Integrand size = 24, antiderivative size = 128 \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=-\frac {2 \sqrt {a+b x}}{b}+\frac {2}{5} b^4 \text {RootSum}\left [a^5-b^5-5 a^4 \text {$\#$1}^2+10 a^3 \text {$\#$1}^4-10 a^2 \text {$\#$1}^6+5 a \text {$\#$1}^8-\text {$\#$1}^{10}\&,\frac {\log \left (\sqrt {a+b x}-\text {$\#$1}\right )}{a^4 \text {$\#$1}-4 a^3 \text {$\#$1}^3+6 a^2 \text {$\#$1}^5-4 a \text {$\#$1}^7+\text {$\#$1}^9}\&\right ] \] Output:
Unintegrable
Leaf count is larger than twice the leaf count of optimal. \(455\) vs. \(2(128)=256\).
Time = 0.28 (sec) , antiderivative size = 455, normalized size of antiderivative = 3.55 \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\frac {2}{5} \left (-\frac {5 \sqrt {a+b x}}{b}+\frac {2 \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {-a+b}}\right )}{\sqrt {-a+b}}+\text {RootSum}\left [a^4+a^3 b+a^2 b^2+a b^3+b^4-4 a^3 \text {$\#$1}^2-3 a^2 b \text {$\#$1}^2-2 a b^2 \text {$\#$1}^2-b^3 \text {$\#$1}^2+6 a^2 \text {$\#$1}^4+3 a b \text {$\#$1}^4+b^2 \text {$\#$1}^4-4 a \text {$\#$1}^6-b \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-a^3 \log \left (\sqrt {a+b x}-\text {$\#$1}\right )-2 a^2 b \log \left (\sqrt {a+b x}-\text {$\#$1}\right )-3 a b^2 \log \left (\sqrt {a+b x}-\text {$\#$1}\right )-4 b^3 \log \left (\sqrt {a+b x}-\text {$\#$1}\right )+3 a^2 \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^2+4 a b \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^2+3 b^2 \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^2-3 a \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^4-2 b \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^6}{4 a^3 \text {$\#$1}+3 a^2 b \text {$\#$1}+2 a b^2 \text {$\#$1}+b^3 \text {$\#$1}-12 a^2 \text {$\#$1}^3-6 a b \text {$\#$1}^3-2 b^2 \text {$\#$1}^3+12 a \text {$\#$1}^5+3 b \text {$\#$1}^5-4 \text {$\#$1}^7}\&\right ]\right ) \] Input:
Integrate[(1 - x^5)/(Sqrt[a + b*x]*(1 + x^5)),x]
Output:
(2*((-5*Sqrt[a + b*x])/b + (2*ArcTan[Sqrt[a + b*x]/Sqrt[-a + b]])/Sqrt[-a + b] + RootSum[a^4 + a^3*b + a^2*b^2 + a*b^3 + b^4 - 4*a^3*#1^2 - 3*a^2*b* #1^2 - 2*a*b^2*#1^2 - b^3*#1^2 + 6*a^2*#1^4 + 3*a*b*#1^4 + b^2*#1^4 - 4*a* #1^6 - b*#1^6 + #1^8 & , (-(a^3*Log[Sqrt[a + b*x] - #1]) - 2*a^2*b*Log[Sqr t[a + b*x] - #1] - 3*a*b^2*Log[Sqrt[a + b*x] - #1] - 4*b^3*Log[Sqrt[a + b* x] - #1] + 3*a^2*Log[Sqrt[a + b*x] - #1]*#1^2 + 4*a*b*Log[Sqrt[a + b*x] - #1]*#1^2 + 3*b^2*Log[Sqrt[a + b*x] - #1]*#1^2 - 3*a*Log[Sqrt[a + b*x] - #1 ]*#1^4 - 2*b*Log[Sqrt[a + b*x] - #1]*#1^4 + Log[Sqrt[a + b*x] - #1]*#1^6)/ (4*a^3*#1 + 3*a^2*b*#1 + 2*a*b^2*#1 + b^3*#1 - 12*a^2*#1^3 - 6*a*b*#1^3 - 2*b^2*#1^3 + 12*a*#1^5 + 3*b*#1^5 - 4*#1^7) & ]))/5
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.
| \(\displaystyle \int \frac {1-x^5}{\left (x^5+1\right ) \sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 \int \frac {1-x^5}{x^5+1}d\sqrt {a+b x}}{b}\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \frac {2 \int \left (\frac {2 b}{5 (x b+b)}+\frac {2 \left (\left (\frac {b \left (2 a^2+3 b a+4 b^2\right )}{a^3}+1\right ) a^3-3 \left (\frac {b (4 a+3 b)}{3 a^2}+1\right ) (a+b x) a^2+3 \left (\frac {2 b}{3 a}+1\right ) (a+b x)^2 a-(a+b x)^3\right ) b}{5 \left (\left (\frac {b \left (a^3+b a^2+b^2 a+b^3\right )}{a^4}+1\right ) a^4-4 \left (\frac {b \left (3 a^2+2 b a+b^2\right )}{4 a^3}+1\right ) (a+b x) a^3+6 \left (\frac {b (3 a+b)}{6 a^2}+1\right ) (a+b x)^2 a^2-4 \left (\frac {b}{4 a}+1\right ) (a+b x)^3 a+(a+b x)^4\right )}-1\right )d\sqrt {a+b x}}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {2}{5} b \int \frac {(a+b x)^3}{-\left (\left (\frac {b \left (a^3+b a^2+b^2 a+b^3\right )}{a^4}+1\right ) a^4\right )+4 \left (\frac {b \left (3 a^2+2 b a+b^2\right )}{4 a^3}+1\right ) (a+b x) a^3-6 \left (\frac {b (3 a+b)}{6 a^2}+1\right ) (a+b x)^2 a^2+4 \left (\frac {b}{4 a}+1\right ) (a+b x)^3 a-(a+b x)^4}d\sqrt {a+b x}+\frac {2}{5} b \left (a^3+2 a^2 b+3 a b^2+4 b^3\right ) \int \frac {1}{\left (\frac {b \left (a^3+b a^2+b^2 a+b^3\right )}{a^4}+1\right ) a^4-4 \left (\frac {b \left (3 a^2+2 b a+b^2\right )}{4 a^3}+1\right ) (a+b x) a^3+6 \left (\frac {b (3 a+b)}{6 a^2}+1\right ) (a+b x)^2 a^2-4 \left (\frac {b}{4 a}+1\right ) (a+b x)^3 a+(a+b x)^4}d\sqrt {a+b x}-\frac {2}{5} b \left (3 a^2+4 a b+3 b^2\right ) \int \frac {a+b x}{\left (\frac {b \left (a^3+b a^2+b^2 a+b^3\right )}{a^4}+1\right ) a^4-4 \left (\frac {b \left (3 a^2+2 b a+b^2\right )}{4 a^3}+1\right ) (a+b x) a^3+6 \left (\frac {b (3 a+b)}{6 a^2}+1\right ) (a+b x)^2 a^2-4 \left (\frac {b}{4 a}+1\right ) (a+b x)^3 a+(a+b x)^4}d\sqrt {a+b x}+\frac {2}{5} b (3 a+2 b) \int \frac {(a+b x)^2}{\left (\frac {b \left (a^3+b a^2+b^2 a+b^3\right )}{a^4}+1\right ) a^4-4 \left (\frac {b \left (3 a^2+2 b a+b^2\right )}{4 a^3}+1\right ) (a+b x) a^3+6 \left (\frac {b (3 a+b)}{6 a^2}+1\right ) (a+b x)^2 a^2-4 \left (\frac {b}{4 a}+1\right ) (a+b x)^3 a+(a+b x)^4}d\sqrt {a+b x}-\frac {2 b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b}}\right )}{5 \sqrt {a-b}}-\sqrt {a+b x}\right )}{b}\) |
