3.4.8 \(\int \frac {1-2 x}{x^{2/3} (1+x)^2} \, dx\) [308]

Optimal. Leaf size=12 \[ \frac {3 \sqrt [3]{x}}{1+x} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {75} \begin {gather*} \frac {3 \sqrt [3]{x}}{x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)/(x^(2/3)*(1 + x)^2),x]

[Out]

(3*x^(1/3))/(1 + x)

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {3 \sqrt [3]{x}}{1+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 12, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{x}}{1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x^(1/3))/(1 + x)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs. \(2(10)=20\).
time = 0.09, size = 32, normalized size = 2.67

method	result	size
gosper	\(\frac {3 x^{\frac {1}{3}}}{x +1}\)	\(11\)
trager	\(\frac {3 x^{\frac {1}{3}}}{x +1}\)	\(11\)
risch	\(\frac {3 x^{\frac {1}{3}}}{x +1}\)	\(11\)
derivativedivides	\(-\frac {-x^{\frac {1}{3}}-1}{x^{\frac {2}{3}}-x^{\frac {1}{3}}+1}-\frac {1}{1+x^{\frac {1}{3}}}\)	\(32\)
default	\(-\frac {-x^{\frac {1}{3}}-1}{x^{\frac {2}{3}}-x^{\frac {1}{3}}+1}-\frac {1}{1+x^{\frac {1}{3}}}\)	\(32\)
meijerg	\(\frac {3 x^{\frac {1}{3}}}{3+3 x}+\frac {2 \ln \left (1+x^{\frac {1}{3}}\right )}{3}-\frac {\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )}{3}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x^{\frac {1}{3}}}{2-x^{\frac {1}{3}}}\right )}{3}+\frac {2 x^{\frac {1}{3}}}{x +1}-\frac {2 x^{\frac {1}{3}} \left (\frac {\ln \left (1+x^{\frac {1}{3}}\right )}{x^{\frac {1}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )}{2 x^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x^{\frac {1}{3}}}{2-x^{\frac {1}{3}}}\right )}{x^{\frac {1}{3}}}\right )}{3}\)	\(125\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.34, size = 10, normalized size = 0.83 \begin {gather*} \frac {3 \, x^{\frac {1}{3}}}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^(1/3)/(x + 1)

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Fricas [A]
time = 0.56, size = 10, normalized size = 0.83 \begin {gather*} \frac {3 \, x^{\frac {1}{3}}}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^(1/3)/(x + 1)

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Sympy [A]
time = 0.28, size = 8, normalized size = 0.67 \begin {gather*} \frac {3 \sqrt [3]{x}}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x**(1/3)/(x + 1)

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Giac [A]
time = 0.44, size = 10, normalized size = 0.83 \begin {gather*} \frac {3 \, x^{\frac {1}{3}}}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^(1/3)/(x + 1)

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Mupad [B]
time = 0.08, size = 10, normalized size = 0.83 \begin {gather*} \frac {3\,x^{1/3}}{x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)/(x^(2/3)*(x + 1)^2),x)

[Out]

(3*x^(1/3))/(x + 1)

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

not solved

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