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

Optimal. Leaf size=123 \[ -\frac {a \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}+\frac {a \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}-\frac {a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}} \]

[Out]

1/24*a*arctan(-1+6^(1/4)*x)*6^(3/4)+1/24*a*arctan(1+6^(1/4)*x)*6^(3/4)-1/48*a*ln(-6^(3/4)*x+3*x^2+6^(1/2))*6^(
3/4)+1/48*a*ln(6^(3/4)*x+3*x^2+6^(1/2))*6^(3/4)+1/12*b*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {1876, 211, 1165, 628, 1162, 617, 204, 275, 203} \[ -\frac {a \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}+\frac {a \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}-\frac {a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(2 + 3*x^4),x]

[Out]

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

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 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 211

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

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 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 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 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 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 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

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 107, normalized size = 0.87 \[ \frac {-2 \left (\sqrt [4]{6} a+2 b\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \left (\sqrt [4]{6} a-2 b\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+\sqrt [4]{6} a \left (\log \left (\sqrt {6} x^2+2 \sqrt [4]{6} x+2\right )-\log \left (\sqrt {6} x^2-2 \sqrt [4]{6} x+2\right )\right )}{8 \sqrt {6}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(2 + 3*x^4),x]

[Out]

(-2*(6^(1/4)*a + 2*b)*ArcTan[1 - 6^(1/4)*x] + 2*(6^(1/4)*a - 2*b)*ArcTan[1 + 6^(1/4)*x] + 6^(1/4)*a*(-Log[2 -
2*6^(1/4)*x + Sqrt[6]*x^2] + Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2]))/(8*Sqrt[6])

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

[Out]

Timed out

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giac [A]  time = 0.19, size = 115, normalized size = 0.93 \[ \frac {1}{48} \cdot 6^{\frac {3}{4}} a \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{48} \cdot 6^{\frac {3}{4}} a \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) + \frac {1}{24} \, {\left (6^{\frac {3}{4}} a - 2 \, \sqrt {6} b\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{24} \, {\left (6^{\frac {3}{4}} a + 2 \, \sqrt {6} b\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 129, normalized size = 1.05 \[ \frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{48}+\frac {\sqrt {6}\, b \arctan \left (\frac {\sqrt {6}\, x^{2}}{2}\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*3^(1/2)*6^(1/4)*2^(1/2)*a*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/24*3^(1/2)*6^(1/4)*2^(1/2)*a*arctan(1
/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/48*3^(1/2)*6^(1/4)*2^(1/2)*a*ln((x^2+1/3*3^(1/2)*6^(1/4)*2^(1/2)*x+1/3*6^(1/
2))/(x^2-1/3*3^(1/2)*6^(1/4)*2^(1/2)*x+1/3*6^(1/2)))+1/12*6^(1/2)*b*arctan(1/2*6^(1/2)*x^2)

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maxima [A]  time = 2.91, size = 147, normalized size = 1.20 \[ \frac {1}{48} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} a \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - \frac {1}{48} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} a \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{24} \, \sqrt {3} {\left (3^{\frac {1}{4}} 2^{\frac {3}{4}} a - 2 \, \sqrt {2} b\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{24} \, \sqrt {3} {\left (3^{\frac {1}{4}} 2^{\frac {3}{4}} a + 2 \, \sqrt {2} b\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.20, size = 119, normalized size = 0.97 \[ \frac {2^{3/4}\,3^{3/4}\,a\,\ln \left (x^2+\frac {6^{3/4}\,x}{3}+\frac {\sqrt {6}}{3}\right )}{48}-\frac {2^{3/4}\,3^{3/4}\,a\,\ln \left (x^2-\frac {6^{3/4}\,x}{3}+\frac {\sqrt {6}}{3}\right )}{48}+\frac {2^{3/4}\,3^{3/4}\,a\,\mathrm {atan}\left (6^{1/4}\,x-1\right )}{24}+\frac {2^{3/4}\,3^{3/4}\,a\,\mathrm {atan}\left (6^{1/4}\,x+1\right )}{24}+\frac {\sqrt {2}\,\sqrt {3}\,b\,\mathrm {atan}\left (6^{1/4}\,x-1\right )}{12}-\frac {\sqrt {2}\,\sqrt {3}\,b\,\mathrm {atan}\left (6^{1/4}\,x+1\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(3*x^4 + 2),x)

[Out]

(2^(3/4)*3^(3/4)*a*log((6^(3/4)*x)/3 + 6^(1/2)/3 + x^2))/48 - (2^(3/4)*3^(3/4)*a*log(6^(1/2)/3 - (6^(3/4)*x)/3
+ x^2))/48 + (2^(3/4)*3^(3/4)*a*atan(6^(1/4)*x - 1))/24 + (2^(3/4)*3^(3/4)*a*atan(6^(1/4)*x + 1))/24 + (2^(1/
2)*3^(1/2)*b*atan(6^(1/4)*x - 1))/12 - (2^(1/2)*3^(1/2)*b*atan(6^(1/4)*x + 1))/12

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sympy [A]  time = 0.72, size = 88, normalized size = 0.72 \[ \operatorname {RootSum} {\left (18432 t^{4} + 384 t^{2} b^{2} - 96 t a^{2} b + 3 a^{4} + 2 b^{4}, \left (t \mapsto t \log {\left (x + \frac {3072 t^{3} b^{2} + 192 t^{2} a^{2} b + 24 t a^{4} + 32 t b^{4} - 10 a^{2} b^{3}}{3 a^{5} - 8 a b^{4}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(18432*_t**4 + 384*_t**2*b**2 - 96*_t*a**2*b + 3*a**4 + 2*b**4, Lambda(_t, _t*log(x + (3072*_t**3*b**2
+ 192*_t**2*a**2*b + 24*_t*a**4 + 32*_t*b**4 - 10*a**2*b**3)/(3*a**5 - 8*a*b**4))))

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