\(\int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx\) [736]

Optimal result

Integrand size = 19, antiderivative size = 235 \[ \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{b^{5/6} \sqrt [6]{d}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{b^{5/6} \sqrt [6]{d}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{5/6} \sqrt [6]{d}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x} \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}\right )}{b^{5/6} \sqrt [6]{d}} \] Output:

-3^(1/2)*arctan(1/3*(b^(1/6)-2*d^(1/6)*(b*x+a)^(1/6)/(d*x+c)^(1/6))*3^(1/2 
)/b^(1/6))/b^(5/6)/d^(1/6)+3^(1/2)*arctan(1/3*(b^(1/6)+2*d^(1/6)*(b*x+a)^( 
1/6)/(d*x+c)^(1/6))*3^(1/2)/b^(1/6))/b^(5/6)/d^(1/6)+2*arctanh(d^(1/6)*(b* 
x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6))/b^(5/6)/d^(1/6)+arctanh(b^(1/6)*d^(1/6)* 
(b*x+a)^(1/6)/(d*x+c)^(1/6)/(b^(1/3)+d^(1/3)*(b*x+a)^(1/3)/(d*x+c)^(1/3))) 
/b^(5/6)/d^(1/6)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx=\frac {\sqrt {3} \left (\arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )-\arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )\right )+2 \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{b^{5/6} \sqrt [6]{d}} \] Input:

Integrate[1/((a + b*x)^(5/6)*(c + d*x)^(1/6)),x]
 

Output:

(Sqrt[3]*(ArcTan[(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))/(-2*d^(1/6)*(a + b*x)^( 
1/6) + b^(1/6)*(c + d*x)^(1/6))] - ArcTan[(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6) 
)/(2*d^(1/6)*(a + b*x)^(1/6) + b^(1/6)*(c + d*x)^(1/6))]) + 2*ArcTanh[(b^( 
1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))] + ArcTanh[(d^(1/6)*(a + b 
*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6)) + (b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)* 
(a + b*x)^(1/6))])/(b^(5/6)*d^(1/6))
 

Rubi [A] (warning: unable to verify)

Time = 0.41 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.54, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {73, 770, 754, 27, 221, 1142, 25, 27, 1082, 217, 1103}

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.

\(\displaystyle \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {6 \int \frac {1}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}d\sqrt [6]{a+b x}}{b}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {6 \int \frac {1}{1-\frac {d (a+b x)}{b}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b}\)

\(\Big \downarrow \) 754

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

\(\displaystyle \frac {6 \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {6 \left (\frac {1}{6} \sqrt [6]{b} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}-\frac {\log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}+\frac {\log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b}\)

Input:

Int[1/((a + b*x)^(5/6)*(c + d*x)^(1/6)),x]
 

Output:

(6*((b^(1/6)*ArcTanh[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c - (a*d)/b + (d* 
(a + b*x))/b)^(1/6))])/(3*d^(1/6)) + (b^(1/6)*(-((Sqrt[3]*ArcTan[(1 - (2*d 
^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c - (a*d)/b + (d*(a + b*x))/b)^(1/6)))/S 
qrt[3]])/d^(1/6)) - Log[b^(1/3) + d^(1/3)*(a + b*x)^(1/3) - (b^(1/6)*d^(1/ 
6)*(a + b*x)^(1/6))/(c - (a*d)/b + (d*(a + b*x))/b)^(1/6)]/(2*d^(1/6))))/6 
 + (b^(1/6)*((Sqrt[3]*ArcTan[(1 + (2*d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c 
- (a*d)/b + (d*(a + b*x))/b)^(1/6)))/Sqrt[3]])/d^(1/6) + Log[b^(1/3) + d^( 
1/3)*(a + b*x)^(1/3) + (b^(1/6)*d^(1/6)*(a + b*x)^(1/6))/(c - (a*d)/b + (d 
*(a + b*x))/b)^(1/6)]/(2*d^(1/6))))/6))/b
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 754

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a 
/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* 
Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 
*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) 
 Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n))   Sum[u, {k, 1, (n - 2)/4}], x]] / 
; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
 

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

Maple [F]

\[\int \frac {1}{\left (b x +a \right )^{\frac {5}{6}} \left (x d +c \right )^{\frac {1}{6}}}d x\]

Input:

int(1/(b*x+a)^(5/6)/(d*x+c)^(1/6),x)
 

Output:

int(1/(b*x+a)^(5/6)/(d*x+c)^(1/6),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (169) = 338\).

