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

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

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Rubi [A]  time = 0.19, 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 = {2021, 2024, 2032, 59} \begin {gather*} \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}}+\frac {10 (x+1)^{2/3} x^{4/3} \tan ^{-1}\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}} \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 59

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

Rule 2021

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 2024

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 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[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 [C]  time = 0.01, size = 38, normalized size = 0.32 \begin {gather*} \frac {3 x^3 \sqrt [3]{x^2 (x+1)} \, _2F_1\left (-\frac {1}{3},\frac {11}{3};\frac {14}{3};-x\right )}{11 \sqrt [3]{x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(x^2 + x^3)^(1/3),x]

[Out]

(3*x^3*(x^2*(1 + x))^(1/3)*Hypergeometric2F1[-1/3, 11/3, 14/3, -x])/(11*(1 + x)^(1/3))

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IntegrateAlgebraic [A]  time = 0.35, size = 118, normalized size = 1.00 \begin {gather*} \frac {1}{324} \sqrt [3]{x^2+x^3} \left (20-12 x+9 x^2+81 x^3\right )+\frac {10 \tan ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x^2 + x^3)^(1/3)*(20 - 12*x + 9*x^2 + 81*x^3))/324 + (10*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))])/(81*
Sqrt[3]) + (10*Log[-x + (x^2 + x^3)^(1/3)])/243 - (5*Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)])/243

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fricas [A]  time = 0.45, 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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giac [A]  time = 0.19, 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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maple [C]  time = 0.84, 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 {36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+45 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +57 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+27 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-5 x^{2}-2 x}{x}\right )}{243}-\frac {10 \ln \left (-\frac {36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+45 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +57 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+27 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-5 x^{2}-2 x}{x}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{81}+\frac {10 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-45 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -33 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-51 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -24 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-24 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-20 x^{2}-15 x}{x}\right )}{81}\)	\(496\)
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 {5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \ln \left (-\frac {-5 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}}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +38 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}+36 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+36 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+70 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x +16 x^{2}+36 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+32 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )+28 x +12}{1+x}\right )}{243}-\frac {5 \ln \left (\frac {5 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}}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +58 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}+60 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+60 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+70 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x +80 x^{2}+60 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+12 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )+112 x +32}{1+x}\right ) \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )}{243}-\frac {10 \ln \left (\frac {5 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}}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +58 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}+60 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+60 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+70 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x +80 x^{2}+60 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+12 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )+112 x +32}{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 )}\)	\(669\)

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*} \int {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x^{2}\,{d x} \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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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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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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