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

Optimal. Leaf size=209 \[ \frac {\left (c-\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e-\sqrt {e^2-4 d f}}}\right )}{\sqrt {2} \sqrt {f} \sqrt {e-\sqrt {e^2-4 d f}}}+\frac {\left (\frac {c e-2 a f}{\sqrt {e^2-4 d f}}+c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {\sqrt {e^2-4 d f}+e}}\right )}{\sqrt {2} \sqrt {f} \sqrt {\sqrt {e^2-4 d f}+e}}-\frac {b \tanh ^{-1}\left (\frac {e+2 f x^2}{\sqrt {e^2-4 d f}}\right )}{\sqrt {e^2-4 d f}} \]

[Out]

-b*arctanh((2*f*x^2+e)/(-4*d*f+e^2)^(1/2))/(-4*d*f+e^2)^(1/2)+1/2*arctan(x*2^(1/2)*f^(1/2)/(e-(-4*d*f+e^2)^(1/
2))^(1/2))*(c+(2*a*f-c*e)/(-4*d*f+e^2)^(1/2))*2^(1/2)/f^(1/2)/(e-(-4*d*f+e^2)^(1/2))^(1/2)+1/2*arctan(x*2^(1/2
)*f^(1/2)/(e+(-4*d*f+e^2)^(1/2))^(1/2))*(c+(-2*a*f+c*e)/(-4*d*f+e^2)^(1/2))*2^(1/2)/f^(1/2)/(e+(-4*d*f+e^2)^(1
/2))^(1/2)

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Rubi [A]  time = 0.37, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1673, 1166, 205, 12, 1107, 618, 206} \[ \frac {\left (c-\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e-\sqrt {e^2-4 d f}}}\right )}{\sqrt {2} \sqrt {f} \sqrt {e-\sqrt {e^2-4 d f}}}+\frac {\left (\frac {c e-2 a f}{\sqrt {e^2-4 d f}}+c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {\sqrt {e^2-4 d f}+e}}\right )}{\sqrt {2} \sqrt {f} \sqrt {\sqrt {e^2-4 d f}+e}}-\frac {b \tanh ^{-1}\left (\frac {e+2 f x^2}{\sqrt {e^2-4 d f}}\right )}{\sqrt {e^2-4 d f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

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 1107

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

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 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rubi steps

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

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Mathematica [A]  time = 0.25, size = 234, normalized size = 1.12 \[ \frac {\frac {\sqrt {2} \left (2 a f+c \left (\sqrt {e^2-4 d f}-e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e-\sqrt {e^2-4 d f}}}\right )}{\sqrt {f} \sqrt {e-\sqrt {e^2-4 d f}}}+\frac {\sqrt {2} \left (c \left (\sqrt {e^2-4 d f}+e\right )-2 a f\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {\sqrt {e^2-4 d f}+e}}\right )}{\sqrt {f} \sqrt {\sqrt {e^2-4 d f}+e}}+b \log \left (\sqrt {e^2-4 d f}-e-2 f x^2\right )-b \log \left (\sqrt {e^2-4 d f}+e+2 f x^2\right )}{2 \sqrt {e^2-4 d f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[2]*(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f]))*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e - Sqrt[e^2 - 4*d*f]]])/(Sqrt[
f]*Sqrt[e - Sqrt[e^2 - 4*d*f]]) + (Sqrt[2]*(-2*a*f + c*(e + Sqrt[e^2 - 4*d*f]))*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqr
t[e + Sqrt[e^2 - 4*d*f]]])/(Sqrt[f]*Sqrt[e + Sqrt[e^2 - 4*d*f]]) + b*Log[-e + Sqrt[e^2 - 4*d*f] - 2*f*x^2] - b
*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x^2])/(2*Sqrt[e^2 - 4*d*f])

