\(\int \frac {1}{\sqrt [3]{a+b x-\frac {b^3 x^3}{6 a^2}}} \, dx\) [31]

Optimal result

Integrand size = 22, antiderivative size = 767 \[ \int \frac {1}{\sqrt [3]{a+b x-\frac {b^3 x^3}{6 a^2}}} \, dx=-\frac {3^{5/6} \sqrt [3]{a} (a+b x) \sqrt [3]{\frac {a b^3 \left (6 a+6 b x-\frac {b^3 x^3}{a^2}\right )}{(a+b x)^3}} \arctan \left (\frac {1-\frac {2 b^2 x}{\sqrt [3]{a} (a+b x) \sqrt [3]{-\frac {b^3 \left (2-\frac {(2 a+b x)^3}{(a+b x)^3}\right )}{a}}}}{\sqrt {3}}\right )}{2^{2/3} b^2 \sqrt [3]{6 a+6 b x-\frac {b^3 x^3}{a^2}}}+\frac {\sqrt [3]{a} (a+b x) \sqrt [3]{\frac {a b^3 \left (6 a+6 b x-\frac {b^3 x^3}{a^2}\right )}{(a+b x)^3}} \arctan \left (\frac {1+\frac {2 b (2 a+b x)}{\sqrt [3]{a} (a+b x) \sqrt [3]{-\frac {b^3 \left (2-\frac {(2 a+b x)^3}{(a+b x)^3}\right )}{a}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt [6]{3} b^2 \sqrt [3]{6 a+6 b x-\frac {b^3 x^3}{a^2}}}+\frac {\sqrt [3]{3} \sqrt [3]{a} (a+b x) \sqrt [3]{\frac {a b^3 \left (6 a+6 b x-\frac {b^3 x^3}{a^2}\right )}{(a+b x)^3}} \log (a+b x)}{2^{2/3} b^2 \sqrt [3]{6 a+6 b x-\frac {b^3 x^3}{a^2}}}-\frac {\sqrt [3]{3} \sqrt [3]{a} (a+b x) \sqrt [3]{\frac {a b^3 \left (6 a+6 b x-\frac {b^3 x^3}{a^2}\right )}{(a+b x)^3}} \log \left (-\frac {2 a b+b^2 x-a^{4/3} \sqrt [3]{\frac {b^3 \left (6 a^3+6 a^2 b x-b^3 x^3\right )}{a (a+b x)^3}}-\sqrt [3]{a} b x \sqrt [3]{\frac {b^3 \left (6 a^3+6 a^2 b x-b^3 x^3\right )}{a (a+b x)^3}}}{a+b x}\right )}{2\ 2^{2/3} b^2 \sqrt [3]{6 a+6 b x-\frac {b^3 x^3}{a^2}}}+\frac {3 \sqrt [3]{3} \sqrt [3]{a} (a+b x) \sqrt [3]{\frac {a b^3 \left (6 a+6 b x-\frac {b^3 x^3}{a^2}\right )}{(a+b x)^3}} \log \left (\frac {b^2 x+a^{4/3} \sqrt [3]{\frac {b^3 \left (6 a^3+6 a^2 b x-b^3 x^3\right )}{a (a+b x)^3}}+\sqrt [3]{a} b x \sqrt [3]{\frac {b^3 \left (6 a^3+6 a^2 b x-b^3 x^3\right )}{a (a+b x)^3}}}{a+b x}\right )}{2\ 2^{2/3} b^2 \sqrt [3]{6 a+6 b x-\frac {b^3 x^3}{a^2}}} \] Output:

