3.1436 \(\int \frac{1+x}{\left (-3+2 x+x^2\right )^{2/3}} \, dx\)

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 = 0.009707, 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 \[ \frac{3}{2} \sqrt [3]{x^2+2 x-3} \]

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

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Rubi in Sympy [A]  time = 2.49932, size = 14, normalized size = 0.88 \[ \frac{3 \sqrt [3]{x^{2} + 2 x - 3}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [A]  time = 0.0108682, 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 = 0.006, size = 19, normalized size = 1.2 \[{\frac{ \left ( 9+3\,x \right ) \left ( -1+x \right ) }{2} \left ({x}^{2}+2\,x-3 \right ) ^{-{\frac{2}{3}}}} \]

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 = 0.680852, size = 16, normalized size = 1. \[ \frac{3}{2} \,{\left (x^{2} + 2 \, x - 3\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(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 = 0.203441, size = 16, normalized size = 1. \[ \frac{3}{2} \,{\left (x^{2} + 2 \, x - 3\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(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 = 0.408136, size = 14, normalized size = 0.88 \[ \frac{3 \sqrt [3]{x^{2} + 2 x - 3}}{2} \]

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/XCAS [A]  time = 0.212252, size = 16, normalized size = 1. \[ \frac{3}{2} \,{\left (x^{2} + 2 \, x - 3\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*(x^2 + 2*x - 3)^(1/3)