Integrand size = 21, antiderivative size = 545 \[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}-\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} \sqrt [5]{b c-a d} x}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}+\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sqrt [5]{b c-a d} x}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}-\frac {\log \left (\sqrt [5]{c}-\frac {\sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1-\sqrt {5}\right ) \log \left (\frac {2 (b c-a d)^{2/5} x^2+\sqrt [5]{c} \sqrt [5]{b c-a d} x \sqrt [5]{a+b x^5}-\sqrt {5} \sqrt [5]{c} \sqrt [5]{b c-a d} x \sqrt [5]{a+b x^5}+2 c^{2/5} \left (a+b x^5\right )^{2/5}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1+\sqrt {5}\right ) \log \left (\frac {2 (b c-a d)^{2/5} x^2+\sqrt [5]{c} \sqrt [5]{b c-a d} x \sqrt [5]{a+b x^5}+\sqrt {5} \sqrt [5]{c} \sqrt [5]{b c-a d} x \sqrt [5]{a+b x^5}+2 c^{2/5} \left (a+b x^5\right )^{2/5}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}} \] Output:
1/10*(10+2*5^(1/2))^(1/2)*arctan(-1/5*(25-10*5^(1/2))^(1/2)+2*2^(1/2)/(5+5 ^(1/2))^(1/2)*(-a*d+b*c)^(1/5)*x/c^(1/5)/(b*x^5+a)^(1/5))/c^(4/5)/(-a*d+b* c)^(1/5)+1/10*(10-2*5^(1/2))^(1/2)*arctan(1/5*(25+10*5^(1/2))^(1/2)+1/5*(5 0+10*5^(1/2))^(1/2)*(-a*d+b*c)^(1/5)*x/c^(1/5)/(b*x^5+a)^(1/5))/c^(4/5)/(- a*d+b*c)^(1/5)-1/5*ln(c^(1/5)-(-a*d+b*c)^(1/5)*x/(b*x^5+a)^(1/5))/c^(4/5)/ (-a*d+b*c)^(1/5)+1/20*(-5^(1/2)+1)*ln((2*(-a*d+b*c)^(2/5)*x^2+c^(1/5)*(-a* d+b*c)^(1/5)*x*(b*x^5+a)^(1/5)-5^(1/2)*c^(1/5)*(-a*d+b*c)^(1/5)*x*(b*x^5+a )^(1/5)+2*c^(2/5)*(b*x^5+a)^(2/5))/(b*x^5+a)^(2/5))/c^(4/5)/(-a*d+b*c)^(1/ 5)+1/20*(5^(1/2)+1)*ln((2*(-a*d+b*c)^(2/5)*x^2+c^(1/5)*(-a*d+b*c)^(1/5)*x* (b*x^5+a)^(1/5)+5^(1/2)*c^(1/5)*(-a*d+b*c)^(1/5)*x*(b*x^5+a)^(1/5)+2*c^(2/ 5)*(b*x^5+a)^(2/5))/(b*x^5+a)^(2/5))/c^(4/5)/(-a*d+b*c)^(1/5)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.09 \[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\frac {x \operatorname {Hypergeometric2F1}\left (\frac {1}{5},1,\frac {6}{5},\frac {(b c-a d) x^5}{c \left (a+b x^5\right )}\right )}{c \sqrt [5]{a+b x^5}} \] Input:
Integrate[1/((a + b*x^5)^(1/5)*(c + d*x^5)),x]
Output:
(x*Hypergeometric2F1[1/5, 1, 6/5, ((b*c - a*d)*x^5)/(c*(a + b*x^5))])/(c*( a + b*x^5)^(1/5))
Time = 1.56 (sec) , antiderivative size = 503, normalized size of antiderivative = 0.92, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {902, 752, 16, 27, 1142, 27, 1083, 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.
