3.1.89 \(\int \frac {1}{x^2 \sqrt {x+x^4}} \, dx\)

Optimal. Leaf size=16 \[ -\frac {2 \sqrt {x^4+x}}{3 x^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2014} \begin {gather*} -\frac {2 \sqrt {x^4+x}}{3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[x + x^4]),x]

[Out]

(-2*Sqrt[x + x^4])/(3*x^2)

Rule 2014

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*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {x^4+x}}{3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[x + x^4]),x]

[Out]

(-2*Sqrt[x + x^4])/(3*x^2)

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IntegrateAlgebraic [A]  time = 0.33, size = 16, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {x+x^4}}{3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^2*Sqrt[x + x^4]),x]

[Out]

(-2*Sqrt[x + x^4])/(3*x^2)

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fricas [A]  time = 0.46, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2 \, \sqrt {x^{4} + x}}{3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(x^4 + x)/x^2

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giac [A]  time = 0.39, size = 9, normalized size = 0.56 \begin {gather*} -\frac {2}{3} \, \sqrt {\frac {1}{x^{3}} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(1/x^3 + 1)

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maple [A]  time = 0.22, size = 13, normalized size = 0.81

method	result	size
default	\(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}\)	\(13\)
trager	\(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}\)	\(13\)
meijerg	\(-\frac {2 \sqrt {x^{3}+1}}{3 x^{\frac {3}{2}}}\)	\(13\)
elliptic	\(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}\)	\(13\)
risch	\(-\frac {2 \left (x^{3}+1\right )}{3 x \sqrt {x \left (x^{3}+1\right )}}\)	\(20\)
gosper	\(-\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right )}{3 x \sqrt {x^{4}+x}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.67, size = 25, normalized size = 1.56 \begin {gather*} -\frac {2 \, {\left (x^{4} + x\right )}}{3 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(x^4 + x)/(sqrt(x^2 - x + 1)*sqrt(x + 1)*x^(5/2))

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mupad [B]  time = 0.18, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2\,\sqrt {x^4+x}}{3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(x + x^4)^(1/2)),x)

[Out]

-(2*(x + x^4)^(1/2))/(3*x^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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