Integrand size = 39, antiderivative size = 310 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{2 a}-\frac {\log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 a}+\frac {1}{6} \text {RootSum}\left [a^9-2 a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-a^3 \log (x)-2 b^2 \log (x)+a^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+2 b^2 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{a^3 \text {$\#$1}-\text {$\#$1}^4}\&\right ] \]
Time = 10.42 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.86 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {3 \sqrt [3]{a} x \sqrt [3]{1+\frac {a^3 x}{b^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {a^3 x}{b^2}\right )-x \left (\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {\sqrt [3]{a} \left (a^{8/3}+\sqrt [3]{-2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {\sqrt [3]{a} \left (a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {a^3 x-\sqrt [3]{2} \sqrt [3]{a} b^{5/3} x}{b^2+a^3 x}\right )\right )}{2 \sqrt [3]{a} \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
(3*a^(1/3)*x*(1 + (a^3*x)/b^2)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, -((a ^3*x)/b^2)] - x*((a^(1/3) + (-2)^(2/3)*b^(1/3))*Hypergeometric2F1[1/3, 1, 4/3, (a^(1/3)*(a^(8/3) + (-2)^(1/3)*b^(5/3))*x)/(b^2 + a^3*x)] + (a^(1/3) - (-1)^(1/3)*2^(2/3)*b^(1/3))*Hypergeometric2F1[1/3, 1, 4/3, (a^(1/3)*(a^( 8/3) - (-1)^(2/3)*2^(1/3)*b^(5/3))*x)/(b^2 + a^3*x)] + (a^(1/3) + 2^(2/3)* b^(1/3))*Hypergeometric2F1[1/3, 1, 4/3, (a^3*x - 2^(1/3)*a^(1/3)*b^(5/3)*x )/(b^2 + a^3*x)]))/(2*a^(1/3)*(x^2*(b^2 + a^3*x))^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(4968\) vs. \(2(310)=620\).
Time = 10.96 (sec) , antiderivative size = 4968, normalized size of antiderivative = 16.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {2027, 2467, 2035, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a x^3+b x}{\left (2 a x^3+b\right ) \sqrt [3]{a^3 x^3+b^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x \left (a x^2+b\right )}{\left (2 a x^3+b\right ) \sqrt [3]{a^3 x^3+b^2 x^2}}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {\sqrt [3]{x} \left (a x^2+b\right )}{\sqrt [3]{x a^3+b^2} \left (2 a x^3+b\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {x \left (a x^2+b\right )}{\sqrt [3]{x a^3+b^2} \left (2 a x^3+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {1}{2 \sqrt [3]{x a^3+b^2}}-\frac {b-2 b x}{2 \sqrt [3]{x a^3+b^2} \left (2 a x^3+b\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \left (-\frac {(-1)^{7/9} \left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {\sqrt [3]{-2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{8/9} a^{2/9} \sqrt [9]{b} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{4/9} \left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {\sqrt [3]{-2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{8/9} a^{2/9} \sqrt [9]{b} \sqrt [3]{x a^3+b^2}}-\frac {\sqrt [9]{-1} \left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {\sqrt [3]{-2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{8/9} a^{2/9} \sqrt [9]{b} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{2/3} \left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {\sqrt [3]{2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{8/9} a^{2/9} \sqrt [9]{b} \sqrt [3]{x a^3+b^2}}-\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {\sqrt [3]{2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{8/9} a^{2/9} \sqrt [9]{b} \sqrt [3]{x a^3+b^2}}+\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {\sqrt [3]{2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{8/9} a^{2/9} \sqrt [9]{b} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{2/9} \left ((-1)^{2/3} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{8/9} a^{2/9} \sqrt [9]{b} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{8/9} \sqrt [9]{a} \left (\frac {2^{2/3} \sqrt [3]{b}}{\sqrt [3]{a}}+(-1)^{2/3}\right ) x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{8/9} \sqrt [9]{b} \sqrt [3]{x a^3+b^2}}-\frac {(-1)^{5/9} \sqrt [9]{a} \left (\frac {2^{2/3} \sqrt [3]{b}}{\sqrt [3]{a}}+(-1)^{2/3}\right ) x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{8/9} \sqrt [9]{b} \sqrt [3]{x a^3+b^2}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{x} a}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a}-\frac {\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{-2} b^{5/3}}}-\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}}}-\frac {\left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}}}+\frac {(-1)^{2/3} \left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{7/9} \sqrt {3} a^{4/9} \sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{7/9} \sqrt {3} a^{4/9} \sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}}+\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{7/9} \sqrt {3} a^{4/9} \sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {(-1)^{8/9} \left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{7/9} \sqrt {3} a^{4/9} \sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}}+\frac {(-1)^{5/9} \left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{7/9} \sqrt {3} a^{4/9} \sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}}-\frac {(-1)^{2/9} \left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{7/9} \sqrt {3} a^{4/9} \sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}}+\frac {(-1)^{7/9} \left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{7/9} \sqrt {3} a^{4/9} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}}-\frac {(-1)^{4/9} \left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{7/9} \sqrt {3} a^{4/9} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}}+\frac {\sqrt [9]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{7/9} \sqrt {3} a^{4/9} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}}+\frac {(-1)^{8/9} \left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{b}-\sqrt [3]{-2} \sqrt [3]{a} x\right )}{54\ 2^{7/9} a^{4/9} \sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}}-\frac {(-1)^{5/9} \left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{b}-\sqrt [3]{-2} \sqrt [3]{a} x\right )}{54\ 2^{7/9} a^{4/9} \sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}}+\frac {(-1)^{2/9} \left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{b}-\sqrt [3]{-2} \sqrt [3]{a} x\right )}{54\ 2^{7/9} a^{4/9} \sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}}-\frac {\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{-2} \sqrt [3]{a} x-\sqrt [3]{b}\right )}{36 a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{-2} b^{5/3}}}-\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{a} x-\sqrt [3]{b}\right )}{36 a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}}}-\frac {(-1)^{2/3} \left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{54\ 2^{7/9} a^{4/9} \sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}}+\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{54\ 2^{7/9} a^{4/9} \sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{54\ 2^{7/9} a^{4/9} \sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {\left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x-\sqrt [3]{b}\right )}{36 a^{4/9} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}}}-\frac {\sqrt [9]{-1} \left ((-1)^{2/3} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{54\ 2^{7/9} a^{4/9} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}}+\frac {(-1)^{4/9} \left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \log \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{54\ 2^{7/9} a^{4/9} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}}-\frac {\sqrt [9]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \log \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{54\ 2^{7/9} a^{4/9} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}}+\frac {\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{-2} b^{5/3}}}+\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}}}+\frac {\left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}}}-\frac {\log \left (\sqrt [3]{x a^3+b^2}-a \sqrt [3]{x}\right )}{4 a}-\frac {(-1)^{8/9} \left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}-\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{7/9} a^{4/9} \sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}}+\frac {(-1)^{5/9} \left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}-\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{7/9} a^{4/9} \sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}}-\frac {(-1)^{2/9} \left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}-\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{7/9} a^{4/9} \sqrt [3]{2 b^{5/3}-(-2)^{2/3} a^{8/3}}}+\frac {\sqrt [9]{-1} \left ((-1)^{2/3} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}-\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{7/9} a^{4/9} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}}-\frac {(-1)^{4/9} \left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}-\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{7/9} a^{4/9} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}}+\frac {\sqrt [9]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}-\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{7/9} a^{4/9} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}+2 b^{5/3}}}+\frac {(-1)^{2/3} \left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}+\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{7/9} a^{4/9} \sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}+\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{7/9} a^{4/9} \sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}}+\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}+\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{7/9} a^{4/9} \sqrt [3]{2^{2/3} a^{8/3}-2 b^{5/3}}}\right )}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
(3*x^(2/3)*(b^2 + a^3*x)^(1/3)*(-1/18*((-1)^(1/9)*(a^(1/3) + (-2)^(2/3)*b^ (1/3))*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, ((-2)^(1 /3)*a^(1/3)*x)/b^(1/3), -((a^3*x)/b^2)])/(2^(8/9)*a^(2/9)*b^(1/9)*(b^2 + a ^3*x)^(1/3)) + ((-1)^(4/9)*(a^(1/3) + (-2)^(2/3)*b^(1/3))*x^(2/3)*(1 + (a^ 3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, ((-2)^(1/3)*a^(1/3)*x)/b^(1/3), -((a^3*x)/b^2)])/(18*2^(8/9)*a^(2/9)*b^(1/9)*(b^2 + a^3*x)^(1/3)) - ((-1) ^(7/9)*(a^(1/3) + (-2)^(2/3)*b^(1/3))*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*Appe llF1[2/3, 1, 1/3, 5/3, ((-2)^(1/3)*a^(1/3)*x)/b^(1/3), -((a^3*x)/b^2)])/(1 8*2^(8/9)*a^(2/9)*b^(1/9)*(b^2 + a^3*x)^(1/3)) + ((a^(1/3) + 2^(2/3)*b^(1/ 3))*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -((2^(1/3)* a^(1/3)*x)/b^(1/3)), -((a^3*x)/b^2)])/(18*2^(8/9)*a^(2/9)*b^(1/9)*(b^2 + a ^3*x)^(1/3)) - ((-1)^(1/3)*(a^(1/3) + 2^(2/3)*b^(1/3))*x^(2/3)*(1 + (a^3*x )/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -((2^(1/3)*a^(1/3)*x)/b^(1/3)), -( (a^3*x)/b^2)])/(18*2^(8/9)*a^(2/9)*b^(1/9)*(b^2 + a^3*x)^(1/3)) + ((-1)^(2 /3)*(a^(1/3) + 2^(2/3)*b^(1/3))*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*AppellF1[2 /3, 1, 1/3, 5/3, -((2^(1/3)*a^(1/3)*x)/b^(1/3)), -((a^3*x)/b^2)])/(18*2^(8 /9)*a^(2/9)*b^(1/9)*(b^2 + a^3*x)^(1/3)) + ((-1)^(2/9)*((-1)^(2/3)*a^(1/3) + 2^(2/3)*b^(1/3))*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -(((-1)^(2/3)*2^(1/3)*a^(1/3)*x)/b^(1/3)), -((a^3*x)/b^2)])/(18*2^(8/ 9)*a^(2/9)*b^(1/9)*(b^2 + a^3*x)^(1/3)) - ((-1)^(5/9)*a^(1/9)*((-1)^(2/...
3.29.78.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.54 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.68
| method | result | size |
| pseudoelliptic | \(\frac {-6 \sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 a^{3} \textit {\_Z}^{6}+3 a^{6} \textit {\_Z}^{3}-a^{9}+2 a \,b^{5}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right ) \left (\textit {\_R}^{3}-a^{3}-2 b^{2}\right )}{\textit {\_R} \left (\textit {\_R}^{3}-a^{3}\right )}\right ) a -6 \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )+3 \ln \left (\frac {a^{2} x^{2}+a \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{12 a}\) | \(212\) |
1/12*(-6*3^(1/2)*arctan(1/3*(a*x+2*(x^2*(a^3*x+b^2))^(1/3))*3^(1/2)/a/x)+2 *sum(ln((-_R*x+(x^2*(a^3*x+b^2))^(1/3))/x)*(_R^3-a^3-2*b^2)/_R/(_R^3-a^3), _R=RootOf(_Z^9-3*_Z^6*a^3+3*_Z^3*a^6-a^9+2*a*b^5))*a-6*ln((-a*x+(x^2*(a^3* x+b^2))^(1/3))/x)+3*ln((a^2*x^2+a*(x^2*(a^3*x+b^2))^(1/3)*x+(x^2*(a^3*x+b^ 2))^(2/3))/x^2))/a
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 5.39 (sec) , antiderivative size = 66610, normalized size of antiderivative = 214.87 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Too large to display} \]
Not integrable
Time = 23.38 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.10 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {x \left (a x^{2} + b\right )}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a x^{3} + b\right )}\, dx \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.13 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {a x^{3} + b x}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{3} + b\right )}} \,d x } \]
Timed out. \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Timed out} \]
Not integrable
Time = 7.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.13 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {a\,x^3+b\,x}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (2\,a\,x^3+b\right )} \,d x \]