3.1.25 \(\int (1+3 x^2) \sqrt [3]{x+x^3} \, dx\)

Optimal. Leaf size=13 \[ \frac {3}{4} \left (x^3+x\right )^{4/3} \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1588} \begin {gather*} \frac {3}{4} \left (x^3+x\right )^{4/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 3*x^2)*(x + x^3)^(1/3),x]

[Out]

(3*(x + x^3)^(4/3))/4

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx &=\frac {3}{4} \left (x+x^3\right )^{4/3}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 19, normalized size = 1.46 \begin {gather*} \frac {3}{4} x \left (x^2+1\right ) \sqrt [3]{x^3+x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*(1 + x^2)*(x + x^3)^(1/3))/4

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IntegrateAlgebraic [A]  time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {3}{4} \left (x+x^3\right )^{4/3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(x + x^3)^(4/3))/4

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fricas [A]  time = 0.48, size = 9, normalized size = 0.69 \begin {gather*} \frac {3}{4} \, {\left (x^{3} + x\right )}^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^3 + x)^(4/3)

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giac [A]  time = 0.29, size = 9, normalized size = 0.69 \begin {gather*} \frac {3}{4} \, {\left (x^{3} + x\right )}^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^3 + x)^(4/3)

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maple [A]  time = 0.08, size = 10, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {3 \left (x^{3}+x \right )^{\frac {4}{3}}}{4}\)	\(10\)
default	\(\frac {3 \left (x^{3}+x \right )^{\frac {4}{3}}}{4}\)	\(10\)
gosper	\(\frac {3 \left (x^{2}+1\right ) x \left (x^{3}+x \right )^{\frac {1}{3}}}{4}\)	\(16\)
trager	\(\frac {3 \left (x^{2}+1\right ) x \left (x^{3}+x \right )^{\frac {1}{3}}}{4}\)	\(16\)
risch	\(\frac {3 \left (\left (x^{2}+1\right ) x \right )^{\frac {1}{3}} \left (x^{2}+1\right ) x}{4}\)	\(18\)
meijerg	\(\frac {3 \hypergeom \left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{2}\right ) x^{\frac {4}{3}}}{4}+\frac {9 \hypergeom \left (\left [-\frac {1}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], -x^{2}\right ) x^{\frac {10}{3}}}{10}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^3+x)^(4/3)

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maxima [A]  time = 0.37, size = 9, normalized size = 0.69 \begin {gather*} \frac {3}{4} \, {\left (x^{3} + x\right )}^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^3 + x)^(4/3)

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mupad [B]  time = 0.09, size = 9, normalized size = 0.69 \begin {gather*} \frac {3\,{\left (x^3+x\right )}^{4/3}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 + 1)*(x + x^3)^(1/3),x)

[Out]

(3*(x + x^3)^(4/3))/4

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sympy [B]  time = 0.17, size = 27, normalized size = 2.08 \begin {gather*} \frac {3 x^{3} \sqrt [3]{x^{3} + x}}{4} + \frac {3 x \sqrt [3]{x^{3} + x}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x**3*(x**3 + x)**(1/3)/4 + 3*x*(x**3 + x)**(1/3)/4

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