3.2.88 \(\int \frac {(d+e x^2)^2}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=238 \[ \frac {\left (\frac {-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt {b^2-4 a c}}+e (2 c d-b e)\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (e (2 c d-b e)-\frac {-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} c^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {e^2 x}{c} \]

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Rubi [A]  time = 0.64, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1170, 1166, 205} \begin {gather*} \frac {\left (\frac {-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt {b^2-4 a c}}+e (2 c d-b e)\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (e (2 c d-b e)-\frac {-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} c^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {e^2 x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1170

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

Rubi steps

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

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Mathematica [A]  time = 0.32, size = 269, normalized size = 1.13 \begin {gather*} \frac {\frac {\sqrt {2} \left (-2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+2 c^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (-2 c e \left (d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}+2 \sqrt {c} e^2 x}{2 c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 3.22, size = 4690, normalized size = 19.71

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*e^2*x - sqrt(1/2)*c*sqrt(-(b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b^2*c - 2*a^2*c^2)*d*e^
3 + (a*b^3 - 3*a^2*b*c)*e^4 + (a*b^2*c^3 - 4*a^2*c^4)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 4
8*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*
c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))/(a*b^2*c^3 - 4*a^2*c
^4))*log(2*(c^5*d^8 - 2*b*c^4*d^7*e + 14*a*b*c^3*d^5*e^3 + (b^2*c^3 - 4*a*c^4)*d^6*e^2 - 5*(3*a*b^2*c^2 + 2*a^
2*c^3)*d^4*e^4 + 6*(a*b^3*c + 3*a^2*b*c^2)*d^3*e^5 - (a*b^4 + 9*a^2*b^2*c + 4*a^3*c^2)*d^2*e^6 + 2*(a^2*b^3 +
a^3*b*c)*d*e^7 - (a^3*b^2 - a^4*c)*e^8)*x + sqrt(1/2)*((b^2*c^4 - 4*a*c^5)*d^6 - 7*(a*b^2*c^3 - 4*a^2*c^4)*d^4
*e^2 + 4*(a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^3 - (a*b^4*c - 11*a^2*b^2*c^2 + 28*a^3*c^3)*d^2*e^4 - 4*(a^2*b^3*c -
4*a^3*b*c^2)*d*e^5 + (a^2*b^4 - 5*a^3*b^2*c + 4*a^4*c^2)*e^6 - ((a*b^3*c^4 - 4*a^2*b*c^5)*d^2 - 4*(a^2*b^2*c^4
- 4*a^3*c^5)*d*e + (a^2*b^3*c^3 - 4*a^3*b*c^4)*e^2)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48
*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c
- a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))*sqrt(-(b*c^3*d^4 - 8*
a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b^2*c - 2*a^2*c^2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e^4 + (a*b^2*c^3 - 4*a^2
*c^4)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)
*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c +
a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))/(a*b^2*c^3 - 4*a^2*c^4))) + sqrt(1/2)*c*sqrt(-(b*c^3*d^4 - 8*a*c^3*d
^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b^2*c - 2*a^2*c^2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e^4 + (a*b^2*c^3 - 4*a^2*c^4)*s
qrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^
4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2
)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))/(a*b^2*c^3 - 4*a^2*c^4))*log(2*(c^5*d^8 - 2*b*c^4*d^7*e + 14*a*b*c^3*d^5*e^
3 + (b^2*c^3 - 4*a*c^4)*d^6*e^2 - 5*(3*a*b^2*c^2 + 2*a^2*c^3)*d^4*e^4 + 6*(a*b^3*c + 3*a^2*b*c^2)*d^3*e^5 - (a
*b^4 + 9*a^2*b^2*c + 4*a^3*c^2)*d^2*e^6 + 2*(a^2*b^3 + a^3*b*c)*d*e^7 - (a^3*b^2 - a^4*c)*e^8)*x - sqrt(1/2)*(
(b^2*c^4 - 4*a*c^5)*d^6 - 7*(a*b^2*c^3 - 4*a^2*c^4)*d^4*e^2 + 4*(a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^3 - (a*b^4*c -
