3.164 \(\int \frac{a+c x^2+d x^3}{2+3 x^4} \, dx\)

Optimal. Leaf size=154 \[ -\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}+\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}-\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

[Out]

-((Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x])/(4*6^(3/4)) + ((Sqrt[6]*a + 2*c)*ArcTan[1 + 6^(1/4)*x])/(4*6^(3/4))
- ((Sqrt[6]*a - 2*c)*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(8*6^(3/4)) + ((Sqrt[6]*a - 2*c)*Log[Sqrt[6] + 6^(3/4)
*x + 3*x^2])/(8*6^(3/4)) + (d*Log[2 + 3*x^4])/12

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Rubi [A]  time = 0.117019, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {1876, 260, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}+\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}-\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2 + d*x^3)/(2 + 3*x^4),x]

[Out]

-((Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x])/(4*6^(3/4)) + ((Sqrt[6]*a + 2*c)*ArcTan[1 + 6^(1/4)*x])/(4*6^(3/4))
- ((Sqrt[6]*a - 2*c)*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(8*6^(3/4)) + ((Sqrt[6]*a - 2*c)*Log[Sqrt[6] + 6^(3/4)
*x + 3*x^2])/(8*6^(3/4)) + (d*Log[2 + 3*x^4])/12

Rule 1876

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1168

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

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[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, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1165

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

Rule 628

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

Rubi steps

\begin{align*} \int \frac{a+c x^2+d x^3}{2+3 x^4} \, dx &=\int \left (\frac{d x^3}{2+3 x^4}+\frac{a+c x^2}{2+3 x^4}\right ) \, dx\\ &=d \int \frac{x^3}{2+3 x^4} \, dx+\int \frac{a+c x^2}{2+3 x^4} \, dx\\ &=\frac{1}{12} d \log \left (2+3 x^4\right )+\frac{1}{12} \left (\sqrt{6} a-2 c\right ) \int \frac{\sqrt{6}-3 x^2}{2+3 x^4} \, dx+\frac{1}{12} \left (\sqrt{6} a+2 c\right ) \int \frac{\sqrt{6}+3 x^2}{2+3 x^4} \, dx\\ &=\frac{1}{12} d \log \left (2+3 x^4\right )-\frac{\left (\sqrt{6} a-2 c\right ) \int \frac{\frac{2^{3/4}}{\sqrt [4]{3}}+2 x}{-\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{8\ 6^{3/4}}-\frac{\left (\sqrt{6} a-2 c\right ) \int \frac{\frac{2^{3/4}}{\sqrt [4]{3}}-2 x}{-\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{8\ 6^{3/4}}+\frac{1}{24} \left (\sqrt{6} a+2 c\right ) \int \frac{1}{\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac{1}{24} \left (\sqrt{6} a+2 c\right ) \int \frac{1}{\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx\\ &=-\frac{\left (\sqrt{6} a-2 c\right ) \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac{\left (\sqrt{6} a-2 c\right ) \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac{1}{12} d \log \left (2+3 x^4\right )+\frac{\left (\sqrt{6} a+2 c\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}-\frac{\left (\sqrt{6} a+2 c\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{4\ 6^{3/4}}\\ &=-\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{4\ 6^{3/4}}-\frac{\left (\sqrt{6} a-2 c\right ) \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac{\left (\sqrt{6} a-2 c\right ) \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac{1}{12} d \log \left (2+3 x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.092163, size = 148, normalized size = 0.96 \[ \frac{1}{48} \left (-\sqrt [4]{6} \left (\sqrt{6} a-2 c\right ) \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+\sqrt [4]{6} \left (\sqrt{6} a-2 c\right ) \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \sqrt [4]{6} \left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} \left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+4 d \log \left (3 x^4+2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2 + d*x^3)/(2 + 3*x^4),x]

[Out]

(-2*6^(1/4)*(Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*(Sqrt[6]*a + 2*c)*ArcTan[1 + 6^(1/4)*x] - 6^(1
/4)*(Sqrt[6]*a - 2*c)*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + 6^(1/4)*(Sqrt[6]*a - 2*c)*Log[2 + 2*6^(1/4)*x + Sqr
t[6]*x^2] + 4*d*Log[2 + 3*x^4])/48

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Maple [B]  time = 0.041, size = 237, normalized size = 1.5 \begin{align*}{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({ \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{144}\ln \left ({ \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c*x^2+a)/(3*x^4+2),x)

[Out]

1/24*a*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/48*a*3^(1/2)*6^(1/4)*2^(1/2)*ln((x^2+
1/3*3^(1/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2))/(x^2-1/3*3^(1/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2)))+1/24*a*3^(1/2)*6^(
1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/72*c*3^(1/2)*6^(3/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)
*6^(3/4)*x-1)+1/144*c*3^(1/2)*6^(3/4)*2^(1/2)*ln((x^2-1/3*3^(1/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2))/(x^2+1/3*3^(1
/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2)))+1/72*c*3^(1/2)*6^(3/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/1
2*d*ln(3*x^4+2)

