e-c*d)*c*e
)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-
2*c*d)/c)^(1/2)*b+e*c^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-
(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)*((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2
*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2
)*d+1/2*e^(3/2)/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c
*d)*c*e)^(1/2))*ln((-(-(b*e-c*d)*c*e)^(1/2)/c+(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)*e)/
e^(1/2)+((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e
-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))*b-1/2*e^3/((-d*e)^(1/2)*c+(-(b*e-c*d
)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(-(
b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*
d)*c*e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-
2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/
(x+(-(b*e-c*d)*c*e)^(1/2)/c/e))*b^2+2*e^2*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/
2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(-(b*e-2*c*d)
/c)^(1/2)*ln((-2*(b*e-2*c*d)/c-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1
/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c
*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x+(-(b*e-
c*d)*c*e)^(1/2)/c/e))*b*d-2*e*c^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e
)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(-(b*e-2*c*d)/c)^(1/2)*
ln((-2*(b*e-2*c*d)/c-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)+2
*(-(b*e-2*c*d)/c)^(1/2)*((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(
1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x+(-(b*e-c*d)*c*e)^
(1/2)/c/e))*d^2
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)^(3/2)/(c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.385129, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)^(3/2)/(c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e),x, algorithm="fricas")
[Out]
[1/4*(sqrt(e)*sqrt((2*c*d - b*e)/(c*d*e - b*e^2))*log((c^2*d^4 - 2*b*c*d^3*e + b
^2*d^2*e^2 + (17*c^2*d^2*e^2 - 24*b*c*d*e^3 + 8*b^2*e^4)*x^4 + 2*(7*c^2*d^3*e -
11*b*c*d^2*e^2 + 4*b^2*d*e^3)*x^2 - 4*((3*c^2*d^2*e^2 - 5*b*c*d*e^3 + 2*b^2*e^4)
*x^3 + (c^2*d^3*e - 2*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(e*x^2 + d)*sqrt((2*c*d -
b*e)/(c*d*e - b*e^2)))/(c^2*e^2*x^4 + c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*(c^2*d*e
- b*c*e^2)*x^2)) + 2*log(-2*sqrt(e*x^2 + d)*e*x - (2*e*x^2 + d)*sqrt(e)))/(c*sq
rt(e)), 1/4*(sqrt(-e)*sqrt((2*c*d - b*e)/(c*d*e - b*e^2))*log((c^2*d^4 - 2*b*c*d
^3*e + b^2*d^2*e^2 + (17*c^2*d^2*e^2 - 24*b*c*d*e^3 + 8*b^2*e^4)*x^4 + 2*(7*c^2*
d^3*e - 11*b*c*d^2*e^2 + 4*b^2*d*e^3)*x^2 - 4*((3*c^2*d^2*e^2 - 5*b*c*d*e^3 + 2*
b^2*e^4)*x^3 + (c^2*d^3*e - 2*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(e*x^2 + d)*sqrt((
2*c*d - b*e)/(c*d*e - b*e^2)))/(c^2*e^2*x^4 + c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*
(c^2*d*e - b*c*e^2)*x^2)) + 4*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)))/(c*sqrt(-e)),
-1/2*(sqrt(e)*sqrt(-(2*c*d - b*e)/(c*d*e - b*e^2))*arctan(1/2*(c*d^2 - b*d*e + (
3*c*d*e - 2*b*e^2)*x^2)/((c*d*e - b*e^2)*sqrt(e*x^2 + d)*x*sqrt(-(2*c*d - b*e)/(
c*d*e - b*e^2)))) - log(-2*sqrt(e*x^2 + d)*e*x - (2*e*x^2 + d)*sqrt(e)))/(c*sqrt
(e)), -1/2*(sqrt(-e)*sqrt(-(2*c*d - b*e)/(c*d*e - b*e^2))*arctan(1/2*(c*d^2 - b*
d*e + (3*c*d*e - 2*b*e^2)*x^2)/((c*d*e - b*e^2)*sqrt(e*x^2 + d)*x*sqrt(-(2*c*d -
b*e)/(c*d*e - b*e^2)))) - 2*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)))/(c*sqrt(-e))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x^{2}}}{b e - c d + c e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d)**(3/2)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)
[Out]
Integral(sqrt(d + e*x**2)/(b*e - c*d + c*e*x**2), x)
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)^(3/2)/(c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e),x, algorithm="giac")
[Out]
Timed out