3.2.60 \(\int \frac {1}{(d+e x^2) (-c d^2+b d e+b e^2 x^2+c e^2 x^4)} \, dx\)

Optimal. Leaf size=136 \[ -\frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^2}-\frac {(4 c d-b e) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} (2 c d-b e)^2}-\frac {x}{2 d \left (d+e x^2\right ) (2 c d-b e)} \]

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Rubi [A]  time = 0.18, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {1149, 414, 522, 205, 208} \begin {gather*} -\frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^2}-\frac {(4 c d-b e) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} (2 c d-b e)^2}-\frac {x}{2 d \left (d+e x^2\right ) (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x^2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

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

Rule 205

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

Rule 208

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

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 1149

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p +
q)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2
, 0] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 133, normalized size = 0.98 \begin {gather*} \frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {b e-c d}}\right )}{\sqrt {e} (b e-2 c d)^2 \sqrt {b e-c d}}+\frac {(b e-4 c d) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} (2 c d-b e)^2}-\frac {x}{2 d \left (d+e x^2\right ) (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x^2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

-1/2*x/(d*(2*c*d - b*e)*(d + e*x^2)) + ((-4*c*d + b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[e]*(2*c*d
- b*e)^2) + (c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(Sqrt[e]*(-2*c*d + b*e)^2*Sqrt[-(c*d) + b
*e])

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((d + e*x^2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

IntegrateAlgebraic[1/((d + e*x^2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)), x]

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fricas [A]  time = 1.29, size = 895, normalized size = 6.58 \begin {gather*} \left [\frac {2 \, {\left (c d^{2} e^{2} x^{2} + c d^{3} e\right )} \sqrt {\frac {c}{c d e - b e^{2}}} \log \left (\frac {c e x^{2} - 2 \, {\left (c d e - b e^{2}\right )} x \sqrt {\frac {c}{c d e - b e^{2}}} + c d - b e}{c e x^{2} - c d + b e}\right ) + {\left (4 \, c d^{2} - b d e + {\left (4 \, c d e - b e^{2}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} x}{4 \, {\left (4 \, c^{2} d^{5} e - 4 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3} + {\left (4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2}\right )}}, -\frac {{\left (4 \, c d^{2} - b d e + {\left (4 \, c d e - b e^{2}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - {\left (c d^{2} e^{2} x^{2} + c d^{3} e\right )} \sqrt {\frac {c}{c d e - b e^{2}}} \log \left (\frac {c e x^{2} - 2 \, {\left (c d e - b e^{2}\right )} x \sqrt {\frac {c}{c d e - b e^{2}}} + c d - b e}{c e x^{2} - c d + b e}\right ) + {\left (2 \, c d^{2} e - b d e^{2}\right )} x}{2 \, {\left (4 \, c^{2} d^{5} e - 4 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3} + {\left (4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2}\right )}}, \frac {4 \, {\left (c d^{2} e^{2} x^{2} + c d^{3} e\right )} \sqrt {-\frac {c}{c d e - b e^{2}}} \arctan \left (e x \sqrt {-\frac {c}{c d e - b e^{2}}}\right ) + {\left (4 \, c d^{2} - b d e + {\left (4 \, c d e - b e^{2}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} x}{4 \, {\left (4 \, c^{2} d^{5} e - 4 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3} + {\left (4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2}\right )}}, \frac {2 \, {\left (c d^{2} e^{2} x^{2} + c d^{3} e\right )} \sqrt {-\frac {c}{c d e - b e^{2}}} \arctan \left (e x \sqrt {-\frac {c}{c d e - b e^{2}}}\right ) - {\left (4 \, c d^{2} - b d e + {\left (4 \, c d e - b e^{2}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - {\left (2 \, c d^{2} e - b d e^{2}\right )} x}{2 \, {\left (4 \, c^{2} d^{5} e - 4 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3} + {\left (4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")

[Out]

