\(\int \frac {d+e x^2}{b x^2+c (\frac {d^2}{e^2}+x^4)} \, dx\) [36]

Optimal result

Integrand size = 29, antiderivative size = 130 \[ \int \frac {d+e x^2}{b x^2+c \left (\frac {d^2}{e^2}+x^4\right )} \, dx=-\frac {e^{3/2} \arctan \left (\frac {\sqrt {2 c d-b e}-2 \sqrt {c} \sqrt {e} x}{\sqrt {2 c d+b e}}\right )}{\sqrt {c} \sqrt {2 c d+b e}}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {2 c d-b e}+2 \sqrt {c} \sqrt {e} x}{\sqrt {2 c d+b e}}\right )}{\sqrt {c} \sqrt {2 c d+b e}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2015, 1175, 632, 210} \[ \int \frac {d+e x^2}{b x^2+c \left (\frac {d^2}{e^2}+x^4\right )} \, dx=\frac {e^{3/2} \arctan \left (\frac {\sqrt {2 c d-b e}+2 \sqrt {c} \sqrt {e} x}{\sqrt {b e+2 c d}}\right )}{\sqrt {c} \sqrt {b e+2 c d}}-\frac {e^{3/2} \arctan \left (\frac {\sqrt {2 c d-b e}-2 \sqrt {c} \sqrt {e} x}{\sqrt {b e+2 c d}}\right )}{\sqrt {c} \sqrt {b e+2 c d}} \]

[In]

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

[Out]

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

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 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1175

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), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/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[2*(d/e) - b/c, 0] || ( !Lt
Q[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rule 2015

Int[(u_)^(q_.)*(v_)^(p_.), x_Symbol] :> Int[ExpandToSum[u, x]^q*ExpandToSum[v, x]^p, x] /; FreeQ[{p, q}, x] &&
BinomialQ[u, x] && TrinomialQ[v, x] &&  !(BinomialMatchQ[u, x] && TrinomialMatchQ[v, x])

Rubi steps \begin{align*} \text {integral}& = \int \frac {d+e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx \\ & = \frac {e \int \frac {1}{\frac {d}{e}-\frac {\sqrt {2 c d-b e} x}{\sqrt {c} \sqrt {e}}+x^2} \, dx}{2 c}+\frac {e \int \frac {1}{\frac {d}{e}+\frac {\sqrt {2 c d-b e} x}{\sqrt {c} \sqrt {e}}+x^2} \, dx}{2 c} \\ & = -\frac {e \text {Subst}\left (\int \frac {1}{-\frac {b}{c}-\frac {2 d}{e}-x^2} \, dx,x,-\frac {\sqrt {2 c d-b e}}{\sqrt {c} \sqrt {e}}+2 x\right )}{c}-\frac {e \text {Subst}\left (\int \frac {1}{-\frac {b}{c}-\frac {2 d}{e}-x^2} \, dx,x,\frac {\sqrt {2 c d-b e}}{\sqrt {c} \sqrt {e}}+2 x\right )}{c} \\ & = -\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2 c d-b e}-2 \sqrt {c} \sqrt {e} x}{\sqrt {2 c d+b e}}\right )}{\sqrt {c} \sqrt {2 c d+b e}}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2 c d-b e}+2 \sqrt {c} \sqrt {e} x}{\sqrt {2 c d+b e}}\right )}{\sqrt {c} \sqrt {2 c d+b e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.91 \[ \int \frac {d+e x^2}{b x^2+c \left (\frac {d^2}{e^2}+x^4\right )} \, dx=\frac {e^{3/2} \left (\frac {\left (2 c d-b e+\sqrt {-4 c^2 d^2+b^2 e^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {e} x}{\sqrt {b e-\sqrt {-4 c^2 d^2+b^2 e^2}}}\right )}{\sqrt {b e-\sqrt {-4 c^2 d^2+b^2 e^2}}}+\frac {\left (-2 c d+b e+\sqrt {-4 c^2 d^2+b^2 e^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {e} x}{\sqrt {b e+\sqrt {-4 c^2 d^2+b^2 e^2}}}\right )}{\sqrt {b e+\sqrt {-4 c^2 d^2+b^2 e^2}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {-4 c^2 d^2+b^2 e^2}} \]

