3.14.28 \(\int \frac {\sqrt [3]{x+x^3}}{x^2} \, dx\) [1328]

Optimal. Leaf size=96 \[ -\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {1}{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )-\frac {1}{2} \log \left (-x+\sqrt [3]{x+x^3}\right )+\frac {1}{4} \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right ) \]

[Out]

-3/2*(x^3+x)^(1/3)/x-1/2*arctan(3^(1/2)*x/(x+2*(x^3+x)^(1/3)))*3^(1/2)-1/2*ln(-x+(x^3+x)^(1/3))+1/4*ln(x^2+x*(
x^3+x)^(1/3)+(x^3+x)^(2/3))

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Rubi [A]
time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2045, 2057, 335, 281, 337} \begin {gather*} -\frac {\sqrt {3} x^{2/3} \left (x^2+1\right )^{2/3} \text {ArcTan}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \left (x^3+x\right )^{2/3}}-\frac {3 \sqrt [3]{x^3+x}}{2 x}-\frac {3 x^{2/3} \left (x^2+1\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 \left (x^3+x\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + x^3)^(1/3)/x^2,x]

[Out]

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

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 337

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

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*} \int \frac {\sqrt [3]{x+x^3}}{x^2} \, dx &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\int \frac {x}{\left (x+x^3\right )^{2/3}} \, dx\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (x^{2/3} \left (1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3}} \, dx}{\left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}-\frac {\left (x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \left (x+x^3\right )^{2/3}}-\frac {\left (3 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}+\frac {x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \left (x+x^3\right )^{2/3}}+\frac {\left (3 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {\sqrt {3} x^{2/3} \left (1+x^2\right )^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \left (x+x^3\right )^{2/3}}-\frac {x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}+\frac {x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \left (x+x^3\right )^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 142, normalized size = 1.48 \begin {gather*} \frac {\sqrt [3]{x+x^3} \left (-6 \sqrt [3]{1+x^2}-2 \sqrt {3} x^{2/3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )-2 x^{2/3} \log \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )+x^{2/3} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )}{4 x \sqrt [3]{1+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + x^3)^(1/3)/x^2,x]

[Out]

((x + x^3)^(1/3)*(-6*(1 + x^2)^(1/3) - 2*Sqrt[3]*x^(2/3)*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2*(1 + x^2)^(1/3)
)] - 2*x^(2/3)*Log[-x^(2/3) + (1 + x^2)^(1/3)] + x^(2/3)*Log[x^(4/3) + x^(2/3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/
3)]))/(4*x*(1 + x^2)^(1/3))

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 3.
time = 2.30, size = 17, normalized size = 0.18

method	result	size
meijerg	\(-\frac {3 \hypergeom \left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x^{2}\right )}{2 x^{\frac {2}{3}}}\)	\(17\)
trager	\(-\frac {3 \left (x^{3}+x \right )^{\frac {1}{3}}}{2 x}+\frac {\ln \left (45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-72 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-72 \left (x^{3}+x \right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -87 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+15 x \left (x^{3}+x \right )^{\frac {1}{3}}+20 x^{2}-18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+8\right )}{2}-\frac {3 \ln \left (45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-72 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-72 \left (x^{3}+x \right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -87 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+15 x \left (x^{3}+x \right )^{\frac {1}{3}}+20 x^{2}-18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+8\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{2}+\frac {3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+72 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}+72 \left (x^{3}+x \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}-45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-9 \left (x^{3}+x \right )^{\frac {2}{3}}-9 x \left (x^{3}+x \right )^{\frac {1}{3}}-4 x^{2}+48 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-3\right )}{2}\)	\(432\)
risch	\(\text {Expression too large to display}\)	\(731\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((x^3 + x)^(1/3)/x^2, x)

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Fricas [A]
time = 0.44, size = 95, normalized size = 0.99 \begin {gather*} -\frac {2 \, \sqrt {3} x \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\right ) + x \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) + 6 \, {\left (x^{3} + x\right )}^{\frac {1}{3}}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(3)*x*arctan(-(196*sqrt(3)*(x^3 + x)^(1/3)*x - sqrt(3)*(539*x^2 + 507) - 1274*sqrt(3)*(x^3 + x)^(2
/3))/(2205*x^2 + 2197)) + x*log(3*(x^3 + x)^(1/3)*x - 3*(x^3 + x)^(2/3) + 1) + 6*(x^3 + x)^(1/3))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.43, size = 64, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {3}{2} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + \frac {1}{4} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{2} \, \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(1/x^2 + 1)^(1/3) + 1)) - 3/2*(1/x^2 + 1)^(1/3) + 1/4*log((1/x^2 + 1)^(2/3)
+ (1/x^2 + 1)^(1/3) + 1) - 1/2*log(abs((1/x^2 + 1)^(1/3) - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x^3)^(1/3)/x^2,x)

[Out]

int((x + x^3)^(1/3)/x^2, x)

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