\(\int \frac {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx\) [34]

Optimal result

Integrand size = 30, antiderivative size = 134 \[ \int \frac {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx=-\frac {e^{3/2} \log \left (\sqrt {c} d-\sqrt {e} \sqrt {2 c d-b e} x+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}}+\frac {e^{3/2} \log \left (\sqrt {c} d+\sqrt {e} \sqrt {2 c d-b e} x+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}} \]

[Out]

-1/2*e^(3/2)*ln(d*c^(1/2)+e*x^2*c^(1/2)-x*e^(1/2)*(-b*e+2*c*d)^(1/2))/c^(1/2)/(-b*e+2*c*d)^(1/2)+1/2*e^(3/2)*l
n(d*c^(1/2)+e*x^2*c^(1/2)+x*e^(1/2)*(-b*e+2*c*d)^(1/2))/c^(1/2)/(-b*e+2*c*d)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1178, 642} \[ \int \frac {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx=\frac {e^{3/2} \log \left (\sqrt {e} x \sqrt {2 c d-b e}+\sqrt {c} d+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}}-\frac {e^{3/2} \log \left (-\sqrt {e} x \sqrt {2 c d-b e}+\sqrt {c} d+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}} \]

[In]

Int[(d - e*x^2)/((c*d^2)/e^2 + b*x^2 + c*x^4),x]

[Out]

-1/2*(e^(3/2)*Log[Sqrt[c]*d - Sqrt[e]*Sqrt[2*c*d - b*e]*x + Sqrt[c]*e*x^2])/(Sqrt[c]*Sqrt[2*c*d - b*e]) + (e^(
3/2)*Log[Sqrt[c]*d + Sqrt[e]*Sqrt[2*c*d - b*e]*x + Sqrt[c]*e*x^2])/(2*Sqrt[c]*Sqrt[2*c*d - b*e])

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1178

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e) - b/c, 2]},
Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x
- x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] &&  !GtQ[b^2
- 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{3/2} \int \frac {\frac {\sqrt {2 c d-b e}}{\sqrt {c} \sqrt {e}}+2 x}{-\frac {d}{e}-\frac {\sqrt {2 c d-b e} x}{\sqrt {c} \sqrt {e}}-x^2} \, dx}{2 \sqrt {c} \sqrt {2 c d-b e}}-\frac {e^{3/2} \int \frac {\frac {\sqrt {2 c d-b e}}{\sqrt {c} \sqrt {e}}-2 x}{-\frac {d}{e}+\frac {\sqrt {2 c d-b e} x}{\sqrt {c} \sqrt {e}}-x^2} \, dx}{2 \sqrt {c} \sqrt {2 c d-b e}} \\ & = -\frac {e^{3/2} \log \left (\sqrt {c} d-\sqrt {e} \sqrt {2 c d-b e} x+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}}+\frac {e^{3/2} \log \left (\sqrt {c} d+\sqrt {e} \sqrt {2 c d-b e} x+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.87 \[ \int \frac {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx=\frac {e^{3/2} \left (-\frac {\left (-2 c d-b e+\sqrt {-4 c^2 d^2+b^2 e^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {e} x}{\sqrt {b e-\sqrt {-4 c^2 d^2+b^2 e^2}}}\right )}{\sqrt {b e-\sqrt {-4 c^2 d^2+b^2 e^2}}}-\frac {\left (2 c d+b e+\sqrt {-4 c^2 d^2+b^2 e^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {e} x}{\sqrt {b e+\sqrt {-4 c^2 d^2+b^2 e^2}}}\right )}{\sqrt {b e+\sqrt {-4 c^2 d^2+b^2 e^2}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {-4 c^2 d^2+b^2 e^2}} \]

[In]

Integrate[(d - e*x^2)/((c*d^2)/e^2 + b*x^2 + c*x^4),x]

[Out]

