Integrand size = 37, antiderivative size = 49 \[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx=-\frac {\arctan \left (\frac {x}{\sqrt [4]{a} \sqrt {1+x^5}}\right )}{a^{3/4}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [4]{a} \sqrt {1+x^5}}\right )}{a^{3/4}} \]
Time = 5.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx=-\frac {\arctan \left (\frac {x}{\sqrt [4]{a} \sqrt {1+x^5}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{a} \sqrt {1+x^5}}\right )}{a^{3/4}} \]
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.
\(\displaystyle \int \frac {\sqrt {x^5+1} \left (3 x^5-2\right )}{a x^{10}+2 a x^5+a-x^4} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 x^5 \sqrt {x^5+1}}{a x^{10}+2 a x^5+a-x^4}-\frac {2 \sqrt {x^5+1}}{a x^{10}+2 a x^5+a-x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \int \frac {x^5 \sqrt {x^5+1}}{a x^{10}+2 a x^5-x^4+a}dx-2 \int \frac {\sqrt {x^5+1}}{a x^{10}+2 a x^5-x^4+a}dx\) |
3.7.33.3.1 Defintions of rubi rules used
Time = 2.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.47
method | result | size |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\sqrt {x^{5}+1}}{x \left (\frac {1}{a}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {-\left (\frac {1}{a}\right )^{\frac {1}{4}} x -\sqrt {x^{5}+1}}{\left (\frac {1}{a}\right )^{\frac {1}{4}} x -\sqrt {x^{5}+1}}\right )}{2 \left (\frac {1}{a}\right )^{\frac {1}{4}} a}\) | \(72\) |
1/2/(1/a)^(1/4)*(2*arctan((x^5+1)^(1/2)/x/(1/a)^(1/4))-ln((-(1/a)^(1/4)*x- (x^5+1)^(1/2))/((1/a)^(1/4)*x-(x^5+1)^(1/2))))/a
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 431, normalized size of antiderivative = 8.80 \[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx=-\frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {a x^{10} + 2 \, a x^{5} + x^{4} + 2 \, \sqrt {x^{5} + 1} {\left (a \frac {1}{a^{3}}^{\frac {1}{4}} x^{3} + {\left (a^{3} x^{6} + a^{3} x\right )} \frac {1}{a^{3}}^{\frac {3}{4}}\right )} + 2 \, {\left (a^{2} x^{7} + a^{2} x^{2}\right )} \sqrt {\frac {1}{a^{3}}} + a}{a x^{10} + 2 \, a x^{5} - x^{4} + a}\right ) + \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {a x^{10} + 2 \, a x^{5} + x^{4} - 2 \, \sqrt {x^{5} + 1} {\left (a \frac {1}{a^{3}}^{\frac {1}{4}} x^{3} + {\left (a^{3} x^{6} + a^{3} x\right )} \frac {1}{a^{3}}^{\frac {3}{4}}\right )} + 2 \, {\left (a^{2} x^{7} + a^{2} x^{2}\right )} \sqrt {\frac {1}{a^{3}}} + a}{a x^{10} + 2 \, a x^{5} - x^{4} + a}\right ) + \frac {1}{4} i \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {a x^{10} + 2 \, a x^{5} + x^{4} - 2 \, \sqrt {x^{5} + 1} {\left (i \, a \frac {1}{a^{3}}^{\frac {1}{4}} x^{3} + {\left (-i \, a^{3} x^{6} - i \, a^{3} x\right )} \frac {1}{a^{3}}^{\frac {3}{4}}\right )} - 2 \, {\left (a^{2} x^{7} + a^{2} x^{2}\right )} \sqrt {\frac {1}{a^{3}}} + a}{a x^{10} + 2 \, a x^{5} - x^{4} + a}\right ) - \frac {1}{4} i \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {a x^{10} + 2 \, a x^{5} + x^{4} - 2 \, \sqrt {x^{5} + 1} {\left (-i \, a \frac {1}{a^{3}}^{\frac {1}{4}} x^{3} + {\left (i \, a^{3} x^{6} + i \, a^{3} x\right )} \frac {1}{a^{3}}^{\frac {3}{4}}\right )} - 2 \, {\left (a^{2} x^{7} + a^{2} x^{2}\right )} \sqrt {\frac {1}{a^{3}}} + a}{a x^{10} + 2 \, a x^{5} - x^{4} + a}\right ) \]
