\(\int \frac {1}{x^2 \sqrt {x+x^4}} \, dx\) [89]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2 \sqrt {x+x^4}}{3 x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2039} \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2 \sqrt {x^4+x}}{3 x^2} \]

[In]

Int[1/(x^2*Sqrt[x + x^4]),x]

[Out]

(-2*Sqrt[x + x^4])/(3*x^2)

Rule 2039

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] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {x+x^4}}{3 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2 \sqrt {x+x^4}}{3 x^2} \]

[In]

Integrate[1/(x^2*Sqrt[x + x^4]),x]

[Out]

(-2*Sqrt[x + x^4])/(3*x^2)

Maple [A] (verified)

Time = 3.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 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	\(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}\)	\(13\)
pseudoelliptic	\(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}\)	\(13\)
risch	\(-\frac {2 \left (x^{3}+1\right )}{3 x \sqrt {x \left (x^{3}+1\right )}}\)	\(20\)
gosper	\(-\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right )}{3 x \sqrt {x^{4}+x}}\)	\(24\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2 \, \sqrt {x^{4} + x}}{3 \, x^{2}} \]

[In]

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

[Out]

-2/3*sqrt(x^4 + x)/x^2

Sympy [F]

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

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2 \, {\left (x^{4} + x\right )}}{3 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} x^{\frac {5}{2}}} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2}{3} \, \sqrt {\frac {1}{x^{3}} + 1} \]

[In]

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

[Out]

-2/3*sqrt(1/x^3 + 1)

Mupad [B] (verification not implemented)

Time = 5.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2\,\sqrt {x^4+x}}{3\,x^2} \]

[In]

int(1/(x^2*(x + x^4)^(1/2)),x)

[Out]

-(2*(x + x^4)^(1/2))/(3*x^2)