(e^(3/2)*(-(((-2*c*d - b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[e]*x)/Sqrt[b*e - Sqrt[-4
*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e - Sqrt[-4*c^2*d^2 + b^2*e^2]]) - ((2*c*d + b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*A
rcTan[(Sqrt[2]*Sqrt[c]*Sqrt[e]*x)/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^
2]]))/(Sqrt[2]*Sqrt[c]*Sqrt[-4*c^2*d^2 + b^2*e^2])

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.90

method	result	size
risch	\(\frac {\sqrt {-c \left (b e -2 c d \right ) e}\, e \ln \left (-c \,x^{2} e +\sqrt {-c \left (b e -2 c d \right ) e}\, x -c d \right )}{2 c \left (b e -2 c d \right )}-\frac {\sqrt {-c \left (b e -2 c d \right ) e}\, e \ln \left (-c \,x^{2} e -\sqrt {-c \left (b e -2 c d \right ) e}\, x -c d \right )}{2 c \left (b e -2 c d \right )}\)	\(121\)
default	\(4 e^{4} c \left (\frac {\left (-b \,e^{2}-2 d c e -\sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\right ) \sqrt {2}\, \arctan \left (\frac {c x e \sqrt {2}}{\sqrt {c \left (b \,e^{2}+\sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\right )}}\right )}{8 \sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\, c \,e^{2} \sqrt {c \left (b \,e^{2}+\sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\right )}}-\frac {\left (b \,e^{2}+2 d c e -\sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x e \sqrt {2}}{\sqrt {c \left (-b \,e^{2}+\sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\right )}}\right )}{8 \sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\, c \,e^{2} \sqrt {c \left (-b \,e^{2}+\sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\right )}}\right )\)	\(291\)

[In]

int((-e*x^2+d)/(c*d^2/e^2+b*x^2+c*x^4),x,method=_RETURNVERBOSE)

[Out]

1/2*(-c*(b*e-2*c*d)*e)^(1/2)/c/(b*e-2*c*d)*e*ln(-c*x^2*e+(-c*(b*e-2*c*d)*e)^(1/2)*x-c*d)-1/2*(-c*(b*e-2*c*d)*e
)^(1/2)/c/(b*e-2*c*d)*e*ln(-c*x^2*e-(-c*(b*e-2*c*d)*e)^(1/2)*x-c*d)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.82 \[ \int \frac {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx=\left [\frac {1}{2} \, e \sqrt {\frac {e}{2 \, c^{2} d - b c e}} \log \left (\frac {c e^{2} x^{4} + c d^{2} + {\left (4 \, c d e - b e^{2}\right )} x^{2} + 2 \, {\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} x^{3} + {\left (2 \, c^{2} d^{2} - b c d e\right )} x\right )} \sqrt {\frac {e}{2 \, c^{2} d - b c e}}}{c e^{2} x^{4} + b e^{2} x^{2} + c d^{2}}\right ), -e \sqrt {-\frac {e}{2 \, c^{2} d - b c e}} \arctan \left (c x \sqrt {-\frac {e}{2 \, c^{2} d - b c e}}\right ) + e \sqrt {-\frac {e}{2 \, c^{2} d - b c e}} \arctan \left (\frac {{\left (c e x^{3} - {\left (c d - b e\right )} x\right )} \sqrt {-\frac {e}{2 \, c^{2} d - b c e}}}{d}\right )\right ] \]

[In]

integrate((-e*x^2+d)/(c*d^2/e^2+b*x^2+c*x^4),x, algorithm="fricas")

[Out]

[1/2*e*sqrt(e/(2*c^2*d - b*c*e))*log((c*e^2*x^4 + c*d^2 + (4*c*d*e - b*e^2)*x^2 + 2*((2*c^2*d*e - b*c*e^2)*x^3
+ (2*c^2*d^2 - b*c*d*e)*x)*sqrt(e/(2*c^2*d - b*c*e)))/(c*e^2*x^4 + b*e^2*x^2 + c*d^2)), -e*sqrt(-e/(2*c^2*d -
b*c*e))*arctan(c*x*sqrt(-e/(2*c^2*d - b*c*e))) + e*sqrt(-e/(2*c^2*d - b*c*e))*arctan((c*e*x^3 - (c*d - b*e)*x
)*sqrt(-e/(2*c^2*d - b*c*e))/d)]