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Maxima [A]  time = 1.45753, size = 263, normalized size = 1.71 \begin{align*} -\frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (\sqrt{3} \sqrt{2} c - 2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d - 3 \, a\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (\sqrt{3} \sqrt{2} c + 2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d - 3 \, a\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{72} \, \sqrt{3}{\left (3 \cdot 3^{\frac{1}{4}} 2^{\frac{3}{4}} a + 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} c\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{72} \, \sqrt{3}{\left (3 \cdot 3^{\frac{1}{4}} 2^{\frac{3}{4}} a + 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} c\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c*x^2+a)/(3*x^4+2),x, algorithm="maxima")

[Out]

-1/144*3^(3/4)*2^(3/4)*(sqrt(3)*sqrt(2)*c - 2*3^(1/4)*2^(1/4)*d - 3*a)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + s
qrt(2)) + 1/144*3^(3/4)*2^(3/4)*(sqrt(3)*sqrt(2)*c + 2*3^(1/4)*2^(1/4)*d - 3*a)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3
/4)*x + sqrt(2)) + 1/72*sqrt(3)*(3*3^(1/4)*2^(3/4)*a + 2*3^(3/4)*2^(1/4)*c)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt
(3)*x + 3^(1/4)*2^(3/4))) + 1/72*sqrt(3)*(3*3^(1/4)*2^(3/4)*a + 2*3^(3/4)*2^(1/4)*c)*arctan(1/6*3^(3/4)*2^(1/4
)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4)))

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Fricas [B]  time = 1.95114, size = 5162, normalized size = 33.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c*x^2+a)/(3*x^4+2),x, algorithm="fricas")

[Out]

1/144*(2*sqrt(6)*sqrt(2)*(54*a^4 + 72*a^2*c^2 + 24*c^4)^(3/4)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4)*sqrt((9*a^4 + 1
2*a^2*c^2 + 4*c^4 + 2*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*a*c)/(9*a^4 - 12*a^2*c^2 + 4*c^4))*arctan(-1/12*(sqrt
(2)*sqrt(1/3)*(54*a^4 + 72*a^2*c^2 + 24*c^4)^(3/4)*(sqrt(6)*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*sqrt(9*a^4 - 12
*a^2*c^2 + 4*c^4)*a - 2*sqrt(6)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4)*(3*a^2*c + 2*c^3))*sqrt((9*a^4 + 12*a^2*c^2 +
4*c^4 + 2*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*a*c)/(9*a^4 - 12*a^2*c^2 + 4*c^4))*sqrt((3*(9*a^4 + 12*a^2*c^2 +
4*c^4)*x^2 + sqrt(2)*(54*a^4 + 72*a^2*c^2 + 24*c^4)^(1/4)*(sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*c*x - 3*(3*a^3
+ 2*a*c^2)*x)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 + 2*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*a*c)/(9*a^4 - 12*a^2*c^2
+ 4*c^4)) + sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*(3*a^2 + 2*c^2))/(9*a^4 + 12*a^2*c^2 + 4*c^4)) - sqrt(2)*(54*a
^4 + 72*a^2*c^2 + 24*c^4)^(3/4)*(sqrt(6)*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4)*a
*x - 2*sqrt(6)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4)*(3*a^2*c + 2*c^3)*x)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 + 2*sqrt
(54*a^4 + 72*a^2*c^2 + 24*c^4)*a*c)/(9*a^4 - 12*a^2*c^2 + 4*c^4)) + 2*sqrt(6)*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^
4)*(9*a^4 + 12*a^2*c^2 + 4*c^4)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4))/(81*a^8 + 108*a^6*c^2 - 48*a^2*c^6 - 16*c^8)
) + 2*sqrt(6)*sqrt(2)*(54*a^4 + 72*a^2*c^2 + 24*c^4)^(3/4)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4)*sqrt((9*a^4 + 12*a
^2*c^2 + 4*c^4 + 2*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*a*c)/(9*a^4 - 12*a^2*c^2 + 4*c^4))*arctan(-1/12*(sqrt(2)
*sqrt(1/3)*(54*a^4 + 72*a^2*c^2 + 24*c^4)^(3/4)*(sqrt(6)*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*sqrt(9*a^4 - 12*a^
2*c^2 + 4*c^4)*a - 2*sqrt(6)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4)*(3*a^2*c + 2*c^3))*sqrt((9*a^4 + 12*a^2*c^2 + 4*
c^4 + 2*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*a*c)/(9*a^4 - 12*a^2*c^2 + 4*c^4))*sqrt((3*(9*a^4 + 12*a^2*c^2 + 4*
c^4)*x^2 - sqrt(2)*(54*a^4 + 72*a^2*c^2 + 24*c^4)^(1/4)*(sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*c*x - 3*(3*a^3 + 2
*a*c^2)*x)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 + 2*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*a*c)/(9*a^4 - 12*a^2*c^2 +
4*c^4)) + sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*(3*a^2 + 2*c^2))/(9*a^4 + 12*a^2*c^2 + 4*c^4)) - sqrt(2)*(54*a^4
+ 72*a^2*c^2 + 24*c^4)^(3/4)*(sqrt(6)*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4)*a*x
- 2*sqrt(6)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4)*(3*a^2*c + 2*c^3)*x)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 + 2*sqrt(54
*a^4 + 72*a^2*c^2 + 24*c^4)*a*c)/(9*a^4 - 12*a^2*c^2 + 4*c^4)) - 2*sqrt(6)*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*
(9*a^4 + 12*a^2*c^2 + 4*c^4)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4))/(81*a^8 + 108*a^6*c^2 - 48*a^2*c^6 - 16*c^8)) -
3*(sqrt(2)*(54*a^4 + 72*a^2*c^2 + 24*c^4)^(1/4)*(9*a^4 + 12*a^2*c^2 + 4*c^4 - 2*sqrt(54*a^4 + 72*a^2*c^2 + 24
*c^4)*a*c)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 + 2*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*a*c)/(9*a^4 - 12*a^2*c^2 +
4*c^4)) - 4*(9*a^4 + 12*a^2*c^2 + 4*c^4)*d)*log(3*(9*a^4 + 12*a^2*c^2 + 4*c^4)*x^2 + sqrt(2)*(54*a^4 + 72*a^2*
c^2 + 24*c^4)^(1/4)*(sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*c*x - 3*(3*a^3 + 2*a*c^2)*x)*sqrt((9*a^4 + 12*a^2*c^2
+ 4*c^4 + 2*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*a*c)/(9*a^4 - 12*a^2*c^2 + 4*c^4)) + sqrt(54*a^4 + 72*a^2*c^2 +
24*c^4)*(3*a^2 + 2*c^2)) + 3*(sqrt(2)*(54*a^4 + 72*a^2*c^2 + 24*c^4)^(1/4)*(9*a^4 + 12*a^2*c^2 + 4*c^4 - 2*sq
rt(54*a^4 + 72*a^2*c^2 + 24*c^4)*a*c)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 + 2*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*
a*c)/(9*a^4 - 12*a^2*c^2 + 4*c^4)) + 4*(9*a^4 + 12*a^2*c^2 + 4*c^4)*d)*log(3*(9*a^4 + 12*a^2*c^2 + 4*c^4)*x^2
- sqrt(2)*(54*a^4 + 72*a^2*c^2 + 24*c^4)^(1/4)*(sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*c*x - 3*(3*a^3 + 2*a*c^2)*x
)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 + 2*sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*a*c)/(9*a^4 - 12*a^2*c^2 + 4*c^4)) +
sqrt(54*a^4 + 72*a^2*c^2 + 24*c^4)*(3*a^2 + 2*c^2)))/(9*a^4 + 12*a^2*c^2 + 4*c^4)

