3.9.7 \(\int x \sqrt {3+2 x^4} \, dx\) [807]

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

[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 = {281, 201, 221} \begin {gather*} \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 \end {gather*}

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 201

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 221

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 281

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} \text {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} \text {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 = 0.12, size = 44, normalized size = 1.10 \begin {gather*} \frac {1}{8} \left (2 x^2 \sqrt {3+2 x^4}+3 \sqrt {2} \tanh ^{-1}\left (\frac {x^2}{\sqrt {\frac {3}{2}+x^4}}\right )\right ) \end {gather*}

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]*ArcTanh[x^2/Sqrt[3/2 + x^4]])/8

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Maple [A]
time = 0.24, size = 30, normalized size = 0.75

method	result	size
default	\(\frac {3 \arcsinh \left (\frac {x^{2} \sqrt {6}}{3}\right ) \sqrt {2}}{8}+\frac {x^{2} \sqrt {2 x^{4}+3}}{4}\)	\(30\)
risch	\(\frac {3 \arcsinh \left (\frac {x^{2} \sqrt {6}}{3}\right ) \sqrt {2}}{8}+\frac {x^{2} \sqrt {2 x^{4}+3}}{4}\)	\(30\)
elliptic	\(\frac {3 \arcsinh \left (\frac {x^{2} \sqrt {6}}{3}\right ) \sqrt {2}}{8}+\frac {x^{2} \sqrt {2 x^{4}+3}}{4}\)	\(30\)
trager	\(\frac {x^{2} \sqrt {2 x^{4}+3}}{4}+\frac {3 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}-2\right ) \sqrt {2 x^{4}+3}+2 x^{2}\right )}{8}\)	\(47\)
meijerg	\(-\frac {3 \sqrt {2}\, \left (-\frac {2 \sqrt {\pi }\, \sqrt {2}\, \sqrt {3}\, x^{2} \sqrt {\frac {2 x^{4}}{3}+1}}{3}-2 \sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {2}\, \sqrt {3}\, x^{2}}{3}\right )\right )}{16 \sqrt {\pi }}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (29) = 58\).
time = 0.50, size = 75, normalized size = 1.88 \begin {gather*} -\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 {gather*}

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 = 0.37, size = 45, normalized size = 1.12 \begin {gather*} \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 {gather*}

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 = 0.71, size = 51, normalized size = 1.28 \begin {gather*} \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 {gather*}

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 = 0.97, size = 39, normalized size = 0.98 \begin {gather*} \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 {gather*}

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

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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