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [B]  time = 4.70, size = 1587, normalized size = 7.59 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*d*f^3 - f^4 + 2*f^3*e - f^2*e^2)*sqrt(-4*d*f + e^2)*b*log(x^2 + 1/2*(sqrt(-4*d*f + e^2) + e)/f)/((16*d^
2*f^2 - 4*d*f^3 + 8*d*f^2*e - 8*d*f*e^2 + f^2*e^2 - 2*f*e^3 + e^4)*f^2) - 1/2*(4*d*f^3 - f^4 + 2*f^3*e - f^2*e
^2)*sqrt(-4*d*f + e^2)*b*log(x^2 - 1/2*(sqrt(-4*d*f + e^2) - e)/f)/((16*d^2*f^2 - 4*d*f^3 + 8*d*f^2*e - 8*d*f*
e^2 + f^2*e^2 - 2*f*e^3 + e^4)*f^2) + 1/4*((16*sqrt(2)*sqrt(f*e + sqrt(-4*d*f + e^2)*f)*d^2*f^2 - 4*sqrt(2)*sq
rt(f*e + sqrt(-4*d*f + e^2)*f)*d*f^3 - 32*d^2*f^3 + 8*sqrt(2)*sqrt(f*e + sqrt(-4*d*f + e^2)*f)*d*f^2*e - 8*d*f
^3*e + 4*sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(f*e + sqrt(-4*d*f + e^2)*f)*d*f*e - sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(f
*e + sqrt(-4*d*f + e^2)*f)*f^2*e + 8*(4*d*f - e^2)*d*f^2 - 8*sqrt(2)*sqrt(f*e + sqrt(-4*d*f + e^2)*f)*d*f*e^2
+ sqrt(2)*sqrt(f*e + sqrt(-4*d*f + e^2)*f)*f^2*e^2 + 16*d*f^2*e^2 + 2*(4*d*f - e^2)*f^2*e + 2*sqrt(2)*sqrt(-4*
d*f + e^2)*sqrt(f*e + sqrt(-4*d*f + e^2)*f)*f*e^2 - 2*sqrt(2)*sqrt(f*e + sqrt(-4*d*f + e^2)*f)*f*e^3 + 2*f^2*e
^3 - 2*(4*d*f - e^2)*f*e^2 - sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(f*e + sqrt(-4*d*f + e^2)*f)*e^3 + sqrt(2)*sqrt(f*
e + sqrt(-4*d*f + e^2)*f)*e^4 - 2*f*e^4)*a + 2*(8*d^2*f^3 - 4*sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(f*e + sqrt(-4*d*
f + e^2)*f)*d^2*f + sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(f*e + sqrt(-4*d*f + e^2)*f)*d*f^2 - 2*sqrt(2)*sqrt(-4*d*f
+ e^2)*sqrt(f*e + sqrt(-4*d*f + e^2)*f)*d*f*e - 2*(4*d*f - e^2)*d*f^2 - 2*d*f^2*e^2 + sqrt(2)*sqrt(-4*d*f + e^
2)*sqrt(f*e + sqrt(-4*d*f + e^2)*f)*d*e^2)*c)*arctan(2*sqrt(1/2)*x/sqrt((sqrt(-4*d*f + e^2) + e)/f))/((16*d^3*
f^2 - 4*d^2*f^3 + 8*d^2*f^2*e - 8*d^2*f*e^2 + d*f^2*e^2 - 2*d*f*e^3 + d*e^4)*abs(f)) + 1/4*((16*sqrt(2)*sqrt(f
*e - sqrt(-4*d*f + e^2)*f)*d^2*f^2 - 4*sqrt(2)*sqrt(f*e - sqrt(-4*d*f + e^2)*f)*d*f^3 + 32*d^2*f^3 + 8*sqrt(2)
*sqrt(f*e - sqrt(-4*d*f + e^2)*f)*d*f^2*e + 8*d*f^3*e - 4*sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(f*e - sqrt(-4*d*f +
e^2)*f)*d*f*e + sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(f*e - sqrt(-4*d*f + e^2)*f)*f^2*e - 8*(4*d*f - e^2)*d*f^2 - 8*
sqrt(2)*sqrt(f*e - sqrt(-4*d*f + e^2)*f)*d*f*e^2 + sqrt(2)*sqrt(f*e - sqrt(-4*d*f + e^2)*f)*f^2*e^2 - 16*d*f^2
*e^2 - 2*(4*d*f - e^2)*f^2*e - 2*sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(f*e - sqrt(-4*d*f + e^2)*f)*f*e^2 - 2*sqrt(2)
*sqrt(f*e - sqrt(-4*d*f + e^2)*f)*f*e^3 - 2*f^2*e^3 + 2*(4*d*f - e^2)*f*e^2 + sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(
f*e - sqrt(-4*d*f + e^2)*f)*e^3 + sqrt(2)*sqrt(f*e - sqrt(-4*d*f + e^2)*f)*e^4 + 2*f*e^4)*a - 2*(8*d^2*f^3 - 4
*sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(f*e - sqrt(-4*d*f + e^2)*f)*d^2*f + sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(f*e - sqr
t(-4*d*f + e^2)*f)*d*f^2 - 2*sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(f*e - sqrt(-4*d*f + e^2)*f)*d*f*e - 2*(4*d*f - e^
2)*d*f^2 - 2*d*f^2*e^2 + sqrt(2)*sqrt(-4*d*f + e^2)*sqrt(f*e - sqrt(-4*d*f + e^2)*f)*d*e^2)*c)*arctan(2*sqrt(1
/2)*x/sqrt(-(sqrt(-4*d*f + e^2) - e)/f))/((16*d^3*f^2 - 4*d^2*f^3 + 8*d^2*f^2*e - 8*d^2*f*e^2 + d*f^2*e^2 - 2*
d*f*e^3 + d*e^4)*abs(f))

