Integrand size = 26, antiderivative size = 360 \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {5 x}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{108 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{81 a^2 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {10 \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {10 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
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Time = 0.13 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1369, 294, 205, 206, 31, 648, 631, 210, 642} \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {x}{81 a^2 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{108 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {10 \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {10 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 x}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
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Rule 31
Rule 205
Rule 206
Rule 210
Rule 294
Rule 631
Rule 642
Rule 648
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x^3\right )\right ) \int \frac {x^3}{\left (a b+b^2 x^3\right )^5} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {x}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (b^2 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{\left (a b+b^2 x^3\right )^4} \, dx}{12 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {x}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{108 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (2 b \left (a b+b^2 x^3\right )\right ) \int \frac {1}{\left (a b+b^2 x^3\right )^3} \, dx}{27 a \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {x}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{108 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{81 a^2 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (5 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{\left (a b+b^2 x^3\right )^2} \, dx}{81 a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {5 x}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{108 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{81 a^2 b \left (a+b x^3\right ) \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 {1}{a b+b^2 x^3} \, dx}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {5 x}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{108 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{81 a^2 b \left (a+b x^3\right ) \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 {1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{729 a^{11/3} b^{5/3} \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 {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}{729 a^{11/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {5 x}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{108 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{81 a^2 b \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 ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (5 \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}{729 a^{11/3} b^{7/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (5 \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}{243 a^{10/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {5 x}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{108 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{81 a^2 b \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 ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (10 \left (a b+b^2 x^3\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{243 a^{11/3} b^{7/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {5 x}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{108 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{81 a^2 b \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 ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {10 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ \end{align*}
Time = 1.09 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.61 \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {\left (a+b x^3\right ) \left (-243 a^{11/3} \sqrt [3]{b} x+27 a^{8/3} \sqrt [3]{b} x \left (a+b x^3\right )+36 a^{5/3} \sqrt [3]{b} x \left (a+b x^3\right )^2+60 a^{2/3} \sqrt [3]{b} x \left (a+b x^3\right )^3+40 \sqrt {3} \left (a+b x^3\right )^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )+40 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-20 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{2916 a^{11/3} b^{4/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.81 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.30
method | result | size |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {5 b^{2} x^{10}}{243 a^{3}}+\frac {2 b \,x^{7}}{27 a^{2}}+\frac {31 x^{4}}{324 a}-\frac {10 x}{243 b}\right )}{\left (b \,x^{3}+a \right )^{5}}+\frac {10 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{729 \left (b \,x^{3}+a \right ) a^{3} b^{2}}\) | \(107\) |
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^{4} x^{12}+40 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{4} x^{12}-20 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) b^{4} x^{12}+60 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4} x^{10}-160 \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^{3} x^{9}+160 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a \,b^{3} x^{9}-80 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a \,b^{3} x^{9}+216 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{3} x^{7}-240 \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} b^{2} x^{6}+240 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} b^{2} x^{6}-120 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{2} b^{2} x^{6}+279 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b^{2} x^{4}-160 \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^{3} b \,x^{3}+160 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{3} b \,x^{3}-80 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{3} b \,x^{3}-120 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b x -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^{4}+40 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{4}-20 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{4}\right ) \left (b \,x^{3}+a \right )}{2916 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2} a^{3} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}\) | \(519\) |
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Time = 0.29 (sec) , antiderivative size = 723, normalized size of antiderivative = 2.01 \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\left [\frac {60 \, a^{2} b^{4} x^{10} + 216 \, a^{3} b^{3} x^{7} + 279 \, a^{4} b^{2} x^{4} - 120 \, a^{5} b x + 60 \, \sqrt {\frac {1}{3}} {\left (a b^{5} x^{12} + 4 \, a^{2} b^{4} x^{9} + 6 \, a^{3} b^{3} x^{6} + 4 \, a^{4} b^{2} x^{3} + a^{5} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 20 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 40 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{2916 \, {\left (a^{5} b^{6} x^{12} + 4 \, a^{6} b^{5} x^{9} + 6 \, a^{7} b^{4} x^{6} + 4 \, a^{8} b^{3} x^{3} + a^{9} b^{2}\right )}}, \frac {60 \, a^{2} b^{4} x^{10} + 216 \, a^{3} b^{3} x^{7} + 279 \, a^{4} b^{2} x^{4} - 120 \, a^{5} b x + 120 \, \sqrt {\frac {1}{3}} {\left (a b^{5} x^{12} + 4 \, a^{2} b^{4} x^{9} + 6 \, a^{3} b^{3} x^{6} + 4 \, a^{4} b^{2} x^{3} + a^{5} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 20 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 40 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{2916 \, {\left (a^{5} b^{6} x^{12} + 4 \, a^{6} b^{5} x^{9} + 6 \, a^{7} b^{4} x^{6} + 4 \, a^{8} b^{3} x^{3} + a^{9} b^{2}\right )}}\right ] \]
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\[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {x^{3}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.54 \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {20 \, b^{3} x^{10} + 72 \, a b^{2} x^{7} + 93 \, a^{2} b x^{4} - 40 \, a^{3} x}{972 \, {\left (a^{3} b^{5} x^{12} + 4 \, a^{4} b^{4} x^{9} + 6 \, a^{5} b^{3} x^{6} + 4 \, a^{6} b^{2} x^{3} + a^{7} b\right )}} + \frac {10 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {10 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.29 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.55 \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {10 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {5 \, \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {10 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{4} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {20 \, b^{3} x^{10} + 72 \, a b^{2} x^{7} + 93 \, a^{2} b x^{4} - 40 \, a^{3} x}{972 \, {\left (b x^{3} + a\right )}^{4} a^{3} b \mathrm {sgn}\left (b x^{3} + a\right )} \]
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Timed out. \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {x^3}{{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}} \,d x \]
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