3.18.47 \(\int x^2 \sqrt [3]{x^2+x^3} \, dx\) [1747]

Optimal. Leaf size=118 \[ \frac {1}{324} \sqrt [3]{x^2+x^3} \left (20-12 x+9 x^2+81 x^3\right )+\frac {10 \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )}{81 \sqrt {3}}+\frac {10}{243} \log \left (-x+\sqrt [3]{x^2+x^3}\right )-\frac {5}{243} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right ) \]

[Out]

1/324*(x^3+x^2)^(1/3)*(81*x^3+9*x^2-12*x+20)+10/243*3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^3+x^2)^(1/3)))+10/243*ln(
-x+(x^3+x^2)^(1/3))-5/243*ln(x^2+x*(x^3+x^2)^(1/3)+(x^3+x^2)^(2/3))

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Rubi [A]
time = 0.12, antiderivative size = 200, normalized size of antiderivative = 1.69, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2046, 2049, 2057, 61} \begin {gather*} \frac {10 (x+1)^{2/3} x^{4/3} \text {ArcTan}\left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )}{81 \sqrt {3} \left (x^3+x^2\right )^{2/3}}+\frac {1}{4} \sqrt [3]{x^3+x^2} x^3+\frac {1}{36} \sqrt [3]{x^3+x^2} x^2-\frac {1}{27} \sqrt [3]{x^3+x^2} x+\frac {5}{81} \sqrt [3]{x^3+x^2}+\frac {5 (x+1)^{2/3} x^{4/3} \log (x+1)}{243 \left (x^3+x^2\right )^{2/3}}+\frac {5 (x+1)^{2/3} x^{4/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x+1}}-1\right )}{81 \left (x^3+x^2\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(x^2 + x^3)^(1/3),x]

[Out]

(5*(x^2 + x^3)^(1/3))/81 - (x*(x^2 + x^3)^(1/3))/27 + (x^2*(x^2 + x^3)^(1/3))/36 + (x^3*(x^2 + x^3)^(1/3))/4 +
(10*x^(4/3)*(1 + x)^(2/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(1 + x)^(1/3))])/(81*Sqrt[3]*(x^2 + x^3)^(2
/3)) + (5*x^(4/3)*(1 + x)^(2/3)*Log[1 + x])/(243*(x^2 + x^3)^(2/3)) + (5*x^(4/3)*(1 + x)^(2/3)*Log[-1 + x^(1/3
)/(1 + x)^(1/3)])/(81*(x^2 + x^3)^(2/3))

Rule 61

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt
[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*
((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] /; FreeQ[{a, b, c, d}, x] && Ne
Q[b*c - a*d, 0] && PosQ[d/b]

Rule 2046

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b
*x^n)^p/(c*(m + n*p + 1))), x] + Dist[a*(n - j)*(p/(c^j*(m + n*p + 1))), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2049

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n +
1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rubi steps

\begin {align*} \int x^2 \sqrt [3]{x^2+x^3} \, dx &=\frac {1}{4} x^3 \sqrt [3]{x^2+x^3}+\frac {1}{12} \int \frac {x^4}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{36} x^2 \sqrt [3]{x^2+x^3}+\frac {1}{4} x^3 \sqrt [3]{x^2+x^3}-\frac {2}{27} \int \frac {x^3}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{27} x \sqrt [3]{x^2+x^3}+\frac {1}{36} x^2 \sqrt [3]{x^2+x^3}+\frac {1}{4} x^3 \sqrt [3]{x^2+x^3}+\frac {5}{81} \int \frac {x^2}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=\frac {5}{81} \sqrt [3]{x^2+x^3}-\frac {1}{27} x \sqrt [3]{x^2+x^3}+\frac {1}{36} x^2 \sqrt [3]{x^2+x^3}+\frac {1}{4} x^3 \sqrt [3]{x^2+x^3}-\frac {10}{243} \int \frac {x}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=\frac {5}{81} \sqrt [3]{x^2+x^3}-\frac {1}{27} x \sqrt [3]{x^2+x^3}+\frac {1}{36} x^2 \sqrt [3]{x^2+x^3}+\frac {1}{4} x^3 \sqrt [3]{x^2+x^3}-\frac {\left (10 x^{4/3} (1+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (1+x)^{2/3}} \, dx}{243 \left (x^2+x^3\right )^{2/3}}\\ &=\frac {5}{81} \sqrt [3]{x^2+x^3}-\frac {1}{27} x \sqrt [3]{x^2+x^3}+\frac {1}{36} x^2 \sqrt [3]{x^2+x^3}+\frac {1}{4} x^3 \sqrt [3]{x^2+x^3}+\frac {10 x^{4/3} (1+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{81 \sqrt {3} \left (x^2+x^3\right )^{2/3}}+\frac {5 x^{4/3} (1+x)^{2/3} \log (1+x)}{243 \left (x^2+x^3\right )^{2/3}}+\frac {5 x^{4/3} (1+x)^{2/3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{81 \left (x^2+x^3\right )^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 160, normalized size = 1.36 \begin {gather*} \frac {60 x^2+24 x^3-9 x^4+270 x^5+243 x^6+40 \sqrt {3} x^{4/3} (1+x)^{2/3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )+40 x^{4/3} (1+x)^{2/3} \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )-20 x^{4/3} (1+x)^{2/3} \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )}{972 \left (x^2 (1+x)\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(x^2 + x^3)^(1/3),x]

