3.12.49 \(\int \frac {1+x}{(-3+2 x+x^2)^{2/3}} \, dx\)

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

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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 = {629} \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 629

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 = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {3}{2} \sqrt [3]{x^2+2 x-3} \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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IntegrateAlgebraic [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {3}{2} \sqrt [3]{x^2+2 x-3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-3 + 2*x + x^2)^(1/3))/2

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fricas [A]  time = 0.40, 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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giac [A]  time = 0.16, 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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maple [A]  time = 0.04, size = 19, normalized size = 1.19 \begin {gather*} \frac {3 \left (x +3\right ) \left (x -1\right )}{2 \left (x^{2}+2 x -3\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.29, 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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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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sympy [A]  time = 0.17, 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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