-1/2*3^(5/6)*a^(1/3)*(b*x+a)*(a*b^3*(6*a+6*b*x-b^3*x^3/a^2)/(b*x+a)^3)^(1/ 
3)*arctan(1/3*(1-2*b^2*x/a^(1/3)/(b*x+a)/(-b^3*(2-(b*x+2*a)^3/(b*x+a)^3)/a 
)^(1/3))*3^(1/2))*2^(1/3)/b^2/(6*a+6*b*x-b^3*x^3/a^2)^(1/3)+1/6*a^(1/3)*(b 
*x+a)*(a*b^3*(6*a+6*b*x-b^3*x^3/a^2)/(b*x+a)^3)^(1/3)*arctan(1/3*(1+2*b*(b 
*x+2*a)/a^(1/3)/(b*x+a)/(-b^3*(2-(b*x+2*a)^3/(b*x+a)^3)/a)^(1/3))*3^(1/2)) 
*2^(1/3)*3^(5/6)/b^2/(6*a+6*b*x-b^3*x^3/a^2)^(1/3)+1/2*3^(1/3)*a^(1/3)*(b* 
x+a)*(a*b^3*(6*a+6*b*x-b^3*x^3/a^2)/(b*x+a)^3)^(1/3)*ln(b*x+a)*2^(1/3)/b^2 
/(6*a+6*b*x-b^3*x^3/a^2)^(1/3)-1/4*3^(1/3)*a^(1/3)*(b*x+a)*(a*b^3*(6*a+6*b 
*x-b^3*x^3/a^2)/(b*x+a)^3)^(1/3)*ln(-(2*a*b+b^2*x-a^(4/3)*(b^3*(-b^3*x^3+6 
*a^2*b*x+6*a^3)/a/(b*x+a)^3)^(1/3)-a^(1/3)*b*x*(b^3*(-b^3*x^3+6*a^2*b*x+6* 
a^3)/a/(b*x+a)^3)^(1/3))/(b*x+a))*2^(1/3)/b^2/(6*a+6*b*x-b^3*x^3/a^2)^(1/3 
)+3/4*3^(1/3)*a^(1/3)*(b*x+a)*(a*b^3*(6*a+6*b*x-b^3*x^3/a^2)/(b*x+a)^3)^(1 
/3)*ln((b^2*x+a^(4/3)*(b^3*(-b^3*x^3+6*a^2*b*x+6*a^3)/a/(b*x+a)^3)^(1/3)+a 
^(1/3)*b*x*(b^3*(-b^3*x^3+6*a^2*b*x+6*a^3)/a/(b*x+a)^3)^(1/3))/(b*x+a))*2^ 
(1/3)/b^2/(6*a+6*b*x-b^3*x^3/a^2)^(1/3)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.31 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\sqrt [3]{a+b x-\frac {b^3 x^3}{6 a^2}}} \, dx=-\frac {3 \left (\left (\sqrt [3]{2}+2^{2/3}\right ) a-b x\right ) \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {\left (\sqrt [3]{2}+2^{2/3}\right ) a-b x}{a \left (\sqrt [3]{2}+2^{2/3}-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}-6\&,2\right ]\right )},\frac {\left (\sqrt [3]{2}+2^{2/3}\right ) a-b x}{a \left (\sqrt [3]{2}+2^{2/3}-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}-6\&,3\right ]\right )}\right ) \sqrt [3]{\frac {3 b x-3 a \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}-6\&,2\right ]}{a \left (\sqrt [3]{2}+2^{2/3}-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}-6\&,2\right ]\right )}} \sqrt [3]{\frac {b x-a \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}-6\&,3\right ]}{a \left (\sqrt [3]{2}+2^{2/3}-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}-6\&,3\right ]\right )}}}{2^{2/3} b \sqrt [3]{6 a+6 b x-\frac {b^3 x^3}{a^2}}} \] Input:

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

Output:

(-3*((2^(1/3) + 2^(2/3))*a - b*x)*AppellF1[2/3, 1/3, 1/3, 5/3, ((2^(1/3) + 
 2^(2/3))*a - b*x)/(a*(2^(1/3) + 2^(2/3) - Root[-6 - 6*#1 + #1^3 & , 2, 0] 
)), ((2^(1/3) + 2^(2/3))*a - b*x)/(a*(2^(1/3) + 2^(2/3) - Root[-6 - 6*#1 + 
 #1^3 & , 3, 0]))]*((3*b*x - 3*a*Root[-6 - 6*#1 + #1^3 & , 2, 0])/(a*(2^(1 
/3) + 2^(2/3) - Root[-6 - 6*#1 + #1^3 & , 2, 0])))^(1/3)*((b*x - a*Root[-6 
 - 6*#1 + #1^3 & , 3, 0])/(a*(2^(1/3) + 2^(2/3) - Root[-6 - 6*#1 + #1^3 & 
, 3, 0])))^(1/3))/(2^(2/3)*b*(6*a + 6*b*x - (b^3*x^3)/a^2)^(1/3))
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 3.35 (sec) , antiderivative size = 871, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2475, 27, 1179, 150}

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.