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.18 \[ \int \frac {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx=\frac {\sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}} \log {\left (\frac {d}{e} + x^{2} + \frac {x \left (- b e \sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}} + 2 c d \sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}}\right )}{e^{2}} \right )}}{2} - \frac {\sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}} \log {\left (\frac {d}{e} + x^{2} + \frac {x \left (b e \sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}} - 2 c d \sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}}\right )}{e^{2}} \right )}}{2} \]

[In]

integrate((-e*x**2+d)/(c*d**2/e**2+b*x**2+c*x**4),x)

[Out]

sqrt(-e**3/(c*(b*e - 2*c*d)))*log(d/e + x**2 + x*(-b*e*sqrt(-e**3/(c*(b*e - 2*c*d))) + 2*c*d*sqrt(-e**3/(c*(b*
e - 2*c*d))))/e**2)/2 - sqrt(-e**3/(c*(b*e - 2*c*d)))*log(d/e + x**2 + x*(b*e*sqrt(-e**3/(c*(b*e - 2*c*d))) -
2*c*d*sqrt(-e**3/(c*(b*e - 2*c*d))))/e**2)/2

Maxima [F]

\[ \int \frac {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx=\int { -\frac {e x^{2} - d}{c x^{4} + b x^{2} + \frac {c d^{2}}{e^{2}}} \,d x } \]

[In]

integrate((-e*x^2+d)/(c*d^2/e^2+b*x^2+c*x^4),x, algorithm="maxima")

[Out]

-integrate((e*x^2 - d)/(c*x^4 + b*x^2 + c*d^2/e^2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6331 vs. \(2 (102) = 204\).

Time = 1.01 (sec) , antiderivative size = 6331, normalized size of antiderivative = 47.25 \[ \int \frac {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx=\text {Too large to display} \]

[In]

integrate((-e*x^2+d)/(c*d^2/e^2+b*x^2+c*x^4),x, algorithm="giac")

[Out]