[1/4*(2*(c*d^2*e^2*x^2 + c*d^3*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 - 2*(c*d*e - b*e^2)*x*sqrt(c/(c*d*e - b
*e^2)) + c*d - b*e)/(c*e*x^2 - c*d + b*e)) + (4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*sqrt(-d*e)*log((e*x^2 -
2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - 2*(2*c*d^2*e - b*d*e^2)*x)/(4*c^2*d^5*e - 4*b*c*d^4*e^2 + b^2*d^3*e^3 + (4
*c^2*d^4*e^2 - 4*b*c*d^3*e^3 + b^2*d^2*e^4)*x^2), -1/2*((4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*sqrt(d*e)*ar
ctan(sqrt(d*e)*x/d) - (c*d^2*e^2*x^2 + c*d^3*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 - 2*(c*d*e - b*e^2)*x*sqr
t(c/(c*d*e - b*e^2)) + c*d - b*e)/(c*e*x^2 - c*d + b*e)) + (2*c*d^2*e - b*d*e^2)*x)/(4*c^2*d^5*e - 4*b*c*d^4*e
^2 + b^2*d^3*e^3 + (4*c^2*d^4*e^2 - 4*b*c*d^3*e^3 + b^2*d^2*e^4)*x^2), 1/4*(4*(c*d^2*e^2*x^2 + c*d^3*e)*sqrt(-
c/(c*d*e - b*e^2))*arctan(e*x*sqrt(-c/(c*d*e - b*e^2))) + (4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*sqrt(-d*e)
*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - 2*(2*c*d^2*e - b*d*e^2)*x)/(4*c^2*d^5*e - 4*b*c*d^4*e^2 + b^2
*d^3*e^3 + (4*c^2*d^4*e^2 - 4*b*c*d^3*e^3 + b^2*d^2*e^4)*x^2), 1/2*(2*(c*d^2*e^2*x^2 + c*d^3*e)*sqrt(-c/(c*d*e
- b*e^2))*arctan(e*x*sqrt(-c/(c*d*e - b*e^2))) - (4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*sqrt(d*e)*arctan(s
qrt(d*e)*x/d) - (2*c*d^2*e - b*d*e^2)*x)/(4*c^2*d^5*e - 4*b*c*d^4*e^2 + b^2*d^3*e^3 + (4*c^2*d^4*e^2 - 4*b*c*d
^3*e^3 + b^2*d^2*e^4)*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [b,c,d,exp(1),exp(2)]=[-95,-68,60,-66,8]Warning, need to choose a branch for the root of a polyn
omial with parameters. This might be wrong.The choice was done assuming [b,c,d,exp(1),exp(2)]=[79,32,2,-92,39]
sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(cons
t gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const ind
ex_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vect
eur & l) Error: Bad Argument ValueEvaluation time: 10.77Done

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maple [A]  time = 0.01, size = 155, normalized size = 1.14 \begin {gather*} \frac {c^{2} \arctan \left (\frac {c e x}{\sqrt {\left (b e -c d \right ) c e}}\right )}{\left (b e -2 c d \right )^{2} \sqrt {\left (b e -c d \right ) c e}}+\frac {b e x}{2 \left (b e -2 c d \right )^{2} \left (e \,x^{2}+d \right ) d}+\frac {b e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \left (b e -2 c d \right )^{2} \sqrt {d e}\, d}-\frac {c x}{\left (b e -2 c d \right )^{2} \left (e \,x^{2}+d \right )}-\frac {2 c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\left (b e -2 c d \right )^{2} \sqrt {d e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

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mupad [B]  time = 5.40, size = 3901, normalized size = 28.68

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x^2)*(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e)),x)

[Out]

