\(\int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx\) [1391]

Optimal result

Integrand size = 15, antiderivative size = 100 \[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=-\frac {3 \sqrt [3]{x^2+x^3}}{x}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right ) \]

[Out]

-3*(x^3+x^2)^(1/3)/x-3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^3+x^2)^(1/3)))-ln(-x+(x^3+x^2)^(1/3))+1/2*ln(x^2+x*(x^3+
x^2)^(1/3)+(x^3+x^2)^(2/3))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.47, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2045, 2057, 61} \[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=-\frac {\sqrt {3} (x+1)^{2/3} x^{4/3} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )}{\left (x^3+x^2\right )^{2/3}}-\frac {3 \sqrt [3]{x^3+x^2}}{x}-\frac {(x+1)^{2/3} x^{4/3} \log (x+1)}{2 \left (x^3+x^2\right )^{2/3}}-\frac {3 (x+1)^{2/3} x^{4/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x+1}}-1\right )}{2 \left (x^3+x^2\right )^{2/3}} \]

[In]

Int[(x^2 + x^3)^(1/3)/x^2,x]

[Out]

(-3*(x^2 + x^3)^(1/3))/x - (Sqrt[3]*x^(4/3)*(1 + x)^(2/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(1 + x)^(1/3
))])/(x^2 + x^3)^(2/3) - (x^(4/3)*(1 + x)^(2/3)*Log[1 + x])/(2*(x^2 + x^3)^(2/3)) - (3*x^(4/3)*(1 + x)^(2/3)*L
og[-1 + x^(1/3)/(1 + x)^(1/3)])/(2*(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 2045

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 + j*p + 1))), x] - Dist[b*p*((n - j)/(c^n*(m + j*p + 1))), Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p -
1), x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p
, 0] && LtQ[m + j*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*} \text {integral}& = -\frac {3 \sqrt [3]{x^2+x^3}}{x}+\int \frac {x}{\left (x^2+x^3\right )^{2/3}} \, dx \\ & = -\frac {3 \sqrt [3]{x^2+x^3}}{x}+\frac {\left (x^{4/3} (1+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (1+x)^{2/3}} \, dx}{\left (x^2+x^3\right )^{2/3}} \\ & = -\frac {3 \sqrt [3]{x^2+x^3}}{x}-\frac {\sqrt {3} x^{4/3} (1+x)^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{\left (x^2+x^3\right )^{2/3}}-\frac {x^{4/3} (1+x)^{2/3} \log (1+x)}{2 \left (x^2+x^3\right )^{2/3}}-\frac {3 x^{4/3} (1+x)^{2/3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{2 \left (x^2+x^3\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=-\frac {x (1+x)^{2/3} \left (6 \sqrt [3]{1+x}+2 \sqrt {3} \sqrt [3]{x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )+2 \sqrt [3]{x} \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )-\sqrt [3]{x} \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )\right )}{2 \left (x^2 (1+x)\right )^{2/3}} \]

[In]

Integrate[(x^2 + x^3)^(1/3)/x^2,x]

[Out]

-1/2*(x*(1 + x)^(2/3)*(6*(1 + x)^(1/3) + 2*Sqrt[3]*x^(1/3)*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*(1 + x)^(1/3)
)] + 2*x^(1/3)*Log[-x^(1/3) + (1 + x)^(1/3)] - x^(1/3)*Log[x^(2/3) + x^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)]))/
(x^2*(1 + x))^(2/3)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3.

Time = 0.70 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.15

method	result	size
meijerg	\(-\frac {3 \operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x \right )}{x^{\frac {1}{3}}}\)	\(15\)
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right ) x +\ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right ) x -2 \ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}-x}{x}\right ) x -6 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{2 x}\)	\(98\)
risch	\(-\frac {3 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}+\frac {\left (-\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-48 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}+30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+14 \operatorname {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 \operatorname {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 )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+24 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -19 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}-28 \operatorname {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 \operatorname {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 )}{2}\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 )}\)	\(450\)
trager	\(-\frac {3 \left (x^{3}+x^{2}\right )^{\frac {1}{3}}}{x}-3 \ln \left (-\frac {36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -45 \operatorname {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}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-27 \operatorname {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 ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-\frac {36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +45 \operatorname {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}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +33 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+51 \operatorname {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 )+\ln \left (-\frac {36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -45 \operatorname {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}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-27 \operatorname {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 )\)	\(483\)

[In]

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

[Out]

-3/x^(1/3)*hypergeom([-1/3,-1/3],[2/3],-x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) - 2 \, x \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + x \log \left (\frac {x^{2} + {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 6 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{2 \, x} \]

[In]

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

[Out]

1/2*(2*sqrt(3)*x*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + x^2)^(1/3))/x) - 2*x*log(-(x - (x^3 + x^2)^(1/3))/x)
+ x*log((x^2 + (x^3 + x^2)^(1/3)*x + (x^3 + x^2)^(2/3))/x^2) - 6*(x^3 + x^2)^(1/3))/x

Sympy [F]

\[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=\int \frac {\sqrt [3]{x^{2} \left (x + 1\right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=\int { \frac {{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate((x^3 + x^2)^(1/3)/x^2, x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - 3 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + \frac {1}{2} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]

[In]

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

[Out]

sqrt(3)*arctan(1/3*sqrt(3)*(2*(1/x + 1)^(1/3) + 1)) - 3*(1/x + 1)^(1/3) + 1/2*log((1/x + 1)^(2/3) + (1/x + 1)^
(1/3) + 1) - log(abs((1/x + 1)^(1/3) - 1))

Mupad [B] (verification not implemented)

Time = 6.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=-\frac {3\,{\left (x^2\,\left (x+1\right )\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},-\frac {1}{3};\ \frac {2}{3};\ -x\right )}{x\,{\left (x+1\right )}^{1/3}} \]

[In]

int((x^2 + x^3)^(1/3)/x^2,x)

[Out]

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