Integrand size = 34, antiderivative size = 305 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=-4 \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-b c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+c^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-3 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3+3 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^5-\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^7}{a^2 c+8 b c \text {$\#$1}-8 c^3 \text {$\#$1}-3 a^2 \text {$\#$1}^2-8 b \text {$\#$1}^3+24 c^2 \text {$\#$1}^3-24 c \text {$\#$1}^5+8 \text {$\#$1}^7}\&\right ] \]
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\[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x \left (-b+x^2\right )}{b^2-2 b x^2+x^4-a^2 x \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {x \left (-c+x^2\right ) \left (-b+\left (c-x^2\right )^2\right )}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right )^4-a^2 x \left (-c+x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \frac {1}{2} \log \left (b^2-2 b (b+a x)+(b+a x)^2-a^2 \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {-a^2 c+3 a^2 x^2}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right )^4-a^2 x \left (-c+x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \frac {1}{2} \log \left (b^2-2 b (b+a x)+(b+a x)^2-a^2 \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}\right )+\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2 c}{-b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )-a^2 c x-4 b c \left (1-\frac {c^2}{b}\right ) x^2+a^2 x^3+2 b \left (1-\frac {3 c^2}{b}\right ) x^4+4 c x^6-x^8}+\frac {3 a^2 x^2}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \frac {1}{2} \log \left (b^2-2 b (b+a x)+(b+a x)^2-a^2 \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}\right )+\frac {1}{2} \left (3 a^2\right ) \text {Subst}\left (\int \frac {x^2}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )+\frac {1}{2} \left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{-b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )-a^2 c x-4 b c \left (1-\frac {c^2}{b}\right ) x^2+a^2 x^3+2 b \left (1-\frac {3 c^2}{b}\right ) x^4+4 c x^6-x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \frac {1}{2} \log \left (b^2-2 b (b+a x)+(b+a x)^2-a^2 \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}\right )+\frac {1}{2} \left (3 a^2\right ) \text {Subst}\left (\int \frac {x^2}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right ) \left (c^3+a^2 x-3 c^2 x^2+3 c x^4-x^6\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )+\frac {1}{2} \left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{-b^2+2 b \left (c-x^2\right )^2-\left (c-x^2\right ) \left (c^3+a^2 x-3 c^2 x^2+3 c x^4-x^6\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=-4 \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-b c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+c^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-3 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3+3 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^5-\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^7}{a^2 c+8 b c \text {$\#$1}-8 c^3 \text {$\#$1}-3 a^2 \text {$\#$1}^2-8 b \text {$\#$1}^3+24 c^2 \text {$\#$1}^3-24 c \text {$\#$1}^5+8 \text {$\#$1}^7}\&\right ] \]
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Time = 0.06 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.56
method | result | size |
derivativedivides | \(4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{7}-3 c \,\textit {\_R}^{5}+\left (3 c^{2}-b \right ) \textit {\_R}^{3}+c \left (-c^{2}+b \right ) \textit {\_R} \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 c \,\textit {\_R}^{5}+24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\) | \(172\) |
default | \(4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{7}-3 c \,\textit {\_R}^{5}+\left (3 c^{2}-b \right ) \textit {\_R}^{3}+c \left (-c^{2}+b \right ) \textit {\_R} \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 c \,\textit {\_R}^{5}+24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\) | \(172\) |
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Timed out. \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {x}{x^{2} - \sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}} \,d x } \]
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Not integrable
Time = 2.43 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {x}{x^{2} - \sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.11 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=-\int \frac {x}{\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}-x^2} \,d x \]
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