1/8*(2*sqrt(2*c^2*d*e + b*c*e^2)*b*c^3*d*e^6*sgn(c)*sgn(e) - sqrt(2*c^2*d*e + b*c*e^2)*b^2*c^2*e^7*sgn(c)*sgn(
e) - 12*b*c^4*d^2*e^6 + 3*b^3*c^2*e^8 + 4*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*b*c^2*d^2*e^4*s
gn(c)*sgn(e) + 4*sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*d^2*e^4*sgn(c)*sgn(e) - sqrt(-4*c^
2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*b^3*e^6*sgn(c)*sgn(e) - sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e +
b*c*e^2)*b^3*e^6*sgn(c)*sgn(e) + 2*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*b^2*c*e^6*sgn(c)*sgn(
e) + 2*sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c*e^6*sgn(c)*sgn(e) - 4*sqrt(-4*c^2*d^2 + b^2*
e^2)*b*c^3*d^2*e^5 - 2*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^3*d*e^6 + sqrt(-4*c^2*d^2 + b^2*e^2)*b^3*c*e^7 - 2*sqrt(
-4*c^2*d^2 + b^2*e^2)*b^2*c^2*e^7 - sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c^2*e^7 - 3*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt
(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c*e^5*sgn(c)*sgn(e) + 4*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*
c^2*d*e + b*c*e^2)*b*c^2*d^2*e^4 - sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^3*e^6 + 2*sqrt(-4*c
^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c*e^6 + (4*sqrt(2*c^2*d*e + b*c*e^2)*c^4*d^2*e^3*sgn(c)*sgn(e
) - 4*sqrt(2*c^2*d*e + b*c*e^2)*b*c^3*d*e^4*sgn(c)*sgn(e) + sqrt(2*c^2*d*e + b*c*e^2)*b^2*c^2*e^5*sgn(c)*sgn(e
) - 24*c^5*d^3*e^3 + 12*b*c^4*d^2*e^4 + 6*b^2*c^3*d*e^5 - 3*b^3*c^2*e^6 + 8*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*
c^2*d*e + b*c*e^2)*c^3*d^3*e*sgn(c)*sgn(e) + 8*sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*c^3*d^3*e*
sgn(c)*sgn(e) - 4*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*b*c^2*d^2*e^2*sgn(c)*sgn(e) - 4*sqrt(2*
c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*d^2*e^2*sgn(c)*sgn(e) - 2*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(
2*c^2*d*e + b*c*e^2)*b^2*c*d*e^3*sgn(c)*sgn(e) - 2*sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c*
d*e^3*sgn(c)*sgn(e) + 4*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*b*c^2*d*e^3*sgn(c)*sgn(e) + 4*sqr
t(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*d*e^3*sgn(c)*sgn(e) + sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(
2*c^2*d*e + b*c*e^2)*b^3*e^4*sgn(c)*sgn(e) + sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^3*e^4*sgn(
c)*sgn(e) - 2*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*b^2*c*e^4*sgn(c)*sgn(e) - 2*sqrt(2*c^2*d*e
+ b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c*e^4*sgn(c)*sgn(e) - 8*sqrt(-4*c^2*d^2 + b^2*e^2)*c^4*d^3*e^2 + 4*s
qrt(-4*c^2*d^2 + b^2*e^2)*b*c^3*d^2*e^3 - 4*sqrt(-2*c^2*d*e + b*c*e^2)*c^4*d^2*e^3 + 2*sqrt(-4*c^2*d^2 + b^2*e
^2)*b^2*c^2*d*e^4 - 4*sqrt(-4*c^2*d^2 + b^2*e^2)*b*c^3*d*e^4 - sqrt(-4*c^2*d^2 + b^2*e^2)*b^3*c*e^5 + 2*sqrt(-
4*c^2*d^2 + b^2*e^2)*b^2*c^2*e^5 + sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c^2*e^5 - 6*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(
2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*c^2*d*e^2*sgn(c)*sgn(e) + 3*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*
c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c*e^3*sgn(c)*sgn(e) + 8*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2
*d*e + b*c*e^2)*c^3*d^3*e - 4*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*d^2*e^2 - 2*sqrt(-4*
c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c*d*e^3 + 4*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c
*e^2)*b*c^2*d*e^3 + sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^3*e^4 - 2*sqrt(-4*c^2*d^2 + b^2*e^
2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c*e^4)*e^2 + 2*(8*sqrt(2*c^2*d*e + b*c*e^2)*c^4*d^4*e^2*sgn(c)*sgn(e) - 4*sq
