3.135 \(\int \frac{x}{(1+\sqrt{3}+x) \sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=136 \[ \frac{\sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{x^3+1}}\right )}{3^{3/4}} \]

[Out]

-((Sqrt[2]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]])/3^(3/4)) + (Sqrt[2]*(1 + x)*Sqrt[(1 - x + x^2)
/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(3/4)*Sqrt[(1
+ x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])

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Rubi [A]  time = 0.220413, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2141, 218, 2140, 203} \[ \frac{\sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{x^3+1}}\right )}{3^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[x/((1 + Sqrt[3] + x)*Sqrt[1 + x^3]),x]

[Out]

-((Sqrt[2]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]])/3^(3/4)) + (Sqrt[2]*(1 + x)*Sqrt[(1 - x + x^2)
/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(3/4)*Sqrt[(1
+ x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])

Rule 2141

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> -Dist[(6*a*d^4*e - c*f*
(b*c^3 - 22*a*d^3))/(c*d*(b*c^3 - 28*a*d^3)), Int[1/Sqrt[a + b*x^3], x], x] + Dist[(d*e - c*f)/(c*d*(b*c^3 - 2
8*a*d^3)), Int[(c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[6*a*d^4*e - c*f*(b*c^3 - 2
2*a*d^3), 0]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 2140

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{k = Simplify[(d*e
+ 2*c*f)/(c*f)]}, Dist[((1 + k)*e)/d, Subst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + ((1 + k)*d*x)/c)/Sqrt[a +
b*x^3]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6
, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]

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])

Rubi steps

\begin{align*} \int \frac{x}{\left (1+\sqrt{3}+x\right ) \sqrt{1+x^3}} \, dx &=\frac{\left (-1-\sqrt{3}\right ) \int \frac{\left (1+\sqrt{3}\right ) \left (-22+\left (1+\sqrt{3}\right )^3\right )+6 x}{\left (1+\sqrt{3}+x\right ) \sqrt{1+x^3}} \, dx}{\left (1+\sqrt{3}\right ) \left (-28+\left (1+\sqrt{3}\right )^3\right )}+\frac{\left (-22+\left (1+\sqrt{3}\right )^3\right ) \int \frac{1}{\sqrt{1+x^3}} \, dx}{-28+\left (1+\sqrt{3}\right )^3}\\ &=\frac{\sqrt{2} (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{1+x^3}}-\frac{\left (12 \left (-1-\sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+\left (3+2 \sqrt{3}\right ) x^2} \, dx,x,\frac{1+x}{\sqrt{1+x^3}}\right )}{\left (1+\sqrt{3}\right ) \left (-28+\left (1+\sqrt{3}\right )^3\right )}\\ &=-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1+x)}{\sqrt{1+x^3}}\right )}{3^{3/4}}+\frac{\sqrt{2} (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{1+x^3}}\\ \end{align*}

Mathematica [C]  time = 0.48615, size = 209, normalized size = 1.54 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \left (-\frac{\left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}+\frac{2 i \left (1+\sqrt{3}\right ) \sqrt{x^2-x+1} \Pi \left (\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{3+(2+i) \sqrt{3}}\right )}{\sqrt{x^3+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/((1 + Sqrt[3] + x)*Sqrt[1 + x^3]),x]

[Out]

(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*Elli
pticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))
]) + ((2*I)*(1 + Sqrt[3])*Sqrt[1 - x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[(1 +
(-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(3 + (2 + I)*Sqrt[3])))/Sqrt[1 + x^3]

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Maple [B]  time = 0.022, size = 255, normalized size = 1.9 \begin{align*} 2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }+{\frac{ \left ( -2\,\sqrt{3}-2 \right ) \left ({\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{3}\sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},{\frac{ \left ( -{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{3}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(1+x+3^(1/2))/(x^3+1)^(1/2),x)

[Out]

2*(3/2-1/2*I*3^(1/2))*((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x
-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)*EllipticF(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3
/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+2/3*(-3^(1/2)-1)*(3/2-1/2*I*3^(1/2))*((1+x)/(3/2-1/2*I*3^(1/2))
)^(1/2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/
(x^3+1)^(1/2)*3^(1/2)*EllipticPi(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),1/3*(-3/2+1/2*I*3^(1/2))*3^(1/2),((-3/2+1/2
*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x^{3} + 1}{\left (x + \sqrt{3} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(1+x+3^(1/2))/(x^3+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x/(sqrt(x^3 + 1)*(x + sqrt(3) + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{3} + 1}{\left (x^{2} - \sqrt{3} x + x\right )}}{x^{5} + 2 \, x^{4} - 2 \, x^{3} + x^{2} + 2 \, x - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(1+x+3^(1/2))/(x^3+1)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(x^3 + 1)*(x^2 - sqrt(3)*x + x)/(x^5 + 2*x^4 - 2*x^3 + x^2 + 2*x - 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1 + \sqrt{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(1+x+3**(1/2))/(x**3+1)**(1/2),x)

[Out]

Integral(x/(sqrt((x + 1)*(x**2 - x + 1))*(x + 1 + sqrt(3))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x^{3} + 1}{\left (x + \sqrt{3} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(1+x+3^(1/2))/(x^3+1)^(1/2),x, algorithm="giac")

[Out]

integrate(x/(sqrt(x^3 + 1)*(x + sqrt(3) + 1)), x)