\(\int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} (2 b+a^6 x^6)} \, dx\) [1620]

Optimal result
Mathematica [A] (verified)
Rubi [F]
Maple [N/A] (verified)
Fricas [F(-1)]
Sympy [N/A]
Maxima [N/A]
Giac [N/A]
Mupad [N/A]
Reduce [N/A]

Optimal result

Integrand size = 33, antiderivative size = 110 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=-\frac {\text {RootSum}\left [2 a^{18}+a^6 b^{11}-12 a^{15} \text {$\#$1}^3+30 a^{12} \text {$\#$1}^6-40 a^9 \text {$\#$1}^9+30 a^6 \text {$\#$1}^{12}-12 a^3 \text {$\#$1}^{15}+2 \text {$\#$1}^{18}\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{12 b} \] Output:

Unintegrable
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [2 a^{18}+a^6 b^{11}-12 a^{15} \text {$\#$1}^3+30 a^{12} \text {$\#$1}^6-40 a^9 \text {$\#$1}^9+30 a^6 \text {$\#$1}^{12}-12 a^3 \text {$\#$1}^{15}+2 \text {$\#$1}^{18}\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{12 b \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \] Input:

Integrate[1/((b^2*x^2 + a^3*x^3)^(1/3)*(2*b + a^6*x^6)),x]
 

Output:

-1/12*(x^(2/3)*(b^2 + a^3*x)^(1/3)*RootSum[2*a^18 + a^6*b^11 - 12*a^15*#1^ 
3 + 30*a^12*#1^6 - 40*a^9*#1^9 + 30*a^6*#1^12 - 12*a^3*#1^15 + 2*#1^18 & , 
 (-Log[x^(1/3)] + Log[(b^2 + a^3*x)^(1/3) - x^(1/3)*#1])/#1 & ])/(b*(x^2*( 
b^2 + a^3*x))^(1/3))
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{\sqrt [3]{a^3 x^3+b^2 x^2} \left (a^6 x^6+2 b\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{x^{2/3} \sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b \sqrt [3]{x a^3+b^2} \left (a^3 x^3+\sqrt {2} \sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (a^6 x^6+2 b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

Input:

Int[1/((b^2*x^2 + a^3*x^3)^(1/3)*(2*b + a^6*x^6)),x]
 

Output:

$Aborted
 
Maple [N/A] (verified)

Time = 0.38 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.88

method result size
pseudoelliptic \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{18}-12 a^{3} \textit {\_Z}^{15}+30 a^{6} \textit {\_Z}^{12}-40 a^{9} \textit {\_Z}^{9}+30 a^{12} \textit {\_Z}^{6}-12 a^{15} \textit {\_Z}^{3}+2 a^{18}+a^{6} b^{11}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{12 b}\) \(97\)

Input:

int(1/(a^3*x^3+b^2*x^2)^(1/3)/(a^6*x^6+2*b),x,method=_RETURNVERBOSE)
 

Output:

-1/12*sum(ln((-_R*x+(x^2*(a^3*x+b^2))^(1/3))/x)/_R,_R=RootOf(2*_Z^18-12*_Z 
^15*a^3+30*_Z^12*a^6-40*_Z^9*a^9+30*_Z^6*a^12-12*_Z^3*a^15+2*a^18+a^6*b^11 
))/b
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(a^3*x^3+b^2*x^2)^(1/3)/(a^6*x^6+2*b),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [N/A]

Not integrable

Time = 6.59 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.25 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a^{6} x^{6} + 2 b\right )}\, dx \] Input:

integrate(1/(a**3*x**3+b**2*x**2)**(1/3)/(a**6*x**6+2*b),x)
 

Output:

Integral(1/((x**2*(a**3*x + b**2))**(1/3)*(a**6*x**6 + 2*b)), x)
 

Maxima [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=\int { \frac {1}{{\left (a^{6} x^{6} + 2 \, b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(a^3*x^3+b^2*x^2)^(1/3)/(a^6*x^6+2*b),x, algorithm="maxima")
 

Output:

integrate(1/((a^6*x^6 + 2*b)*(a^3*x^3 + b^2*x^2)^(1/3)), x)
 

Giac [N/A]

Not integrable

Time = 1.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=\int { \frac {1}{{\left (a^{6} x^{6} + 2 \, b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(a^3*x^3+b^2*x^2)^(1/3)/(a^6*x^6+2*b),x, algorithm="giac")
 

Output:

integrate(1/((a^6*x^6 + 2*b)*(a^3*x^3 + b^2*x^2)^(1/3)), x)
 

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=\int \frac {1}{\left (a^6\,x^6+2\,b\right )\,{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}} \,d x \] Input:

int(1/((2*b + a^6*x^6)*(a^3*x^3 + b^2*x^2)^(1/3)),x)
 

Output:

int(1/((2*b + a^6*x^6)*(a^3*x^3 + b^2*x^2)^(1/3)), x)
 

Reduce [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.36 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=\int \frac {1}{x^{\frac {20}{3}} \left (a^{3} x +b^{2}\right )^{\frac {1}{3}} a^{6}+2 x^{\frac {2}{3}} \left (a^{3} x +b^{2}\right )^{\frac {1}{3}} b}d x \] Input:

int(1/(a^3*x^3+b^2*x^2)^(1/3)/(a^6*x^6+2*b),x)
 

Output:

int(1/(x**(2/3)*(a**3*x + b**2)**(1/3)*a**6*x**6 + 2*x**(2/3)*(a**3*x + b* 
*2)**(1/3)*b),x)