3.307 \(\int \sqrt{(1+\sqrt [3]{x}) x} \, dx\)

Optimal. Leaf size=126 \[ \frac{3}{40} \sqrt{\left (\sqrt [3]{x}+1\right ) x} x^{2/3}+\frac{21}{128} \tanh ^{-1}\left (\frac{x^{2/3}}{\sqrt{\left (\sqrt [3]{x}+1\right ) x}}\right )+\frac{3}{5} \sqrt{\left (\sqrt [3]{x}+1\right ) x} x-\frac{7}{80} \sqrt{\left (\sqrt [3]{x}+1\right ) x} \sqrt [3]{x}+\frac{7}{64} \sqrt{\left (\sqrt [3]{x}+1\right ) x}-\frac{21 \sqrt{\left (\sqrt [3]{x}+1\right ) x}}{128 \sqrt [3]{x}} \]

[Out]

(7*Sqrt[(1 + x^(1/3))*x])/64 - (21*Sqrt[(1 + x^(1/3))*x])/(128*x^(1/3)) - (7*x^(1/3)*Sqrt[(1 + x^(1/3))*x])/80
+ (3*x^(2/3)*Sqrt[(1 + x^(1/3))*x])/40 + (3*x*Sqrt[(1 + x^(1/3))*x])/5 + (21*ArcTanh[x^(2/3)/Sqrt[(1 + x^(1/3
))*x]])/128

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Rubi [A]  time = 0.115334, antiderivative size = 114, normalized size of antiderivative = 0.9, number of steps used = 8, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {1979, 2004, 2024, 2010, 2029, 206} \[ \frac{3}{5} \sqrt{x^{4/3}+x} x+\frac{3}{40} \sqrt{x^{4/3}+x} x^{2/3}-\frac{7}{80} \sqrt{x^{4/3}+x} \sqrt [3]{x}+\frac{7}{64} \sqrt{x^{4/3}+x}-\frac{21 \sqrt{x^{4/3}+x}}{128 \sqrt [3]{x}}+\frac{21}{128} \tanh ^{-1}\left (\frac{x^{2/3}}{\sqrt{x^{4/3}+x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(1 + x^(1/3))*x],x]

[Out]

(7*Sqrt[x + x^(4/3)])/64 - (21*Sqrt[x + x^(4/3)])/(128*x^(1/3)) - (7*x^(1/3)*Sqrt[x + x^(4/3)])/80 + (3*x^(2/3
)*Sqrt[x + x^(4/3)])/40 + (3*x*Sqrt[x + x^(4/3)])/5 + (21*ArcTanh[x^(2/3)/Sqrt[x + x^(4/3)]])/128

Rule 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2004

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(x*(a*x^j + b*x^n)^p)/(n*p + 1), x] + Dist[(
a*(n - j)*p)/(n*p + 1), Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] &&  !IntegerQ[p] && LtQ[0,
j, n] && GtQ[p, 0] && NeQ[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 2010

Int[1/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[(-2*Sqrt[a*x^j + b*x^n])/(b*(n - 2)*x^(n -
1)), x] - Dist[(a*(2*n - j - 2))/(b*(n - 2)), Int[1/(x^(n - j)*Sqrt[a*x^j + b*x^n]), x], x] /; FreeQ[{a, b}, x
] && LtQ[2*(n - 1), j, n]