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maple [B]  time = 0.06, size = 616, normalized size = 2.95 \[ -\frac {2 \sqrt {2}\, c d f \arctanh \left (\frac {\sqrt {2}\, f x}{\sqrt {\left (-e +\sqrt {-4 d f +e^{2}}\right ) f}}\right )}{\left (4 d f -e^{2}\right ) \sqrt {\left (-e +\sqrt {-4 d f +e^{2}}\right ) f}}+\frac {2 \sqrt {2}\, c d f \arctan \left (\frac {\sqrt {2}\, f x}{\sqrt {\left (e +\sqrt {-4 d f +e^{2}}\right ) f}}\right )}{\left (4 d f -e^{2}\right ) \sqrt {\left (e +\sqrt {-4 d f +e^{2}}\right ) f}}+\frac {\sqrt {2}\, c \,e^{2} \arctanh \left (\frac {\sqrt {2}\, f x}{\sqrt {\left (-e +\sqrt {-4 d f +e^{2}}\right ) f}}\right )}{2 \left (4 d f -e^{2}\right ) \sqrt {\left (-e +\sqrt {-4 d f +e^{2}}\right ) f}}-\frac {\sqrt {2}\, c \,e^{2} \arctan \left (\frac {\sqrt {2}\, f x}{\sqrt {\left (e +\sqrt {-4 d f +e^{2}}\right ) f}}\right )}{2 \left (4 d f -e^{2}\right ) \sqrt {\left (e +\sqrt {-4 d f +e^{2}}\right ) f}}+\frac {\sqrt {-4 d f +e^{2}}\, \sqrt {2}\, a f \arctanh \left (\frac {\sqrt {2}\, f x}{\sqrt {\left (-e +\sqrt {-4 d f +e^{2}}\right ) f}}\right )}{\left (4 d f -e^{2}\right ) \sqrt {\left (-e +\sqrt {-4 d f +e^{2}}\right ) f}}+\frac {\sqrt {-4 d f +e^{2}}\, \sqrt {2}\, a f \arctan \left (\frac {\sqrt {2}\, f x}{\sqrt {\left (e +\sqrt {-4 d f +e^{2}}\right ) f}}\right )}{\left (4 d f -e^{2}\right ) \sqrt {\left (e +\sqrt {-4 d f +e^{2}}\right ) f}}-\frac {\sqrt {-4 d f +e^{2}}\, \sqrt {2}\, c e \arctanh \left (\frac {\sqrt {2}\, f x}{\sqrt {\left (-e +\sqrt {-4 d f +e^{2}}\right ) f}}\right )}{2 \left (4 d f -e^{2}\right ) \sqrt {\left (-e +\sqrt {-4 d f +e^{2}}\right ) f}}-\frac {\sqrt {-4 d f +e^{2}}\, \sqrt {2}\, c e \arctan \left (\frac {\sqrt {2}\, f x}{\sqrt {\left (e +\sqrt {-4 d f +e^{2}}\right ) f}}\right )}{2 \left (4 d f -e^{2}\right ) \sqrt {\left (e +\sqrt {-4 d f +e^{2}}\right ) f}}-\frac {\sqrt {-4 d f +e^{2}}\, b \ln \left (-2 f \,x^{2}-e +\sqrt {-4 d f +e^{2}}\right )}{2 \left (4 d f -e^{2}\right )}+\frac {\sqrt {-4 d f +e^{2}}\, b \ln \left (2 f \,x^{2}+e +\sqrt {-4 d f +e^{2}}\right )}{8 d f -2 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-4*d*f+e^2)^(1/2)/(4*d*f-e^2)*b*ln(-2*f*x^2+(-4*d*f+e^2)^(1/2)-e)-2*f/(4*d*f-e^2)*2^(1/2)/(((-4*d*f+e^2)
^(1/2)-e)*f)^(1/2)*arctanh(f*x*2^(1/2)/(((-4*d*f+e^2)^(1/2)-e)*f)^(1/2))*c*d+1/2/(4*d*f-e^2)*2^(1/2)/(((-4*d*f
+e^2)^(1/2)-e)*f)^(1/2)*arctanh(f*x*2^(1/2)/(((-4*d*f+e^2)^(1/2)-e)*f)^(1/2))*c*e^2+f*(-4*d*f+e^2)^(1/2)/(4*d*
f-e^2)*2^(1/2)/(((-4*d*f+e^2)^(1/2)-e)*f)^(1/2)*arctanh(f*x*2^(1/2)/(((-4*d*f+e^2)^(1/2)-e)*f)^(1/2))*a-1/2*(-
4*d*f+e^2)^(1/2)/(4*d*f-e^2)*2^(1/2)/(((-4*d*f+e^2)^(1/2)-e)*f)^(1/2)*arctanh(f*x*2^(1/2)/(((-4*d*f+e^2)^(1/2)
-e)*f)^(1/2))*c*e+1/2*(-4*d*f+e^2)^(1/2)/(4*d*f-e^2)*b*ln(2*f*x^2+(-4*d*f+e^2)^(1/2)+e)+2*f/(4*d*f-e^2)*2^(1/2
)/((e+(-4*d*f+e^2)^(1/2))*f)^(1/2)*arctan(f*x*2^(1/2)/((e+(-4*d*f+e^2)^(1/2))*f)^(1/2))*c*d-1/2/(4*d*f-e^2)*2^
(1/2)/((e+(-4*d*f+e^2)^(1/2))*f)^(1/2)*arctan(f*x*2^(1/2)/((e+(-4*d*f+e^2)^(1/2))*f)^(1/2))*c*e^2+f*(-4*d*f+e^
2)^(1/2)/(4*d*f-e^2)*2^(1/2)/((e+(-4*d*f+e^2)^(1/2))*f)^(1/2)*arctan(f*x*2^(1/2)/((e+(-4*d*f+e^2)^(1/2))*f)^(1
/2))*a-1/2*(-4*d*f+e^2)^(1/2)/(4*d*f-e^2)*2^(1/2)/((e+(-4*d*f+e^2)^(1/2))*f)^(1/2)*arctan(f*x*2^(1/2)/((e+(-4*
d*f+e^2)^(1/2))*f)^(1/2))*c*e

