\(\int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx\) [710]

Optimal result

Integrand size = 19, antiderivative size = 284 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx=-\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}-\frac {\sqrt {3} b^{7/6} \arctan \left (\frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{d^{13/6}}+\frac {\sqrt {3} b^{7/6} \arctan \left (\frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{d^{13/6}}+\frac {2 b^{7/6} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}+\frac {b^{7/6} \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x} \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}\right )}{d^{13/6}} \] Output:

-6/7*(b*x+a)^(7/6)/d/(d*x+c)^(7/6)-6*b*(b*x+a)^(1/6)/d^2/(d*x+c)^(1/6)-3^( 
1/2)*b^(7/6)*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))/d^(13/6)+3^(1/2)*b^(7/6)*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))/d^(13/6)+2*b^(7/6)*arctanh(d^(1/ 
6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6))/d^(13/6)+b^(7/6)*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)))/d^(13/6)
 

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx=\frac {-\frac {6 \sqrt [6]{d} \sqrt [6]{a+b x} (7 b c+a d+8 b d x)}{(c+d x)^{7/6}}+7 \sqrt {3} b^{7/6} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )-7 \sqrt {3} b^{7/6} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+14 b^{7/6} \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+7 b^{7/6} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{7 d^{13/6}} \] Input:

Integrate[(a + b*x)^(7/6)/(c + d*x)^(13/6),x]
 

Output:

((-6*d^(1/6)*(a + b*x)^(1/6)*(7*b*c + a*d + 8*b*d*x))/(c + d*x)^(7/6) + 7* 
Sqrt[3]*b^(7/6)*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))] - 7*Sqrt[3]*b^(7/6)*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))] + 14*b^(7/6)*ArcTanh[(b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^( 
1/6))] + 7*b^(7/6)*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))])/(7*d^(13/6))
 

Rubi [A] (warning: unable to verify)

Time = 0.45 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.47, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {57, 57, 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.

\(\displaystyle \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx\)

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 770

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

\(\Big \downarrow \) 754

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

Int[(a + b*x)^(7/6)/(c + d*x)^(13/6),x]
 

Output:

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

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 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[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]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {7}{6}}}{\left (x d +c \right )^{\frac {13}{6}}}d x\]

Input:

int((b*x+a)^(7/6)/(d*x+c)^(13/6),x)
 

Output:

int((b*x+a)^(7/6)/(d*x+c)^(13/6),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 711 vs. \(2 (208) = 416\).

Time = 0.13 (sec) , antiderivative size = 711, normalized size of antiderivative = 2.50 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(7/6)/(d*x+c)^(13/6),x, algorithm="fricas")
 

Output:

1/14*(7*(d^4*x^2 + 2*c*d^3*x + c^2*d^2 + sqrt(-3)*(d^4*x^2 + 2*c*d^3*x + c 
^2*d^2))*(b^7/d^13)^(1/6)*log((2*(b*x + a)^(1/6)*(d*x + c)^(5/6)*b + (d^3* 
x + c*d^2 + sqrt(-3)*(d^3*x + c*d^2))*(b^7/d^13)^(1/6))/(d*x + c)) - 7*(d^ 
4*x^2 + 2*c*d^3*x + c^2*d^2 + sqrt(-3)*(d^4*x^2 + 2*c*d^3*x + c^2*d^2))*(b 
^7/d^13)^(1/6)*log((2*(b*x + a)^(1/6)*(d*x + c)^(5/6)*b - (d^3*x + c*d^2 + 
 sqrt(-3)*(d^3*x + c*d^2))*(b^7/d^13)^(1/6))/(d*x + c)) + 7*(d^4*x^2 + 2*c 
*d^3*x + c^2*d^2 - sqrt(-3)*(d^4*x^2 + 2*c*d^3*x + c^2*d^2))*(b^7/d^13)^(1 
/6)*log((2*(b*x + a)^(1/6)*(d*x + c)^(5/6)*b + (d^3*x + c*d^2 - sqrt(-3)*( 
d^3*x + c*d^2))*(b^7/d^13)^(1/6))/(d*x + c)) - 7*(d^4*x^2 + 2*c*d^3*x + c^ 
2*d^2 - sqrt(-3)*(d^4*x^2 + 2*c*d^3*x + c^2*d^2))*(b^7/d^13)^(1/6)*log((2* 
(b*x + a)^(1/6)*(d*x + c)^(5/6)*b - (d^3*x + c*d^2 - sqrt(-3)*(d^3*x + c*d 
^2))*(b^7/d^13)^(1/6))/(d*x + c)) + 14*(d^4*x^2 + 2*c*d^3*x + c^2*d^2)*(b^ 
7/d^13)^(1/6)*log(((b*x + a)^(1/6)*(d*x + c)^(5/6)*b + (d^3*x + c*d^2)*(b^ 
7/d^13)^(1/6))/(d*x + c)) - 14*(d^4*x^2 + 2*c*d^3*x + c^2*d^2)*(b^7/d^13)^ 
(1/6)*log(((b*x + a)^(1/6)*(d*x + c)^(5/6)*b - (d^3*x + c*d^2)*(b^7/d^13)^ 
(1/6))/(d*x + c)) - 12*(8*b*d*x + 7*b*c + a*d)*(b*x + a)^(1/6)*(d*x + c)^( 
5/6))/(d^4*x^2 + 2*c*d^3*x + c^2*d^2)
 