- x/(2*(d + e*x^2)*(2*c*d^2 - b*d*e)) - (atan(((((((96*c^7*d^6*e^6 - 224*b*c^6*d^5*e^7 - 2*b^5*c^2*d*e^11 + 20
8*b^2*c^5*d^4*e^8 - 96*b^3*c^4*d^3*e^9 + 22*b^4*c^3*d^2*e^10)/(2*(8*c^3*d^5 - b^3*d^2*e^3 + 6*b^2*c*d^3*e^2 -
12*b*c^2*d^4*e)) - (x*(-c^3*e*(b*e - c*d))^(1/2)*(256*b*c^6*d^6*e^8 - 512*b^2*c^5*d^5*e^9 + 384*b^3*c^4*d^4*e^
10 - 128*b^4*c^3*d^3*e^11 + 16*b^5*c^2*d^2*e^12))/(8*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)*(b^3*e^4 - 4*c^3*
d^3*e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^3)))*(-c^3*e*(b*e - c*d))^(1/2))/(2*(b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2
*e^2 - 5*b^2*c*d*e^3)) - (x*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4*d*e^7))/(4*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*
c*d^3*e)))*(-c^3*e*(b*e - c*d))^(1/2)*1i)/(b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^3) - (((((96*
c^7*d^6*e^6 - 224*b*c^6*d^5*e^7 - 2*b^5*c^2*d*e^11 + 208*b^2*c^5*d^4*e^8 - 96*b^3*c^4*d^3*e^9 + 22*b^4*c^3*d^2
*e^10)/(2*(8*c^3*d^5 - b^3*d^2*e^3 + 6*b^2*c*d^3*e^2 - 12*b*c^2*d^4*e)) + (x*(-c^3*e*(b*e - c*d))^(1/2)*(256*b
*c^6*d^6*e^8 - 512*b^2*c^5*d^5*e^9 + 384*b^3*c^4*d^4*e^10 - 128*b^4*c^3*d^3*e^11 + 16*b^5*c^2*d^2*e^12))/(8*(4
*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)*(b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^3)))*(-c^3*e*(b*e
- c*d))^(1/2))/(2*(b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^3)) + (x*(b^2*c^3*e^8 + 20*c^5*d^2*e
^6 - 8*b*c^4*d*e^7))/(4*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)))*(-c^3*e*(b*e - c*d))^(1/2)*1i)/(b^3*e^4 - 4*
c^3*d^3*e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^3))/((((((96*c^7*d^6*e^6 - 224*b*c^6*d^5*e^7 - 2*b^5*c^2*d*e^11 + 20
8*b^2*c^5*d^4*e^8 - 96*b^3*c^4*d^3*e^9 + 22*b^4*c^3*d^2*e^10)/(2*(8*c^3*d^5 - b^3*d^2*e^3 + 6*b^2*c*d^3*e^2 -
12*b*c^2*d^4*e)) - (x*(-c^3*e*(b*e - c*d))^(1/2)*(256*b*c^6*d^6*e^8 - 512*b^2*c^5*d^5*e^9 + 384*b^3*c^4*d^4*e^
10 - 128*b^4*c^3*d^3*e^11 + 16*b^5*c^2*d^2*e^12))/(8*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)*(b^3*e^4 - 4*c^3*
d^3*e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^3)))*(-c^3*e*(b*e - c*d))^(1/2))/(2*(b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2
*e^2 - 5*b^2*c*d*e^3)) - (x*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4*d*e^7))/(4*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*
c*d^3*e)))*(-c^3*e*(b*e - c*d))^(1/2))/(b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^3) - ((b*c^4*e^6
)/2 - 2*c^5*d*e^5)/(8*c^3*d^5 - b^3*d^2*e^3 + 6*b^2*c*d^3*e^2 - 12*b*c^2*d^4*e) + (((((96*c^7*d^6*e^6 - 224*b*
c^6*d^5*e^7 - 2*b^5*c^2*d*e^11 + 208*b^2*c^5*d^4*e^8 - 96*b^3*c^4*d^3*e^9 + 22*b^4*c^3*d^2*e^10)/(2*(8*c^3*d^5
- b^3*d^2*e^3 + 6*b^2*c*d^3*e^2 - 12*b*c^2*d^4*e)) + (x*(-c^3*e*(b*e - c*d))^(1/2)*(256*b*c^6*d^6*e^8 - 512*b
^2*c^5*d^5*e^9 + 384*b^3*c^4*d^4*e^10 - 128*b^4*c^3*d^3*e^11 + 16*b^5*c^2*d^2*e^12))/(8*(4*c^2*d^4 + b^2*d^2*e
^2 - 4*b*c*d^3*e)*(b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^3)))*(-c^3*e*(b*e - c*d))^(1/2))/(2*(
b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^3)) + (x*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4*d*e^7))
/(4*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)))*(-c^3*e*(b*e - c*d))^(1/2))/(b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2
