3.18.72 \(\int \sqrt [6]{a+b x} (c+d x)^{5/6} \, dx\) [1772]

3.18.72.1 Optimal result

Integrand size = 19, antiderivative size = 427 \[ \int \sqrt [6]{a+b x} (c+d x)^{5/6} \, dx=\frac {5 (b c-a d) \sqrt [6]{a+b x} (c+d x)^{5/6}}{12 b d}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}+\frac {5 (b c-a d)^2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{24 \sqrt {3} b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{24 \sqrt {3} b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{11/6} d^{7/6}}+\frac {5 (b c-a d)^2 \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{144 b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{144 b^{11/6} d^{7/6}} \]

output

5/12*(-a*d+b*c)*(b*x+a)^(1/6)*(d*x+c)^(5/6)/b/d+1/2*(b*x+a)^(7/6)*(d*x+c)^ 
(5/6)/b-5/36*(-a*d+b*c)^2*arctanh(d^(1/6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^(1 
/6))/b^(11/6)/d^(7/6)+5/144*(-a*d+b*c)^2*ln(b^(1/3)+d^(1/3)*(b*x+a)^(1/3)/ 
(d*x+c)^(1/3)-b^(1/6)*d^(1/6)*(b*x+a)^(1/6)/(d*x+c)^(1/6))/b^(11/6)/d^(7/6 
)-5/144*(-a*d+b*c)^2*ln(b^(1/3)+d^(1/3)*(b*x+a)^(1/3)/(d*x+c)^(1/3)+b^(1/6 
)*d^(1/6)*(b*x+a)^(1/6)/(d*x+c)^(1/6))/b^(11/6)/d^(7/6)-5/72*(-a*d+b*c)^2* 
arctan(-1/3*3^(1/2)+2/3*d^(1/6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6)*3^(1/2 
))/b^(11/6)/d^(7/6)*3^(1/2)-5/72*(-a*d+b*c)^2*arctan(1/3*3^(1/2)+2/3*d^(1/ 
6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6)*3^(1/2))/b^(11/6)/d^(7/6)*3^(1/2)
 

3.18.72.2 Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.74 \[ \int \sqrt [6]{a+b x} (c+d x)^{5/6} \, dx=\frac {6 b^{5/6} \sqrt [6]{d} \sqrt [6]{a+b x} (c+d x)^{5/6} (5 b c+a d+6 b d x)-5 \sqrt {3} (b c-a d)^2 \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+5 \sqrt {3} (b c-a d)^2 \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )-10 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )-5 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{72 b^{11/6} d^{7/6}} \]

input

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

output

(6*b^(5/6)*d^(1/6)*(a + b*x)^(1/6)*(c + d*x)^(5/6)*(5*b*c + a*d + 6*b*d*x) 
 - 5*Sqrt[3]*(b*c - a*d)^2*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))] + 5*Sqrt[3]*(b*c - a*d)^ 
2*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))] - 10*(b*c - a*d)^2*ArcTanh[(b^(1/6)*(c + d*x)^(1/6 
))/(d^(1/6)*(a + b*x)^(1/6))] - 5*(b*c - a*d)^2*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))])/(72*b^(11/6)*d^(7/6))
 

3.18.72.3 Rubi [A] (warning: unable to verify)

Time = 0.49 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {60, 60, 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.

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

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 (b c-a d) \int \frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}dx}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 (b c-a d) \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}}dx}{6 d}\right )}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {5 (b c-a d) \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}d\sqrt [6]{a+b x}}{b d}\right )}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {5 (b c-a d) \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \int \frac {1}{1-\frac {d (a+b x)}{b}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b d}\right )}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

\(\Big \downarrow \) 754

\(\displaystyle \frac {5 (b c-a d) \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{3} \sqrt [3]{b} \int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{3} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \left (\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}\right )}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{3} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \left (\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}\right )}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{b d}\right )}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 (b c-a d) \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{3} \sqrt [3]{b} \int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{6} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{6} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{b d}\right )}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {5 (b c-a d) \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{6} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {5 (b c-a d) \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\int -\frac {\sqrt [6]{d} \left (\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {5 (b c-a d) \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 (b c-a d) \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {5 (b c-a d) \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {3 \int \frac {1}{-\sqrt [3]{a+b x}-3}d\left (1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [6]{d}}+\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {3 \int \frac {1}{-\sqrt [3]{a+b x}-3}d\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+1\right )}{\sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {5 (b c-a d) \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {5 (b c-a d) \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}-\frac {\log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}+\frac {\log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{12 b}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}\)

input

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

output

((a + b*x)^(7/6)*(c + d*x)^(5/6))/(2*b) + (5*(b*c - a*d)*(((a + b*x)^(1/6) 
*(c + d*x)^(5/6))/d - ((b*c - a*d)*((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)))/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^(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 
*d)))/(12*b)
 

3.18.72.3.1 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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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]
 

3.18.72.4 Maple [F]

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

input

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

output

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

3.18.72.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2599 vs. \(2 (321) = 642\).

Time = 0.28 (sec) , antiderivative size = 2599, normalized size of antiderivative = 6.09 \[ \int \sqrt [6]{a+b x} (c+d x)^{5/6} \, dx=\text {Too large to display} \]

input

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

output

-1/144*(10*b*d*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220 
*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6 
*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 
 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^11*d^7))^(1/6)* 
log(5*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(1/6)*(d*x + c)^(5/6) + ( 
b^2*d^2*x + b^2*c*d)*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 
 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a 
^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c 
^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^11*d^7))^ 
(1/6))/(d*x + c)) - 10*b*d*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^ 
10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 
 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9 
*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^11* 
d^7))^(1/6)*log(5*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(1/6)*(d*x + 
c)^(5/6) - (b^2*d^2*x + b^2*c*d)*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b 
^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7 
*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 2 
20*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/ 
(b^11*d^7))^(1/6))/(d*x + c)) + 5*(sqrt(-3)*b*d + b*d)*((b^12*c^12 - 12*a* 
b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*...
 

3.18.72.6 Sympy [F]

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

input

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

output

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

3.18.72.7 Maxima [F]

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

input

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

output

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

3.18.72.8 Giac [F]

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

input

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

output

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

3.18.72.9 Mupad [F(-1)]

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

input

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

output

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

3.18.72.10 Reduce [F]

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

input

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

output

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