\(\int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx\) [742]

Optimal result

Integrand size = 19, antiderivative size = 324 \[ \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx=\frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {7 \sqrt [6]{d} (b c-a d) \arctan \left (\frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{2 \sqrt {3} b^{13/6}}-\frac {7 \sqrt [6]{d} (b c-a d) \arctan \left (\frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{2 \sqrt {3} b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x} \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}\right )}{6 b^{13/6}} \] Output:

7*d*(b*x+a)^(5/6)*(d*x+c)^(1/6)/b^2-6*(d*x+c)^(7/6)/b/(b*x+a)^(1/6)+7/6*d^ 
(1/6)*(-a*d+b*c)*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))*3^(1/2)/b^(13/6)-7/6*d^(1/6)*(-a*d+b*c)*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))*3^(1/2)/b^(13 
/6)+7/3*d^(1/6)*(-a*d+b*c)*arctanh(d^(1/6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^( 
1/6))/b^(13/6)+7/6*d^(1/6)*(-a*d+b*c)*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^(13/6)
 

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.98 \[ \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx=\frac {\frac {6 \sqrt [6]{b} \sqrt [6]{c+d x} (-6 b c+7 a d+b d x)}{\sqrt [6]{a+b x}}+7 \sqrt {3} \sqrt [6]{d} (-b c+a d) \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+7 \sqrt {3} \sqrt [6]{d} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+14 \sqrt [6]{d} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+7 \sqrt [6]{d} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{6 b^{13/6}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.47 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.33, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {57, 60, 73, 854, 27, 825, 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.

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 854

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 825

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(-6*(c + d*x)^(7/6))/(b*(a + b*x)^(1/6)) + (7*d*(((a + b*x)^(5/6)*(c + d*x 
)^(1/6))/b + ((b*c - a*d)*(ArcTanh[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c - 
 (a*d)/b + (d*(a + b*x))/b)^(1/6))]/(3*b^(1/6)*d^(5/6)) - (-((Sqrt[3]*ArcT 
an[(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)*d^(2/3)) - ((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)*d^(2/3)) 
))/b))/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 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 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 825

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

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 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)] && IntegersQ[m, p + (m + 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]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1562 vs. \(2 (242) = 484\).

Time = 0.14 (sec) , antiderivative size = 1562, normalized size of antiderivative = 4.82 \[ \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/12*(7*(b^3*x + a*b^2 + sqrt(-3)*(b^3*x + a*b^2))*((b^6*c^6*d - 6*a*b^5*c 
^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6* 
a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6)*log(-7/2*(2*(b*c - a*d)*(b*x + a)^(5/6) 
*(d*x + c)^(1/6) + (b^3*x + a*b^2 + sqrt(-3)*(b^3*x + a*b^2))*((b^6*c^6*d 
- 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c 
^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6))/(b*x + a)) - 7*(b^3*x + a*b 
^2 + sqrt(-3)*(b^3*x + a*b^2))*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4* 
c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^ 
7)/b^13)^(1/6)*log(-7/2*(2*(b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - ( 
b^3*x + a*b^2 + sqrt(-3)*(b^3*x + a*b^2))*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 
15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d 
^6 + a^6*d^7)/b^13)^(1/6))/(b*x + a)) + 7*(b^3*x + a*b^2 - sqrt(-3)*(b^3*x 
 + a*b^2))*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3 
*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6)*log(- 
7/2*(2*(b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) + (b^3*x + a*b^2 - sqrt 
(-3)*(b^3*x + a*b^2))*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 
 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^ 
(1/6))/(b*x + a)) - 7*(b^3*x + a*b^2 - sqrt(-3)*(b^3*x + a*b^2))*((b^6*c^6 
*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^ 
2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6)*log(-7/2*(2*(b*c - a*d...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*d*(c + d*x)**(1/6)*(a + b*x)**(5/6)*a*c**2*d - 6*d*(c + d*x)**(1/6)* 
(a + b*x)**(5/6)*a*c*d**2*x + 6*d*(c + d*x)**(1/6)*(a + b*x)**(5/6)*b*c**3 
 + 6*d*(c + d*x)**(1/6)*(a + b*x)**(5/6)*b*c**2*d*x - 6*(c + d*x)**(7/6)*( 
a + b*x)**(5/6)*log((a + b*x)**(1/6))*a*c*d**2 - 6*(c + d*x)**(7/6)*(a + b 
*x)**(5/6)*log((a + b*x)**(1/6))*b*c*d**2*x + 6*(c + d*x)**(7/6)*(a + b*x) 
**(5/6)*log((c + d*x)**(1/6))*a*c*d**2 + 6*(c + d*x)**(7/6)*(a + b*x)**(5/ 
6)*log((c + d*x)**(1/6))*b*c*d**2*x + int(((c + d*x)**(1/6)*x)/((a + b*x)* 
*(1/6)*a + (a + b*x)**(1/6)*b*x),x)*a**2*b*c*d**3 + int(((c + d*x)**(1/6)* 
x)/((a + b*x)**(1/6)*a + (a + b*x)**(1/6)*b*x),x)*a**2*b*d**4*x - int(((c 
+ d*x)**(1/6)*x)/((a + b*x)**(1/6)*a + (a + b*x)**(1/6)*b*x),x)*a*b**2*c** 
2*d**2 + int(((c + d*x)**(1/6)*x)/((a + b*x)**(1/6)*a + (a + b*x)**(1/6)*b 
*x),x)*a*b**2*d**4*x**2 - int(((c + d*x)**(1/6)*x)/((a + b*x)**(1/6)*a + ( 
a + b*x)**(1/6)*b*x),x)*b**3*c**2*d**2*x - int(((c + d*x)**(1/6)*x)/((a + 
b*x)**(1/6)*a + (a + b*x)**(1/6)*b*x),x)*b**3*c*d**3*x**2)/(b*d*(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))