Rule 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \sqrt{\left (1+\sqrt [3]{x}\right ) x} \, dx &=\int \sqrt{x+x^{4/3}} \, dx\\ &=\frac{3}{5} x \sqrt{x+x^{4/3}}+\frac{1}{10} \int \frac{x}{\sqrt{x+x^{4/3}}} \, dx\\ &=\frac{3}{40} x^{2/3} \sqrt{x+x^{4/3}}+\frac{3}{5} x \sqrt{x+x^{4/3}}-\frac{7}{80} \int \frac{x^{2/3}}{\sqrt{x+x^{4/3}}} \, dx\\ &=-\frac{7}{80} \sqrt [3]{x} \sqrt{x+x^{4/3}}+\frac{3}{40} x^{2/3} \sqrt{x+x^{4/3}}+\frac{3}{5} x \sqrt{x+x^{4/3}}+\frac{7}{96} \int \frac{\sqrt [3]{x}}{\sqrt{x+x^{4/3}}} \, dx\\ &=\frac{7}{64} \sqrt{x+x^{4/3}}-\frac{7}{80} \sqrt [3]{x} \sqrt{x+x^{4/3}}+\frac{3}{40} x^{2/3} \sqrt{x+x^{4/3}}+\frac{3}{5} x \sqrt{x+x^{4/3}}-\frac{7}{128} \int \frac{1}{\sqrt{x+x^{4/3}}} \, dx\\ &=\frac{7}{64} \sqrt{x+x^{4/3}}-\frac{21 \sqrt{x+x^{4/3}}}{128 \sqrt [3]{x}}-\frac{7}{80} \sqrt [3]{x} \sqrt{x+x^{4/3}}+\frac{3}{40} x^{2/3} \sqrt{x+x^{4/3}}+\frac{3}{5} x \sqrt{x+x^{4/3}}+\frac{7}{256} \int \frac{1}{\sqrt [3]{x} \sqrt{x+x^{4/3}}} \, dx\\ &=\frac{7}{64} \sqrt{x+x^{4/3}}-\frac{21 \sqrt{x+x^{4/3}}}{128 \sqrt [3]{x}}-\frac{7}{80} \sqrt [3]{x} \sqrt{x+x^{4/3}}+\frac{3}{40} x^{2/3} \sqrt{x+x^{4/3}}+\frac{3}{5} x \sqrt{x+x^{4/3}}+\frac{21}{128} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x^{2/3}}{\sqrt{x+x^{4/3}}}\right )\\ &=\frac{7}{64} \sqrt{x+x^{4/3}}-\frac{21 \sqrt{x+x^{4/3}}}{128 \sqrt [3]{x}}-\frac{7}{80} \sqrt [3]{x} \sqrt{x+x^{4/3}}+\frac{3}{40} x^{2/3} \sqrt{x+x^{4/3}}+\frac{3}{5} x \sqrt{x+x^{4/3}}+\frac{21}{128} \tanh ^{-1}\left (\frac{x^{2/3}}{\sqrt{x+x^{4/3}}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0488544, size = 83, normalized size = 0.66 \[ \frac{\sqrt{x^{4/3}+x} \left (\sqrt{\sqrt [3]{x}+1} \sqrt [6]{x} \left (384 x^{4/3}-56 x^{2/3}+48 x+70 \sqrt [3]{x}-105\right )+105 \sinh ^{-1}\left (\sqrt [6]{x}\right )\right )}{640 \sqrt{\sqrt [3]{x}+1} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(1 + x^(1/3))*x],x]

[Out]

(Sqrt[x + x^(4/3)]*(Sqrt[1 + x^(1/3)]*x^(1/6)*(-105 + 70*x^(1/3) - 56*x^(2/3) + 48*x + 384*x^(4/3)) + 105*ArcS
inh[x^(1/6)]))/(640*Sqrt[1 + x^(1/3)]*Sqrt[x])

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Maple [A]  time = 0.01, size = 108, normalized size = 0.9 \begin{align*}{\frac{1}{1280}\sqrt{ \left ( \sqrt [3]{x}+1 \right ) x} \left ( 768\,{x}^{2/3} \left ({x}^{2/3}+\sqrt [3]{x} \right ) ^{3/2}-672\,\sqrt [3]{x} \left ({x}^{2/3}+\sqrt [3]{x} \right ) ^{3/2}+560\, \left ({x}^{2/3}+\sqrt [3]{x} \right ) ^{3/2}-420\,\sqrt{{x}^{2/3}+\sqrt [3]{x}}\sqrt [3]{x}-210\,\sqrt{{x}^{2/3}+\sqrt [3]{x}}+105\,\ln \left ( 1/2+\sqrt [3]{x}+\sqrt{{x}^{2/3}+\sqrt [3]{x}} \right ) \right ){\frac{1}{\sqrt [3]{x}}}{\frac{1}{\sqrt{ \left ( \sqrt [3]{x}+1 \right ) \sqrt [3]{x}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1280*((x^(1/3)+1)*x)^(1/2)*(768*x^(2/3)*(x^(2/3)+x^(1/3))^(3/2)-672*x^(1/3)*(x^(2/3)+x^(1/3))^(3/2)+560*(x^(
2/3)+x^(1/3))^(3/2)-420*(x^(2/3)+x^(1/3))^(1/2)*x^(1/3)-210*(x^(2/3)+x^(1/3))^(1/2)+105*ln(1/2+x^(1/3)+(x^(2/3
)+x^(1/3))^(1/2)))/x^(1/3)/((x^(1/3)+1)*x^(1/3))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x{\left (x^{\frac{1}{3}} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x*(x^(1/3) + 1)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(x**(1/3) + 1)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError