\(\int \frac {1}{x \sqrt {x^3+x^4}} \, dx\) [226]

Optimal result

Integrand size = 15, antiderivative size = 23 \[ \int \frac {1}{x \sqrt {x^3+x^4}} \, dx=\frac {2 (-1+2 x) \sqrt {x^3+x^4}}{3 x^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2041, 2025} \[ \int \frac {1}{x \sqrt {x^3+x^4}} \, dx=\frac {4 \sqrt {x^4+x^3}}{3 x^2}-\frac {2 \sqrt {x^4+x^3}}{3 x^3} \]

[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 2025

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 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^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*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {x^3+x^4}} \, dx=\frac {2 (1+x) (-1+2 x)}{3 \sqrt {x^3 (1+x)}} \]

[In]

Integrate[1/(x*Sqrt[x^3 + x^4]),x]

[Out]

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

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 16, normalized size of antiderivative = 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	\(\frac {-\frac {2}{3}+\frac {2}{3} x +\frac {4}{3} x^{2}}{\sqrt {x^{3} \left (1+x \right )}}\)	\(20\)
pseudoelliptic	\(\frac {4 x^{2}+2 x -2}{3 \sqrt {x^{3} \left (1+x \right )}}\)	\(22\)
default	\(\frac {2 \sqrt {\left (1+x \right ) x}\, \sqrt {x^{2}+x}\, \left (-1+2 x \right )}{3 x \sqrt {x^{4}+x^{3}}}\)	\(34\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x \sqrt {x^3+x^4}} \, dx=\frac {2 \, {\left (2 \, x^{3} + \sqrt {x^{4} + x^{3}} {\left (2 \, x - 1\right )}\right )}}{3 \, x^{3}} \]

[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

Sympy [F]

\[ \int \frac {1}{x \sqrt {x^3+x^4}} \, dx=\int \frac {1}{x \sqrt {x^{3} \left (x + 1\right )}}\, dx \]

[In]

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

[Out]

Integral(1/(x*sqrt(x**3*(x + 1))), x)

Maxima [F]

\[ \int \frac {1}{x \sqrt {x^3+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} + x^{3}} x} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt(x^4 + x^3)*x), x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x \sqrt {x^3+x^4}} \, dx=\frac {2 \, {\left (3 \, x - 3 \, \sqrt {x^{2} + x} + 1\right )}}{3 \, {\left (x - \sqrt {x^{2} + x}\right )}^{3} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x \sqrt {x^3+x^4}} \, dx=\frac {4\,x\,\sqrt {x^4+x^3}-2\,\sqrt {x^4+x^3}}{3\,x^3} \]

[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)