Integrand size = 26, antiderivative size = 316 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {4}{9 a^2 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {20 b^{2/3} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {20 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {10 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
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Time = 0.10 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1369, 296, 331, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {4}{9 a^2 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )}+\frac {20 b^{2/3} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {20 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {10 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
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Rule 31
Rule 206
Rule 210
Rule 296
Rule 331
Rule 631
Rule 642
Rule 648
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^3 \left (a b+b^2 x^3\right )^3} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {1}{6 a x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (4 b \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^3 \left (a b+b^2 x^3\right )^2} \, dx}{3 a \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {4}{9 a^2 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (20 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^3 \left (a b+b^2 x^3\right )} \, dx}{9 a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {4}{9 a^2 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (20 b \left (a b+b^2 x^3\right )\right ) \int \frac {1}{a b+b^2 x^3} \, dx}{9 a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {4}{9 a^2 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (20 \sqrt [3]{b} \left (a b+b^2 x^3\right )\right ) \int \frac {1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (20 \sqrt [3]{b} \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}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {4}{9 a^2 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {20 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (10 \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}{27 a^{11/3} \sqrt [3]{b} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (10 b^{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}{9 a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {4}{9 a^2 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {20 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {10 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (20 \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 )}{9 a^{11/3} \sqrt [3]{b} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {4}{9 a^2 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {20 b^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {20 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {10 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ \end{align*}
Time = 1.07 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {-27 a^{8/3}-96 a^{5/3} b x^3-60 a^{2/3} b^2 x^6+40 \sqrt {3} b^{2/3} x^2 \left (a+b x^3\right )^2 \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-40 b^{2/3} x^2 \left (a+b x^3\right )^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+20 a^2 b^{2/3} x^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+40 a b^{5/3} x^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+20 b^{8/3} x^8 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} x^2 \left (a+b x^3\right ) \sqrt {\left (a+b x^3\right )^2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.80 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.37
method | result | size |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-\frac {10 b^{2} x^{6}}{9 a^{3}}-\frac {16 b \,x^{3}}{9 a^{2}}-\frac {1}{2 a}\right )}{\left (b \,x^{3}+a \right )^{3} x^{2}}+\frac {20 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{11}-3 b^{2}\right ) x -a^{4} b \textit {\_R} \right )\right )}{27 \left (b \,x^{3}+a \right )}\) | \(116\) |
default | \(-\frac {\left (-40 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b^{2} x^{8}+40 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{2} x^{8}-20 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) b^{2} x^{8}+60 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2} x^{6}-80 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a b \,x^{5}+80 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a b \,x^{5}-40 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a b \,x^{5}+96 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b \,x^{3}-40 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2} x^{2}+40 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} x^{2}-20 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{2} x^{2}+27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}\right ) \left (b \,x^{3}+a \right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} x^{2} a^{3} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}\) | \(322\) |
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Time = 0.27 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=-\frac {60 \, b^{2} x^{6} + 96 \, a b x^{3} - 40 \, \sqrt {3} {\left (b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 20 \, {\left (b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - 40 \, {\left (b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 27 \, a^{2}}{54 \, {\left (a^{3} b^{2} x^{8} + 2 \, a^{4} b x^{5} + a^{5} x^{2}\right )}} \]
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\[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.47 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=-\frac {20 \, b^{2} x^{6} + 32 \, a b x^{3} + 9 \, a^{2}}{18 \, {\left (a^{3} b^{2} x^{8} + 2 \, a^{4} b x^{5} + a^{5} x^{2}\right )}} - \frac {20 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {10 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {20 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.30 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {20 \, b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {20 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {10 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {20 \, b^{2} x^{6} + 32 \, a b x^{3} + 9 \, a^{2}}{18 \, {\left (b x^{4} + a x\right )}^{2} a^{3} \mathrm {sgn}\left (b x^{3} + a\right )} \]
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Timed out. \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}} \,d x \]
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