3.1.18 \(\int \frac {x^2}{\sqrt {1+x^3}} \, dx\)

Optimal. Leaf size=13 \[ \frac {2 \sqrt {x^3+1}}{3} \]

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {261} \begin {gather*} \frac {2 \sqrt {x^3+1}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[1 + x^3])/3

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x^2}{\sqrt {1+x^3}} \, dx &=\frac {2 \sqrt {1+x^3}}{3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x^3+1}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[1 + x^3])/3

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IntegrateAlgebraic [A]  time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1+x^3}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[1 + x^3])/3

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fricas [A]  time = 0.43, size = 9, normalized size = 0.69 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(x^3 + 1)

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giac [A]  time = 0.75, size = 9, normalized size = 0.69 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(x^3 + 1)

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maple [A]  time = 0.12, size = 10, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {2 \sqrt {x^{3}+1}}{3}\)	\(10\)
default	\(\frac {2 \sqrt {x^{3}+1}}{3}\)	\(10\)
trager	\(\frac {2 \sqrt {x^{3}+1}}{3}\)	\(10\)
risch	\(\frac {2 \sqrt {x^{3}+1}}{3}\)	\(10\)
elliptic	\(\frac {2 \sqrt {x^{3}+1}}{3}\)	\(10\)
gosper	\(\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right )}{3 \sqrt {x^{3}+1}}\)	\(21\)
meijerg	\(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{3}+1}}{3 \sqrt {\pi }}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.71, size = 9, normalized size = 0.69 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(x^3 + 1)

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mupad [B]  time = 0.07, size = 9, normalized size = 0.69 \begin {gather*} \frac {2\,\sqrt {x^3+1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^3 + 1)^(1/2),x)

[Out]

(2*(x^3 + 1)^(1/2))/3

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sympy [A]  time = 0.13, size = 10, normalized size = 0.77 \begin {gather*} \frac {2 \sqrt {x^{3} + 1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x**3 + 1)/3

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