\(\int \frac {x}{\sqrt {x+x^4}} \, dx\) [137]

Optimal result

Integrand size = 11, antiderivative size = 18 \[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\frac {2}{3} \text {arctanh}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2054, 212} \[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\frac {2}{3} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \]

[In]

Int[x/Sqrt[x + x^4],x]

[Out]

(2*ArcTanh[x^2/Sqrt[x + x^4]])/3

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2054

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right ) \\ & = \frac {2}{3} \text {arctanh}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(18)=36\).

Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.39 \[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\frac {2 \sqrt {x} \sqrt {1+x^3} \log \left (x^{3/2}+\sqrt {1+x^3}\right )}{3 \sqrt {x+x^4}} \]

[In]

Integrate[x/Sqrt[x + x^4],x]

[Out]

(2*Sqrt[x]*Sqrt[1 + x^3]*Log[x^(3/2) + Sqrt[1 + x^3]])/(3*Sqrt[x + x^4])

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.39

method	result	size
meijerg	\(\frac {2 \,\operatorname {arcsinh}\left (x^{\frac {3}{2}}\right )}{3}\)	\(7\)
pseudoelliptic	\(\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{4}+x}}{x^{2}}\right )}{3}\)	\(15\)
default	\(\frac {\ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{3}\)	\(21\)
trager	\(\frac {\ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{3}\)	\(21\)
elliptic	\(-\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(290\)

[In]

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

[Out]

2/3*arcsinh(x^(3/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\frac {1}{3} \, \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\int \frac {x}{\sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\int { \frac {x}{\sqrt {x^{4} + x}} \,d x } \]

[In]

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

[Out]

integrate(x/sqrt(x^4 + x), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\frac {1}{3} \, \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) - \frac {1}{3} \, \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \]

[In]

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

[Out]

1/3*log(sqrt(1/x^3 + 1) + 1) - 1/3*log(abs(sqrt(1/x^3 + 1) - 1))

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\int \frac {x}{\sqrt {x^4+x}} \,d x \]

[In]

int(x/(x + x^4)^(1/2),x)

[Out]

int(x/(x + x^4)^(1/2), x)