Integrand size = 40, antiderivative size = 41 \[ \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-2 b-3 x^2+2 a x^3\right )} \, dx=\sqrt {\frac {2}{3}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x \sqrt {-b+a x^3}}{b-a x^3}\right ) \] Output:
1/3*6^(1/2)*arctanh(1/2*6^(1/2)*x*(a*x^3-b)^(1/2)/(-a*x^3+b))
Time = 0.85 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.78 \[ \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-2 b-3 x^2+2 a x^3\right )} \, dx=-\sqrt {\frac {2}{3}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-b+a x^3}}\right ) \] Input:
Integrate[(2*b + a*x^3)/(Sqrt[-b + a*x^3]*(-2*b - 3*x^2 + 2*a*x^3)),x]
Output:
-(Sqrt[2/3]*ArcTanh[(Sqrt[3/2]*x)/Sqrt[-b + a*x^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.
\(\displaystyle \int \frac {a x^3+2 b}{\sqrt {a x^3-b} \left (2 a x^3-2 b-3 x^2\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{2 \sqrt {a x^3-b}}+\frac {3 \left (2 b+x^2\right )}{2 \sqrt {a x^3-b} \left (2 a x^3-2 b-3 x^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 b \int \frac {1}{\left (-2 a x^3+3 x^2+2 b\right ) \sqrt {a x^3-b}}dx+\frac {3}{2} \int \frac {x^2}{\sqrt {a x^3-b} \left (2 a x^3-3 x^2-2 b\right )}dx-\frac {\sqrt {2-\sqrt {3}} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt {\frac {a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{a} \sqrt {-\frac {\sqrt [3]{b} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} \sqrt {a x^3-b}}\) |
Input:
Int[(2*b + a*x^3)/(Sqrt[-b + a*x^3]*(-2*b - 3*x^2 + 2*a*x^3)),x]
Output:
$Aborted
Time = 0.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63
method | result | size |
default | \(-\frac {\sqrt {6}\, \operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}-b}\, \sqrt {6}}{3 x}\right )}{3}\) | \(26\) |
pseudoelliptic | \(-\frac {\sqrt {6}\, \operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}-b}\, \sqrt {6}}{3 x}\right )}{3}\) | \(26\) |
elliptic | \(\text {Expression too large to display}\) | \(790\) |
Input:
int((a*x^3+2*b)/(a*x^3-b)^(1/2)/(2*a*x^3-3*x^2-2*b),x,method=_RETURNVERBOS E)
Output:
-1/3*6^(1/2)*arctanh(1/3*(a*x^3-b)^(1/2)/x*6^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (23) = 46\).
Time = 0.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.85 \[ \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-2 b-3 x^2+2 a x^3\right )} \, dx=\frac {1}{4} \, \sqrt {\frac {2}{3}} \log \left (\frac {4 \, a^{2} x^{6} + 36 \, a x^{5} - 8 \, a b x^{3} + 9 \, x^{4} - 36 \, b x^{2} - 12 \, \sqrt {\frac {2}{3}} {\left (2 \, a x^{4} + 3 \, x^{3} - 2 \, b x\right )} \sqrt {a x^{3} - b} + 4 \, b^{2}}{4 \, a^{2} x^{6} - 12 \, a x^{5} - 8 \, a b x^{3} + 9 \, x^{4} + 12 \, b x^{2} + 4 \, b^{2}}\right ) \] Input:
integrate((a*x^3+2*b)/(a*x^3-b)^(1/2)/(2*a*x^3-3*x^2-2*b),x, algorithm="fr icas")
Output:
1/4*sqrt(2/3)*log((4*a^2*x^6 + 36*a*x^5 - 8*a*b*x^3 + 9*x^4 - 36*b*x^2 - 1 2*sqrt(2/3)*(2*a*x^4 + 3*x^3 - 2*b*x)*sqrt(a*x^3 - b) + 4*b^2)/(4*a^2*x^6 - 12*a*x^5 - 8*a*b*x^3 + 9*x^4 + 12*b*x^2 + 4*b^2))
\[ \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-2 b-3 x^2+2 a x^3\right )} \, dx=\int \frac {a x^{3} + 2 b}{\sqrt {a x^{3} - b} \left (2 a x^{3} - 2 b - 3 x^{2}\right )}\, dx \] Input:
integrate((a*x**3+2*b)/(a*x**3-b)**(1/2)/(2*a*x**3-3*x**2-2*b),x)
Output:
Integral((a*x**3 + 2*b)/(sqrt(a*x**3 - b)*(2*a*x**3 - 2*b - 3*x**2)), x)
\[ \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-2 b-3 x^2+2 a x^3\right )} \, dx=\int { \frac {a x^{3} + 2 \, b}{{\left (2 \, a x^{3} - 3 \, x^{2} - 2 \, b\right )} \sqrt {a x^{3} - b}} \,d x } \] Input:
integrate((a*x^3+2*b)/(a*x^3-b)^(1/2)/(2*a*x^3-3*x^2-2*b),x, algorithm="ma xima")
Output:
integrate((a*x^3 + 2*b)/((2*a*x^3 - 3*x^2 - 2*b)*sqrt(a*x^3 - b)), x)
\[ \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-2 b-3 x^2+2 a x^3\right )} \, dx=\int { \frac {a x^{3} + 2 \, b}{{\left (2 \, a x^{3} - 3 \, x^{2} - 2 \, b\right )} \sqrt {a x^{3} - b}} \,d x } \] Input:
integrate((a*x^3+2*b)/(a*x^3-b)^(1/2)/(2*a*x^3-3*x^2-2*b),x, algorithm="gi ac")
Output:
integrate((a*x^3 + 2*b)/((2*a*x^3 - 3*x^2 - 2*b)*sqrt(a*x^3 - b)), x)
Time = 9.39 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.37 \[ \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-2 b-3 x^2+2 a x^3\right )} \, dx=\frac {\sqrt {6}\,\ln \left (\frac {2\,b-2\,a\,x^3-3\,x^2+2\,\sqrt {6}\,x\,\sqrt {a\,x^3-b}}{-2\,a\,x^3+3\,x^2+2\,b}\right )}{6} \] Input:
int(-(2*b + a*x^3)/((a*x^3 - b)^(1/2)*(2*b - 2*a*x^3 + 3*x^2)),x)
Output:
(6^(1/2)*log((2*b - 2*a*x^3 - 3*x^2 + 2*6^(1/2)*x*(a*x^3 - b)^(1/2))/(2*b - 2*a*x^3 + 3*x^2)))/6
\[ \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-2 b-3 x^2+2 a x^3\right )} \, dx=2 \left (\int \frac {\sqrt {a \,x^{3}-b}}{2 a^{2} x^{6}-3 a \,x^{5}-4 a b \,x^{3}+3 b \,x^{2}+2 b^{2}}d x \right ) b +\left (\int \frac {\sqrt {a \,x^{3}-b}\, x^{3}}{2 a^{2} x^{6}-3 a \,x^{5}-4 a b \,x^{3}+3 b \,x^{2}+2 b^{2}}d x \right ) a \] Input:
int((a*x^3+2*b)/(a*x^3-b)^(1/2)/(2*a*x^3-3*x^2-2*b),x)
Output:
2*int(sqrt(a*x**3 - b)/(2*a**2*x**6 - 4*a*b*x**3 - 3*a*x**5 + 2*b**2 + 3*b *x**2),x)*b + int((sqrt(a*x**3 - b)*x**3)/(2*a**2*x**6 - 4*a*b*x**3 - 3*a* x**5 + 2*b**2 + 3*b*x**2),x)*a