### 3.1447 $$\int \frac{1+x}{(-3+2 x+x^2)^{2/3}} \, dx$$

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

[Out]

(3*(-3 + 2*x + x^2)^(1/3))/2

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Rubi [A]  time = 0.0039318, antiderivative size = 16, normalized size of antiderivative = 1., 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} $\frac{3}{2} \sqrt [3]{x^2+2 x-3}$

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*}

Mathematica [A]  time = 0.0038345, size = 16, normalized size = 1. $\frac{3}{2} \sqrt [3]{x^2+2 x-3}$

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 = 0.04, size = 19, normalized size = 1.2 \begin{align*}{\frac{ \left ( 9+3\,x \right ) \left ( -1+x \right ) }{2} \left ({x}^{2}+2\,x-3 \right ) ^{-{\frac{2}{3}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17557, size = 16, normalized size = 1. \begin{align*} \frac{3}{2} \,{\left (x^{2} + 2 \, x - 3\right )}^{\frac{1}{3}} \end{align*}

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 = 2.02086, size = 36, normalized size = 2.25 \begin{align*} \frac{3}{2} \,{\left (x^{2} + 2 \, x - 3\right )}^{\frac{1}{3}} \end{align*}

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 = 0.15836, size = 14, normalized size = 0.88 \begin{align*} \frac{3 \sqrt [3]{x^{2} + 2 x - 3}}{2} \end{align*}

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 = 1.21356, size = 16, normalized size = 1. \begin{align*} \frac{3}{2} \,{\left (x^{2} + 2 \, x - 3\right )}^{\frac{1}{3}} \end{align*}

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)