Integrand size = 24, antiderivative size = 359 \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {35 x^2}{243 a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{12 a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 x^2}{54 a^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 x^2}{324 a^3 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {35 \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {35 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1369, 296, 298, 31, 648, 631, 210, 642} \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {5 x^2}{54 a^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{12 a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {35 \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {35 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 x^2}{243 a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 x^2}{324 a^3 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
[In]
[Out]
Rule 31
Rule 210
Rule 296
Rule 298
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}{\left (a b+b^2 x^3\right )^5} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {x^2}{12 a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (5 b^3 \left (a b+b^2 x^3\right )\right ) \int \frac {x}{\left (a b+b^2 x^3\right )^4} \, dx}{6 a \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {x^2}{12 a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 x^2}{54 a^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (35 b^2 \left (a b+b^2 x^3\right )\right ) \int \frac {x}{\left (a b+b^2 x^3\right )^3} \, dx}{54 a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {x^2}{12 a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 x^2}{54 a^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 x^2}{324 a^3 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (35 b \left (a b+b^2 x^3\right )\right ) \int \frac {x}{\left (a b+b^2 x^3\right )^2} \, dx}{81 a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {35 x^2}{243 a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{12 a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 x^2}{54 a^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 x^2}{324 a^3 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (35 \left (a b+b^2 x^3\right )\right ) \int \frac {x}{a b+b^2 x^3} \, dx}{243 a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {35 x^2}{243 a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{12 a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 x^2}{54 a^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 x^2}{324 a^3 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (35 \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^{13/3} b \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (35 \left (a b+b^2 x^3\right )\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}{729 a^{13/3} b \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {35 x^2}{243 a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{12 a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 x^2}{54 a^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 x^2}{324 a^3 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {35 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (35 \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}{1458 a^{13/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (35 \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}{486 a^4 b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {35 x^2}{243 a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{12 a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 x^2}{54 a^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 x^2}{324 a^3 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {35 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (35 \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^{13/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {35 x^2}{243 a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{12 a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 x^2}{54 a^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 x^2}{324 a^3 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {35 \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^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {35 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ \end{align*}
Time = 1.09 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.61 \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {\left (a+b x^3\right ) \left (243 a^{10/3} x^2+270 a^{7/3} x^2 \left (a+b x^3\right )+315 a^{4/3} x^2 \left (a+b x^3\right )^2+420 \sqrt [3]{a} x^2 \left (a+b x^3\right )^3+\frac {140 \sqrt {3} \left (a+b x^3\right )^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3}}-\frac {140 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {70 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}\right )}{2916 a^{13/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.87 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.31
method | result | size |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {35 b^{3} x^{11}}{243 a^{4}}+\frac {175 b^{2} x^{8}}{324 a^{3}}+\frac {20 b \,x^{5}}{27 a^{2}}+\frac {104 x^{2}}{243 a}\right )}{\left (b \,x^{3}+a \right )^{5}}+\frac {35 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{729 \left (b \,x^{3}+a \right ) b \,a^{4}}\) | \(112\) |
default | \(\frac {\left (-140 \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}-140 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{4} x^{12}+70 \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}+420 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4} x^{11}-560 \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}-560 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a \,b^{3} x^{9}+280 \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}+1575 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{3} x^{8}-840 \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}-840 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} b^{2} x^{6}+420 \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}+2160 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b^{2} x^{5}-560 \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}-560 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{3} b \,x^{3}+280 \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}+1248 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3} b \,x^{2}-140 \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}-140 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{4}+70 \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 {1}{3}} b \,a^{4} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}\) | \(521\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 734, normalized size of antiderivative = 2.04 \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\left [\frac {420 \, a b^{5} x^{11} + 1575 \, a^{2} b^{4} x^{8} + 2160 \, a^{3} b^{3} x^{5} + 1248 \, a^{4} b^{2} x^{2} + 210 \, \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 b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) + 70 \, {\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 b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 140 \, {\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 b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{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 {420 \, a b^{5} x^{11} + 1575 \, a^{2} b^{4} x^{8} + 2160 \, a^{3} b^{3} x^{5} + 1248 \, a^{4} b^{2} x^{2} + 420 \, \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 b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + 70 \, {\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 b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 140 \, {\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 b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{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 ] \]
[In]
[Out]
\[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {x}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.53 \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {140 \, b^{3} x^{11} + 525 \, a b^{2} x^{8} + 720 \, a^{2} b x^{5} + 416 \, a^{3} x^{2}}{972 \, {\left (a^{4} b^{4} x^{12} + 4 \, a^{5} b^{3} x^{9} + 6 \, a^{6} b^{2} x^{6} + 4 \, a^{7} b x^{3} + a^{8}\right )}} + \frac {35 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {35 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {35 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.55 \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {35 \, \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {35 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {35 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{5} b^{2} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {140 \, b^{3} x^{11} + 525 \, a b^{2} x^{8} + 720 \, a^{2} b x^{5} + 416 \, a^{3} x^{2}}{972 \, {\left (b x^{3} + a\right )}^{4} a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {x}{{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}} \,d x \]
[In]
[Out]