Time = 0.09 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx=\frac {1}{2} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b d x + b c + \sqrt {-3} {\left (b d x + b c\right )}\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} + 2 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) - \frac {1}{2} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b d x + b c + \sqrt {-3} {\left (b d x + b c\right )}\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} - 2 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) - \frac {1}{2} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b d x + b c - \sqrt {-3} {\left (b d x + b c\right )}\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} + 2 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) + \frac {1}{2} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b d x + b c - \sqrt {-3} {\left (b d x + b c\right )}\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} - 2 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) + \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b d x + b c\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} + {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) - \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b d x + b c\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} - {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) \] Input:

integrate(1/(b*x+a)^(5/6)/(d*x+c)^(1/6),x, algorithm="fricas")
 

Output:

1/2*(sqrt(-3) + 1)*(1/(b^5*d))^(1/6)*log(((b*d*x + b*c + sqrt(-3)*(b*d*x + 
 b*c))*(1/(b^5*d))^(1/6) + 2*(b*x + a)^(1/6)*(d*x + c)^(5/6))/(d*x + c)) - 
 1/2*(sqrt(-3) + 1)*(1/(b^5*d))^(1/6)*log(-((b*d*x + b*c + sqrt(-3)*(b*d*x 
 + b*c))*(1/(b^5*d))^(1/6) - 2*(b*x + a)^(1/6)*(d*x + c)^(5/6))/(d*x + c)) 
 - 1/2*(sqrt(-3) - 1)*(1/(b^5*d))^(1/6)*log(((b*d*x + b*c - sqrt(-3)*(b*d* 
x + b*c))*(1/(b^5*d))^(1/6) + 2*(b*x + a)^(1/6)*(d*x + c)^(5/6))/(d*x + c) 
) + 1/2*(sqrt(-3) - 1)*(1/(b^5*d))^(1/6)*log(-((b*d*x + b*c - sqrt(-3)*(b* 
d*x + b*c))*(1/(b^5*d))^(1/6) - 2*(b*x + a)^(1/6)*(d*x + c)^(5/6))/(d*x + 
c)) + (1/(b^5*d))^(1/6)*log(((b*d*x + b*c)*(1/(b^5*d))^(1/6) + (b*x + a)^( 
1/6)*(d*x + c)^(5/6))/(d*x + c)) - (1/(b^5*d))^(1/6)*log(-((b*d*x + b*c)*( 
1/(b^5*d))^(1/6) - (b*x + a)^(1/6)*(d*x + c)^(5/6))/(d*x + c))
 

Sympy [F]

\[ \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{6}} \sqrt [6]{c + d x}}\, dx \] Input:

integrate(1/(b*x+a)**(5/6)/(d*x+c)**(1/6),x)
 

Output:

Integral(1/((a + b*x)**(5/6)*(c + d*x)**(1/6)), x)
 

Maxima [F]

\[ \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}} \,d x } \] Input:

integrate(1/(b*x+a)^(5/6)/(d*x+c)^(1/6),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)^(5/6)*(d*x + c)^(1/6)), x)
 

Giac [F]

\[ \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}} \,d x } \] Input:

integrate(1/(b*x+a)^(5/6)/(d*x+c)^(1/6),x, algorithm="giac")
 

Output:

integrate(1/((b*x + a)^(5/6)*(d*x + c)^(1/6)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{1/6}} \,d x \] Input:

int(1/((a + b*x)^(5/6)*(c + d*x)^(1/6)),x)
 

Output:

int(1/((a + b*x)^(5/6)*(c + d*x)^(1/6)), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx=\int \frac {1}{\left (b x +a \right )^{\frac {5}{6}} \left (d x +c \right )^{\frac {1}{6}}}d x \] Input:

int(1/(b*x+a)^(5/6)/(d*x+c)^(1/6),x)
 

Output:

int(1/(b*x+a)^(5/6)/(d*x+c)^(1/6),x)