11*a^2*b^2*c^2 + 28*a^3*c^3)*d^2*e^4 - 4*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^5 + (a^2*b^4 - 5*a^3*b^2*c + 4*a^4*c^2
)*e^6 - ((a*b^3*c^4 - 4*a^2*b*c^5)*d^2 - 4*(a^2*b^2*c^4 - 4*a^3*c^5)*d*e + (a^2*b^3*c^3 - 4*a^3*b*c^4)*e^2)*sq
rt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4
+ 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)
*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))*sqrt(-(b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b^2*c - 2*a^2*c^
2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e^4 + (a*b^2*c^3 - 4*a^2*c^4)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*
e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a
^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))/(a*b^2*c^3 -
4*a^2*c^4))) - sqrt(1/2)*c*sqrt(-(b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b^2*c - 2*a^2*c^2)*d*e^
3 + (a*b^3 - 3*a^2*b*c)*e^4 - (a*b^2*c^3 - 4*a^2*c^4)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 4
8*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*
c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))/(a*b^2*c^3 - 4*a^2*c
^4))*log(2*(c^5*d^8 - 2*b*c^4*d^7*e + 14*a*b*c^3*d^5*e^3 + (b^2*c^3 - 4*a*c^4)*d^6*e^2 - 5*(3*a*b^2*c^2 + 2*a^
2*c^3)*d^4*e^4 + 6*(a*b^3*c + 3*a^2*b*c^2)*d^3*e^5 - (a*b^4 + 9*a^2*b^2*c + 4*a^3*c^2)*d^2*e^6 + 2*(a^2*b^3 +
a^3*b*c)*d*e^7 - (a^3*b^2 - a^4*c)*e^8)*x + sqrt(1/2)*((b^2*c^4 - 4*a*c^5)*d^6 - 7*(a*b^2*c^3 - 4*a^2*c^4)*d^4
*e^2 + 4*(a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^3 - (a*b^4*c - 11*a^2*b^2*c^2 + 28*a^3*c^3)*d^2*e^4 - 4*(a^2*b^3*c -
4*a^3*b*c^2)*d*e^5 + (a^2*b^4 - 5*a^3*b^2*c + 4*a^4*c^2)*e^6 + ((a*b^3*c^4 - 4*a^2*b*c^5)*d^2 - 4*(a^2*b^2*c^4
- 4*a^3*c^5)*d*e + (a^2*b^3*c^3 - 4*a^3*b*c^4)*e^2)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48
*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c
- a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))*sqrt(-(b*c^3*d^4 - 8*
a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b^2*c - 2*a^2*c^2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e^4 - (a*b^2*c^3 - 4*a^2
*c^4)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)
*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c +
a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))/(a*b^2*c^3 - 4*a^2*c^4))) + sqrt(1/2)*c*sqrt(-(b*c^3*d^4 - 8*a*c^3*d
^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b^2*c - 2*a^2*c^2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e^4 - (a*b^2*c^3 - 4*a^2*c^4)*s
qrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^
4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2
)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))/(a*b^2*c^3 - 4*a^2*c^4))*log(2*(c^5*d^8 - 2*b*c^4*d^7*e + 14*a*b*c^3*d^5*e^
3 + (b^2*c^3 - 4*a*c^4)*d^6*e^2 - 5*(3*a*b^2*c^2 + 2*a^2*c^3)*d^4*e^4 + 6*(a*b^3*c + 3*a^2*b*c^2)*d^3*e^5 - (a
*b^4 + 9*a^2*b^2*c + 4*a^3*c^2)*d^2*e^6 + 2*(a^2*b^3 + a^3*b*c)*d*e^7 - (a^3*b^2 - a^4*c)*e^8)*x - sqrt(1/2)*(
(b^2*c^4 - 4*a*c^5)*d^6 - 7*(a*b^2*c^3 - 4*a^2*c^4)*d^4*e^2 + 4*(a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^3 - (a*b^4*c -
11*a^2*b^2*c^2 + 28*a^3*c^3)*d^2*e^4 - 4*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^5 + (a^2*b^4 - 5*a^3*b^2*c + 4*a^4*c^2
)*e^6 + ((a*b^3*c^4 - 4*a^2*b*c^5)*d^2 - 4*(a^2*b^2*c^4 - 4*a^3*c^5)*d*e + (a^2*b^3*c^3 - 4*a^3*b*c^4)*e^2)*sq
rt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4
+ 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)
*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))*sqrt(-(b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b^2*c - 2*a^2*c^
2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e^4 - (a*b^2*c^3 - 4*a^2*c^4)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*
e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a