[In]

Integrate[(d + e*x^2)/(b*x^2 + c*(d^2/e^2 + 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])*ArcT
an[(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 = 0.05 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.92

method	result	size
risch	\(\frac {\sqrt {-c \left (b e +2 c d \right ) e}\, e \ln \left (-c \,x^{2} e -\sqrt {-c \left (b e +2 c d \right ) e}\, x +c d \right )}{2 c \left (b e +2 c d \right )}-\frac {\sqrt {-c \left (b e +2 c d \right ) e}\, e \ln \left (-c \,x^{2} e +\sqrt {-c \left (b e +2 c d \right ) e}\, x +c d \right )}{2 c \left (b e +2 c d \right )}\)	\(119\)
default	\(4 e^{4} c \left (\frac {\left (b \,e^{2}-2 d c e +\sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\right ) \sqrt {2}\, \arctan \left (\frac {c x e \sqrt {2}}{\sqrt {c \left (b \,e^{2}+\sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\right )}}\right )}{8 \sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\, c \,e^{2} \sqrt {c \left (b \,e^{2}+\sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\right )}}-\frac {\left (-b \,e^{2}+2 d c e +\sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x e \sqrt {2}}{\sqrt {c \left (-b \,e^{2}+\sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\right )}}\right )}{8 \sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\, c \,e^{2} \sqrt {c \left (-b \,e^{2}+\sqrt {e^{2} \left (b e -2 c d \right ) \left (b e +2 c d \right )}\right )}}\right )\)	\(287\)

[In]

int((e*x^2+d)/(b*x^2+c*(d^2/e^2+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 = 0.26 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.78 \[ \int \frac {d+e x^2}{b x^2+c \left (\frac {d^2}{e^2}+x^4\right )} \, dx=\left [\frac {1}{2} \, e \sqrt {-\frac {e}{2 \, c^{2} d + b c e}} \log \left (\frac {c e^{2} x^{4} + c d^{2} - {\left (4 \, c d e + b e^{2}\right )} x^{2} + 2 \, {\left ({\left (2 \, c^{2} d e + b c e^{2}\right )} x^{3} - {\left (2 \, c^{2} d^{2} + b c d e\right )} x\right )} \sqrt {-\frac {e}{2 \, c^{2} d + b c e}}}{c e^{2} x^{4} + b e^{2} x^{2} + c d^{2}}\right ), e \sqrt {\frac {e}{2 \, c^{2} d + b c e}} \arctan \left (c x \sqrt {\frac {e}{2 \, c^{2} d + b c e}}\right ) + e \sqrt {\frac {e}{2 \, c^{2} d + b c e}} \arctan \left (\frac {{\left (c e x^{3} + {\left (c d + b e\right )} x\right )} \sqrt {\frac {e}{2 \, c^{2} d + b c e}}}{d}\right )\right ] \]

[In]

integrate((e*x^2+d)/(b*x^2+c*(d^2/e^2+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 = 0.38 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.23 \[ \int \frac {d+e x^2}{b x^2+c \left (\frac {d^2}{e^2}+x^4\right )} \, dx=- \frac {\sqrt {- \frac {e^{3}}{c \left (b e + 2 c d\right )}} \log {\left (- \frac {d}{e} + x^{2} + \frac {x \left (- b e \sqrt {- \frac {e^{3}}{c \left (b e + 2 c d\right )}} - 2 c d \sqrt {- \frac {e^{3}}{c \left (b e + 2 c d\right )}}\right )}{e^{2}} \right )}}{2} + \frac {\sqrt {- \frac {e^{3}}{c \left (b e + 2 c d\right )}} \log {\left (- \frac {d}{e} + x^{2} + \frac {x \left (b e \sqrt {- \frac {e^{3}}{c \left (b e + 2 c d\right )}} + 2 c d \sqrt {- \frac {e^{3}}{c \left (b e + 2 c d\right )}}\right )}{e^{2}} \right )}}{2} \]

[In]

integrate((e*x**2+d)/(b*x**2+c*(d**2/e**2+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]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6341 vs. \(2 (105) = 210\).

