Integrand size = 21, antiderivative size = 176 \[ \int \frac {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=x-4 \sqrt {c+\sqrt {b+a x}}+\frac {4 \text {RootSum}\left [b-c^2-a \text {$\#$1}+2 c \text {$\#$1}^2-\text {$\#$1}^4\&,\frac {-a b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+a c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+a^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}-a c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{a-4 c \text {$\#$1}+4 \text {$\#$1}^3}\&\right ]}{a} \]
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\[ \int \frac {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x \left (-b+x^2\right )}{-b+x^2+a \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )}{a} \\ & = \frac {4 \text {Subst}\left (\int \frac {x \left (-c+x^2\right ) \left (-b+\left (c-x^2\right )^2\right )}{-b+a x+\left (c-x^2\right )^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a} \\ & = \frac {4 \text {Subst}\left (\int \frac {x \left (c-x^2\right ) \left (-b+c^2-2 c x^2+x^4\right )}{b-c^2-a x+2 c x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a} \\ & = \frac {4 \text {Subst}\left (\int \left (-a-c x+x^3+\frac {a \left (b-c^2\right )-a^2 x+a c x^2}{b-c^2-a x+2 c x^2-x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a} \\ & = -\frac {2 c \sqrt {b+a x}}{a}-4 \sqrt {c+\sqrt {b+a x}}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+\frac {4 \text {Subst}\left (\int \frac {a \left (b-c^2\right )-a^2 x+a c x^2}{b-c^2-a x+2 c x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a} \\ & = -\frac {2 c \sqrt {b+a x}}{a}-4 \sqrt {c+\sqrt {b+a x}}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+\frac {4 \text {Subst}\left (\int \left (-\frac {a \left (b-c^2\right )}{-b+c^2+a x-2 c x^2+x^4}+\frac {a^2 x}{-b+c^2+a x-2 c x^2+x^4}-\frac {a c x^2}{-b+c^2+a x-2 c x^2+x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a} \\ & = -\frac {2 c \sqrt {b+a x}}{a}-4 \sqrt {c+\sqrt {b+a x}}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+(4 a) \text {Subst}\left (\int \frac {x}{-b+c^2+a x-2 c x^2+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-(4 c) \text {Subst}\left (\int \frac {x^2}{-b+c^2+a x-2 c x^2+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-\left (4 \left (b-c^2\right )\right ) \text {Subst}\left (\int \frac {1}{-b+c^2+a x-2 c x^2+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.04 \[ \int \frac {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\frac {b-c^2+a x}{a}-4 \sqrt {c+\sqrt {b+a x}}+4 \text {RootSum}\left [b-c^2-a \text {$\#$1}+2 c \text {$\#$1}^2-\text {$\#$1}^4\&,\frac {b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-a \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-a+4 c \text {$\#$1}-4 \text {$\#$1}^3}\&\right ] \]
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Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.66
method | result | size |
default | \(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{2}}{2}+c \left (c +\sqrt {a x +b}\right )+2 \sqrt {c +\sqrt {a x +b}}\, a -2 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 c \,\textit {\_Z}^{2}+\textit {\_Z} a +c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{2} c +\textit {\_R} a +c^{2}-b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-4 \textit {\_R} c +a}\right )\right )}{a}\) | \(117\) |
derivativedivides | \(\frac {\left (c +\sqrt {a x +b}\right )^{2}-2 c \left (c +\sqrt {a x +b}\right )-4 \sqrt {c +\sqrt {a x +b}}\, a +4 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 c \,\textit {\_Z}^{2}+\textit {\_Z} a +c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{2} c +\textit {\_R} a +c^{2}-b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-4 \textit {\_R} c +a}\right )}{a}\) | \(118\) |
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Exception generated. \[ \int \frac {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\text {Exception raised: AttributeError} \]
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Not integrable
Time = 118.35 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.10 \[ \int \frac {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {x}{x + \sqrt {c + \sqrt {a x + b}}}\, dx \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.11 \[ \int \frac {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {x}{x + \sqrt {c + \sqrt {a x + b}}} \,d x } \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.11 \[ \int \frac {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {x}{x + \sqrt {c + \sqrt {a x + b}}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.11 \[ \int \frac {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {x}{x+\sqrt {c+\sqrt {b+a\,x}}} \,d x \]
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