^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))/(a*b^2*c^3 -
4*a^2*c^4))))/c

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giac [B]  time = 1.14, size = 4107, normalized size = 17.26

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^2/c + 1/8*(2*(2*b^4*c^3 - 16*a*b^2*c^4 + 32*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*
c)*c)*b^4*c + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a
*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2
*c^3 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*b^2*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^4 - 2*(b^2
- 4*a*c)*b^2*c^3 + 8*(b^2 - 4*a*c)*a*c^4)*c^2*d*e + 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c^3 - 8*sqr
t(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^4 + 2*b^4*c^4
+ 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*c^5 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^5 + sq
rt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^5 - 16*a*b^2*c^5 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^6
+ 32*a^2*c^6 - 2*(b^2 - 4*a*c)*b^2*c^4 + 8*(b^2 - 4*a*c)*a*c^5)*d^2*abs(c) - (2*b^5*c^2 - 16*a*b^3*c^3 + 32*a
^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c - 16*sqrt
(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(
b^2 - 4*a*c)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt
(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*b^3*c^2 + 8*(b^2 - 4*a*c)*a*b*c^3)*c^2
*e^2 + 2*(2*b^3*c^6 - 8*a*b*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^4 + 4*sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^5 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2
- 4*a*c)*c)*b^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b*c^6 - 2*(b^2 - 4*a*c)*b*c^6)
*d^2 - 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^
2*c^3 - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + 2*a*b^4*c^3 + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4
*a*c)*c)*a^3*c^4 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*
c)*a*b^2*c^4 - 16*a^2*b^2*c^4 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*c^5 + 32*a^3*c^5 - 2*(b^2 - 4*a*
c)*a*b^2*c^3 + 8*(b^2 - 4*a*c)*a^2*c^4)*abs(c)*e^2 - 2*(2*b^4*c^5 - 8*a*b^2*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sq
rt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4
+ 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - s
qrt(b^2 - 4*a*c)*c)*b^2*c^5 - 2*(b^2 - 4*a*c)*b^2*c^5)*d*e + (2*b^5*c^4 - 12*a*b^3*c^5 + 16*a^2*b*c^6 - sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2
- 4*a*c)*c)*a*b^3*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c^3 - 8*sqrt(2)*sqrt(b
^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*
c)*c)*a*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a
*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^5 - 2*(b^2 - 4*a*c)*b^3*c^4 + 4*(b^2 - 4*a*c)*a*b*c^5)*e^2)*arctan(2
*sqrt(1/2)*x/sqrt((b*c + sqrt(b^2*c^2 - 4*a*c^3))/c^2))/((a*b^4*c^3 - 8*a^2*b^2*c^4 - 2*a*b^3*c^4 + 16*a^3*c^5
+ 8*a^2*b*c^5 + a*b^2*c^5 - 4*a^2*c^6)*c^2) - 1/8*(2*(2*b^4*c^3 - 16*a*b^2*c^4 + 32*a^2*c^5 - sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)
*a*b^2*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c
)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^
3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a*c^4 - 2*(b^2 - 4*a*c)*b^2*c^3 + 8*(b^2 - 4*a*c)*a*c^4)*c^2*d*e - 2*(sqrt(2)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*b^4*c^3 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - 2*sqrt(2)*sqrt(b*c + sqr
t(b^2 - 4*a*c)*c)*b^3*c^4 - 2*b^4*c^4 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^5 + 8*sqrt(2)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a*b*c^5 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^5 + 16*a*b^2*c^5 - 4*sqrt(2)*
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^6 - 32*a^2*c^6 + 2*(b^2 - 4*a*c)*b^2*c^4 - 8*(b^2 - 4*a*c)*a*c^5)*d^2*abs(