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Sympy [A]  time = 1.07802, size = 148, normalized size = 0.96 \begin{align*} \operatorname{RootSum}{\left (165888 t^{4} - 55296 t^{3} d + t^{2} \left (6912 a c + 6912 d^{2}\right ) + t \left (- 1152 a c d - 384 d^{3}\right ) + 27 a^{4} + 36 a^{2} c^{2} + 48 a c d^{2} + 12 c^{4} + 8 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 13824 t^{3} c + 3456 t^{2} c d + 216 t a^{3} - 432 t a c^{2} - 288 t c d^{2} - 18 a^{3} d + 36 a c^{2} d + 8 c d^{3}}{27 a^{4} - 12 c^{4}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c*x**2+a)/(3*x**4+2),x)

[Out]

RootSum(165888*_t**4 - 55296*_t**3*d + _t**2*(6912*a*c + 6912*d**2) + _t*(-1152*a*c*d - 384*d**3) + 27*a**4 +
36*a**2*c**2 + 48*a*c*d**2 + 12*c**4 + 8*d**4, Lambda(_t, _t*log(x + (-13824*_t**3*c + 3456*_t**2*c*d + 216*_t
*a**3 - 432*_t*a*c**2 - 288*_t*c*d**2 - 18*a**3*d + 36*a*c**2*d + 8*c*d**3)/(27*a**4 - 12*c**4))))

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Giac [A]  time = 1.10467, size = 185, normalized size = 1.2 \begin{align*} \frac{1}{24} \,{\left (6^{\frac{3}{4}} a + 2 \cdot 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \,{\left (6^{\frac{3}{4}} a + 2 \cdot 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 2 \cdot 6^{\frac{1}{4}} c + 4 \, d\right )} \log \left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 2 \cdot 6^{\frac{1}{4}} c - 4 \, d\right )} \log \left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c*x^2+a)/(3*x^4+2),x, algorithm="giac")

[Out]

1/24*(6^(3/4)*a + 2*6^(1/4)*c)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/24*(6^(3/4)*a +
2*6^(1/4)*c)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 1/48*(6^(3/4)*a - 2*6^(1/4)*c + 4*
d)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/48*(6^(3/4)*a - 2*6^(1/4)*c - 4*d)*log(x^2 - sqrt(2)*(2/3)
^(1/4)*x + sqrt(2/3))