Input:
Int[(1 - x^5)/(Sqrt[a + b*x]*(1 + x^5)),x]
Output:
$Aborted
Time = 0.79 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.01
| method | result | size |
| risch | \(-\frac {2 \sqrt {b x +a}}{b}-\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+\left (-4 a -b \right ) \textit {\_Z}^{6}+\left (6 a^{2}+3 a b +b^{2}\right ) \textit {\_Z}^{4}+\left (-4 a^{3}-3 a^{2} b -2 a \,b^{2}-b^{3}\right ) \textit {\_Z}^{2}+a^{4}+a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+\left (3 a +2 b \right ) \textit {\_R}^{4}+\left (-3 a^{2}-4 a b -3 b^{2}\right ) \textit {\_R}^{2}+a^{3}+2 a^{2} b +3 a \,b^{2}+4 b^{3}\right ) \ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{\textit {\_R} \left (-4 \textit {\_R}^{6}+3 \left (4 a +b \right ) \textit {\_R}^{4}+2 \left (-6 a^{2}-3 a b -b^{2}\right ) \textit {\_R}^{2}+4 a^{3}+3 a^{2} b +2 a \,b^{2}+b^{3}\right )}\right )}{5}+\frac {4 \arctan \left (\frac {\sqrt {b x +a}}{\sqrt {-a +b}}\right )}{5 \sqrt {-a +b}}\) | \(257\) |
| derivativedivides | \(-\frac {2 \left (\sqrt {b x +a}-\frac {2 b \arctan \left (\frac {\sqrt {b x +a}}{\sqrt {-a +b}}\right )}{5 \sqrt {-a +b}}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+\left (-4 a -b \right ) \textit {\_Z}^{6}+\left (6 a^{2}+3 a b +b^{2}\right ) \textit {\_Z}^{4}+\left (-4 a^{3}-3 a^{2} b -2 a \,b^{2}-b^{3}\right ) \textit {\_Z}^{2}+a^{4}+a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+\left (3 a +2 b \right ) \textit {\_R}^{4}+\left (-3 a^{2}-4 a b -3 b^{2}\right ) \textit {\_R}^{2}+a^{3}+2 a^{2} b +3 a \,b^{2}+4 b^{3}\right ) \ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-12 \textit {\_R}^{5} a -3 \textit {\_R}^{5} b +12 \textit {\_R}^{3} a^{2}+6 \textit {\_R}^{3} a b +2 \textit {\_R}^{3} b^{2}-4 \textit {\_R} \,a^{3}-3 \textit {\_R} \,a^{2} b -2 \textit {\_R} a \,b^{2}-\textit {\_R} \,b^{3}}\right )}{5}\right )}{b}\) | \(267\) |
| default | \(-\frac {2 \left (\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+\left (-4 a -b \right ) \textit {\_Z}^{6}+\left (6 a^{2}+3 a b +b^{2}\right ) \textit {\_Z}^{4}+\left (-4 a^{3}-3 a^{2} b -2 a \,b^{2}-b^{3}\right ) \textit {\_Z}^{2}+a^{4}+a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}\right )}{\sum }\frac {\ln \left (\sqrt {b x +a}-\textit {\_R} \right ) \left (-\textit {\_R}^{6}+\left (3 a +2 b \right ) \textit {\_R}^{4}+\left (-3 a^{2}-4 a b -3 b^{2}\right ) \textit {\_R}^{2}+a^{3}+2 a^{2} b +3 a \,b^{2}+4 b^{3}\right )}{4 \textit {\_R} \left (-\textit {\_R}^{6}+\left (3 a +\frac {3 b}{4}\right ) \textit {\_R}^{4}+\left (-3 a^{2}-\frac {3}{2} a b -\frac {1}{2} b^{2}\right ) \textit {\_R}^{2}+a^{3}+\frac {3 a^{2} b}{4}+\frac {a \,b^{2}}{2}+\frac {b^{3}}{4}\right )}\right ) b \sqrt {-a +b}+5 \sqrt {b x +a}\, \sqrt {-a +b}-2 \arctan \left (\frac {\sqrt {b x +a}}{\sqrt {-a +b}}\right ) b \right )}{5 \sqrt {-a +b}\, b}\) | \(275\) |
| pseudoelliptic | \(-\frac {2 \left (\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+\left (-4 a -b \right ) \textit {\_Z}^{6}+\left (6 a^{2}+3 a b +b^{2}\right ) \textit {\_Z}^{4}+\left (-4 a^{3}-3 a^{2} b -2 a \,b^{2}-b^{3}\right ) \textit {\_Z}^{2}+a^{4}+a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}\right )}{\sum }\frac {\ln \left (\sqrt {b x +a}-\textit {\_R} \right ) \left (-\textit {\_R}^{6}+\left (3 a +2 b \right ) \textit {\_R}^{4}+\left (-3 a^{2}-4 a b -3 b^{2}\right ) \textit {\_R}^{2}+a^{3}+2 a^{2} b +3 a \,b^{2}+4 b^{3}\right )}{4 \textit {\_R} \left (-\textit {\_R}^{6}+\left (3 a +\frac {3 b}{4}\right ) \textit {\_R}^{4}+\left (-3 a^{2}-\frac {3}{2} a b -\frac {1}{2} b^{2}\right ) \textit {\_R}^{2}+a^{3}+\frac {3 a^{2} b}{4}+\frac {a \,b^{2}}{2}+\frac {b^{3}}{4}\right )}\right ) b \sqrt {-a +b}+5 \sqrt {b x +a}\, \sqrt {-a +b}-2 \arctan \left (\frac {\sqrt {b x +a}}{\sqrt {-a +b}}\right ) b \right )}{5 \sqrt {-a +b}\, b}\) | \(275\) |
Input:
int((-x^5+1)/(b*x+a)^(1/2)/(x^5+1),x,method=_RETURNVERBOSE)
Output:
-2/b*(b*x+a)^(1/2)-2/5*sum((-_R^6+(3*a+2*b)*_R^4+(-3*a^2-4*a*b-3*b^2)*_R^2 +a^3+2*a^2*b+3*a*b^2+4*b^3)/_R/(-4*_R^6+3*(4*a+b)*_R^4+2*(-6*a^2-3*a*b-b^2 )*_R^2+4*a^3+3*a^2*b+2*a*b^2+b^3)*ln((b*x+a)^(1/2)-_R),_R=RootOf(_Z^8+(-4* a-b)*_Z^6+(6*a^2+3*a*b+b^2)*_Z^4+(-4*a^3-3*a^2*b-2*a*b^2-b^3)*_Z^2+a^4+a^3 *b+a^2*b^2+a*b^3+b^4))+4/5/(-a+b)^(1/2)*arctan((b*x+a)^(1/2)/(-a+b)^(1/2))
Timed out. \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\text {Timed out} \] Input:
integrate((-x^5+1)/(b*x+a)^(1/2)/(x^5+1),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\text {Timed out} \] Input:
integrate((-x**5+1)/(b*x+a)**(1/2)/(x**5+1),x)