c) - (2*b^5*c^2 - 16*a*b^3*c^3 + 32*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5
+ 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*b^4*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 8*sqrt(
2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2
- 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*b
^3*c^2 + 8*(b^2 - 4*a*c)*a*b*c^3)*c^2*e^2 + 2*(2*b^3*c^6 - 8*a*b*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq
rt(b^2 - 4*a*c)*c)*b^3*c^4 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^5 + 2*sqrt(2)*s
qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a
*c)*c)*b*c^6 - 2*(b^2 - 4*a*c)*b*c^6)*d^2 + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 - 8*sqrt(2)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 - 2*a*b^4*c^3
+ 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 +
sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 + 16*a^2*b^2*c^4 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)
*a^2*c^5 - 32*a^3*c^5 + 2*(b^2 - 4*a*c)*a*b^2*c^3 - 8*(b^2 - 4*a*c)*a^2*c^4)*abs(c)*e^2 - 2*(2*b^4*c^5 - 8*a*b
^2*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(
b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^4 - s
qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^5 - 2*(b^2 - 4*a*c)*b^2*c^5)*d*e + (2*b^5*c^4 -
12*a*b^3*c^5 + 16*a^2*b*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^2 + 6*sqrt(2)*s
qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*b^4*c^3 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 - 4*sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*
c)*b^3*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^5 - 2*(b^2 - 4*a*c)*b^3*c^4 + 4
*(b^2 - 4*a*c)*a*b*c^5)*e^2)*arctan(2*sqrt(1/2)*x/sqrt((b*c - sqrt(b^2*c^2 - 4*a*c^3))/c^2))/((a*b^4*c^3 - 8*a
^2*b^2*c^4 - 2*a*b^3*c^4 + 16*a^3*c^5 + 8*a^2*b*c^5 + a*b^2*c^5 - 4*a^2*c^6)*c^2)

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maple [B]  time = 0.03, size = 695, normalized size = 2.92 \begin {gather*} \frac {\sqrt {2}\, a \,e^{2} \arctanh \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\sqrt {2}\, a \,e^{2} \arctan \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\sqrt {2}\, b^{2} e^{2} \arctanh \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, c}-\frac {\sqrt {2}\, b^{2} e^{2} \arctan \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, c}+\frac {\sqrt {2}\, b d e \arctanh \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\sqrt {2}\, b d e \arctan \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\sqrt {2}\, c \,d^{2} \arctanh \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\sqrt {2}\, c \,d^{2} \arctan \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\sqrt {2}\, b \,e^{2} \arctanh \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, c}-\frac {\sqrt {2}\, b \,e^{2} \arctan \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, c}-\frac {\sqrt {2}\, d e \arctanh \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\sqrt {2}\, d e \arctan \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {e^{2} x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*e^2*x+1/2/c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*
x)*b*e^2-2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*d*e+
1/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/
2)*c*x)*a*e^2-1/2/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+
b^2)^(1/2))*c)^(1/2)*c*x)*b^2*e^2+1/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/
2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b*d*e-c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)
*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*d^2-1/2/c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arc
tan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b*e^2+2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2