rt(2*c^2*d*e + b*c*e^2)*b*c^3*d^3*e^3*sgn(c)*sgn(e) + 4*sqrt(2*c^2*d*e + b*c*e^2)*c^4*d^3*e^3*sgn(c)*sgn(e) -
2*sqrt(2*c^2*d*e + b*c*e^2)*b^2*c^2*d^2*e^4*sgn(c)*sgn(e) - 2*sqrt(2*c^2*d*e + b*c*e^2)*c^4*d^2*e^4*sgn(c)*sgn
(e) + sqrt(2*c^2*d*e + b*c*e^2)*b^3*c*d*e^5*sgn(c)*sgn(e) - sqrt(2*c^2*d*e + b*c*e^2)*b^2*c^2*d*e^5*sgn(c)*sgn
(e) + sqrt(2*c^2*d*e + b*c*e^2)*b*c^3*d*e^5*sgn(c)*sgn(e) - 24*c^5*d^4*e^3 + 12*b*c^4*d^3*e^4 + 6*b^2*c^3*d^2*
e^5 - 3*b^3*c^2*d*e^6 + 2*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*c^3*d^2*e^3*sgn(c)*sgn(e) + 2*s
qrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*c^3*d^2*e^3*sgn(c)*sgn(e) - sqrt(-4*c^2*d^2 + b^2*e^2)*sqr
t(2*c^2*d*e + b*c*e^2)*b*c^2*d*e^4*sgn(c)*sgn(e) - sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*
d*e^4*sgn(c)*sgn(e) + 8*sqrt(-2*c^2*d*e + b*c*e^2)*c^4*d^4*e^2 - 4*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^3*d^3*e^3 -
4*sqrt(-2*c^2*d*e + b*c*e^2)*c^4*d^3*e^3 - 2*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c^2*d^2*e^4 + 4*sqrt(-2*c^2*d*e +
b*c*e^2)*b*c^3*d^2*e^4 - 2*sqrt(-4*c^2*d^2 + b^2*e^2)*c^4*d^2*e^4 - 2*sqrt(-2*c^2*d*e + b*c*e^2)*c^4*d^2*e^4 +
sqrt(-2*c^2*d*e + b*c*e^2)*b^3*c*d*e^5 - sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c^2*d*e^5 + sqrt(-4*c^2*d^2 + b^2*e^2
)*b*c^3*d*e^5 + sqrt(-2*c^2*d*e + b*c*e^2)*b*c^3*d*e^5 - 6*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2
)*sqrt(-2*c^2*d*e + b*c*e^2)*c^2*d^2*e^2*sgn(c)*sgn(e) + 3*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2
)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c*d*e^3*sgn(c)*sgn(e) + 2*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)
*c^3*d^2*e^3 - sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*d*e^4)*abs(e))*arctan(2*sqrt(1/2)*x
/sqrt((b*e^2 + sqrt(-4*c^2*d^2*e^2 + b^2*e^4))/(c*e^2)))/((8*c^5*d^5 - 4*b*c^4*d^4*e - 2*b^2*c^3*d^3*e^2 + 4*b
*c^4*d^3*e^2 - 2*c^5*d^3*e^2 + b^3*c^2*d^2*e^3 - 2*b^2*c^3*d^2*e^3 + b*c^4*d^2*e^3)*e^2*abs(c)) - 1/8*(2*sqrt(
2*c^2*d*e + b*c*e^2)*b*c^3*d*e^6*sgn(c)*sgn(e) - sqrt(2*c^2*d*e + b*c*e^2)*b^2*c^2*e^7*sgn(c)*sgn(e) - 12*b*c^
4*d^2*e^6 + 3*b^3*c^2*e^8 - 4*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*b*c^2*d^2*e^4*sgn(c)*sgn(e)
- 4*sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*d^2*e^4*sgn(c)*sgn(e) + sqrt(-4*c^2*d^2 + b^2*
e^2)*sqrt(2*c^2*d*e + b*c*e^2)*b^3*e^6*sgn(c)*sgn(e) + sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^
3*e^6*sgn(c)*sgn(e) - 2*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*b^2*c*e^6*sgn(c)*sgn(e) - 2*sqrt(
2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c*e^6*sgn(c)*sgn(e) + 4*sqrt(-4*c^2*d^2 + b^2*e^2)*b*c^3*d
^2*e^5 + 2*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^3*d*e^6 - sqrt(-4*c^2*d^2 + b^2*e^2)*b^3*c*e^7 + 2*sqrt(-4*c^2*d^2 +
b^2*e^2)*b^2*c^2*e^7 + sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c^2*e^7 - 3*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e +
b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c*e^5*sgn(c)*sgn(e) + 4*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*
c*e^2)*b*c^2*d^2*e^4 - sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^3*e^6 + 2*sqrt(-4*c^2*d^2 + b^2
*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c*e^6 + (4*sqrt(2*c^2*d*e + b*c*e^2)*c^4*d^2*e^3*sgn(c)*sgn(e) - 4*sqrt(2
*c^2*d*e + b*c*e^2)*b*c^3*d*e^4*sgn(c)*sgn(e) + sqrt(2*c^2*d*e + b*c*e^2)*b^2*c^2*e^5*sgn(c)*sgn(e) - 24*c^5*d
^3*e^3 + 12*b*c^4*d^2*e^4 + 6*b^2*c^3*d*e^5 - 3*b^3*c^2*e^6 - 8*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*
c*e^2)*c^3*d^3*e*sgn(c)*sgn(e) - 8*sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*c^3*d^3*e*sgn(c)*sgn(e
) + 4*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*b*c^2*d^2*e^2*sgn(c)*sgn(e) + 4*sqrt(2*c^2*d*e + b*
c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*d^2*e^2*sgn(c)*sgn(e) + 2*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e +
