3.807 \(\int x \sqrt{3+2 x^4} \, dx\)

Optimal. Leaf size=40 \[ \frac{1}{4} \sqrt{2 x^4+3} x^2+\frac{3 \sinh ^{-1}\left (\sqrt{\frac{2}{3}} x^2\right )}{4 \sqrt{2}} \]

[Out]

(x^2*Sqrt[3 + 2*x^4])/4 + (3*ArcSinh[Sqrt[2/3]*x^2])/(4*Sqrt[2])

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Rubi [A]  time = 0.0138261, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 195, 215} \[ \frac{1}{4} \sqrt{2 x^4+3} x^2+\frac{3 \sinh ^{-1}\left (\sqrt{\frac{2}{3}} x^2\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[3 + 2*x^4],x]

[Out]

(x^2*Sqrt[3 + 2*x^4])/4 + (3*ArcSinh[Sqrt[2/3]*x^2])/(4*Sqrt[2])

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 215

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

Rubi steps

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

Mathematica [A]  time = 0.0124788, size = 40, normalized size = 1. \[ \frac{1}{8} \left (2 \sqrt{2 x^4+3} x^2+3 \sqrt{2} \sinh ^{-1}\left (\sqrt{\frac{2}{3}} x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[3 + 2*x^4],x]

[Out]

(2*x^2*Sqrt[3 + 2*x^4] + 3*Sqrt[2]*ArcSinh[Sqrt[2/3]*x^2])/8

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Maple [A]  time = 0.007, size = 30, normalized size = 0.8 \begin{align*}{\frac{3\,\sqrt{2}}{8}{\it Arcsinh} \left ({\frac{{x}^{2}\sqrt{6}}{3}} \right ) }+{\frac{{x}^{2}}{4}\sqrt{2\,{x}^{4}+3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*x^4+3)^(1/2),x)

[Out]

3/8*arcsinh(1/3*x^2*6^(1/2))*2^(1/2)+1/4*x^2*(2*x^4+3)^(1/2)

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Maxima [B]  time = 1.45058, size = 101, normalized size = 2.52 \begin{align*} -\frac{3}{16} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{\sqrt{2 \, x^{4} + 3}}{x^{2}}}{\sqrt{2} + \frac{\sqrt{2 \, x^{4} + 3}}{x^{2}}}\right ) + \frac{3 \, \sqrt{2 \, x^{4} + 3}}{4 \, x^{2}{\left (\frac{2 \, x^{4} + 3}{x^{4}} - 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*x^4+3)^(1/2),x, algorithm="maxima")

[Out]

-3/16*sqrt(2)*log(-(sqrt(2) - sqrt(2*x^4 + 3)/x^2)/(sqrt(2) + sqrt(2*x^4 + 3)/x^2)) + 3/4*sqrt(2*x^4 + 3)/(x^2
*((2*x^4 + 3)/x^4 - 2))

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Fricas [A]  time = 1.47075, size = 119, normalized size = 2.98 \begin{align*} \frac{1}{4} \, \sqrt{2 \, x^{4} + 3} x^{2} + \frac{3}{16} \, \sqrt{2} \log \left (-4 \, x^{4} - 2 \, \sqrt{2} \sqrt{2 \, x^{4} + 3} x^{2} - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*x^4+3)^(1/2),x, algorithm="fricas")

[Out]

1/4*sqrt(2*x^4 + 3)*x^2 + 3/16*sqrt(2)*log(-4*x^4 - 2*sqrt(2)*sqrt(2*x^4 + 3)*x^2 - 3)

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Sympy [A]  time = 2.03145, size = 51, normalized size = 1.27 \begin{align*} \frac{x^{6}}{2 \sqrt{2 x^{4} + 3}} + \frac{3 x^{2}}{4 \sqrt{2 x^{4} + 3}} + \frac{3 \sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{6} x^{2}}{3} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*x**4+3)**(1/2),x)

[Out]

x**6/(2*sqrt(2*x**4 + 3)) + 3*x**2/(4*sqrt(2*x**4 + 3)) + 3*sqrt(2)*asinh(sqrt(6)*x**2/3)/8

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Giac [A]  time = 1.10158, size = 53, normalized size = 1.32 \begin{align*} \frac{1}{4} \, \sqrt{2 \, x^{4} + 3} x^{2} - \frac{3}{8} \, \sqrt{2} \log \left (-\sqrt{2} x^{2} + \sqrt{2 \, x^{4} + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*x^4+3)^(1/2),x, algorithm="giac")

[Out]

1/4*sqrt(2*x^4 + 3)*x^2 - 3/8*sqrt(2)*log(-sqrt(2)*x^2 + sqrt(2*x^4 + 3))