Integrand size = 24, antiderivative size = 128 \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=-\frac {2 \sqrt {a+b x}}{b}+\frac {2}{5} b^4 \text {RootSum}\left [a^5-b^5-5 a^4 \text {$\#$1}^2+10 a^3 \text {$\#$1}^4-10 a^2 \text {$\#$1}^6+5 a \text {$\#$1}^8-\text {$\#$1}^{10}\&,\frac {\log \left (\sqrt {a+b x}-\text {$\#$1}\right )}{a^4 \text {$\#$1}-4 a^3 \text {$\#$1}^3+6 a^2 \text {$\#$1}^5-4 a \text {$\#$1}^7+\text {$\#$1}^9}\&\right ] \]
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\[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {1+\frac {\left (a-x^2\right )^5}{b^5}}{1-\frac {\left (a-x^2\right )^5}{b^5}} \, dx,x,\sqrt {a+b x}\right )}{b} \\ & = \frac {2 \text {Subst}\left (\int \left (-1+\frac {2 b}{5 \left (-a+b+x^2\right )}+\frac {2 b \left (a^3 \left (1+\frac {b \left (2 a^2+3 a b+4 b^2\right )}{a^3}\right )-3 a^2 \left (1+\frac {b (4 a+3 b)}{3 a^2}\right ) x^2+3 a \left (1+\frac {2 b}{3 a}\right ) x^4-x^6\right )}{5 \left (a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8\right )}\right ) \, dx,x,\sqrt {a+b x}\right )}{b} \\ & = -\frac {2 \sqrt {a+b x}}{b}+\frac {4}{5} \text {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b x}\right )+\frac {4}{5} \text {Subst}\left (\int \frac {a^3 \left (1+\frac {b \left (2 a^2+3 a b+4 b^2\right )}{a^3}\right )-3 a^2 \left (1+\frac {b (4 a+3 b)}{3 a^2}\right ) x^2+3 a \left (1+\frac {2 b}{3 a}\right ) x^4-x^6}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8} \, dx,x,\sqrt {a+b x}\right ) \\ & = -\frac {2 \sqrt {a+b x}}{b}-\frac {4 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b}}\right )}{5 \sqrt {a-b}}+\frac {4}{5} \text {Subst}\left (\int \left (\frac {x^6}{-a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )+4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2-6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4+4 a \left (1+\frac {b}{4 a}\right ) x^6-x^8}+\frac {a^3+2 a^2 b+3 a b^2+4 b^3}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8}+\frac {\left (-3 a^2-4 a b-3 b^2\right ) x^2}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8}+\frac {(3 a+2 b) x^4}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8}\right ) \, dx,x,\sqrt {a+b x}\right ) \\ & = -\frac {2 \sqrt {a+b x}}{b}-\frac {4 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b}}\right )}{5 \sqrt {a-b}}+\frac {4}{5} \text {Subst}\left (\int \frac {x^6}{-a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )+4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2-6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4+4 a \left (1+\frac {b}{4 a}\right ) x^6-x^8} \, dx,x,\sqrt {a+b x}\right )+\frac {1}{5} (4 (3 a+2 b)) \text {Subst}\left (\int \frac {x^4}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8} \, dx,x,\sqrt {a+b x}\right )-\frac {1}{5} \left (4 \left (3 a^2+4 a b+3 b^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8} \, dx,x,\sqrt {a+b x}\right )+\frac {1}{5} \left (4 \left (a^3+2 a^2 b+3 a b^2+4 b^3\right )\right ) \text {Subst}\left (\int \frac {1}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8} \, dx,x,\sqrt {a+b x}\right ) \\ \end{align*}
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 \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\frac {2}{5} \left (-\frac {5 \sqrt {a+b x}}{b}+\frac {2 \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {-a+b}}\right )}{\sqrt {-a+b}}+\text {RootSum}\left [a^4+a^3 b+a^2 b^2+a b^3+b^4-4 a^3 \text {$\#$1}^2-3 a^2 b \text {$\#$1}^2-2 a b^2 \text {$\#$1}^2-b^3 \text {$\#$1}^2+6 a^2 \text {$\#$1}^4+3 a b \text {$\#$1}^4+b^2 \text {$\#$1}^4-4 a \text {$\#$1}^6-b \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-a^3 \log \left (\sqrt {a+b x}-\text {$\#$1}\right )-2 a^2 b \log \left (\sqrt {a+b x}-\text {$\#$1}\right )-3 a b^2 \log \left (\sqrt {a+b x}-\text {$\#$1}\right )-4 b^3 \log \left (\sqrt {a+b x}-\text {$\#$1}\right )+3 a^2 \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^2+4 a b \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^2+3 b^2 \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^2-3 a \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^4-2 b \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^6}{4 a^3 \text {$\#$1}+3 a^2 b \text {$\#$1}+2 a b^2 \text {$\#$1}+b^3 \text {$\#$1}-12 a^2 \text {$\#$1}^3-6 a b \text {$\#$1}^3-2 b^2 \text {$\#$1}^3+12 a \text {$\#$1}^5+3 b \text {$\#$1}^5-4 \text {$\#$1}^7}\&\right ]\right ) \]