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

\(\Big \downarrow \) 2475

\(\displaystyle \frac {\sqrt [3]{\frac {b \left (\left (a \sqrt {\frac {b^6}{a^2}}+3 b^3\right )^{2/3}+2 b^2\right )}{a \sqrt [3]{a \sqrt {\frac {b^6}{a^2}}+3 b^3}}-\frac {b^3 x}{a^2}} \sqrt [3]{\frac {b^6 x^2}{a^4}-\frac {b^2 \left (-\left (a \sqrt {\frac {b^6}{a^2}}+3 b^3\right )^{2/3}-\frac {4 b^4}{\left (a \sqrt {\frac {b^6}{a^2}}+3 b^3\right )^{2/3}}+2 b^2\right )}{a^2}+\frac {b^4 x \left (\left (a \sqrt {\frac {b^6}{a^2}}+3 b^3\right )^{2/3}+2 b^2\right )}{a^3 \sqrt [3]{a \sqrt {\frac {b^6}{a^2}}+3 b^3}}} \int \frac {6}{\sqrt [3]{\frac {b \left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )}{a \sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}}}-\frac {b^3 x}{a^2}} \sqrt [3]{\frac {x^2 b^6}{a^4}+\frac {\left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right ) x b^4}{a^3 \sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}}}-\frac {\left (-\frac {4 b^4}{\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}}+2 b^2-\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right ) b^2}{a^2}}}dx}{6^{2/3} \sqrt [3]{-\frac {b^3 x^3}{a^2}+6 a+6 b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt [3]{6} \sqrt [3]{\frac {b \left (\left (a \sqrt {\frac {b^6}{a^2}}+3 b^3\right )^{2/3}+2 b^2\right )}{a \sqrt [3]{a \sqrt {\frac {b^6}{a^2}}+3 b^3}}-\frac {b^3 x}{a^2}} \sqrt [3]{\frac {b^6 x^2}{a^4}-\frac {b^2 \left (-\left (a \sqrt {\frac {b^6}{a^2}}+3 b^3\right )^{2/3}-\frac {4 b^4}{\left (a \sqrt {\frac {b^6}{a^2}}+3 b^3\right )^{2/3}}+2 b^2\right )}{a^2}+\frac {b^4 x \left (\left (a \sqrt {\frac {b^6}{a^2}}+3 b^3\right )^{2/3}+2 b^2\right )}{a^3 \sqrt [3]{a \sqrt {\frac {b^6}{a^2}}+3 b^3}}} \int \frac {1}{\sqrt [3]{\frac {b \left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )}{a \sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}}}-\frac {b^3 x}{a^2}} \sqrt [3]{\frac {x^2 b^6}{a^4}+\frac {\left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right ) x b^4}{a^3 \sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}}}-\frac {\left (-\frac {4 b^4}{\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}}+2 b^2-\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right ) b^2}{a^2}}}dx}{\sqrt [3]{-\frac {b^3 x^3}{a^2}+6 a+6 b x}}\)

\(\Big \downarrow \) 1179

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

\(\Big \downarrow \) 150

\(\displaystyle -\frac {3 \sqrt [3]{3} a^2 \left (\frac {b \left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )}{a \sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}}}-\frac {b^3 x}{a^2}\right ) \sqrt [3]{\frac {2 \sqrt {-a^6} \sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}} \left (\frac {a \left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )}{\sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}}}-b^2 x\right )}{a \left (\sqrt {3} a^3 \left (2 b^2-\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )-3 \sqrt {-a^6} \left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )\right )}+1} \sqrt [3]{1-\frac {2 \sqrt {-a^6} \sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}} \left (\frac {a \left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )}{\sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}}}-b^2 x\right )}{a \left (\sqrt {3} \left (2 b^2-\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right ) a^3+3 \sqrt {-a^6} \left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )\right )}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},-\frac {2 a \sqrt {-a^6} \sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}} \left (\frac {b \left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )}{a \sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}}}-\frac {b^3 x}{a^2}\right )}{b \left (\sqrt {3} a^3 \left (2 b^2-\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )-3 \sqrt {-a^6} \left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )\right )},\frac {2 a \sqrt {-a^6} \sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}} \left (\frac {b \left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )}{a \sqrt [3]{3 b^3+a \sqrt {\frac {b^6}{a^2}}}}-\frac {b^3 x}{a^2}\right )}{b \left (\sqrt {3} \left (2 b^2-\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right ) a^3+3 \sqrt {-a^6} \left (2 b^2+\left (3 b^3+a \sqrt {\frac {b^6}{a^2}}\right )^{2/3}\right )\right )}\right )}{2^{2/3} b^3 \sqrt [3]{-\frac {b^3 x^3}{a^2}+6 b x+6 a}}\)

