3.445 \(\int \frac{x+3 x^2}{\sqrt{x^2+2 x^3}} \, dx\)

Optimal. Leaf size=13 \[ \sqrt{2 x^3+x^2} \]

[Out]

Sqrt[x^2 + 2*x^3]

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Rubi [A]  time = 0.0072265, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \sqrt{2 x^3+x^2} \]

Antiderivative was successfully verified.

[In]  Int[(x + 3*x^2)/Sqrt[x^2 + 2*x^3],x]

[Out]

Sqrt[x^2 + 2*x^3]

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Rubi in Sympy [A]  time = 1.91949, size = 10, normalized size = 0.77 \[ \sqrt{2 x^{3} + x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2+x)/(2*x**3+x**2)**(1/2),x)

[Out]

sqrt(2*x**3 + x**2)

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Mathematica [A]  time = 0.0132076, size = 13, normalized size = 1. \[ \sqrt{x^2 (2 x+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(x + 3*x^2)/Sqrt[x^2 + 2*x^3],x]

[Out]

Sqrt[x^2*(1 + 2*x)]

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Maple [A]  time = 0.004, size = 21, normalized size = 1.6 \[{{x}^{2} \left ( 1+2\,x \right ){\frac{1}{\sqrt{2\,{x}^{3}+{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+x)/(2*x^3+x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.692179, size = 15, normalized size = 1.15 \[ \sqrt{2 \, x^{3} + x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + x)/sqrt(2*x^3 + x^2),x, algorithm="maxima")

[Out]

sqrt(2*x^3 + x^2)

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Fricas [A]  time = 0.272712, size = 15, normalized size = 1.15 \[ \sqrt{2 \, x^{3} + x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + x)/sqrt(2*x^3 + x^2),x, algorithm="fricas")

[Out]

sqrt(2*x^3 + x^2)

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Sympy [A]  time = 0.340961, size = 10, normalized size = 0.77 \[ \sqrt{2 x^{3} + x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2+x)/(2*x**3+x**2)**(1/2),x)

[Out]

sqrt(2*x**3 + x**2)

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GIAC/XCAS [A]  time = 0.27028, size = 15, normalized size = 1.15 \[ \sqrt{2 \, x^{3} + x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2*x^3 + x^2)