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

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

[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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Rubi [A]  time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.03, size = 40, normalized size = 1.00 \[ \frac {1}{8} \left (3 \sqrt {2} \sinh ^{-1}\left (\sqrt {\frac {2}{3}} x^2\right )+2 \sqrt {2 x^4+3} x^2\right ) \]

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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fricas [A]  time = 1.08, size = 45, normalized size = 1.12 \[ \frac {1}{4} \, \sqrt {2 \, x^{4} + 3} x^{2} + \frac {3}{16} \, \sqrt {2} \log \left (-4 \, x^{4} - 2 \, \sqrt {2} \sqrt {2 \, x^{4} + 3} x^{2} - 3\right ) \]

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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giac [A]  time = 0.18, size = 39, normalized size = 0.98 \[ \frac {1}{4} \, \sqrt {2 \, x^{4} + 3} x^{2} - \frac {3}{8} \, \sqrt {2} \log \left (-\sqrt {2} x^{2} + \sqrt {2 \, x^{4} + 3}\right ) \]

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))

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maple [A]  time = 0.01, size = 30, normalized size = 0.75 \[ \frac {\sqrt {2 x^{4}+3}\, x^{2}}{4}+\frac {3 \sqrt {2}\, \arcsinh \left (\frac {\sqrt {6}\, x^{2}}{3}\right )}{8} \]

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

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maxima [B]  time = 2.98, size = 75, normalized size = 1.88 \[ -\frac {3}{16} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {\sqrt {2 \, x^{4} + 3}}{x^{2}}}{\sqrt {2} + \frac {\sqrt {2 \, x^{4} + 3}}{x^{2}}}\right ) + \frac {3 \, \sqrt {2 \, x^{4} + 3}}{4 \, x^{2} {\left (\frac {2 \, x^{4} + 3}{x^{4}} - 2\right )}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\sqrt {2\,x^4+3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*x^4 + 3)^(1/2),x)

[Out]

int(x*(2*x^4 + 3)^(1/2), x)

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sympy [A]  time = 2.76, size = 51, normalized size = 1.28 \[ \frac {x^{6}}{2 \sqrt {2 x^{4} + 3}} + \frac {3 x^{2}}{4 \sqrt {2 x^{4} + 3}} + \frac {3 \sqrt {2} \operatorname {asinh}{\left (\frac {\sqrt {6} x^{2}}{3} \right )}}{8} \]

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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