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

Optimal. Leaf size=123 \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}} \]

[Out]

(b*ArcTan[Sqrt[3/2]*x^2])/(2*Sqrt[6]) - (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))

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Rubi [A]  time = 0.117672, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {1593, 1831, 275, 203, 297, 1162, 617, 204, 1165, 628} \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)/(2 + 3*x^4),x]

[Out]

(b*ArcTan[Sqrt[3/2]*x^2])/(2*Sqrt[6]) - (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))

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 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 203

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0436847, size = 99, normalized size = 0.8 \[ \frac{-2 \left (\sqrt [4]{6} b+c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \left (c-\sqrt [4]{6} b\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+c \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-c \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )}{4\ 6^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.022, size = 129, normalized size = 1.1 \begin{align*}{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b*arctan(1/2*x^2*6^(1/2))*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)+
1/144*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)

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Maxima [A]  time = 1.45964, size = 198, normalized size = 1.61 \begin{align*} \frac{1}{24} \, \sqrt{2}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} c - 2 \, \sqrt{3} b\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}{24} \, \sqrt{2}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} c + 2 \, \sqrt{3} b\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}{24} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} c \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{24} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} c \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [A]  time = 0.458956, size = 85, normalized size = 0.69 \begin{align*} \operatorname{RootSum}{\left (27648 t^{4} + 576 t^{2} b^{2} + 96 t b c^{2} + 3 b^{4} + 2 c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 1152 t^{3} c^{2} + 288 t^{2} b^{3} - 36 t b^{2} c^{2} + 3 b^{5} - 3 b c^{4}}{6 b^{4} c - c^{5}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27648*_t**4 + 576*_t**2*b**2 + 96*_t*b*c**2 + 3*b**4 + 2*c**4, Lambda(_t, _t*log(x + (-1152*_t**3*c**2
+ 288*_t**2*b**3 - 36*_t*b**2*c**2 + 3*b**5 - 3*b*c**4)/(6*b**4*c - c**5))))

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Giac [A]  time = 1.11023, size = 154, normalized size = 1.25 \begin{align*} -\frac{1}{24} \cdot 6^{\frac{1}{4}} c \log \left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{24} \cdot 6^{\frac{1}{4}} c \log \left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{12} \,{\left (\sqrt{6} b - 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}{12} \,{\left (\sqrt{6} b + 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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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