Optimal. Leaf size=126 \[ \frac {3}{40} \sqrt {\left (\sqrt [3]{x}+1\right ) x} x^{2/3}+\frac {21}{128} \tanh ^{-1}\left (\frac {x^{2/3}}{\sqrt {\left (\sqrt [3]{x}+1\right ) x}}\right )+\frac {3}{5} \sqrt {\left (\sqrt [3]{x}+1\right ) x} x-\frac {7}{80} \sqrt {\left (\sqrt [3]{x}+1\right ) x} \sqrt [3]{x}+\frac {7}{64} \sqrt {\left (\sqrt [3]{x}+1\right ) x}-\frac {21 \sqrt {\left (\sqrt [3]{x}+1\right ) x}}{128 \sqrt [3]{x}} \]
________________________________________________________________________________________
Rubi [A] time = 0.12, 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 = {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]
[Out]
Rule 206
Rule 1979
Rule 2004
Rule 2010
Rule 2024
Rule 2029
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 = 0.05, 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]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.50, size = 69, normalized size = 0.55 \[ \frac {\sqrt {x^{4/3}+x} \left (384 x^{4/3}-56 x^{2/3}+48 x+70 \sqrt [3]{x}-105\right )}{640 \sqrt [3]{x}}+\frac {21}{128} \tanh ^{-1}\left (\frac {x^{2/3}}{\sqrt {x^{4/3}+x}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 74.92, size = 87, normalized size = 0.69 \[ \frac {35 \, x \log \left (\frac {32 \, x^{2} + 48 \, x^{\frac {5}{3}} + 2 \, {\left (16 \, x^{\frac {4}{3}} + 16 \, x + 3 \, x^{\frac {2}{3}}\right )} \sqrt {x^{\frac {4}{3}} + x} + 18 \, x^{\frac {4}{3}} + x}{x}\right ) + 2 \, {\left (384 \, x^{2} + 3 \, {\left (16 \, x - 35\right )} x^{\frac {2}{3}} - 56 \, x^{\frac {4}{3}} + 70 \, x\right )} \sqrt {x^{\frac {4}{3}} + x}}{1280 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.93, size = 66, normalized size = 0.52 \[ \frac {1}{1280} \, {\left (2 \, {\left (2 \, {\left (4 \, {\left (6 \, x^{\frac {1}{3}} {\left (8 \, x^{\frac {1}{3}} + 1\right )} - 7\right )} x^{\frac {1}{3}} + 35\right )} x^{\frac {1}{3}} - 105\right )} \sqrt {x^{\frac {2}{3}} + x^{\frac {1}{3}}} - 105 \, \log \left ({\left | 2 \, \sqrt {x^{\frac {2}{3}} + x^{\frac {1}{3}}} - 2 \, x^{\frac {1}{3}} - 1 \right |}\right )\right )} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.29, size = 51, normalized size = 0.40
method | result | size |
meijerg | \(-\frac {3 \left (\frac {\sqrt {\pi }\, x^{\frac {1}{6}} \left (-1152 x^{\frac {4}{3}}-144 x +168 x^{\frac {2}{3}}-210 x^{\frac {1}{3}}+315\right ) \sqrt {1+x^{\frac {1}{3}}}}{2880}-\frac {7 \sqrt {\pi }\, \arcsinh \left (x^{\frac {1}{6}}\right )}{64}\right )}{2 \sqrt {\pi }}\) | \(51\) |
derivativedivides | \(\frac {\sqrt {\left (1+x^{\frac {1}{3}}\right ) x}\, \left (768 x^{\frac {2}{3}} \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}\right )^{\frac {3}{2}}-672 x^{\frac {1}{3}} \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}\right )^{\frac {3}{2}}+560 \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}\right )^{\frac {3}{2}}-420 \sqrt {x^{\frac {2}{3}}+x^{\frac {1}{3}}}\, x^{\frac {1}{3}}+105 \ln \left (\frac {1}{2}+x^{\frac {1}{3}}+\sqrt {x^{\frac {2}{3}}+x^{\frac {1}{3}}}\right )-210 \sqrt {x^{\frac {2}{3}}+x^{\frac {1}{3}}}\right )}{1280 x^{\frac {1}{3}} \sqrt {\left (1+x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}}\) | \(108\) |
default | \(\frac {\sqrt {\left (1+x^{\frac {1}{3}}\right ) x}\, \left (768 x^{\frac {2}{3}} \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}\right )^{\frac {3}{2}}-672 x^{\frac {1}{3}} \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}\right )^{\frac {3}{2}}+560 \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}\right )^{\frac {3}{2}}-420 \sqrt {x^{\frac {2}{3}}+x^{\frac {1}{3}}}\, x^{\frac {1}{3}}+105 \ln \left (\frac {1}{2}+x^{\frac {1}{3}}+\sqrt {x^{\frac {2}{3}}+x^{\frac {1}{3}}}\right )-210 \sqrt {x^{\frac {2}{3}}+x^{\frac {1}{3}}}\right )}{1280 x^{\frac {1}{3}} \sqrt {\left (1+x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x {\left (x^{\frac {1}{3}} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.29, size = 27, normalized size = 0.21 \[ \frac {2\,x\,\sqrt {x+x^{4/3}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {9}{2};\ \frac {11}{2};\ -x^{1/3}\right )}{3\,\sqrt {x^{1/3}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x \left (\sqrt [3]{x} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________