| \(\displaystyle \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \int \frac {1}{c-\frac {x^5 (b c-a d)}{a+b x^5}}d\frac {x}{\sqrt [5]{a+b x^5}}\) |
| \(\Big \downarrow \) 752 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt [5]{c}-\frac {\sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}}d\frac {x}{\sqrt [5]{b x^5+a}}}{5 c^{4/5}}+\frac {2 \int \frac {\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+4 \sqrt [5]{c}}{2 \left (\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}\right )}d\frac {x}{\sqrt [5]{b x^5+a}}}{5 c^{4/5}}+\frac {2 \int \frac {\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+4 \sqrt [5]{c}}{2 \left (\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}\right )}d\frac {x}{\sqrt [5]{b x^5+a}}}{5 c^{4/5}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 \int \frac {\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+4 \sqrt [5]{c}}{2 \left (\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}\right )}d\frac {x}{\sqrt [5]{b x^5+a}}}{5 c^{4/5}}+\frac {2 \int \frac {\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+4 \sqrt [5]{c}}{2 \left (\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}\right )}d\frac {x}{\sqrt [5]{b x^5+a}}}{5 c^{4/5}}-\frac {\log \left (\sqrt [5]{c}-\frac {x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+4 \sqrt [5]{c}}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}}{5 c^{4/5}}+\frac {\int \frac {\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+4 \sqrt [5]{c}}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}}{5 c^{4/5}}-\frac {\log \left (\sqrt [5]{c}-\frac {x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {1}{2} \left (5+\sqrt {5}\right ) \sqrt [5]{c} \int \frac {1}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}+\frac {\left (1-\sqrt {5}\right ) \int \frac {\sqrt [5]{b c-a d} \left (\frac {4 \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+\left (1-\sqrt {5}\right ) \sqrt [5]{c}\right )}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}}{4 \sqrt [5]{b c-a d}}}{5 c^{4/5}}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) \sqrt [5]{c} \int \frac {1}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}+\frac {\left (1+\sqrt {5}\right ) \int \frac {\sqrt [5]{b c-a d} \left (\frac {4 \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+\left (1+\sqrt {5}\right ) \sqrt [5]{c}\right )}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}}{4 \sqrt [5]{b c-a d}}}{5 c^{4/5}}-\frac {\log \left (\sqrt [5]{c}-\frac {x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \left (5+\sqrt {5}\right ) \sqrt [5]{c} \int \frac {1}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}+\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\frac {4 \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+\left (1-\sqrt {5}\right ) \sqrt [5]{c}}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}}{5 c^{4/5}}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) \sqrt [5]{c} \int \frac {1}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}+\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\frac {4 \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+\left (1+\sqrt {5}\right ) \sqrt [5]{c}}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}}{5 