Integrand size = 30, antiderivative size = 115 \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=4 \text {RootSum}\left [b-c^2-a c \text {$\#$1}+2 c \text {$\#$1}^2+a \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a c-4 c \text {$\#$1}-3 a \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
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\[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {x}{b-x^2+a x \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )\right ) \\ & = -\left (4 \text {Subst}\left (\int \frac {x \left (c-x^2\right )}{-b+(c+(a-x) x) \left (c-x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\right ) \\ & = \log \left (b+\sqrt {b+a x} \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )\right )-\text {Subst}\left (\int \frac {a c-3 a x^2}{-b+(c+(a-x) x) \left (c-x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \log \left (b+\sqrt {b+a x} \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )\right )-\text {Subst}\left (\int \frac {-a c+3 a x^2}{b-c^2-a c x+2 c x^2+a x^3-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \log \left (b+\sqrt {b+a x} \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )\right )-\text {Subst}\left (\int \left (\frac {3 a x^2}{b-c^2-a c x+2 c x^2+a x^3-x^4}+\frac {a c}{-b+c^2+a c x-2 c x^2-a x^3+x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = \log \left (b+\sqrt {b+a x} \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )\right )-(3 a) \text {Subst}\left (\int \frac {x^2}{b-c^2-a c x+2 c x^2+a x^3-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-(a c) \text {Subst}\left (\int \frac {1}{-b+c^2+a c x-2 c x^2-a x^3+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=4 \text {RootSum}\left [b-c^2-a c \text {$\#$1}+2 c \text {$\#$1}^2+a \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a c-4 c \text {$\#$1}-3 a \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
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Time = 0.52 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(-4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}-2 c \,\textit {\_Z}^{2}+a c \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{3}+\textit {\_R} c \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a -4 \textit {\_R} c +a c}\right )\) | \(80\) |
default | \(\text {Expression too large to display}\) | \(3139\) |
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Exception generated. \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Exception raised: AttributeError} \]
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Not integrable
Time = 20.85 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {1}{x - \sqrt {c + \sqrt {a x + b}} \sqrt {a x + b}}\, dx \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { -\frac {1}{\sqrt {a x + b} \sqrt {c + \sqrt {a x + b}} - x} \,d x } \]
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Not integrable
Time = 0.51 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { -\frac {1}{\sqrt {a x + b} \sqrt {c + \sqrt {a x + b}} - x} \,d x } \]
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Not integrable
Time = 5.89 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {1}{x-\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}} \,d x \]
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