Integrand size = 26, antiderivative size = 238 \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {a+b x^3}{a x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{b} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 238, 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, 331, 298, 31, 648, 631, 210, 642} \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {a+b x^3}{a x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{b} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
[In]
[Out]
Rule 31
Rule 210
Rule 298
Rule 331
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 {1}{x^2 \left (a b+b^2 x^3\right )} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {a+b x^3}{a x \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (b \left (a b+b^2 x^3\right )\right ) \int \frac {x}{a b+b^2 x^3} \, dx}{a \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {a+b x^3}{a x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\right ) \int \frac {1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{3 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a b+b^2 x^3\right ) \int \frac {\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 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {a+b x^3}{a x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a b+b^2 x^3\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 a^{4/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (\sqrt [3]{b} \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 a \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {a+b x^3}{a x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\sqrt [3]{b} \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 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a b+b^2 x^3\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{4/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {a+b x^3}{a x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{b} \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} a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\sqrt [3]{b} \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 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {\left (a+b x^3\right ) \left (6 \sqrt [3]{a}-2 \sqrt {3} \sqrt [3]{b} x \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \sqrt [3]{b} x \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\sqrt [3]{b} x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{6 a^{4/3} x \sqrt {\left (a+b x^3\right )^2}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.74 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.39
method | result | size |
risch | \(-\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}}{\left (b \,x^{3}+a \right ) a x}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{4} \textit {\_Z}^{3}-b \right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{4}+3 b \right ) x -a^{3} \textit {\_R}^{2}\right )\right )}{3 b \,x^{3}+3 a}\) | \(93\) |
default | \(-\frac {\left (b \,x^{3}+a \right ) \left (-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) x +\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) x -2 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) x +6 \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {a}{b}\right )^{\frac {1}{3}} a x}\) | \(111\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.43 \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {2 \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, x \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 6}{6 \, a x} \]
[In]
[Out]
\[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\int \frac {1}{x^{2} \sqrt {\left (a + b x^{3}\right )^{2}}}\, dx \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.45 \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {1}{a x} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {1}{6} \, {\left (\frac {2 \, b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a^{2}} + \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2} b} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{2} b} - \frac {6}{a x}\right )} \mathrm {sgn}\left (b x^{3} + a\right ) \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\int \frac {1}{x^2\,\sqrt {{\left (b\,x^3+a\right )}^2}} \,d x \]
[In]
[Out]