*e^2 - 5*b^2*c*d*e^3)))*(-c^3*e*(b*e - c*d))^(1/2)*1i)/(b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^
3) - (atan(((((x*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4*d*e^7))/(2*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)) -
((-d^3*e)^(1/2)*((96*c^7*d^6*e^6 - 224*b*c^6*d^5*e^7 - 2*b^5*c^2*d*e^11 + 208*b^2*c^5*d^4*e^8 - 96*b^3*c^4*d^
3*e^9 + 22*b^4*c^3*d^2*e^10)/(8*c^3*d^5 - b^3*d^2*e^3 + 6*b^2*c*d^3*e^2 - 12*b*c^2*d^4*e) - (x*(-d^3*e)^(1/2)*
(b*e - 4*c*d)*(256*b*c^6*d^6*e^8 - 512*b^2*c^5*d^5*e^9 + 384*b^3*c^4*d^4*e^10 - 128*b^4*c^3*d^3*e^11 + 16*b^5*
c^2*d^2*e^12))/(8*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)))*(b*e -
4*c*d))/(4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)))*(-d^3*e)^(1/2)*(b*e - 4*c*d)*1i)/(4*(4*c^2*d^5*e + b
^2*d^3*e^3 - 4*b*c*d^4*e^2)) + (((x*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4*d*e^7))/(2*(4*c^2*d^4 + b^2*d^2*e^
2 - 4*b*c*d^3*e)) + ((-d^3*e)^(1/2)*((96*c^7*d^6*e^6 - 224*b*c^6*d^5*e^7 - 2*b^5*c^2*d*e^11 + 208*b^2*c^5*d^4*
e^8 - 96*b^3*c^4*d^3*e^9 + 22*b^4*c^3*d^2*e^10)/(8*c^3*d^5 - b^3*d^2*e^3 + 6*b^2*c*d^3*e^2 - 12*b*c^2*d^4*e) +
(x*(-d^3*e)^(1/2)*(b*e - 4*c*d)*(256*b*c^6*d^6*e^8 - 512*b^2*c^5*d^5*e^9 + 384*b^3*c^4*d^4*e^10 - 128*b^4*c^3
*d^3*e^11 + 16*b^5*c^2*d^2*e^12))/(8*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*
c*d^4*e^2)))*(b*e - 4*c*d))/(4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)))*(-d^3*e)^(1/2)*(b*e - 4*c*d)*1i)/
(4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)))/(((b*c^4*e^6)/2 - 2*c^5*d*e^5)/(8*c^3*d^5 - b^3*d^2*e^3 + 6*b
^2*c*d^3*e^2 - 12*b*c^2*d^4*e) + (((x*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4*d*e^7))/(2*(4*c^2*d^4 + b^2*d^2*
e^2 - 4*b*c*d^3*e)) - ((-d^3*e)^(1/2)*((96*c^7*d^6*e^6 - 224*b*c^6*d^5*e^7 - 2*b^5*c^2*d*e^11 + 208*b^2*c^5*d^
4*e^8 - 96*b^3*c^4*d^3*e^9 + 22*b^4*c^3*d^2*e^10)/(8*c^3*d^5 - b^3*d^2*e^3 + 6*b^2*c*d^3*e^2 - 12*b*c^2*d^4*e)
- (x*(-d^3*e)^(1/2)*(b*e - 4*c*d)*(256*b*c^6*d^6*e^8 - 512*b^2*c^5*d^5*e^9 + 384*b^3*c^4*d^4*e^10 - 128*b^4*c
^3*d^3*e^11 + 16*b^5*c^2*d^2*e^12))/(8*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*
b*c*d^4*e^2)))*(b*e - 4*c*d))/(4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)))*(-d^3*e)^(1/2)*(b*e - 4*c*d))/(
4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)) - (((x*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4*d*e^7))/(2*(4*c^
2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)) + ((-d^3*e)^(1/2)*((96*c^7*d^6*e^6 - 224*b*c^6*d^5*e^7 - 2*b^5*c^2*d*e^11
+ 208*b^2*c^5*d^4*e^8 - 96*b^3*c^4*d^3*e^9 + 22*b^4*c^3*d^2*e^10)/(8*c^3*d^5 - b^3*d^2*e^3 + 6*b^2*c*d^3*e^2 -
12*b*c^2*d^4*e) + (x*(-d^3*e)^(1/2)*(b*e - 4*c*d)*(256*b*c^6*d^6*e^8 - 512*b^2*c^5*d^5*e^9 + 384*b^3*c^4*d^4*
e^10 - 128*b^4*c^3*d^3*e^11 + 16*b^5*c^2*d^2*e^12))/(8*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)*(4*c^2*d^5*e +
b^2*d^3*e^3 - 4*b*c*d^4*e^2)))*(b*e - 4*c*d))/(4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)))*(-d^3*e)^(1/2)*
(b*e - 4*c*d))/(4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2))))*(-d^3*e)^(1/2)*(b*e - 4*c*d)*1i)/(2*(4*c^2*d^
5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)

[Out]

Timed out

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