Output:
Timed out
Exception generated. \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((-x^5+1)/(b*x+a)^(1/2)/(x^5+1),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a-4*b>0)', see `assume?` for m ore detail
Not integrable
Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.18 \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\int { -\frac {x^{5} - 1}{{\left (x^{5} + 1\right )} \sqrt {b x + a}} \,d x } \] Input:
integrate((-x^5+1)/(b*x+a)^(1/2)/(x^5+1),x, algorithm="giac")
Output:
integrate(-(x^5 - 1)/((x^5 + 1)*sqrt(b*x + a)), x)
Time = 8.37 (sec) , antiderivative size = 1163, normalized size of antiderivative = 9.09 \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\text {Too large to display} \] Input:
int(-(x^5 - 1)/((x^5 + 1)*(a + b*x)^(1/2)),x)
Output:
(2*log(655360*b^32*(a + b*x)^(1/2) + (2*(1638400*b^33 - (16*((2*((2*(40000 0000*a^2*b^34 - (400000000*a*b^35*(a + b*x)^(1/2))/(a - b)^(1/2)))/(5*(a - b)^(1/2)) - 320000000*a^2*b^33*(a + b*x)^(1/2)))/(5*(a - b)^(1/2)) + 6400 0000*a^3*b^32))/(625*(a - b)^2)))/(5*(a - b)^(1/2))))/(5*(a - b)^(1/2)) - (2*log(655360*b^32*(a + b*x)^(1/2) - (2*(1638400*b^33 - (16*((2*((2*(40000 0000*a^2*b^34 + (400000000*a*b^35*(a + b*x)^(1/2))/(a - b)^(1/2)))/(5*(a - b)^(1/2)) + 320000000*a^2*b^33*(a + b*x)^(1/2)))/(5*(a - b)^(1/2)) + 6400 0000*a^3*b^32))/(625*(a - b)^2)))/(5*(a - b)^(1/2))))/(5*(a - b)^(1/2)) - (2*(a + b*x)^(1/2))/b + symsum(log(root(390625*a^2*b^2*z^8 + 390625*a^3*b* z^8 + 390625*a*b^3*z^8 + 390625*b^4*z^8 + 390625*a^4*z^8 + 187500*a*b^2*z^ 6 + 125000*a^2*b*z^6 - 62500*b^3*z^6 + 62500*a^3*z^6 - 20000*a*b*z^4 + 100 00*b^2*z^4 + 10000*a^2*z^4 - 1600*b*z^2 + 1600*a*z^2 + 256, z, k)*(root(39 0625*a^2*b^2*z^8 + 390625*a^3*b*z^8 + 390625*a*b^3*z^8 + 390625*b^4*z^8 + 390625*a^4*z^8 + 187500*a*b^2*z^6 + 125000*a^2*b*z^6 - 62500*b^3*z^6 + 625 00*a^3*z^6 - 20000*a*b*z^4 + 10000*b^2*z^4 + 10000*a^2*z^4 - 1600*b*z^2 + 1600*a*z^2 + 256, z, k)*(root(390625*a^2*b^2*z^8 + 390625*a^3*b*z^8 + 3906 25*a*b^3*z^8 + 390625*b^4*z^8 + 390625*a^4*z^8 + 187500*a*b^2*z^6 + 125000 *a^2*b*z^6 - 62500*b^3*z^6 + 62500*a^3*z^6 - 20000*a*b*z^4 + 10000*b^2*z^4 + 10000*a^2*z^4 - 1600*b*z^2 + 1600*a*z^2 + 256, z, k)^4*(root(390625*a^2 *b^2*z^8 + 390625*a^3*b*z^8 + 390625*a*b^3*z^8 + 390625*b^4*z^8 + 39062...
