3.15.47 \(\int \frac {1+x}{(-3+2 x+x^2)^{2/3}} \, dx\) [1447]

Optimal. Leaf size=16 \[ \frac {3}{2} \sqrt [3]{-3+2 x+x^2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, 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 = {643} \begin {gather*} \frac {3}{2} \sqrt [3]{x^2+2 x-3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)/(-3 + 2*x + x^2)^(2/3),x]

[Out]

(3*(-3 + 2*x + x^2)^(1/3))/2

Rule 643

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

Rubi steps

\begin {align*} \int \frac {1+x}{\left (-3+2 x+x^2\right )^{2/3}} \, dx &=\frac {3}{2} \sqrt [3]{-3+2 x+x^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {3}{2} \sqrt [3]{-3+2 x+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-3 + 2*x + x^2)^(1/3))/2

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Maple [A]
time = 0.62, size = 13, normalized size = 0.81

method	result	size
default	\(\frac {3 \left (x^{2}+2 x -3\right )^{\frac {1}{3}}}{2}\)	\(13\)
trager	\(\frac {3 \left (x^{2}+2 x -3\right )^{\frac {1}{3}}}{2}\)	\(13\)
risch	\(\frac {3 \left (x^{2}+2 x -3\right )^{\frac {1}{3}}}{2}\)	\(13\)
gosper	\(\frac {3 \left (3+x \right ) \left (x -1\right )}{2 \left (x^{2}+2 x -3\right )^{\frac {2}{3}}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 12, normalized size = 0.75 \begin {gather*} \frac {3}{2} \, {\left (x^{2} + 2 \, x - 3\right )}^{\frac {1}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^2 + 2*x - 3)^(1/3)

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Fricas [A]
time = 2.20, size = 12, normalized size = 0.75 \begin {gather*} \frac {3}{2} \, {\left (x^{2} + 2 \, x - 3\right )}^{\frac {1}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^2 + 2*x - 3)^(1/3)

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Sympy [A]
time = 0.06, size = 14, normalized size = 0.88 \begin {gather*} \frac {3 \sqrt [3]{x^{2} + 2 x - 3}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(x**2 + 2*x - 3)**(1/3)/2

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Giac [A]
time = 1.73, size = 12, normalized size = 0.75 \begin {gather*} \frac {3}{2} \, {\left (x^{2} + 2 \, x - 3\right )}^{\frac {1}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^2 + 2*x - 3)^(1/3)

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Mupad [B]
time = 0.60, size = 11, normalized size = 0.69 \begin {gather*} \frac {3\,{\left (\left (x-1\right )\,\left (x+3\right )\right )}^{1/3}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(2*x + x^2 - 3)^(2/3),x)

[Out]

(3*((x - 1)*(x + 3))^(1/3))/2

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