)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*d*e+1/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
an(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*a*e^2-1/2/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b^2*e^2+1/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a
*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b*d*e-c/(-4*a*c+b^2)^(1/2)*2^(1/2
)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*d^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 6.48, size = 9600, normalized size = 40.34

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((16*a*c^4*d^2 - 16*a^2*c^3*e^2 - 4*b^2*c^3*d^2 + 4*a*b^2*c^2*e^2)/c - (2*x*(4*b^3*c^3 - 16*a*b*c^4)*(-
(a*b^5*e^4 + b^3*c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c
*e^4 + 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2*c^4*d^3*e - 32*a^3*c^3*d*e^3 - 4*a*b*c^4
*d^4 - 4*a*b^4*c*d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b^3*c^2*d^2*e^2 - 24*a^2*b*c^3*d^2*e^2 + 24*a^2*b^2*c^2*d*e^3
- 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c
^3 - 8*a^2*b^2*c^4)))^(1/2))/c)*(-(a*b^5*e^4 + b^3*c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - a*b^2*e^4*(-(4
*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e^4 + 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2*c^4*d^
3*e - 32*a^3*c^3*d*e^3 - 4*a*b*c^4*d^4 - 4*a*b^4*c*d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b^3*c^2*d^2*e^2 - 24*a^2*b*
c^3*d^2*e^2 + 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^
3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2) - (2*x*(b^4*e^4 + 2*c^4*d^4 + 2*a^2*c^2*e^4 - 12
*a*c^3*d^2*e^2 + 6*b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^4 - 4*b*c^3*d^3*e - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))/c)*(-(
a*b^5*e^4 + b^3*c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*
e^4 + 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2*c^4*d^3*e - 32*a^3*c^3*d*e^3 - 4*a*b*c^4*
d^4 - 4*a*b^4*c*d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b^3*c^2*d^2*e^2 - 24*a^2*b*c^3*d^2*e^2 + 24*a^2*b^2*c^2*d*e^3
- 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^
3 - 8*a^2*b^2*c^4)))^(1/2)*1i - (((16*a*c^4*d^2 - 16*a^2*c^3*e^2 - 4*b^2*c^3*d^2 + 4*a*b^2*c^2*e^2)/c + (2*x*(
4*b^3*c^3 - 16*a*b*c^4)*(-(a*b^5*e^4 + b^3*c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - a*b^2*e^4*(-(4*a*c - b
^2)^3)^(1/2) - 7*a^2*b^3*c*e^4 + 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2*c^4*d^3*e - 32
*a^3*c^3*d*e^3 - 4*a*b*c^4*d^4 - 4*a*b^4*c*d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b^3*c^2*d^2*e^2 - 24*a^2*b*c^3*d^2*
e^2 + 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2)
)/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2))/c)*(-(a*b^5*e^4 + b^3*c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)
^3)^(1/2) - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e^4 + 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2
)^3)^(1/2) + 32*a^2*c^4*d^3*e - 32*a^3*c^3*d*e^3 - 4*a*b*c^4*d^4 - 4*a*b^4*c*d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b
^3*c^2*d^2*e^2 - 24*a^2*b*c^3*d^2*e^2 + 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*
b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2) + (2*x*(b^4*e^4 + 2*c^
4*d^4 + 2*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 + 6*b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^4 - 4*b*c^3*d^3*e - 4*b^3*c*d*e^3 +
12*a*b*c^2*d*e^3))/c)*(-(a*b^5*e^4 + b^3*c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - a*b^2*e^4*(-(4*a*c - b^
2)^3)^(1/2) - 7*a^2*b^3*c*e^4 + 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2*c^4*d^3*e - 32*
a^3*c^3*d*e^3 - 4*a*b*c^4*d^4 - 4*a*b^4*c*d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b^3*c^2*d^2*e^2 - 24*a^2*b*c^3*d^2*e
^2 + 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))