Input:

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

Output:

(-3*3^(1/3)*a^2*((b*(2*b^2 + (3*b^3 + a*Sqrt[b^6/a^2])^(2/3)))/(a*(3*b^3 + 
 a*Sqrt[b^6/a^2])^(1/3)) - (b^3*x)/a^2)*(1 + (2*Sqrt[-a^6]*(3*b^3 + a*Sqrt 
[b^6/a^2])^(1/3)*((a*(2*b^2 + (3*b^3 + a*Sqrt[b^6/a^2])^(2/3)))/(3*b^3 + a 
*Sqrt[b^6/a^2])^(1/3) - b^2*x))/(a*(Sqrt[3]*a^3*(2*b^2 - (3*b^3 + a*Sqrt[b 
^6/a^2])^(2/3)) - 3*Sqrt[-a^6]*(2*b^2 + (3*b^3 + a*Sqrt[b^6/a^2])^(2/3)))) 
)^(1/3)*(1 - (2*Sqrt[-a^6]*(3*b^3 + a*Sqrt[b^6/a^2])^(1/3)*((a*(2*b^2 + (3 
*b^3 + a*Sqrt[b^6/a^2])^(2/3)))/(3*b^3 + a*Sqrt[b^6/a^2])^(1/3) - b^2*x))/ 
(a*(Sqrt[3]*a^3*(2*b^2 - (3*b^3 + a*Sqrt[b^6/a^2])^(2/3)) + 3*Sqrt[-a^6]*( 
2*b^2 + (3*b^3 + a*Sqrt[b^6/a^2])^(2/3)))))^(1/3)*AppellF1[2/3, 1/3, 1/3, 
5/3, (-2*a*Sqrt[-a^6]*(3*b^3 + a*Sqrt[b^6/a^2])^(1/3)*((b*(2*b^2 + (3*b^3 
+ a*Sqrt[b^6/a^2])^(2/3)))/(a*(3*b^3 + a*Sqrt[b^6/a^2])^(1/3)) - (b^3*x)/a 
^2))/(b*(Sqrt[3]*a^3*(2*b^2 - (3*b^3 + a*Sqrt[b^6/a^2])^(2/3)) - 3*Sqrt[-a 
^6]*(2*b^2 + (3*b^3 + a*Sqrt[b^6/a^2])^(2/3)))), (2*a*Sqrt[-a^6]*(3*b^3 + 
a*Sqrt[b^6/a^2])^(1/3)*((b*(2*b^2 + (3*b^3 + a*Sqrt[b^6/a^2])^(2/3)))/(a*( 
3*b^3 + a*Sqrt[b^6/a^2])^(1/3)) - (b^3*x)/a^2))/(b*(Sqrt[3]*a^3*(2*b^2 - ( 
3*b^3 + a*Sqrt[b^6/a^2])^(2/3)) + 3*Sqrt[-a^6]*(2*b^2 + (3*b^3 + a*Sqrt[b^ 
6/a^2])^(2/3))))])/(2^(2/3)*b^3*(6*a + 6*b*x - (b^3*x^3)/a^2)^(1/3))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1179

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(a + b*x + c*x^2)^p/(e*(1 - ( 
d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) 
^p)   Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d 
- e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m 
, p}, x]
 

rule 2475

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9* 
a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Simp[(a + b*x + d*x^3)^p 
/(Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3 
)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/1 
8^(1/3))*x + d^2*x^2, x]^p)   Int[Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + 
d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d* 
(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; Free 
Q[{a, b, d, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{a+b x-\frac {b^3 x^3}{6 a^2}}} \, dx=\sqrt [3]{6} \int \frac {1}{\sqrt [3]{6 a + 6 b x - \frac {b^{3} x^{3}}{a^{2}}}}\, dx \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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