Time = 1.10 (sec) , antiderivative size = 6341, normalized size of antiderivative = 48.78 \[ \int \frac {d+e x^2}{b x^2+c \left (\frac {d^2}{e^2}+x^4\right )} \, dx=\text {Too large to display} \]

[In]

integrate((e*x^2+d)/(b*x^2+c*(d^2/e^2+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
) - 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)*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c*e^4*sgn(c)*s
gn(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 - 4*sqrt(-2*c^2*d*e + b*c*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) + 4*sqrt(2*c^2*d*e + b*c*e^2)*b*c^3*d^2*e^4*sgn(c)*sgn(e) - 2*sqr
t(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*s
gn(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 - 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) - 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)*s
gn(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)*sqrt(-2*c^2*d*e + b*c*e^2)*b^2*c*e^4*sgn(c)*sgn(e) + 8*sq
rt(-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 + 4*sqr
t(-2*c^2*d*e + b*c*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) + 4*sqrt(2*c^2*d*e + b*c*e^2)*b*c^3*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 + 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 = 13.83 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.78 \[ \int \frac {d+e x^2}{b x^2+c \left (\frac {d^2}{e^2}+x^4\right )} \, dx=\frac {e^{3/2}\,\left (\mathrm {atan}\left (\frac {c\,\sqrt {e}\,x}{\sqrt {c\,\left (b\,e+2\,c\,d\right )}}\right )-\mathrm {atan}\left (\frac {\left (2\,d\,c^2+b\,e\,c\right )\,\left (x\,\left (\frac {\sqrt {e}\,\left (c\,d\,e^7-\frac {4\,c^3\,d^2\,e^7}{2\,d\,c^2+b\,e\,c}\right )}{d\,\sqrt {c\,\left (b\,e+2\,c\,d\right )}\,\left (b\,e-2\,c\,d\right )}+\frac {e^{3/2}\,\left (2\,c^2\,d\,e^6-b\,c\,e^7\right )}{c\,d\,\sqrt {2\,d\,c^2+b\,e\,c}\,\left (b\,e-2\,c\,d\right )}\right )+\frac {\sqrt {e}\,x^3\,\left (c\,e^8-\frac {2\,b\,c^2\,e^9}{2\,d\,c^2+b\,e\,c}\right )}{d\,\sqrt {c\,\left (b\,e+2\,c\,d\right )}\,\left (b\,e-2\,c\,d\right )}\right )}{c\,e^7}\right )\right )}{\sqrt {2\,d\,c^2+b\,e\,c}} \]

[In]

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

[Out]

(e^(3/2)*(atan((c*e^(1/2)*x)/(c*(b*e + 2*c*d))^(1/2)) - atan(((2*c^2*d + b*c*e)*(x*((e^(1/2)*(c*d*e^7 - (4*c^3
*d^2*e^7)/(2*c^2*d + b*c*e)))/(d*(c*(b*e + 2*c*d))^(1/2)*(b*e - 2*c*d)) + (e^(3/2)*(2*c^2*d*e^6 - b*c*e^7))/(c
*d*(2*c^2*d + b*c*e)^(1/2)*(b*e - 2*c*d))) + (e^(1/2)*x^3*(c*e^8 - (2*b*c^2*e^9)/(2*c^2*d + b*c*e)))/(d*(c*(b*
e + 2*c*d))^(1/2)*(b*e - 2*c*d))))/(c*e^7))))/(2*c^2*d + b*c*e)^(1/2)