\(\int \frac {(2-x^{2/3}) (\sqrt {x}+x)}{x^{3/2}} \, dx\) [25]

Optimal result

Integrand size = 22, antiderivative size = 30 \[ \int \frac {\left (2-x^{2/3}\right ) \left (\sqrt {x}+x\right )}{x^{3/2}} \, dx=4 \sqrt {x}-\frac {3 x^{2/3}}{2}-\frac {6 x^{7/6}}{7}+2 \log (x) \]

[Out]

-3/2*x^(2/3)-6/7*x^(7/6)+2*ln(x)+4*x^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1598, 1834} \[ \int \frac {\left (2-x^{2/3}\right ) \left (\sqrt {x}+x\right )}{x^{3/2}} \, dx=-\frac {6 x^{7/6}}{7}-\frac {3 x^{2/3}}{2}+4 \sqrt {x}+2 \log (x) \]

[In]

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

[Out]

4*Sqrt[x] - (3*x^(2/3))/2 - (6*x^(7/6))/7 + 2*Log[x]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1834

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (1+\sqrt {x}\right ) \left (2-x^{2/3}\right )}{x} \, dx \\ & = -\left (6 \text {Subst}\left (\int \frac {\left (1+x^3\right ) \left (-2+x^4\right )}{x} \, dx,x,\sqrt [6]{x}\right )\right ) \\ & = -\left (6 \text {Subst}\left (\int \left (-\frac {2}{x}-2 x^2+x^3+x^6\right ) \, dx,x,\sqrt [6]{x}\right )\right ) \\ & = 4 \sqrt {x}-\frac {3 x^{2/3}}{2}-\frac {6 x^{7/6}}{7}+2 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2-x^{2/3}\right ) \left (\sqrt {x}+x\right )}{x^{3/2}} \, dx=4 \sqrt {x}-\frac {3 x^{2/3}}{2}-\frac {6 x^{7/6}}{7}+2 \log (x) \]

[In]

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

[Out]

4*Sqrt[x] - (3*x^(2/3))/2 - (6*x^(7/6))/7 + 2*Log[x]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(-\frac {3 x^{\frac {2}{3}}}{2}-\frac {6 x^{\frac {7}{6}}}{7}+2 \ln \left (x \right )+4 \sqrt {x}\)	\(21\)
default	\(-\frac {3 x^{\frac {2}{3}}}{2}-\frac {6 x^{\frac {7}{6}}}{7}+2 \ln \left (x \right )+4 \sqrt {x}\)	\(21\)

[In]

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

[Out]

-3/2*x^(2/3)-6/7*x^(7/6)+2*ln(x)+4*x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {\left (2-x^{2/3}\right ) \left (\sqrt {x}+x\right )}{x^{3/2}} \, dx=-\frac {6}{7} \, x^{\frac {7}{6}} - \frac {3}{2} \, x^{\frac {2}{3}} + 4 \, \sqrt {x} + 12 \, \log \left (x^{\frac {1}{6}}\right ) \]

[In]

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

[Out]

-6/7*x^(7/6) - 3/2*x^(2/3) + 4*sqrt(x) + 12*log(x^(1/6))

Sympy [A] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {\left (2-x^{2/3}\right ) \left (\sqrt {x}+x\right )}{x^{3/2}} \, dx=- \frac {6 x^{\frac {7}{6}}}{7} - \frac {3 x^{\frac {2}{3}}}{2} + 4 \sqrt {x} + 6 \log {\left (\sqrt [3]{x} \right )} \]

[In]

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

[Out]

-6*x**(7/6)/7 - 3*x**(2/3)/2 + 4*sqrt(x) + 6*log(x**(1/3))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \frac {\left (2-x^{2/3}\right ) \left (\sqrt {x}+x\right )}{x^{3/2}} \, dx=-\frac {6}{7} \, x^{\frac {7}{6}} - \frac {3}{2} \, x^{\frac {2}{3}} + 4 \, \sqrt {x} + 2 \, \log \left (x\right ) \]

[In]

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

[Out]

-6/7*x^(7/6) - 3/2*x^(2/3) + 4*sqrt(x) + 2*log(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int \frac {\left (2-x^{2/3}\right ) \left (\sqrt {x}+x\right )}{x^{3/2}} \, dx=-\frac {6}{7} \, x^{\frac {7}{6}} - \frac {3}{2} \, x^{\frac {2}{3}} + 4 \, \sqrt {x} + 2 \, \log \left ({\left | x \right |}\right ) \]

[In]

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

[Out]

-6/7*x^(7/6) - 3/2*x^(2/3) + 4*sqrt(x) + 2*log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {\left (2-x^{2/3}\right ) \left (\sqrt {x}+x\right )}{x^{3/2}} \, dx=12\,\ln \left (x^{1/6}\right )+4\,\sqrt {x}-\frac {3\,x^{2/3}}{2}-\frac {6\,x^{7/6}}{7} \]

[In]

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

[Out]

12*log(x^(1/6)) + 4*x^(1/2) - (3*x^(2/3))/2 - (6*x^(7/6))/7