/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2)*1i)/((2*(2*c^3*d^5*e - a^2*b*e^6 - b^3*d^2*e^4 + 4*a*c^2*
d^3*e^3 - 5*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 + 2*a*b^2*d*e^5 + 2*a^2*c*d*e^5 - 6*a*b*c*d^2*e^4))/c + (((16*a*c^
4*d^2 - 16*a^2*c^3*e^2 - 4*b^2*c^3*d^2 + 4*a*b^2*c^2*e^2)/c - (2*x*(4*b^3*c^3 - 16*a*b*c^4)*(-(a*b^5*e^4 + b^3
*c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e^4 + 12*a^3*b*
c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2*c^4*d^3*e - 32*a^3*c^3*d*e^3 - 4*a*b*c^4*d^4 - 4*a*b^4*c
*d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b^3*c^2*d^2*e^2 - 24*a^2*b*c^3*d^2*e^2 + 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e
^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c
^4)))^(1/2))/c)*(-(a*b^5*e^4 + b^3*c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - a*b^2*e^4*(-(4*a*c - b^2)^3)^(
1/2) - 7*a^2*b^3*c*e^4 + 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2*c^4*d^3*e - 32*a^3*c^3
*d*e^3 - 4*a*b*c^4*d^4 - 4*a*b^4*c*d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b^3*c^2*d^2*e^2 - 24*a^2*b*c^3*d^2*e^2 + 24
*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16
*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2) - (2*x*(b^4*e^4 + 2*c^4*d^4 + 2*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 +
6*b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^4 - 4*b*c^3*d^3*e - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))/c)*(-(a*b^5*e^4 + b^3*
c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e^4 + 12*a^3*b*c
^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2*c^4*d^3*e - 32*a^3*c^3*d*e^3 - 4*a*b*c^4*d^4 - 4*a*b^4*c*
d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b^3*c^2*d^2*e^2 - 24*a^2*b*c^3*d^2*e^2 + 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^
2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^
4)))^(1/2) + (((16*a*c^4*d^2 - 16*a^2*c^3*e^2 - 4*b^2*c^3*d^2 + 4*a*b^2*c^2*e^2)/c + (2*x*(4*b^3*c^3 - 16*a*b*
c^4)*(-(a*b^5*e^4 + b^3*c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) - 7*a^
2*b^3*c*e^4 + 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2*c^4*d^3*e - 32*a^3*c^3*d*e^3 - 4*
a*b*c^4*d^4 - 4*a*b^4*c*d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b^3*c^2*d^2*e^2 - 24*a^2*b*c^3*d^2*e^2 + 24*a^2*b^2*c^
2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 +
a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2))/c)*(-(a*b^5*e^4 + b^3*c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - a*b^2*e
^4*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e^4 + 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2
*c^4*d^3*e - 32*a^3*c^3*d*e^3 - 4*a*b*c^4*d^4 - 4*a*b^4*c*d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b^3*c^2*d^2*e^2 - 24
*a^2*b*c^3*d^2*e^2 + 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c
- b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2) + (2*x*(b^4*e^4 + 2*c^4*d^4 + 2*a^2*c^2*e
^4 - 12*a*c^3*d^2*e^2 + 6*b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^4 - 4*b*c^3*d^3*e - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))
/c)*(-(a*b^5*e^4 + b^3*c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2
*b^3*c*e^4 + 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2*c^4*d^3*e - 32*a^3*c^3*d*e^3 - 4*a
*b*c^4*d^4 - 4*a*b^4*c*d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b^3*c^2*d^2*e^2 - 24*a^2*b*c^3*d^2*e^2 + 24*a^2*b^2*c^2
*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a
*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2)))*(-(a*b^5*e^4 + b^3*c^3*d^4 + c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - a*b^2*e^4*
(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e^4 + 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 32*a^2*c^
4*d^3*e - 32*a^3*c^3*d*e^3 - 4*a*b*c^4*d^4 - 4*a*b^4*c*d*e^3 - 8*a*b^2*c^3*d^3*e + 6*a*b^3*c^2*d^2*e^2 - 24*a^
2*b*c^3*d^2*e^2 + 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b
