Integrand size = 34, antiderivative size = 286 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=-2 \log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}\right )-2 \text {RootSum}\left [1-8 b+16 b^2+4 \text {$\#$1}-16 b \text {$\#$1}+6 \text {$\#$1}^2+8 b \text {$\#$1}^2+4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right )+4 b \log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right )-2 \log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}-\log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}^2}{1-4 b+3 \text {$\#$1}+4 b \text {$\#$1}+3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \]
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Time = 0.78 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.57, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {1004, 635, 212, 1050, 1044, 214, 211} \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {b}-\sqrt {a x-b}}{\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt [4]{b}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a x-b}+\sqrt {b}}{\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt [4]{b}}+2 \text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right ) \]
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Rule 211
Rule 212
Rule 214
Rule 635
Rule 1004
Rule 1044
Rule 1050
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\sqrt {b+x+x^2}}{b+x^2} \, dx,x,\sqrt {-b+a x}\right ) \\ & = 2 \text {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )+2 \text {Subst}\left (\int \frac {x}{\left (b+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )-\frac {\text {Subst}\left (\int \frac {-b-\sqrt {b} x}{\left (b+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{\sqrt {b}}+\frac {\text {Subst}\left (\int \frac {-b+\sqrt {b} x}{\left (b+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{\sqrt {b}} \\ & = 2 \text {arctanh}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )+\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{-2 b^{7/2}+b x^2} \, dx,x,\frac {b^{3/2}+b \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )+\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{2 b^{7/2}+b x^2} \, dx,x,\frac {-b^{3/2}+b \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right ) \\ & = -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {b}-\sqrt {-b+a x}}{\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{\sqrt [4]{b}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {b}+\sqrt {-b+a x}}{\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{\sqrt [4]{b}}+2 \text {arctanh}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=-2 \log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}\right )-\text {RootSum}\left [b+b^2-4 b \text {$\#$1}+2 b \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )-\log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b+b \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]
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Time = 0.36 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.85
method | result | size |
derivativedivides | \(-\frac {\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}+\frac {\left (1-2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}\right )}{2}+\frac {\sqrt {-b}\, \ln \left (\frac {-2 \sqrt {-b}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )+2 \sqrt {-\sqrt {-b}}\, \sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}}{\sqrt {a x -b}+\sqrt {-b}}\right )}{\sqrt {-\sqrt {-b}}}}{\sqrt {-b}}+\frac {\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}+\frac {\left (1+2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}\right )}{2}-\left (-b \right )^{\frac {1}{4}} \ln \left (\frac {2 \sqrt {-b}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+2 \left (-b \right )^{\frac {1}{4}} \sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}}{\sqrt {a x -b}-\sqrt {-b}}\right )}{\sqrt {-b}}\) | \(529\) |
default | \(-\frac {\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}+\frac {\left (1-2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}\right )}{2}+\frac {\sqrt {-b}\, \ln \left (\frac {-2 \sqrt {-b}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )+2 \sqrt {-\sqrt {-b}}\, \sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}}{\sqrt {a x -b}+\sqrt {-b}}\right )}{\sqrt {-\sqrt {-b}}}}{\sqrt {-b}}+\frac {\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}+\frac {\left (1+2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}\right )}{2}-\left (-b \right )^{\frac {1}{4}} \ln \left (\frac {2 \sqrt {-b}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+2 \left (-b \right )^{\frac {1}{4}} \sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}}{\sqrt {a x -b}-\sqrt {-b}}\right )}{\sqrt {-b}}\) | \(529\) |
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Timed out. \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.71 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=\int \frac {\sqrt {a x + \sqrt {a x - b}}}{x \sqrt {a x - b}}\, dx \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=\int { \frac {\sqrt {a x + \sqrt {a x - b}}}{\sqrt {a x - b} x} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=\text {Timed out} \]
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Not integrable
Time = 7.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=\int \frac {\sqrt {a\,x+\sqrt {a\,x-b}}}{x\,\sqrt {a\,x-b}} \,d x \]
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