Integrand size = 26, antiderivative size = 147 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=-\frac {3 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{b^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{2 b \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Time = 0.05 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 660, 45} \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=-\frac {3 a \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{b^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 x^{2/3} \left (a+b \sqrt [3]{x}\right )}{2 b \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Rule 45
Rule 660
Rule 1355
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {\left (3 b \left (a+b \sqrt [3]{x}\right )\right ) \text {Subst}\left (\int \frac {x^2}{a b+b^2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \\ & = \frac {\left (3 b \left (a+b \sqrt [3]{x}\right )\right ) \text {Subst}\left (\int \left (-\frac {a}{b^3}+\frac {x}{b^2}+\frac {a^2}{b^3 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \\ & = -\frac {3 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{b^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{2 b \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=\frac {3 \left (a+b \sqrt [3]{x}\right ) \left (b \left (-2 a+b \sqrt [3]{x}\right ) \sqrt [3]{x}+2 a^2 \log \left (a+b \sqrt [3]{x}\right )\right )}{2 b^3 \sqrt {\left (a+b \sqrt [3]{x}\right )^2}} \]
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Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.35
method | result | size |
derivativedivides | \(\frac {3 \left (a +b \,x^{\frac {1}{3}}\right ) \left (b^{2} x^{\frac {2}{3}}+2 a^{2} \ln \left (a +b \,x^{\frac {1}{3}}\right )-2 a b \,x^{\frac {1}{3}}\right )}{2 \sqrt {\left (a +b \,x^{\frac {1}{3}}\right )^{2}}\, b^{3}}\) | \(52\) |
default | \(\frac {\left (a +b \,x^{\frac {1}{3}}\right ) \left (3 b^{2} x^{\frac {2}{3}}-6 a b \,x^{\frac {1}{3}}+2 a^{2} \ln \left (b^{3} x +a^{3}\right )-2 a^{2} \ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )+4 a^{2} \ln \left (a +b \,x^{\frac {1}{3}}\right )\right )}{2 \sqrt {a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}}\, b^{3}}\) | \(101\) |
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.22 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=\frac {3 \, {\left (2 \, a^{2} \log \left (b x^{\frac {1}{3}} + a\right ) + b^{2} x^{\frac {2}{3}} - 2 \, a b x^{\frac {1}{3}}\right )}}{2 \, b^{3}} \]
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Time = 0.43 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=3 \left (\begin {cases} \frac {a^{2} \left (\frac {a}{b} + \sqrt [3]{x}\right ) \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{b^{2} \sqrt {b^{2} \left (\frac {a}{b} + \sqrt [3]{x}\right )^{2}}} + \left (- \frac {3 a}{2 b^{3}} + \frac {\sqrt [3]{x}}{2 b^{2}}\right ) \sqrt {a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}} & \text {for}\: b^{2} \neq 0 \\\frac {a^{4} \sqrt {a^{2} + 2 a b \sqrt [3]{x}} - \frac {2 a^{2} \left (a^{2} + 2 a b \sqrt [3]{x}\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b \sqrt [3]{x}\right )^{\frac {5}{2}}}{5}}{4 a^{3} b^{3}} & \text {for}\: a b \neq 0 \\\frac {x}{3 \sqrt {a^{2}}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.24 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=\frac {3 \, a^{2} \log \left (x^{\frac {1}{3}} + \frac {a}{b}\right )}{b^{3}} + \frac {3 \, x^{\frac {2}{3}}}{2 \, b} - \frac {3 \, a x^{\frac {1}{3}}}{b^{2}} \]
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Time = 0.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=\frac {3 \, {\left (b x^{\frac {2}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) - 2 \, a x^{\frac {1}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )\right )}}{2 \, b^{2}} + \frac {3 \, a^{2} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{3} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )} \]
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Timed out. \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=\int \frac {1}{\sqrt {a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}} \,d x \]
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