^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2)*2i + atan(((((16*a*c^4*d^2 - 16*a^2*c^3*e^2
- 4*b^2*c^3*d^2 + 4*a*b^2*c^2*e^2)/c - (2*x*(4*b^3*c^3 - 16*a*b*c^4)*((c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - b^3*
c^3*d^4 - a*b^5*e^4 - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) + 7*a^2*b^3*c*e^4 - 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4
*a*c - b^2)^3)^(1/2) - 32*a^2*c^4*d^3*e + 32*a^3*c^3*d*e^3 + 4*a*b*c^4*d^4 + 4*a*b^4*c*d*e^3 + 8*a*b^2*c^3*d^3
*e - 6*a*b^3*c^2*d^2*e^2 + 24*a^2*b*c^3*d^2*e^2 - 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1
/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2))/c)*((c^3*d^
4*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^4 - a*b^5*e^4 - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) + 7*a^2*b^3*c*e^4 -
12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^2*c^4*d^3*e + 32*a^3*c^3*d*e^3 + 4*a*b*c^4*d^4 +
4*a*b^4*c*d*e^3 + 8*a*b^2*c^3*d^3*e - 6*a*b^3*c^2*d^2*e^2 + 24*a^2*b*c^3*d^2*e^2 - 24*a^2*b^2*c^2*d*e^3 - 6*a*
c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*
a^2*b^2*c^4)))^(1/2) - (2*x*(b^4*e^4 + 2*c^4*d^4 + 2*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 + 6*b^2*c^2*d^2*e^2 - 4*a*
b^2*c*e^4 - 4*b*c^3*d^3*e - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))/c)*((c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3
*d^4 - a*b^5*e^4 - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) + 7*a^2*b^3*c*e^4 - 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*
c - b^2)^3)^(1/2) - 32*a^2*c^4*d^3*e + 32*a^3*c^3*d*e^3 + 4*a*b*c^4*d^4 + 4*a*b^4*c*d*e^3 + 8*a*b^2*c^3*d^3*e
- 6*a*b^3*c^2*d^2*e^2 + 24*a^2*b*c^3*d^2*e^2 - 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2)
+ 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2)*1i - (((16*a*c^
4*d^2 - 16*a^2*c^3*e^2 - 4*b^2*c^3*d^2 + 4*a*b^2*c^2*e^2)/c + (2*x*(4*b^3*c^3 - 16*a*b*c^4)*((c^3*d^4*(-(4*a*c
- b^2)^3)^(1/2) - b^3*c^3*d^4 - a*b^5*e^4 - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) + 7*a^2*b^3*c*e^4 - 12*a^3*b*c
^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^2*c^4*d^3*e + 32*a^3*c^3*d*e^3 + 4*a*b*c^4*d^4 + 4*a*b^4*c*
d*e^3 + 8*a*b^2*c^3*d^3*e - 6*a*b^3*c^2*d^2*e^2 + 24*a^2*b*c^3*d^2*e^2 - 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^
2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^
4)))^(1/2))/c)*((c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^4 - a*b^5*e^4 - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/
2) + 7*a^2*b^3*c*e^4 - 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^2*c^4*d^3*e + 32*a^3*c^3*d
*e^3 + 4*a*b*c^4*d^4 + 4*a*b^4*c*d*e^3 + 8*a*b^2*c^3*d^3*e - 6*a*b^3*c^2*d^2*e^2 + 24*a^2*b*c^3*d^2*e^2 - 24*a
^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a
^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2) + (2*x*(b^4*e^4 + 2*c^4*d^4 + 2*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 + 6
*b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^4 - 4*b*c^3*d^3*e - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))/c)*((c^3*d^4*(-(4*a*c -
b^2)^3)^(1/2) - b^3*c^3*d^4 - a*b^5*e^4 - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) + 7*a^2*b^3*c*e^4 - 12*a^3*b*c^2*
e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^2*c^4*d^3*e + 32*a^3*c^3*d*e^3 + 4*a*b*c^4*d^4 + 4*a*b^4*c*d*e
^3 + 8*a*b^2*c^3*d^3*e - 6*a*b^3*c^2*d^2*e^2 + 24*a^2*b*c^3*d^2*e^2 - 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(
-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4))
)^(1/2)*1i)/((2*(2*c^3*d^5*e - a^2*b*e^6 - b^3*d^2*e^4 + 4*a*c^2*d^3*e^3 - 5*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 +
2*a*b^2*d*e^5 + 2*a^2*c*d*e^5 - 6*a*b*c*d^2*e^4))/c + (((16*a*c^4*d^2 - 16*a^2*c^3*e^2 - 4*b^2*c^3*d^2 + 4*a*
b^2*c^2*e^2)/c - (2*x*(4*b^3*c^3 - 16*a*b*c^4)*((c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^4 - a*b^5*e^4 -