-1/4*(a^(-3))^(1/4)*log((a*x^10 + 2*a*x^5 + x^4 + 2*sqrt(x^5 + 1)*(a*(a^(- 3))^(1/4)*x^3 + (a^3*x^6 + a^3*x)*(a^(-3))^(3/4)) + 2*(a^2*x^7 + a^2*x^2)* sqrt(a^(-3)) + a)/(a*x^10 + 2*a*x^5 - x^4 + a)) + 1/4*(a^(-3))^(1/4)*log(( a*x^10 + 2*a*x^5 + x^4 - 2*sqrt(x^5 + 1)*(a*(a^(-3))^(1/4)*x^3 + (a^3*x^6 + a^3*x)*(a^(-3))^(3/4)) + 2*(a^2*x^7 + a^2*x^2)*sqrt(a^(-3)) + a)/(a*x^10 + 2*a*x^5 - x^4 + a)) + 1/4*I*(a^(-3))^(1/4)*log((a*x^10 + 2*a*x^5 + x^4 - 2*sqrt(x^5 + 1)*(I*a*(a^(-3))^(1/4)*x^3 + (-I*a^3*x^6 - I*a^3*x)*(a^(-3) )^(3/4)) - 2*(a^2*x^7 + a^2*x^2)*sqrt(a^(-3)) + a)/(a*x^10 + 2*a*x^5 - x^4 + a)) - 1/4*I*(a^(-3))^(1/4)*log((a*x^10 + 2*a*x^5 + x^4 - 2*sqrt(x^5 + 1 )*(-I*a*(a^(-3))^(1/4)*x^3 + (I*a^3*x^6 + I*a^3*x)*(a^(-3))^(3/4)) - 2*(a^ 2*x^7 + a^2*x^2)*sqrt(a^(-3)) + a)/(a*x^10 + 2*a*x^5 - x^4 + a))
\[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx=\int \frac {\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (3 x^{5} - 2\right )}{a x^{10} + 2 a x^{5} + a - x^{4}}\, dx \]
Integral(sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1))*(3*x**5 - 2)/(a*x**10 + 2*a*x**5 + a - x**4), x)
\[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx=\int { \frac {{\left (3 \, x^{5} - 2\right )} \sqrt {x^{5} + 1}}{a x^{10} + 2 \, a x^{5} - x^{4} + a} \,d x } \]
\[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx=\int { \frac {{\left (3 \, x^{5} - 2\right )} \sqrt {x^{5} + 1}}{a x^{10} + 2 \, a x^{5} - x^{4} + a} \,d x } \]
Time = 22.65 (sec) , antiderivative size = 309, normalized size of antiderivative = 6.31 \[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx=\frac {\ln \left (\frac {\left (x\,\sqrt {a^3}+a^2\,x^4-2\,a\,\sqrt {x^5+1}\,{\left (a^3\right )}^{1/4}\right )\,\left (a\,x^3-2\,x^6\,\sqrt {a^3}-3\,x\,\sqrt {a^3}+a^2\,x^4+a^2\,x^9+2\,a\,\sqrt {x^5+1}\,{\left (a^3\right )}^{1/4}\right )}{\left (a^2-x^2\,\sqrt {a^3}+a^2\,x^5\right )\,\left (4\,\sqrt {a^3}+2\,x^5\,\sqrt {a^3}-a\,x^2-a^2\,x^8\right )}\right )}{2\,{\left (a^3\right )}^{1/4}}+\frac {\ln \left (\frac {\left (2\,a\,\sqrt {x^5+1}\,{\left (a^3\right )}^{1/4}+x\,\sqrt {a^3}\,1{}\mathrm {i}-a^2\,x^4\,1{}\mathrm {i}\right )\,\left (2\,a\,\sqrt {x^5+1}\,{\left (a^3\right )}^{1/4}+x^6\,\sqrt {a^3}\,2{}\mathrm {i}+a\,x^3\,1{}\mathrm {i}+x\,\sqrt {a^3}\,3{}\mathrm {i}+a^2\,x^4\,1{}\mathrm {i}+a^2\,x^9\,1{}\mathrm {i}\right )}{\left (x^2\,\sqrt {a^3}+a^2+a^2\,x^5\right )\,\left (4\,\sqrt {a^3}+2\,x^5\,\sqrt {a^3}+a\,x^2+a^2\,x^8\right )}\right )\,1{}\mathrm {i}}{2\,{\left (a^3\right )}^{1/4}} \]
log(((x*(a^3)^(1/2) + a^2*x^4 - 2*a*(x^5 + 1)^(1/2)*(a^3)^(1/4))*(a*x^3 - 2*x^6*(a^3)^(1/2) - 3*x*(a^3)^(1/2) + a^2*x^4 + a^2*x^9 + 2*a*(x^5 + 1)^(1 /2)*(a^3)^(1/4)))/((a^2 - x^2*(a^3)^(1/2) + a^2*x^5)*(4*(a^3)^(1/2) + 2*x^ 5*(a^3)^(1/2) - a*x^2 - a^2*x^8)))/(2*(a^3)^(1/4)) + (log(((x*(a^3)^(1/2)* 1i - a^2*x^4*1i + 2*a*(x^5 + 1)^(1/2)*(a^3)^(1/4))*(x^6*(a^3)^(1/2)*2i + a *x^3*1i + x*(a^3)^(1/2)*3i + a^2*x^4*1i + a^2*x^9*1i + 2*a*(x^5 + 1)^(1/2) *(a^3)^(1/4)))/((x^2*(a^3)^(1/2) + a^2 + a^2*x^5)*(4*(a^3)^(1/2) + 2*x^5*( a^3)^(1/2) + a*x^2 + a^2*x^8)))*1i)/(2*(a^3)^(1/4))