Integrand size = 32, antiderivative size = 136 \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\frac {(15-12 b+8 a x) \sqrt {a x+\sqrt {-b+a x}}}{96 a^2}+\frac {\sqrt {-b+a x} (-5+36 b+24 a x) \sqrt {a x+\sqrt {-b+a x}}}{48 a^2}+\frac {\left (5-8 b-48 b^2\right ) \log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}\right )}{64 a^2} \]
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Time = 0.34 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1675, 654, 626, 635, 212} \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=-\frac {(1-4 b) (12 b+5) \text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{64 a^2}+\frac {\sqrt {a x-b} \left (\sqrt {a x-b}+a x\right )^{3/2}}{2 a^2}-\frac {5 \left (\sqrt {a x-b}+a x\right )^{3/2}}{12 a^2}+\frac {(12 b+5) \left (2 \sqrt {a x-b}+1\right ) \sqrt {\sqrt {a x-b}+a x}}{32 a^2} \]
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 1675
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \left (b+x^2\right ) \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a^2} \\ & = \frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a^2}+\frac {\text {Subst}\left (\int \left (3 b-\frac {5 x}{2}\right ) \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{2 a^2} \\ & = -\frac {5 \left (a x+\sqrt {-b+a x}\right )^{3/2}}{12 a^2}+\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a^2}+\frac {(5+12 b) \text {Subst}\left (\int \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{8 a^2} \\ & = -\frac {5 \left (a x+\sqrt {-b+a x}\right )^{3/2}}{12 a^2}+\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a^2}+\frac {(5+12 b) \sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{32 a^2}-\frac {((1-4 b) (5+12 b)) \text {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{64 a^2} \\ & = -\frac {5 \left (a x+\sqrt {-b+a x}\right )^{3/2}}{12 a^2}+\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a^2}+\frac {(5+12 b) \sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{32 a^2}-\frac {((1-4 b) (5+12 b)) \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 )}{32 a^2} \\ & = -\frac {5 \left (a x+\sqrt {-b+a x}\right )^{3/2}}{12 a^2}+\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a^2}+\frac {(5+12 b) \sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{32 a^2}-\frac {(1-4 b) (5+12 b) \text {arctanh}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{64 a^2} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96 \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\frac {2 \sqrt {a x+\sqrt {-b+a x}} \left (15-10 \sqrt {-b+a x}+12 b \left (-1+6 \sqrt {-b+a x}\right )+8 a \left (x+6 x \sqrt {-b+a x}\right )\right )-3 \left (-5+8 b+48 b^2\right ) \log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}\right )}{192 a^2} \]
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Time = 0.16 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.34
method | result | size |
derivativedivides | \(\frac {\frac {3 b \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{4}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{8}\right )}{2}+\frac {\sqrt {a x -b}\, \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{2}-\frac {5 \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{12}+\frac {5 \left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{32}+\frac {5 \left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{64}}{a^{2}}\) | \(182\) |
default | \(\frac {\frac {3 b \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{4}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{8}\right )}{2}+\frac {\sqrt {a x -b}\, \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{2}-\frac {5 \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{12}+\frac {5 \left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{32}+\frac {5 \left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{64}}{a^{2}}\) | \(182\) |
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Timed out. \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\text {Timed out} \]
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Time = 0.50 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.23 \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\begin {cases} \frac {2 \left (\sqrt {a x + \sqrt {a x - b}} \left (\frac {a x}{24} - \frac {b}{16} + \left (\frac {5 b}{8} - \frac {5}{96}\right ) \sqrt {a x - b} + \frac {\left (a x - b\right )^{\frac {3}{2}}}{4} + \frac {5}{64}\right ) + \left (b^{2} - b \left (\frac {5 b}{8} - \frac {5}{96}\right ) + \frac {b}{96} - \frac {5}{128}\right ) \left (\begin {cases} \log {\left (2 \sqrt {a x - b} + 2 \sqrt {a x + \sqrt {a x - b}} + 1 \right )} & \text {for}\: b \neq \frac {1}{4} \\\frac {\left (\sqrt {a x - b} + \frac {1}{2}\right ) \log {\left (\sqrt {a x - b} + \frac {1}{2} \right )}}{\sqrt {\left (\sqrt {a x - b} + \frac {1}{2}\right )^{2}}} & \text {otherwise} \end {cases}\right )\right )}{a^{2}} & \text {for}\: a \neq 0 \\\frac {x^{2}}{2 \sqrt [4]{- b}} & \text {otherwise} \end {cases} \]
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\[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\int { \frac {\sqrt {a x + \sqrt {a x - b}} x}{\sqrt {a x - b}} \,d x } \]
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Time = 0.61 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=-\frac {3 \, {\left (48 \, b^{2} + 8 \, b - 5\right )} \log \left ({\left | -2 \, \sqrt {a x - b} + 2 \, \sqrt {a x + \sqrt {a x - b}} - 1 \right |}\right ) - 2 \, \sqrt {a x + \sqrt {a x - b}} {\left (2 \, \sqrt {a x - b} {\left (4 \, \sqrt {a x - b} {\left (6 \, \sqrt {a x - b} + 1\right )} + 60 \, b - 5\right )} - 4 \, b + 15\right )}}{192 \, a^{2}} \]
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Timed out. \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\int \frac {x\,\sqrt {a\,x+\sqrt {a\,x-b}}}{\sqrt {a\,x-b}} \,d x \]
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