Integrand size = 42, antiderivative size = 310 \[ \int \frac {-b x+a x^3}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{2 a}-\frac {\log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 a}+\frac {1}{6} \text {RootSum}\left [a^9+2 a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-a^3 \log (x)-2 b^2 \log (x)+a^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+2 b^2 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{a^3 \text {$\#$1}-\text {$\#$1}^4}\&\right ] \] Output:
Unintegrable
Time = 0.19 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.86 \[ \int \frac {-b x+a x^3}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {3 \sqrt [3]{a} x \sqrt [3]{1+\frac {a^3 x}{b^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {a^3 x}{b^2}\right )-x \left (\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {\sqrt [3]{a} \left (a^{8/3}-\sqrt [3]{-2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {\sqrt [3]{a} \left (a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {a^3 x+\sqrt [3]{2} \sqrt [3]{a} b^{5/3} x}{b^2+a^3 x}\right )\right )}{2 \sqrt [3]{a} \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \] Input:
Integrate[(-(b*x) + a*x^3)/((-b + 2*a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]
Output:
(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))
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 |
\(\displaystyle \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {\sqrt [3]{x} \left (b-a x^2\right )}{\sqrt [3]{x a^3+b^2} \left (b-2 a x^3\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {x \left (b-a x^2\right )}{\sqrt [3]{x a^3+b^2} \left (b-2 a x^3\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {1}{2 \sqrt [3]{x a^3+b^2}}-\frac {b-2 b x}{2 \sqrt [3]{x a^3+b^2} \left (b-2 a x^3\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {b (2 x-1)}{2 \sqrt [3]{x a^3+b^2} \left (b-2 a x^3\right )}+\frac {1}{2 \sqrt [3]{x a^3+b^2}}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {b (2 x-1)}{2 \sqrt [3]{x a^3+b^2} \left (b-2 a x^3\right )}+\frac {1}{2 \sqrt [3]{x a^3+b^2}}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
Input:
Int[(-(b*x) + a*x^3)/((-b + 2*a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]
Output:
$Aborted
Time = 0.20 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 a^{3} \textit {\_Z}^{6}+3 a^{6} \textit {\_Z}^{3}-a^{9}-2 a \,b^{5}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right ) \left (\textit {\_R}^{3}-a^{3}-2 b^{2}\right )}{\textit {\_R} \left (\textit {\_R}^{3}-a^{3}\right )}\right ) a -6 \sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )-6 \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )+3 \ln \left (\frac {a^{2} x^{2}+a \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{12 a}\) | \(212\) |
Input:
int((a*x^3-b*x)/(2*a*x^3-b)/(a^3*x^3+b^2*x^2)^(1/3),x,method=_RETURNVERBOS E)
Output:
1/12*(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*3^(1/2)*arct an(1/3*(a*x+2*(x^2*(a^3*x+b^2))^(1/3))*3^(1/2)/a/x)-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 = 5.12 (sec) , antiderivative size = 66601, normalized size of antiderivative = 214.84 \[ \int \frac {-b x+a x^3}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Too large to display} \] Input:
integrate((a*x^3-b*x)/(2*a*x^3-b)/(a^3*x^3+b^2*x^2)^(1/3),x, algorithm="fr icas")
Output:
Too large to include
Not integrable
Time = 21.37 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.10 \[ \int \frac {-b x+a x^3}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {x \left (a x^{2} - b\right )}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a x^{3} - b\right )}\, dx \] Input:
integrate((a*x**3-b*x)/(2*a*x**3-b)/(a**3*x**3+b**2*x**2)**(1/3),x)
Output:
Integral(x*(a*x**2 - b)/((x**2*(a**3*x + b**2))**(1/3)*(2*a*x**3 - b)), x)
Not integrable
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.14 \[ \int \frac {-b x+a x^3}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {a x^{3} - b x}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{3} - b\right )}} \,d x } \] Input:
integrate((a*x^3-b*x)/(2*a*x^3-b)/(a^3*x^3+b^2*x^2)^(1/3),x, algorithm="ma xima")
Output:
integrate((a*x^3 - b*x)/((a^3*x^3 + b^2*x^2)^(1/3)*(2*a*x^3 - b)), x)
Timed out. \[ \int \frac {-b x+a x^3}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Timed out} \] Input:
integrate((a*x^3-b*x)/(2*a*x^3-b)/(a^3*x^3+b^2*x^2)^(1/3),x, algorithm="gi ac")
Output:
Timed out
Not integrable
Time = 0.00 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.13 \[ \int \frac {-b x+a x^3}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {b\,x-a\,x^3}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-2\,a\,x^3\right )} \,d x \] Input:
int((b*x - a*x^3)/((a^3*x^3 + b^2*x^2)^(1/3)*(b - 2*a*x^3)),x)
Output:
int((b*x - a*x^3)/((a^3*x^3 + b^2*x^2)^(1/3)*(b - 2*a*x^3)), x)
Not integrable
Time = 1.87 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.29 \[ \int \frac {-b x+a x^3}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\left (\int \frac {x^{3}}{2 x^{\frac {11}{3}} \left (a^{3} x +b^{2}\right )^{\frac {1}{3}} a -x^{\frac {2}{3}} \left (a^{3} x +b^{2}\right )^{\frac {1}{3}} b}d x \right ) a -\left (\int \frac {x}{2 x^{\frac {11}{3}} \left (a^{3} x +b^{2}\right )^{\frac {1}{3}} a -x^{\frac {2}{3}} \left (a^{3} x +b^{2}\right )^{\frac {1}{3}} b}d x \right ) b \] Input:
int((a*x^3-b*x)/(2*a*x^3-b)/(a^3*x^3+b^2*x^2)^(1/3),x)
Output:
int(x**3/(2*x**(2/3)*(a**3*x + b**2)**(1/3)*a*x**3 - x**(2/3)*(a**3*x + b* *2)**(1/3)*b),x)*a - int(x/(2*x**(2/3)*(a**3*x + b**2)**(1/3)*a*x**3 - x** (2/3)*(a**3*x + b**2)**(1/3)*b),x)*b