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

Optimal. Leaf size=114 \[ \frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

[Out]

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

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Rubi [A]  time = 0.120906, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {1593, 1831, 297, 1162, 617, 204, 1165, 628, 260} \[ \frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[(c*x^2 + d*x^3)/(2 + 3*x^4),x]

[Out]

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1831

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

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.0276467, size = 108, normalized size = 0.95 \[ \frac{1}{24} \left (\sqrt [4]{6} c \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\sqrt [4]{6} c \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \sqrt [4]{6} c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+2 d \log \left (3 x^4+2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*6^(1/4)*c*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*c*ArcTan[1 + 6^(1/4)*x] + 6^(1/4)*c*Log[2 - 2*6^(1/4)*x + Sqrt
[6]*x^2] - 6^(1/4)*c*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2] + 2*d*Log[2 + 3*x^4])/24

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Maple [A]  time = 0.001, size = 125, normalized size = 1.1 \begin{align*}{\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)/(3*x^4+2),x)

[Out]

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)+1/12*d*ln(3*x^4+2)

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Maxima [A]  time = 1.45013, size = 205, normalized size = 1.8 \begin{align*} \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} d - \sqrt{3} c\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} d + \sqrt{3} c\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{12} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} c \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}{12} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} c \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)/(3*x^4+2),x, algorithm="maxima")

[Out]

1/72*3^(3/4)*2^(1/4)*(3^(1/4)*2^(3/4)*d - sqrt(3)*c)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/72*3^(
3/4)*2^(1/4)*(3^(1/4)*2^(3/4)*d + sqrt(3)*c)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/12*3^(1/4)*2^(
1/4)*c*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4))) + 1/12*3^(1/4)*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 = 1.64733, size = 743, normalized size = 6.52 \begin{align*} -\frac{4 \cdot 6^{\frac{1}{4}}{\left (c^{4}\right )}^{\frac{1}{4}} c^{4} \arctan \left (-\frac{c^{5} + 6^{\frac{1}{4}}{\left (c^{4}\right )}^{\frac{5}{4}} x - 6^{\frac{1}{4}} \sqrt{\frac{1}{3}}{\left (c^{4}\right )}^{\frac{5}{4}} \sqrt{\frac{3 \, c^{3} x^{2} + 6^{\frac{3}{4}}{\left (c^{4}\right )}^{\frac{3}{4}} x + \sqrt{6} \sqrt{c^{4}} c}{c^{3}}}}{c^{5}}\right ) + 4 \cdot 6^{\frac{1}{4}}{\left (c^{4}\right )}^{\frac{1}{4}} c^{4} \arctan \left (\frac{c^{5} - 6^{\frac{1}{4}}{\left (c^{4}\right )}^{\frac{5}{4}} x + 6^{\frac{1}{4}} \sqrt{\frac{1}{3}}{\left (c^{4}\right )}^{\frac{5}{4}} \sqrt{\frac{3 \, c^{3} x^{2} - 6^{\frac{3}{4}}{\left (c^{4}\right )}^{\frac{3}{4}} x + \sqrt{6} \sqrt{c^{4}} c}{c^{3}}}}{c^{5}}\right ) -{\left (2 \, c^{4} d - 6^{\frac{1}{4}}{\left (c^{4}\right )}^{\frac{1}{4}} c^{4}\right )} \log \left (3 \, c^{3} x^{2} + 6^{\frac{3}{4}}{\left (c^{4}\right )}^{\frac{3}{4}} x + \sqrt{6} \sqrt{c^{4}} c\right ) -{\left (2 \, c^{4} d + 6^{\frac{1}{4}}{\left (c^{4}\right )}^{\frac{1}{4}} c^{4}\right )} \log \left (3 \, c^{3} x^{2} - 6^{\frac{3}{4}}{\left (c^{4}\right )}^{\frac{3}{4}} x + \sqrt{6} \sqrt{c^{4}} c\right )}{24 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(4*6^(1/4)*(c^4)^(1/4)*c^4*arctan(-(c^5 + 6^(1/4)*(c^4)^(5/4)*x - 6^(1/4)*sqrt(1/3)*(c^4)^(5/4)*sqrt((3*
c^3*x^2 + 6^(3/4)*(c^4)^(3/4)*x + sqrt(6)*sqrt(c^4)*c)/c^3))/c^5) + 4*6^(1/4)*(c^4)^(1/4)*c^4*arctan((c^5 - 6^
(1/4)*(c^4)^(5/4)*x + 6^(1/4)*sqrt(1/3)*(c^4)^(5/4)*sqrt((3*c^3*x^2 - 6^(3/4)*(c^4)^(3/4)*x + sqrt(6)*sqrt(c^4
)*c)/c^3))/c^5) - (2*c^4*d - 6^(1/4)*(c^4)^(1/4)*c^4)*log(3*c^3*x^2 + 6^(3/4)*(c^4)^(3/4)*x + sqrt(6)*sqrt(c^4
)*c) - (2*c^4*d + 6^(1/4)*(c^4)^(1/4)*c^4)*log(3*c^3*x^2 - 6^(3/4)*(c^4)^(3/4)*x + sqrt(6)*sqrt(c^4)*c))/c^4

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Sympy [A]  time = 0.273372, size = 70, normalized size = 0.61 \begin{align*} \operatorname{RootSum}{\left (41472 t^{4} - 13824 t^{3} d + 1728 t^{2} d^{2} - 96 t d^{3} + 3 c^{4} + 2 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{3456 t^{3} - 864 t^{2} d + 72 t d^{2} - 2 d^{3}}{3 c^{3}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(41472*_t**4 - 13824*_t**3*d + 1728*_t**2*d**2 - 96*_t*d**3 + 3*c**4 + 2*d**4, Lambda(_t, _t*log(x + (3
456*_t**3 - 864*_t**2*d + 72*_t*d**2 - 2*d**3)/(3*c**3))))

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Giac [A]  time = 1.1086, size = 147, normalized size = 1.29 \begin{align*} \frac{1}{12} \cdot 6^{\frac{1}{4}} c \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}{12} \cdot 6^{\frac{1}{4}} c \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{1}{4}} c - 2 \, d\right )} \log \left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{24} \,{\left (6^{\frac{1}{4}} c + 2 \, 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)/(3*x^4+2),x, algorithm="giac")

[Out]

1/12*6^(1/4)*c*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/12*6^(1/4)*c*arctan(3/4*sqrt(2)
*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) - 1/24*(6^(1/4)*c - 2*d)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)
) + 1/24*(6^(1/4)*c + 2*d)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3))