c^{4/5}}-\frac {\log \left (\sqrt [5]{c}-\frac {x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\frac {4 \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+\left (1-\sqrt {5}\right ) \sqrt [5]{c}}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}-\left (5+\sqrt {5}\right ) \sqrt [5]{c} \int \frac {1}{-\frac {x^2}{\left (b x^5+a\right )^{2/5}}-2 \left (5+\sqrt {5}\right ) c^{2/5} (b c-a d)^{2/5}}d\left (\frac {4 (b c-a d)^{2/5} x}{\sqrt [5]{b x^5+a}}+\left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}\right )}{5 c^{4/5}}+\frac {\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\frac {4 \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+\left (1+\sqrt {5}\right ) \sqrt [5]{c}}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}-\left (5-\sqrt {5}\right ) \sqrt [5]{c} \int \frac {1}{-\frac {x^2}{\left (b x^5+a\right )^{2/5}}-2 \left (5-\sqrt {5}\right ) c^{2/5} (b c-a d)^{2/5}}d\left (\frac {4 (b c-a d)^{2/5} x}{\sqrt [5]{b x^5+a}}+\left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}\right )}{5 c^{4/5}}-\frac {\log \left (\sqrt [5]{c}-\frac {x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\frac {4 \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+\left (1-\sqrt {5}\right ) \sqrt [5]{c}}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}+\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {\frac {4 x (b c-a d)^{2/5}}{\sqrt [5]{a+b x^5}}+\left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}}{\sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{c} \sqrt [5]{b c-a d}}\right )}{\sqrt [5]{b c-a d}}}{5 c^{4/5}}+\frac {\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\frac {4 \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+\left (1+\sqrt {5}\right ) \sqrt [5]{c}}{\frac {2 (b c-a d)^{2/5} x^2}{\left (b x^5+a\right )^{2/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{b x^5+a}}+2 c^{2/5}}d\frac {x}{\sqrt [5]{b x^5+a}}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {\frac {4 x (b c-a d)^{2/5}}{\sqrt [5]{a+b x^5}}+\left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}}{\sqrt {2 \left (5-\sqrt {5}\right )} \sqrt [5]{c} \sqrt [5]{b c-a d}}\right )}{\sqrt [5]{b c-a d}}}{5 c^{4/5}}-\frac {\log \left (\sqrt [5]{c}-\frac {x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {\frac {4 x (b c-a d)^{2/5}}{\sqrt [5]{a+b x^5}}+\left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}}{\sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{c} \sqrt [5]{b c-a d}}\right )}{\sqrt [5]{b c-a d}}+\frac {\left (1-\sqrt {5}\right ) \log \left (\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{c} x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}+\frac {2 x^2 (b c-a d)^{2/5}}{\left (a+b x^5\right )^{2/5}}+2 c^{2/5}\right )}{4 \sqrt [5]{b c-a d}}}{5 c^{4/5}}+\frac {\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {\frac {4 x (b c-a d)^{2/5}}{\sqrt [5]{a+b x^5}}+\left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}}{\sqrt {2 \left (5-\sqrt {5}\right )} \sqrt [5]{c} \sqrt [5]{b c-a d}}\right )}{\sqrt [5]{b c-a d}}+\frac {\left (1+\sqrt {5}\right ) \log \left (\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{c} x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}+\frac {2 x^2 (b c-a d)^{2/5}}{\left (a+b x^5\right )^{2/5}}+2 c^{2/5}\right )}{4 \sqrt [5]{b c-a d}}}{5 c^{4/5}}-\frac {\log \left (\sqrt [5]{c}-\frac {x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}\) |