[Out]

(60*x^2 + 24*x^3 - 9*x^4 + 270*x^5 + 243*x^6 + 40*Sqrt[3]*x^(4/3)*(1 + x)^(2/3)*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1
/3) + 2*(1 + x)^(1/3))] + 40*x^(4/3)*(1 + x)^(2/3)*Log[-x^(1/3) + (1 + x)^(1/3)] - 20*x^(4/3)*(1 + x)^(2/3)*Lo
g[x^(2/3) + x^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)])/(972*(x^2*(1 + x))^(2/3))

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 3.
time = 0.86, size = 15, normalized size = 0.13

method	result	size
meijerg	\(\frac {3 x^{\frac {11}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], -x \right )}{11}\)	\(15\)
trager	\(\left (\frac {1}{4} x^{3}+\frac {1}{36} x^{2}-\frac {1}{27} x +\frac {5}{81}\right ) \left (x^{3}+x^{2}\right )^{\frac {1}{3}}+\frac {10 \ln \left (\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-72 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {1}{3}} x +9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +24 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+24 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -16 x^{2}-12 x}{x}\right )}{243}+\frac {10 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {-45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+27 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {1}{3}} x +45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -87 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+24 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -20 x^{2}-8 x}{x}\right )}{81}\)	\(335\)
risch	\(\frac {\left (81 x^{3}+9 x^{2}-12 x +20\right ) \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{324}+\frac {\left (\frac {10 \ln \left (-\frac {x^{2} \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}-30 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -16 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}-30 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}-14 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x +36 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}-96 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +64 x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )-96 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+112 x +48}{1+x}\right )}{243}+\frac {5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \ln \left (-\frac {2 x^{2} \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}-24 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+9 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +19 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+9 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+28 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x -30 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+48 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -10 x^{2}+9 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )+48 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}-14 x -4}{1+x}\right )}{243}\right ) \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} \left (\left (1+x \right )^{2} x \right )^{\frac {1}{3}}}{x \left (1+x \right )}\)	\(462\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/11*x^(11/3)*hypergeom([-1/3,11/3],[14/3],-x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^3 + x^2)^(1/3)*x^2, x)

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Fricas [A]
time = 0.33, size = 110, normalized size = 0.93 \begin {gather*} -\frac {10}{243} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{324} \, {\left (81 \, x^{3} + 9 \, x^{2} - 12 \, x + 20\right )} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} + \frac {10}{243} \, \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {5}{243} \, \log \left (\frac {x^{2} + {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10/243*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + x^2)^(1/3))/x) + 1/324*(81*x^3 + 9*x^2 - 12*x + 20)*(
x^3 + x^2)^(1/3) + 10/243*log(-(x - (x^3 + x^2)^(1/3))/x) - 5/243*log((x^2 + (x^3 + x^2)^(1/3)*x + (x^3 + x^2)
^(2/3))/x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt [3]{x^{2} \left (x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.42, size = 97, normalized size = 0.82 \begin {gather*} \frac {1}{324} \, {\left (20 \, {\left (\frac {1}{x} + 1\right )}^{\frac {10}{3}} - 72 \, {\left (\frac {1}{x} + 1\right )}^{\frac {7}{3}} + 93 \, {\left (\frac {1}{x} + 1\right )}^{\frac {4}{3}} + 40 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )} x^{4} - \frac {10}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {5}{243} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {10}{243} \, \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/324*(20*(1/x + 1)^(10/3) - 72*(1/x + 1)^(7/3) + 93*(1/x + 1)^(4/3) + 40*(1/x + 1)^(1/3))*x^4 - 10/243*sqrt(3
)*arctan(1/3*sqrt(3)*(2*(1/x + 1)^(1/3) + 1)) - 5/243*log((1/x + 1)^(2/3) + (1/x + 1)^(1/3) + 1) + 10/243*log(
abs((1/x + 1)^(1/3) - 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\left (x^3+x^2\right )}^{1/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^2 + x^3)^(1/3),x)

[Out]

int(x^2*(x^2 + x^3)^(1/3), x)

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