3.4.27 \(\int \frac {x}{\sqrt {x^3+x^4}} \, dx\)

Optimal. Leaf size=28 \[ \log (x)-\log \left (-2 x^2+2 \sqrt {x^4+x^3}-x\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 0.64, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2029, 206} \begin {gather*} 2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[x^3 + x^4],x]

[Out]

2*ArcTanh[x^2/Sqrt[x^3 + x^4]]

Rule 206

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

Rule 2029

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

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.11 \begin {gather*} \frac {2 x^{3/2} \sqrt {x+1} \sinh ^{-1}\left (\sqrt {x}\right )}{\sqrt {x^3 (x+1)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[x^3 + x^4],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.10, size = 28, normalized size = 1.00 \begin {gather*} \log (x)-\log \left (-x-2 x^2+2 \sqrt {x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/Sqrt[x^3 + x^4],x]

[Out]

Log[x] - Log[-x - 2*x^2 + 2*Sqrt[x^3 + x^4]]

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fricas [A]  time = 0.47, size = 26, normalized size = 0.93 \begin {gather*} -\log \left (-\frac {2 \, x^{2} + x - 2 \, \sqrt {x^{4} + x^{3}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.37, size = 24, normalized size = 0.86 \begin {gather*} \log \left (\sqrt {\frac {1}{x} + 1} + 1\right ) - \log \left ({\left | \sqrt {\frac {1}{x} + 1} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt(1/x + 1) + 1) - log(abs(sqrt(1/x + 1) - 1))

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maple [A]  time = 0.08, size = 7, normalized size = 0.25

method	result	size
meijerg	\(2 \arcsinh \left (\sqrt {x}\right )\)	\(7\)
trager	\(\ln \left (\frac {2 x^{2}+2 \sqrt {x^{4}+x^{3}}+x}{x}\right )\)	\(24\)
default	\(\frac {x \sqrt {x \left (1+x \right )}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{\sqrt {x^{4}+x^{3}}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arcsinh(x^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{4} + x^{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/sqrt(x^4 + x^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x}{\sqrt {x^4+x^3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^3 + x^4)^(1/2),x)

[Out]

int(x/(x^3 + x^4)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{3} \left (x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(x**3*(x + 1)), x)

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