a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) + 7*a^2*b^3*c*e^4 - 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) -
32*a^2*c^4*d^3*e + 32*a^3*c^3*d*e^3 + 4*a*b*c^4*d^4 + 4*a*b^4*c*d*e^3 + 8*a*b^2*c^3*d^3*e - 6*a*b^3*c^2*d^2*e
^2 + 24*a^2*b*c^3*d^2*e^2 - 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-
(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2))/c)*((c^3*d^4*(-(4*a*c - b^2)^3)^(
1/2) - b^3*c^3*d^4 - a*b^5*e^4 - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) + 7*a^2*b^3*c*e^4 - 12*a^3*b*c^2*e^4 + a^2
*c*e^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^2*c^4*d^3*e + 32*a^3*c^3*d*e^3 + 4*a*b*c^4*d^4 + 4*a*b^4*c*d*e^3 + 8*a*
b^2*c^3*d^3*e - 6*a*b^3*c^2*d^2*e^2 + 24*a^2*b*c^3*d^2*e^2 - 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c -
b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2) -
(2*x*(b^4*e^4 + 2*c^4*d^4 + 2*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 + 6*b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^4 - 4*b*c^3*d^
3*e - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))/c)*((c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^4 - a*b^5*e^4 - a*b
^2*e^4*(-(4*a*c - b^2)^3)^(1/2) + 7*a^2*b^3*c*e^4 - 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) - 32
*a^2*c^4*d^3*e + 32*a^3*c^3*d*e^3 + 4*a*b*c^4*d^4 + 4*a*b^4*c*d*e^3 + 8*a*b^2*c^3*d^3*e - 6*a*b^3*c^2*d^2*e^2
+ 24*a^2*b*c^3*d^2*e^2 - 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*
a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2) + (((16*a*c^4*d^2 - 16*a^2*c^3*e^2 -
4*b^2*c^3*d^2 + 4*a*b^2*c^2*e^2)/c + (2*x*(4*b^3*c^3 - 16*a*b*c^4)*((c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^
3*d^4 - a*b^5*e^4 - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) + 7*a^2*b^3*c*e^4 - 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a
*c - b^2)^3)^(1/2) - 32*a^2*c^4*d^3*e + 32*a^3*c^3*d*e^3 + 4*a*b*c^4*d^4 + 4*a*b^4*c*d*e^3 + 8*a*b^2*c^3*d^3*e
- 6*a*b^3*c^2*d^2*e^2 + 24*a^2*b*c^3*d^2*e^2 - 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2
) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2))/c)*((c^3*d^4*
(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^4 - a*b^5*e^4 - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) + 7*a^2*b^3*c*e^4 - 12
*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^2*c^4*d^3*e + 32*a^3*c^3*d*e^3 + 4*a*b*c^4*d^4 + 4*
a*b^4*c*d*e^3 + 8*a*b^2*c^3*d^3*e - 6*a*b^3*c^2*d^2*e^2 + 24*a^2*b*c^3*d^2*e^2 - 24*a^2*b^2*c^2*d*e^3 - 6*a*c^
2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^
2*b^2*c^4)))^(1/2) + (2*x*(b^4*e^4 + 2*c^4*d^4 + 2*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 + 6*b^2*c^2*d^2*e^2 - 4*a*b^
2*c*e^4 - 4*b*c^3*d^3*e - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))/c)*((c^3*d^4*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d
^4 - a*b^5*e^4 - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) + 7*a^2*b^3*c*e^4 - 12*a^3*b*c^2*e^4 + a^2*c*e^4*(-(4*a*c
- b^2)^3)^(1/2) - 32*a^2*c^4*d^3*e + 32*a^3*c^3*d*e^3 + 4*a*b*c^4*d^4 + 4*a*b^4*c*d*e^3 + 8*a*b^2*c^3*d^3*e -
6*a*b^3*c^2*d^2*e^2 + 24*a^2*b*c^3*d^2*e^2 - 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) +
4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2*c^4)))^(1/2)))*((c^3*d^4*(-(4*
a*c - b^2)^3)^(1/2) - b^3*c^3*d^4 - a*b^5*e^4 - a*b^2*e^4*(-(4*a*c - b^2)^3)^(1/2) + 7*a^2*b^3*c*e^4 - 12*a^3*
b*c^2*e^4 + a^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^2*c^4*d^3*e + 32*a^3*c^3*d*e^3 + 4*a*b*c^4*d^4 + 4*a*b^4
*c*d*e^3 + 8*a*b^2*c^3*d^3*e - 6*a*b^3*c^2*d^2*e^2 + 24*a^2*b*c^3*d^2*e^2 - 24*a^2*b^2*c^2*d*e^3 - 6*a*c^2*d^2
*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^3*c^5 + a*b^4*c^3 - 8*a^2*b^2
*c^4)))^(1/2)*2i + (e^2*x)/c

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

[Out]

Timed out

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