3.3.26 \(\int \frac {1}{x \sqrt {x^3+x^4}} \, dx\)

Optimal. Leaf size=23 \[ \frac {2 (2 x-1) \sqrt {x^4+x^3}}{3 x^3} \]

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Rubi [A]  time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.61, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2016, 2000} \begin {gather*} \frac {4 \sqrt {x^4+x^3}}{3 x^2}-\frac {2 \sqrt {x^4+x^3}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[x^3 + x^4]),x]

[Out]

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

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {x^3+x^4}} \, dx &=-\frac {2 \sqrt {x^3+x^4}}{3 x^3}-\frac {2}{3} \int \frac {1}{\sqrt {x^3+x^4}} \, dx\\ &=-\frac {2 \sqrt {x^3+x^4}}{3 x^3}+\frac {4 \sqrt {x^3+x^4}}{3 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 \left (2 x^2+x-1\right )}{3 \sqrt {x^3 (x+1)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[x^3 + x^4]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.11, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 (-1+2 x) \sqrt {x^3+x^4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*Sqrt[x^3 + x^4]),x]

[Out]

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

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fricas [A]  time = 0.45, size = 26, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left (2 \, x^{3} + \sqrt {x^{4} + x^{3}} {\left (2 \, x - 1\right )}\right )}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.37, size = 19, normalized size = 0.83 \begin {gather*} -\frac {2}{3} \, {\left (\frac {1}{x} + 1\right )}^{\frac {3}{2}} + 2 \, \sqrt {\frac {1}{x} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 16, normalized size = 0.70

method	result	size
meijerg	\(-\frac {2 \left (1-2 x \right ) \sqrt {1+x}}{3 x^{\frac {3}{2}}}\)	\(16\)
gosper	\(\frac {2 \left (1+x \right ) \left (-1+2 x \right )}{3 \sqrt {x^{4}+x^{3}}}\)	\(20\)
trager	\(\frac {2 \left (-1+2 x \right ) \sqrt {x^{4}+x^{3}}}{3 x^{3}}\)	\(20\)
risch	\(\frac {-\frac {2}{3}+\frac {2}{3} x +\frac {4}{3} x^{2}}{\sqrt {x^{3} \left (1+x \right )}}\)	\(20\)
default	\(\frac {2 \sqrt {x \left (1+x \right )}\, \sqrt {x^{2}+x}\, \left (-1+2 x \right )}{3 x \sqrt {x^{4}+x^{3}}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(1-2*x)*(1+x)^(1/2)/x^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x^{4} + x^{3}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x^4 + x^3)*x), x)

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mupad [B]  time = 0.13, size = 29, normalized size = 1.26 \begin {gather*} \frac {4\,x\,\sqrt {x^4+x^3}-2\,\sqrt {x^4+x^3}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^3 + x^4)^(1/2)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt(x**3*(x + 1))), x)

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