Input:
Int[1/((a + b*x^5)^(1/5)*(c + d*x^5)),x]
Output:
-1/5*Log[c^(1/5) - ((b*c - a*d)^(1/5)*x)/(a + b*x^5)^(1/5)]/(c^(4/5)*(b*c - a*d)^(1/5)) + ((Sqrt[(5 + Sqrt[5])/2]*ArcTan[((1 - Sqrt[5])*c^(1/5)*(b*c - a*d)^(1/5) + (4*(b*c - a*d)^(2/5)*x)/(a + b*x^5)^(1/5))/(Sqrt[2*(5 + Sq rt[5])]*c^(1/5)*(b*c - a*d)^(1/5))])/(b*c - a*d)^(1/5) + ((1 - Sqrt[5])*Lo g[2*c^(2/5) + (2*(b*c - a*d)^(2/5)*x^2)/(a + b*x^5)^(2/5) + ((1 - Sqrt[5]) *c^(1/5)*(b*c - a*d)^(1/5)*x)/(a + b*x^5)^(1/5)])/(4*(b*c - a*d)^(1/5)))/( 5*c^(4/5)) + ((Sqrt[(5 - Sqrt[5])/2]*ArcTan[((1 + Sqrt[5])*c^(1/5)*(b*c - a*d)^(1/5) + (4*(b*c - a*d)^(2/5)*x)/(a + b*x^5)^(1/5))/(Sqrt[2*(5 - Sqrt[ 5])]*c^(1/5)*(b*c - a*d)^(1/5))])/(b*c - a*d)^(1/5) + ((1 + Sqrt[5])*Log[2 *c^(2/5) + (2*(b*c - a*d)^(2/5)*x^2)/(a + b*x^5)^(2/5) + ((1 + Sqrt[5])*c^ (1/5)*(b*c - a*d)^(1/5)*x)/(a + b*x^5)^(1/5)])/(4*(b*c - a*d)^(1/5)))/(5*c ^(4/5))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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 - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; r/(a*n ) Int[1/(r - s*x), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 1)/2}], x]] /; F reeQ[{a, b}, x] && IGtQ[(n - 3)/2, 0] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
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]
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]
Time = 13.39 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.75
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {5}\, \left (-\frac {\sqrt {5+\sqrt {5}}\, \sqrt {5-\sqrt {5}}\, \left (\sqrt {5}+1\right ) \ln \left (\frac {2 \left (\frac {a d -b c}{c}\right )^{\frac {2}{5}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} \left (b \,x^{5}+a \right )^{\frac {1}{5}} \left (\sqrt {5}+1\right ) x +2 \left (b \,x^{5}+a \right )^{\frac {2}{5}}}{x^{2}}\right )}{4}+\frac {\sqrt {5+\sqrt {5}}\, \sqrt {5-\sqrt {5}}\, \left (\sqrt {5}-1\right ) \ln \left (\frac {2 \left (\frac {a d -b c}{c}\right )^{\frac {2}{5}} x^{2}+\left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} \left (b \,x^{5}+a \right )^{\frac {1}{5}} \left (\sqrt {5}-1\right ) x +2 \left (b \,x^{5}+a \right )^{\frac {2}{5}}}{x^{2}}\right )}{4}-\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \left (\sqrt {5}-5\right ) \arctan \left (\frac {\sqrt {2}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} \left (\sqrt {5}+1\right ) x -4 \left (b \,x^{5}+a \right )^{\frac {1}{5}}\right )}{2 \sqrt {5-\sqrt {5}}\, \left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} x}\right )}{2}+\sqrt {5-\sqrt {5}}\, \left (-\frac {\sqrt {2}\, \left (5+\sqrt {5}\right ) \arctan \left (\frac {\sqrt {2}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} \left (\sqrt {5}-1\right ) x +4 \left (b \,x^{5}+a \right )^{\frac {1}{5}}\right )}{2 \sqrt {5+\sqrt {5}}\, \left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} x}\right )}{2}+\sqrt {5+\sqrt {5}}\, \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} x +\left (b \,x^{5}+a \right )^{\frac {1}{5}}}{x}\right )\right )\right )}{50 \left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} c}\) | \(408\) |
