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

Optimal. Leaf size=30 \[ -\frac {6 x^{7/6}}{7}-\frac {3 x^{2/3}}{2}+4 \sqrt {x}+2 \log (x) \]

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Rubi [A]  time = 0.09, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1584, 1820} \[ -\frac {6 x^{7/6}}{7}-\frac {3 x^{2/3}}{2}+4 \sqrt {x}+2 \log (x) \]

Antiderivative was successfully verified.

[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 1584

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 1820

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*} \int \frac {\left (2-x^{2/3}\right ) \left (\sqrt {x}+x\right )}{x^{3/2}} \, dx &=\int \frac {\left (1+\sqrt {x}\right ) \left (2-x^{2/3}\right )}{x} \, dx\\ &=-\left (6 \operatorname {Subst}\left (\int \frac {\left (1+x^3\right ) \left (-2+x^4\right )}{x} \, dx,x,\sqrt [6]{x}\right )\right )\\ &=-\left (6 \operatorname {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*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 1.00 \[ -\frac {6 x^{7/6}}{7}-\frac {3 x^{2/3}}{2}+4 \sqrt {x}+2 \log (x) \]

Antiderivative was successfully verified.

[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]

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IntegrateAlgebraic [A]  time = 0.01, size = 30, normalized size = 1.00 \[ -\frac {6 x^{7/6}}{7}-\frac {3 x^{2/3}}{2}+4 \sqrt {x}+2 \log (x) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((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]

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fricas [A]  time = 1.11, size = 22, normalized size = 0.73 \[ -\frac {6}{7} \, x^{\frac {7}{6}} - \frac {3}{2} \, x^{\frac {2}{3}} + 4 \, \sqrt {x} + 12 \, \log \left (x^{\frac {1}{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.97, size = 21, normalized size = 0.70 \[ -\frac {6}{7} \, x^{\frac {7}{6}} - \frac {3}{2} \, x^{\frac {2}{3}} + 4 \, \sqrt {x} + 2 \, \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.02, size = 21, normalized size = 0.70

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.43, size = 20, normalized size = 0.67 \[ -\frac {6}{7} \, x^{\frac {7}{6}} - \frac {3}{2} \, x^{\frac {2}{3}} + 4 \, \sqrt {x} + 2 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.03, size = 22, normalized size = 0.73 \[ 12\,\ln \left (x^{1/6}\right )+4\,\sqrt {x}-\frac {3\,x^{2/3}}{2}-\frac {6\,x^{7/6}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 7.86, size = 31, normalized size = 1.03 \[ - \frac {6 x^{\frac {7}{6}}}{7} - \frac {3 x^{\frac {2}{3}}}{2} + 4 \sqrt {x} + 4 \log {\left (\sqrt {x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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) + 4*log(sqrt(x))

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