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Time = 0.68 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.01
method | result | size |
risch | \(-\frac {2 \sqrt {b x +a}}{b}-\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+\left (-4 a -b \right ) \textit {\_Z}^{6}+\left (6 a^{2}+3 a b +b^{2}\right ) \textit {\_Z}^{4}+\left (-4 a^{3}-3 a^{2} b -2 a \,b^{2}-b^{3}\right ) \textit {\_Z}^{2}+a^{4}+a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+\left (3 a +2 b \right ) \textit {\_R}^{4}+\left (-3 a^{2}-4 a b -3 b^{2}\right ) \textit {\_R}^{2}+a^{3}+2 a^{2} b +3 a \,b^{2}+4 b^{3}\right ) \ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{\left (-4 \textit {\_R}^{6}+3 \left (4 a +b \right ) \textit {\_R}^{4}+2 \left (-6 a^{2}-3 a b -b^{2}\right ) \textit {\_R}^{2}+4 a^{3}+3 a^{2} b +2 a \,b^{2}+b^{3}\right ) \textit {\_R}}\right )}{5}+\frac {4 \arctan \left (\frac {\sqrt {b x +a}}{\sqrt {-a +b}}\right )}{5 \sqrt {-a +b}}\) | \(257\) |
derivativedivides | \(-\frac {2 \left (\sqrt {b x +a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+\left (-4 a -b \right ) \textit {\_Z}^{6}+\left (6 a^{2}+3 a b +b^{2}\right ) \textit {\_Z}^{4}+\left (-4 a^{3}-3 a^{2} b -2 a \,b^{2}-b^{3}\right ) \textit {\_Z}^{2}+a^{4}+a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+\left (3 a +2 b \right ) \textit {\_R}^{4}+\left (-3 a^{2}-4 a b -3 b^{2}\right ) \textit {\_R}^{2}+a^{3}+2 a^{2} b +3 a \,b^{2}+4 b^{3}\right ) \ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-12 \textit {\_R}^{5} a -3 \textit {\_R}^{5} b +12 \textit {\_R}^{3} a^{2}+6 \textit {\_R}^{3} a b +2 \textit {\_R}^{3} b^{2}-4 \textit {\_R} \,a^{3}-3 \textit {\_R} \,a^{2} b -2 \textit {\_R} a \,b^{2}-\textit {\_R} \,b^{3}}\right )}{5}-\frac {2 b \arctan \left (\frac {\sqrt {b x +a}}{\sqrt {-a +b}}\right )}{5 \sqrt {-a +b}}\right )}{b}\) | \(267\) |
default | \(-\frac {2 \left (\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+\left (-4 a -b \right ) \textit {\_Z}^{6}+\left (6 a^{2}+3 a b +b^{2}\right ) \textit {\_Z}^{4}+\left (-4 a^{3}-3 a^{2} b -2 a \,b^{2}-b^{3}\right ) \textit {\_Z}^{2}+a^{4}+a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+\left (3 a +2 b \right ) \textit {\_R}^{4}+\left (-3 a^{2}-4 a b -3 b^{2}\right ) \textit {\_R}^{2}+a^{3}+2 a^{2} b +3 a \,b^{2}+4 b^{3}\right ) \ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{4 \left (-\textit {\_R}^{6}+\left (3 a +\frac {3 b}{4}\right ) \textit {\_R}^{4}+\left (-3 a^{2}-\frac {3}{2} a b -\frac {1}{2} b^{2}\right ) \textit {\_R}^{2}+a^{3}+\frac {3 a^{2} b}{4}+\frac {a \,b^{2}}{2}+\frac {b^{3}}{4}\right ) \textit {\_R}}\right ) b \sqrt {-a +b}+5 \sqrt {b x +a}\, \sqrt {-a +b}-2 \arctan \left (\frac {\sqrt {b x +a}}{\sqrt {-a +b}}\right ) b \right )}{5 b \sqrt {-a +b}}\) | \(275\) |
pseudoelliptic | \(-\frac {2 \left (\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+\left (-4 a -b \right ) \textit {\_Z}^{6}+\left (6 a^{2}+3 a b +b^{2}\right ) \textit {\_Z}^{4}+\left (-4 a^{3}-3 a^{2} b -2 a \,b^{2}-b^{3}\right ) \textit {\_Z}^{2}+a^{4}+a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+\left (3 a +2 b \right ) \textit {\_R}^{4}+\left (-3 a^{2}-4 a b -3 b^{2}\right ) \textit {\_R}^{2}+a^{3}+2 a^{2} b +3 a \,b^{2}+4 b^{3}\right ) \ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{4 \left (-\textit {\_R}^{6}+\left (3 a +\frac {3 b}{4}\right ) \textit {\_R}^{4}+\left (-3 a^{2}-\frac {3}{2} a b -\frac {1}{2} b^{2}\right ) \textit {\_R}^{2}+a^{3}+\frac {3 a^{2} b}{4}+\frac {a \,b^{2}}{2}+\frac {b^{3}}{4}\right ) \textit {\_R}}\right ) b \sqrt {-a +b}+5 \sqrt {b x +a}\, \sqrt {-a +b}-2 \arctan \left (\frac {\sqrt {b x +a}}{\sqrt {-a +b}}\right ) b \right )}{5 b \sqrt {-a +b}}\) | \(275\) |
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Timed out. \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.18 \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\int { -\frac {x^{5} - 1}{{\left (x^{5} + 1\right )} \sqrt {b x + a}} \,d x } \]
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Time = 5.72 (sec) , antiderivative size = 1163, normalized size of antiderivative = 9.09 \[ \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx=\text {Too large to display} \]
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