Not integrable
Time = 0.33 (sec) , antiderivative size = 590, normalized size of antiderivative = 4.61 \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\frac {\frac {2 \sqrt {a -b}\, \mathrm {log}\left (-\sqrt {a -b}+\sqrt {b x +a}\right ) b}{5}-\frac {2 \sqrt {a -b}\, \mathrm {log}\left (\sqrt {a -b}+\sqrt {b x +a}\right ) b}{5}-\frac {6 \sqrt {b x +a}\, a}{5}+\frac {6 \sqrt {b x +a}\, b}{5}+\frac {6 \left (\int \frac {\sqrt {b x +a}}{b \,x^{5}+a \,x^{4}-b \,x^{4}-a \,x^{3}+b \,x^{3}+a \,x^{2}-b \,x^{2}-a x +b x +a}d x \right ) a b}{5}-\frac {6 \left (\int \frac {\sqrt {b x +a}}{b \,x^{5}+a \,x^{4}-b \,x^{4}-a \,x^{3}+b \,x^{3}+a \,x^{2}-b \,x^{2}-a x +b x +a}d x \right ) b^{2}}{5}-\frac {2 \left (\int \frac {\sqrt {b x +a}\, x^{4}}{b \,x^{5}+a \,x^{4}-b \,x^{4}-a \,x^{3}+b \,x^{3}+a \,x^{2}-b \,x^{2}-a x +b x +a}d x \right ) a b}{5}+\frac {2 \left (\int \frac {\sqrt {b x +a}\, x^{4}}{b \,x^{5}+a \,x^{4}-b \,x^{4}-a \,x^{3}+b \,x^{3}+a \,x^{2}-b \,x^{2}-a x +b x +a}d x \right ) b^{2}}{5}+\frac {2 \left (\int \frac {\sqrt {b x +a}\, x^{2}}{b \,x^{5}+a \,x^{4}-b \,x^{4}-a \,x^{3}+b \,x^{3}+a \,x^{2}-b \,x^{2}-a x +b x +a}d x \right ) a b}{5}-\frac {2 \left (\int \frac {\sqrt {b x +a}\, x^{2}}{b \,x^{5}+a \,x^{4}-b \,x^{4}-a \,x^{3}+b \,x^{3}+a \,x^{2}-b \,x^{2}-a x +b x +a}d x \right ) b^{2}}{5}-\frac {4 \left (\int \frac {\sqrt {b x +a}\, x}{b \,x^{5}+a \,x^{4}-b \,x^{4}-a \,x^{3}+b \,x^{3}+a \,x^{2}-b \,x^{2}-a x +b x +a}d x \right ) a b}{5}+\frac {4 \left (\int \frac {\sqrt {b x +a}\, x}{b \,x^{5}+a \,x^{4}-b \,x^{4}-a \,x^{3}+b \,x^{3}+a \,x^{2}-b \,x^{2}-a x +b x +a}d x \right ) b^{2}}{5}}{b \left (a -b \right )} \] Input:
int((-x^5+1)/(b*x+a)^(1/2)/(x^5+1),x)
Output:
(2*(sqrt(a - b)*log( - sqrt(a - b) + sqrt(a + b*x))*b - sqrt(a - b)*log(sq rt(a - b) + sqrt(a + b*x))*b - 3*sqrt(a + b*x)*a + 3*sqrt(a + b*x)*b + 3*i nt(sqrt(a + b*x)/(a*x**4 - a*x**3 + a*x**2 - a*x + a + b*x**5 - b*x**4 + b *x**3 - b*x**2 + b*x),x)*a*b - 3*int(sqrt(a + b*x)/(a*x**4 - a*x**3 + a*x* *2 - a*x + a + b*x**5 - b*x**4 + b*x**3 - b*x**2 + b*x),x)*b**2 - int((sqr t(a + b*x)*x**4)/(a*x**4 - a*x**3 + a*x**2 - a*x + a + b*x**5 - b*x**4 + b *x**3 - b*x**2 + b*x),x)*a*b + int((sqrt(a + b*x)*x**4)/(a*x**4 - a*x**3 + a*x**2 - a*x + a + b*x**5 - b*x**4 + b*x**3 - b*x**2 + b*x),x)*b**2 + int ((sqrt(a + b*x)*x**2)/(a*x**4 - a*x**3 + a*x**2 - a*x + a + b*x**5 - b*x** 4 + b*x**3 - b*x**2 + b*x),x)*a*b - int((sqrt(a + b*x)*x**2)/(a*x**4 - a*x **3 + a*x**2 - a*x + a + b*x**5 - b*x**4 + b*x**3 - b*x**2 + b*x),x)*b**2 - 2*int((sqrt(a + b*x)*x)/(a*x**4 - a*x**3 + a*x**2 - a*x + a + b*x**5 - b *x**4 + b*x**3 - b*x**2 + b*x),x)*a*b + 2*int((sqrt(a + b*x)*x)/(a*x**4 - a*x**3 + a*x**2 - a*x + a + b*x**5 - b*x**4 + b*x**3 - b*x**2 + b*x),x)*b* *2))/(5*b*(a - b))