Input:
int(1/(b*x^5+a)^(1/5)/(d*x^5+c),x,method=_RETURNVERBOSE)
Output:
1/50*5^(1/2)/((a*d-b*c)/c)^(1/5)*(-1/4*(5+5^(1/2))^(1/2)*(5-5^(1/2))^(1/2) *(5^(1/2)+1)*ln((2*((a*d-b*c)/c)^(2/5)*x^2-((a*d-b*c)/c)^(1/5)*(b*x^5+a)^( 1/5)*(5^(1/2)+1)*x+2*(b*x^5+a)^(2/5))/x^2)+1/4*(5+5^(1/2))^(1/2)*(5-5^(1/2 ))^(1/2)*(5^(1/2)-1)*ln((2*((a*d-b*c)/c)^(2/5)*x^2+((a*d-b*c)/c)^(1/5)*(b* x^5+a)^(1/5)*(5^(1/2)-1)*x+2*(b*x^5+a)^(2/5))/x^2)-1/2*2^(1/2)*(5+5^(1/2)) ^(1/2)*(5^(1/2)-5)*arctan(1/2*2^(1/2)*(((a*d-b*c)/c)^(1/5)*(5^(1/2)+1)*x-4 *(b*x^5+a)^(1/5))/(5-5^(1/2))^(1/2)/((a*d-b*c)/c)^(1/5)/x)+(5-5^(1/2))^(1/ 2)*(-1/2*2^(1/2)*(5+5^(1/2))*arctan(1/2*2^(1/2)*(((a*d-b*c)/c)^(1/5)*(5^(1 /2)-1)*x+4*(b*x^5+a)^(1/5))/(5+5^(1/2))^(1/2)/((a*d-b*c)/c)^(1/5)/x)+(5+5^ (1/2))^(1/2)*ln((((a*d-b*c)/c)^(1/5)*x+(b*x^5+a)^(1/5))/x)))/c
Exception generated. \[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(b*x^5+a)^(1/5)/(d*x^5+c),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (trace 0)
\[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\int \frac {1}{\sqrt [5]{a + b x^{5}} \left (c + d x^{5}\right )}\, dx \] Input:
integrate(1/(b*x**5+a)**(1/5)/(d*x**5+c),x)
Output:
Integral(1/((a + b*x**5)**(1/5)*(c + d*x**5)), x)
\[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\int { \frac {1}{{\left (b x^{5} + a\right )}^{\frac {1}{5}} {\left (d x^{5} + c\right )}} \,d x } \] Input:
integrate(1/(b*x^5+a)^(1/5)/(d*x^5+c),x, algorithm="maxima")
Output:
integrate(1/((b*x^5 + a)^(1/5)*(d*x^5 + c)), x)
\[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\int { \frac {1}{{\left (b x^{5} + a\right )}^{\frac {1}{5}} {\left (d x^{5} + c\right )}} \,d x } \] Input:
integrate(1/(b*x^5+a)^(1/5)/(d*x^5+c),x, algorithm="giac")
Output:
integrate(1/((b*x^5 + a)^(1/5)*(d*x^5 + c)), x)
Timed out. \[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\int \frac {1}{{\left (b\,x^5+a\right )}^{1/5}\,\left (d\,x^5+c\right )} \,d x \] Input:
int(1/((a + b*x^5)^(1/5)*(c + d*x^5)),x)
Output:
int(1/((a + b*x^5)^(1/5)*(c + d*x^5)), x)
\[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\frac {\left (b \,x^{5}+a \right )^{\frac {1}{5}} \left (\int \frac {1}{\left (b \,x^{5}+a \right )^{\frac {1}{5}} a c +\left (b \,x^{5}+a \right )^{\frac {1}{5}} a d \,x^{5}+\left (b \,x^{5}+a \right )^{\frac {1}{5}} b c \,x^{5}+\left (b \,x^{5}+a \right )^{\frac {1}{5}} b d \,x^{10}}d x \right ) a^{2} d -\left (b \,x^{5}+a \right )^{\frac {1}{5}} \left (\int \frac {1}{\left (b \,x^{5}+a \right )^{\frac {1}{5}} a c +\left (b \,x^{5}+a \right )^{\frac {1}{5}} a d \,x^{5}+\left (b \,x^{5}+a \right )^{\frac {1}{5}} b c \,x^{5}+\left (b \,x^{5}+a \right )^{\frac {1}{5}} b d \,x^{10}}d x \right ) a b c +b x}{\left (b \,x^{5}+a \right )^{\frac {1}{5}} a d} \] Input:
int(1/(b*x^5+a)^(1/5)/(d*x^5+c),x)
Output:
((a + b*x**5)**(1/5)*int(1/((a + b*x**5)**(1/5)*a*c + (a + b*x**5)**(1/5)* a*d*x**5 + (a + b*x**5)**(1/5)*b*c*x**5 + (a + b*x**5)**(1/5)*b*d*x**10),x )*a**2*d - (a + b*x**5)**(1/5)*int(1/((a + b*x**5)**(1/5)*a*c + (a + b*x** 5)**(1/5)*a*d*x**5 + (a + b*x**5)**(1/5)*b*c*x**5 + (a + b*x**5)**(1/5)*b* d*x**10),x)*a*b*c + b*x)/((a + b*x**5)**(1/5)*a*d)