b*c*e^2)*b^2*c*d*e^3*sgn(c)*sgn(e) + 2*sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c*d*e^3*sgn(c)
*sgn(e) - 4*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*b*c^2*d*e^3*sgn(c)*sgn(e) - 4*sqrt(2*c^2*d*e
+ b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*d*e^3*sgn(c)*sgn(e) - sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e +
b*c*e^2)*b^3*e^4*sgn(c)*sgn(e) - sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^3*e^4*sgn(c)*sgn(e) +
2*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*b^2*c*e^4*sgn(c)*sgn(e) + 2*sqrt(2*c^2*d*e + b*c*e^2)*s
qrt(-2*c^2*d*e + b*c*e^2)*b^2*c*e^4*sgn(c)*sgn(e) + 8*sqrt(-4*c^2*d^2 + b^2*e^2)*c^4*d^3*e^2 - 4*sqrt(-4*c^2*d
^2 + b^2*e^2)*b*c^3*d^2*e^3 + 4*sqrt(-2*c^2*d*e + b*c*e^2)*c^4*d^2*e^3 - 2*sqrt(-4*c^2*d^2 + b^2*e^2)*b^2*c^2*
d*e^4 + 4*sqrt(-4*c^2*d^2 + b^2*e^2)*b*c^3*d*e^4 + sqrt(-4*c^2*d^2 + b^2*e^2)*b^3*c*e^5 - 2*sqrt(-4*c^2*d^2 +
b^2*e^2)*b^2*c^2*e^5 - sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c^2*e^5 - 6*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e +
b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*c^2*d*e^2*sgn(c)*sgn(e) + 3*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*
c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c*e^3*sgn(c)*sgn(e) + 8*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e
^2)*c^3*d^3*e - 4*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*d^2*e^2 - 2*sqrt(-4*c^2*d^2 + b^
2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c*d*e^3 + 4*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*
d*e^3 + sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b^3*e^4 - 2*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c
^2*d*e + b*c*e^2)*b^2*c*e^4)*e^2 + 2*(8*sqrt(2*c^2*d*e + b*c*e^2)*c^4*d^4*e^2*sgn(c)*sgn(e) - 4*sqrt(2*c^2*d*e
+ b*c*e^2)*b*c^3*d^3*e^3*sgn(c)*sgn(e) + 4*sqrt(2*c^2*d*e + b*c*e^2)*c^4*d^3*e^3*sgn(c)*sgn(e) - 2*sqrt(2*c^2
*d*e + b*c*e^2)*b^2*c^2*d^2*e^4*sgn(c)*sgn(e) - 2*sqrt(2*c^2*d*e + b*c*e^2)*c^4*d^2*e^4*sgn(c)*sgn(e) + sqrt(2
*c^2*d*e + b*c*e^2)*b^3*c*d*e^5*sgn(c)*sgn(e) - sqrt(2*c^2*d*e + b*c*e^2)*b^2*c^2*d*e^5*sgn(c)*sgn(e) + sqrt(2
*c^2*d*e + b*c*e^2)*b*c^3*d*e^5*sgn(c)*sgn(e) - 24*c^5*d^4*e^3 + 12*b*c^4*d^3*e^4 + 6*b^2*c^3*d^2*e^5 - 3*b^3*
c^2*d*e^6 - 2*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*c^3*d^2*e^3*sgn(c)*sgn(e) - 2*sqrt(2*c^2*d*
e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*c^3*d^2*e^3*sgn(c)*sgn(e) + sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e
+ b*c*e^2)*b*c^2*d*e^4*sgn(c)*sgn(e) + sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*d*e^4*sgn(c)
*sgn(e) - 8*sqrt(-2*c^2*d*e + b*c*e^2)*c^4*d^4*e^2 + 4*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^3*d^3*e^3 + 4*sqrt(-2*c^
2*d*e + b*c*e^2)*c^4*d^3*e^3 + 2*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c^2*d^2*e^4 - 4*sqrt(-2*c^2*d*e + b*c*e^2)*b*c
^3*d^2*e^4 + 2*sqrt(-4*c^2*d^2 + b^2*e^2)*c^4*d^2*e^4 + 2*sqrt(-2*c^2*d*e + b*c*e^2)*c^4*d^2*e^4 - sqrt(-2*c^2
*d*e + b*c*e^2)*b^3*c*d*e^5 + sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c^2*d*e^5 - sqrt(-4*c^2*d^2 + b^2*e^2)*b*c^3*d*e^
5 - sqrt(-2*c^2*d*e + b*c*e^2)*b*c^3*d*e^5 - 6*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^
2*d*e + b*c*e^2)*c^2*d^2*e^2*sgn(c)*sgn(e) + 3*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(2*c^2*d*e + b*c*e^2)*sqrt(-2*c^
2*d*e + b*c*e^2)*b*c*d*e^3*sgn(c)*sgn(e) + 2*sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*c^3*d^2*e^3
- sqrt(-4*c^2*d^2 + b^2*e^2)*sqrt(-2*c^2*d*e + b*c*e^2)*b*c^2*d*e^4)*abs(e))*arctan(2*sqrt(1/2)*x/sqrt((b*e^2
- sqrt(-4*c^2*d^2*e^2 + b^2*e^4))/(c*e^2)))/((8*c^5*d^5 - 4*b*c^4*d^4*e - 2*b^2*c^3*d^3*e^2 + 4*b*c^4*d^3*e^2
- 2*c^5*d^3*e^2 + b^3*c^2*d^2*e^3 - 2*b^2*c^3*d^2*e^3 + b*c^4*d^2*e^3)*e^2*abs(c))

