Integrand size = 26, antiderivative size = 235 \[ \int \frac {x^3}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {x \left (a+b x^3\right )}{b \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{a} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\sqrt [3]{a} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{a} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
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Time = 0.08 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1369, 327, 206, 31, 648, 631, 210, 642} \[ \int \frac {x^3}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {\sqrt [3]{a} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x \left (a+b x^3\right )}{b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\sqrt [3]{a} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{a} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
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Rule 31
Rule 206
Rule 210
Rule 327
Rule 631
Rule 642
Rule 648
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x^3\right ) \int \frac {x^3}{a b+b^2 x^3} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {x \left (a+b x^3\right )}{b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a \left (a b+b^2 x^3\right )\right ) \int \frac {1}{a b+b^2 x^3} \, dx}{b \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {x \left (a+b x^3\right )}{b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (\sqrt [3]{a} \left (a b+b^2 x^3\right )\right ) \int \frac {1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{3 b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (\sqrt [3]{a} \left (a b+b^2 x^3\right )\right ) \int \frac {2 \sqrt [3]{a} \sqrt [3]{b}-b^{2/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{3 b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {x \left (a+b x^3\right )}{b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\sqrt [3]{a} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (\sqrt [3]{a} \left (a b+b^2 x^3\right )\right ) \int \frac {-\sqrt [3]{a} b+2 b^{4/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{6 b^{7/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a^{2/3} \left (a b+b^2 x^3\right )\right ) \int \frac {1}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{2 b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {x \left (a+b x^3\right )}{b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\sqrt [3]{a} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{a} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (\sqrt [3]{a} \left (a b+b^2 x^3\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b^{7/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {x \left (a+b x^3\right )}{b \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{a} \left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\sqrt [3]{a} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{a} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.54 \[ \int \frac {x^3}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {\left (a+b x^3\right ) \left (6 \sqrt [3]{b} x+2 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{6 b^{4/3} \sqrt {\left (a+b x^3\right )^2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.31
method | result | size |
risch | \(\frac {x \sqrt {\left (b \,x^{3}+a \right )^{2}}}{\left (b \,x^{3}+a \right ) b}-\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3 \left (b \,x^{3}+a \right ) b^{2}}\) | \(74\) |
default | \(\frac {\left (b \,x^{3}+a \right ) \left (6 x b \left (\frac {a}{b}\right )^{\frac {2}{3}}+2 \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, a -2 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a +\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a \right )}{6 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\) | \(110\) |
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Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.45 \[ \int \frac {x^3}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {2 \, \sqrt {3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 6 \, x}{6 \, b} \]
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\[ \int \frac {x^3}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\int \frac {x^{3}}{\sqrt {\left (a + b x^{3}\right )^{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.45 \[ \int \frac {x^3}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {x}{b} - \frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {a \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {a \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.61 \[ \int \frac {x^3}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right )}{3 \, b} + \frac {x \mathrm {sgn}\left (b x^{3} + a\right )}{b} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right ) \mathrm {sgn}\left (b x^{3} + a\right )}{3 \, b^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) \mathrm {sgn}\left (b x^{3} + a\right )}{6 \, b^{2}} \]
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Timed out. \[ \int \frac {x^3}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\int \frac {x^3}{\sqrt {{\left (b\,x^3+a\right )}^2}} \,d x \]
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