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {c x^{2} + b x + a}{f x^{4} + e x^{2} + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.44, size = 3942, normalized size = 18.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(a*b^2*f^2 - a^2*c*f^2 + b^3*f^2*x - c^3*d*f - 8*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3
*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z
^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e
^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*
f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)^3*e^3*f^2*x + a*c^2*e*f - 16*root(16*d*e^4*f*z^4 - 128*d^
2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2
*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*
b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b
^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)^2*a*d*f^3 - 4*root(16*d*e^4*f*z^4
- 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2
*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z
+ 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3
*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)*a^2*f^3*x + 4*root(16*d*e^
4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2
- 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*
d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f -
2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)^2*a*e^2*f^2 + 16*r
oot(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d
*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2
+ 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*
a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)^2*b*d*f
^3*x + 2*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2
- 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*
e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^
2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z,
k)*b^2*e*f^2*x + 4*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d
^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^
2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f
+ 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^
4*f^2, z, k)*c^2*d*f^2*x - 2*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2
- 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^
2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*
b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^
4*d^2 + a^4*f^2, z, k)*c^2*e^2*f*x + 32*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d
*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2
*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f
^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c
^2*e^2 + c^4*d^2 + a^4*f^2, z, k)^3*d*e*f^3*x - 4*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4
- 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b
^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16
*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d
*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)^2*b*e^2*f^2*x + 4*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*
d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^
2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*
d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2
*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)*a*b*e*f^2 - 8*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^
4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*
c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z -
4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e +
a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)*b*c*d*f^2 - 2*a*b*c*f^2*x + b*c^2*e*f*x + 4*ro
ot(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*
e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 +
16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a
^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)*a*c*e*f^
2*x)*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8
*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*
f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*
f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k),
k, 1, 4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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