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.96 \[ \int \frac {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx=-\frac {e^{3/2}\,\left (\mathrm {atan}\left (\frac {\sqrt {e}\,x\,\sqrt {b\,c\,e-2\,c^2\,d}}{b\,e-2\,c\,d}\right )+\mathrm {atan}\left (\frac {c\,e^{3/2}\,x^3\,\sqrt {b\,c\,e-2\,c^2\,d}+b\,e^{3/2}\,x\,\sqrt {b\,c\,e-2\,c^2\,d}-c\,d\,\sqrt {e}\,x\,\sqrt {b\,c\,e-2\,c^2\,d}}{d\,\left (2\,c^2\,d-b\,c\,e\right )}\right )\right )}{\sqrt {b\,c\,e-2\,c^2\,d}} \]

[In]

int((d - e*x^2)/(b*x^2 + c*x^4 + (c*d^2)/e^2),x)

[Out]

-(e^(3/2)*(atan((e^(1/2)*x*(b*c*e - 2*c^2*d)^(1/2))/(b*e - 2*c*d)) + atan((c*e^(3/2)*x^3*(b*c*e - 2*c^2*d)^(1/
2) + b*e^(3/2)*x*(b*c*e - 2*c^2*d)^(1/2) - c*d*e^(1/2)*x*(b*c*e - 2*c^2*d)^(1/2))/(d*(2*c^2*d - b*c*e)))))/(b*
c*e - 2*c^2*d)^(1/2)