3.4.7 \(\int \sqrt {(1+\sqrt [3]{x}) x} \, dx\) [307]

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

[Out]

21/128*arctanh(x^(2/3)/((1+x^(1/3))*x)^(1/2))+7/64*((1+x^(1/3))*x)^(1/2)-21/128*((1+x^(1/3))*x)^(1/2)/x^(1/3)-
7/80*x^(1/3)*((1+x^(1/3))*x)^(1/2)+3/40*x^(2/3)*((1+x^(1/3))*x)^(1/2)+3/5*x*((1+x^(1/3))*x)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 114, normalized size of antiderivative = 0.90, 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 = {2004, 2029, 2049, 2035, 2054, 212} \begin {gather*} \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 ) \end {gather*}

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 212

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

Rule 2004

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

Rule 2029

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 2035

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 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 2054

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]

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} \text {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*}

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Mathematica [A]
time = 0.14, size = 69, normalized size = 0.55 \begin {gather*} \frac {\sqrt {x+x^{4/3}} \left (-105+70 \sqrt [3]{x}-56 x^{2/3}+48 x+384 x^{4/3}\right )}{640 \sqrt [3]{x}}+\frac {21}{128} \tanh ^{-1}\left (\frac {x^{2/3}}{\sqrt {x+x^{4/3}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(1 + x^(1/3))*x],x]

[Out]

(Sqrt[x + x^(4/3)]*(-105 + 70*x^(1/3) - 56*x^(2/3) + 48*x + 384*x^(4/3)))/(640*x^(1/3)) + (21*ArcTanh[x^(2/3)/
Sqrt[x + x^(4/3)]])/128

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Maple [A]
time = 0.06, size = 108, normalized size = 0.86

method	result	size
meijerg	\(-\frac {3 \left (\frac {\sqrt {\pi }\, x^{\frac {1}{6}} \left (-1152 x^{\frac {4}{3}}-144 x +168 x^{\frac {2}{3}}-210 x^{\frac {1}{3}}+315\right ) \sqrt {x^{\frac {1}{3}}+1}}{2880}-\frac {7 \sqrt {\pi }\, \arcsinh \left (x^{\frac {1}{6}}\right )}{64}\right )}{2 \sqrt {\pi }}\)	\(51\)
derivativedivides	\(\frac {\sqrt {\left (x^{\frac {1}{3}}+1\right ) x}\, \left (768 x^{\frac {2}{3}} \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}\right )^{\frac {3}{2}}-672 x^{\frac {1}{3}} \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}\right )^{\frac {3}{2}}+560 \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}\right )^{\frac {3}{2}}-420 \sqrt {x^{\frac {2}{3}}+x^{\frac {1}{3}}}\, x^{\frac {1}{3}}-210 \sqrt {x^{\frac {2}{3}}+x^{\frac {1}{3}}}+105 \ln \left (\frac {1}{2}+x^{\frac {1}{3}}+\sqrt {x^{\frac {2}{3}}+x^{\frac {1}{3}}}\right )\right )}{1280 x^{\frac {1}{3}} \sqrt {\left (x^{\frac {1}{3}}+1\right ) x^{\frac {1}{3}}}}\)	\(108\)
default	\(\frac {\sqrt {\left (x^{\frac {1}{3}}+1\right ) x}\, \left (768 x^{\frac {2}{3}} \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}\right )^{\frac {3}{2}}-672 x^{\frac {1}{3}} \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}\right )^{\frac {3}{2}}+560 \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}\right )^{\frac {3}{2}}-420 \sqrt {x^{\frac {2}{3}}+x^{\frac {1}{3}}}\, x^{\frac {1}{3}}-210 \sqrt {x^{\frac {2}{3}}+x^{\frac {1}{3}}}+105 \ln \left (\frac {1}{2}+x^{\frac {1}{3}}+\sqrt {x^{\frac {2}{3}}+x^{\frac {1}{3}}}\right )\right )}{1280 x^{\frac {1}{3}} \sqrt {\left (x^{\frac {1}{3}}+1\right ) x^{\frac {1}{3}}}}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, 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(((1+x^(1/3))*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x*(x^(1/3) + 1)), x)

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Fricas [A]
time = 20.25, size = 87, normalized size = 0.69 \begin {gather*} \frac {35 \, x \log \left (\frac {32 \, x^{2} + 48 \, x^{\frac {5}{3}} + 2 \, {\left (16 \, x^{\frac {4}{3}} + 16 \, x + 3 \, x^{\frac {2}{3}}\right )} \sqrt {x^{\frac {4}{3}} + x} + 18 \, x^{\frac {4}{3}} + x}{x}\right ) + 2 \, {\left (384 \, x^{2} + 3 \, {\left (16 \, x - 35\right )} x^{\frac {2}{3}} - 56 \, x^{\frac {4}{3}} + 70 \, x\right )} \sqrt {x^{\frac {4}{3}} + x}}{1280 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1280*(35*x*log((32*x^2 + 48*x^(5/3) + 2*(16*x^(4/3) + 16*x + 3*x^(2/3))*sqrt(x^(4/3) + x) + 18*x^(4/3) + x)/
x) + 2*(384*x^2 + 3*(16*x - 35)*x^(2/3) - 56*x^(4/3) + 70*x)*sqrt(x^(4/3) + x))/x

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

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 [A]
time = 0.96, size = 66, normalized size = 0.52 \begin {gather*} \frac {1}{1280} \, {\left (2 \, {\left (2 \, {\left (4 \, {\left (6 \, x^{\frac {1}{3}} {\left (8 \, x^{\frac {1}{3}} + 1\right )} - 7\right )} x^{\frac {1}{3}} + 35\right )} x^{\frac {1}{3}} - 105\right )} \sqrt {x^{\frac {2}{3}} + x^{\frac {1}{3}}} - 105 \, \log \left ({\left | 2 \, \sqrt {x^{\frac {2}{3}} + x^{\frac {1}{3}}} - 2 \, x^{\frac {1}{3}} - 1 \right |}\right )\right )} \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1280*(2*(2*(4*(6*x^(1/3)*(8*x^(1/3) + 1) - 7)*x^(1/3) + 35)*x^(1/3) - 105)*sqrt(x^(2/3) + x^(1/3)) - 105*log
(abs(2*sqrt(x^(2/3) + x^(1/3)) - 2*x^(1/3) - 1)))*sgn(x)

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Mupad [B]
time = 0.29, size = 27, normalized size = 0.21 \begin {gather*} \frac {2\,x\,\sqrt {x+x^{4/3}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {9}{2};\ \frac {11}{2};\ -x^{1/3}\right )}{3\,\sqrt {x^{1/3}+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(x^(1/3) + 1))^(1/2),x)

[Out]

(2*x*(x + x^(4/3))^(1/2)*hypergeom([-1/2, 9/2], 11/2, -x^(1/3)))/(3*(x^(1/3) + 1)^(1/2))

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