Sympy [F]

\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx=\int \frac {\left (a + b x\right )^{\frac {7}{6}}}{\left (c + d x\right )^{\frac {13}{6}}}\, dx \] Input:

integrate((b*x+a)**(7/6)/(d*x+c)**(13/6),x)
 

Output:

Integral((a + b*x)**(7/6)/(c + d*x)**(13/6), x)
 

Maxima [F]

\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {13}{6}}} \,d x } \] Input:

integrate((b*x+a)^(7/6)/(d*x+c)^(13/6),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(7/6)/(d*x + c)^(13/6), x)
 

Giac [F]

\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {13}{6}}} \,d x } \] Input:

integrate((b*x+a)^(7/6)/(d*x+c)^(13/6),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(7/6)/(d*x + c)^(13/6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx=\int \frac {{\left (a+b\,x\right )}^{7/6}}{{\left (c+d\,x\right )}^{13/6}} \,d x \] Input:

int((a + b*x)^(7/6)/(c + d*x)^(13/6),x)
 

Output:

int((a + b*x)^(7/6)/(c + d*x)^(13/6), x)
 

Reduce [B] (verification not implemented)

Time = 13.20 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.56 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx=\frac {-\frac {4 \left (d x +c \right ) \left (b x +a \right )^{\frac {3}{2}} a b d}{3}+\frac {4 \left (d x +c \right ) \left (b x +a \right )^{\frac {3}{2}} b^{2} c}{3}-\frac {6 \sqrt {b x +a}\, a^{3} d^{2}}{7}+\frac {48 \sqrt {b x +a}\, a^{2} b c d}{7}+\frac {30 \sqrt {b x +a}\, a^{2} b \,d^{2} x}{7}-6 \sqrt {b x +a}\, a \,b^{2} c^{2}+\frac {12 \sqrt {b x +a}\, a \,b^{2} c d x}{7}+\frac {36 \sqrt {b x +a}\, a \,b^{2} d^{2} x^{2}}{7}-6 \sqrt {b x +a}\, b^{3} c^{2} x -\frac {36 \sqrt {b x +a}\, b^{3} c d \,x^{2}}{7}+10 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) a \,b^{2} c +10 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) a \,b^{2} d x +10 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{3} c x +10 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b^{3} d \,x^{2}-10 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) a \,b^{2} c -10 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) a \,b^{2} d x -10 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{3} c x -10 \left (d x +c \right ) \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b^{3} d \,x^{2}}{\left (d x +c \right )^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{3}} d^{2} \left (a b \,d^{2} x^{2}-b^{2} c d \,x^{2}+a^{2} d^{2} x -b^{2} c^{2} x +a^{2} c d -a b \,c^{2}\right )} \] Input:

int((b*x+a)^(7/6)/(d*x+c)^(13/6),x)
 

Output:

(2*( - 14*(c + d*x)*(a + b*x)**(3/2)*a*b*d + 14*(c + d*x)*(a + b*x)**(3/2) 
*b**2*c - 9*sqrt(a + b*x)*a**3*d**2 + 72*sqrt(a + b*x)*a**2*b*c*d + 45*sqr 
t(a + b*x)*a**2*b*d**2*x - 63*sqrt(a + b*x)*a*b**2*c**2 + 18*sqrt(a + b*x) 
*a*b**2*c*d*x + 54*sqrt(a + b*x)*a*b**2*d**2*x**2 - 63*sqrt(a + b*x)*b**3* 
c**2*x - 54*sqrt(a + b*x)*b**3*c*d*x**2 + 105*(c + d*x)*sqrt(a + b*x)*log( 
(a + b*x)**(1/6))*a*b**2*c + 105*(c + d*x)*sqrt(a + b*x)*log((a + b*x)**(1 
/6))*a*b**2*d*x + 105*(c + d*x)*sqrt(a + b*x)*log((a + b*x)**(1/6))*b**3*c 
*x + 105*(c + d*x)*sqrt(a + b*x)*log((a + b*x)**(1/6))*b**3*d*x**2 - 105*( 
c + d*x)*sqrt(a + b*x)*log((c + d*x)**(1/6))*a*b**2*c - 105*(c + d*x)*sqrt 
(a + b*x)*log((c + d*x)**(1/6))*a*b**2*d*x - 105*(c + d*x)*sqrt(a + b*x)*l 
og((c + d*x)**(1/6))*b**3*c*x - 105*(c + d*x)*sqrt(a + b*x)*log((c + d*x)* 
*(1/6))*b**3*d*x**2))/(21*(c + d*x)**(1/6)*(a + b*x)**(1/3)*d**2*(a**2*c*d 
 + a**2*d**2*x - a*b*c**2 + a*b*d**2*x**2 - b**2*c**2*x - b**2*c*d*x**2))