\(\int \frac {1}{x^5 \sqrt {x+x^4}} \, dx\) [220]

Optimal result

Integrand size = 13, antiderivative size = 23 \[ \int \frac {1}{x^5 \sqrt {x+x^4}} \, dx=\frac {2 \left (-1+2 x^3\right ) \sqrt {x+x^4}}{9 x^5} \]

[Out]

2/9*(2*x^3-1)*(x^4+x)^(1/2)/x^5

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2041, 2039} \[ \int \frac {1}{x^5 \sqrt {x+x^4}} \, dx=\frac {4 \sqrt {x^4+x}}{9 x^2}-\frac {2 \sqrt {x^4+x}}{9 x^5} \]

[In]

Int[1/(x^5*Sqrt[x + x^4]),x]

[Out]

(-2*Sqrt[x + x^4])/(9*x^5) + (4*Sqrt[x + x^4])/(9*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])

Rule 2041

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*} \text {integral}& = -\frac {2 \sqrt {x+x^4}}{9 x^5}-\frac {2}{3} \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx \\ & = -\frac {2 \sqrt {x+x^4}}{9 x^5}+\frac {4 \sqrt {x+x^4}}{9 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \sqrt {x+x^4}} \, dx=\frac {2 \left (-1+2 x^3\right ) \sqrt {x+x^4}}{9 x^5} \]

[In]

Integrate[1/(x^5*Sqrt[x + x^4]),x]

[Out]

(2*(-1 + 2*x^3)*Sqrt[x + x^4])/(9*x^5)

Maple [A] (verified)

Time = 2.60 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87

method	result	size
trager	\(\frac {2 \left (2 x^{3}-1\right ) \sqrt {x^{4}+x}}{9 x^{5}}\)	\(20\)
meijerg	\(-\frac {2 \left (-2 x^{3}+1\right ) \sqrt {x^{3}+1}}{9 x^{\frac {9}{2}}}\)	\(20\)
pseudoelliptic	\(\frac {2 \left (2 x^{3}-1\right ) \sqrt {x^{4}+x}}{9 x^{5}}\)	\(20\)
risch	\(\frac {\frac {2}{9} x^{3}-\frac {2}{9}+\frac {4}{9} x^{6}}{x^{4} \sqrt {x \left (x^{3}+1\right )}}\)	\(25\)
default	\(-\frac {2 \sqrt {x^{4}+x}}{9 x^{5}}+\frac {4 \sqrt {x^{4}+x}}{9 x^{2}}\)	\(26\)
elliptic	\(-\frac {2 \sqrt {x^{4}+x}}{9 x^{5}}+\frac {4 \sqrt {x^{4}+x}}{9 x^{2}}\)	\(26\)
gosper	\(\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (2 x^{3}-1\right )}{9 x^{4} \sqrt {x^{4}+x}}\)	\(31\)

[In]

int(1/x^5/(x^4+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/9*(2*x^3-1)*(x^4+x)^(1/2)/x^5

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^5 \sqrt {x+x^4}} \, dx=\frac {2 \, \sqrt {x^{4} + x} {\left (2 \, x^{3} - 1\right )}}{9 \, x^{5}} \]

[In]

integrate(1/x^5/(x^4+x)^(1/2),x, algorithm="fricas")

[Out]

2/9*sqrt(x^4 + x)*(2*x^3 - 1)/x^5

Sympy [F]

\[ \int \frac {1}{x^5 \sqrt {x+x^4}} \, dx=\int \frac {1}{x^{5} \sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]

[In]

integrate(1/x**5/(x**4+x)**(1/2),x)

[Out]

Integral(1/(x**5*sqrt(x*(x + 1)*(x**2 - x + 1))), x)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^5 \sqrt {x+x^4}} \, dx=\frac {2 \, {\left (2 \, x^{7} + x^{4} - x\right )}}{9 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} x^{\frac {11}{2}}} \]

[In]

integrate(1/x^5/(x^4+x)^(1/2),x, algorithm="maxima")

[Out]

2/9*(2*x^7 + x^4 - x)/(sqrt(x^2 - x + 1)*sqrt(x + 1)*x^(11/2))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^5 \sqrt {x+x^4}} \, dx=-\frac {2}{9} \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {\frac {1}{x^{3}} + 1} \]

[In]

integrate(1/x^5/(x^4+x)^(1/2),x, algorithm="giac")

[Out]

-2/9*(1/x^3 + 1)^(3/2) + 2/3*sqrt(1/x^3 + 1)

Mupad [B] (verification not implemented)

Time = 5.33 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^5 \sqrt {x+x^4}} \, dx=\frac {2\,\left (2\,x^3-1\right )\,\sqrt {x^4+x}}{9\,x^5} \]

[In]

int(1/(x^5*(x + x^4)^(1/2)),x)

[Out]

(2*(2*x^3 - 1)*(x + x^4)^(1/2))/(9*x^5)