Integrand size = 28, antiderivative size = 118 \[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\frac {\left (-3+2 b \sqrt {b+a x}\right ) \sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}}{2 a b^2}+\frac {\left (-3-4 b^3\right ) \log \left (-1-2 b \sqrt {b+a x}+2 \sqrt {b} \sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}\right )}{4 a b^{5/2}} \]
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Time = 0.16 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {756, 654, 635, 212} \[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\frac {\left (4 b^3+3\right ) \text {arctanh}\left (\frac {2 b \sqrt {a x+b}+1}{2 \sqrt {b} \sqrt {b (a x+b)+\sqrt {a x+b}-b^2}}\right )}{4 a b^{5/2}}+\frac {\sqrt {a x+b} \sqrt {b (a x+b)+\sqrt {a x+b}-b^2}}{a b}-\frac {3 \sqrt {b (a x+b)+\sqrt {a x+b}-b^2}}{2 a b^2} \]
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Rule 212
Rule 635
Rule 654
Rule 756
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^2}{\sqrt {-b^2+x+b x^2}} \, dx,x,\sqrt {b+a x}\right )}{a} \\ & = \frac {\sqrt {b+a x} \sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}}{a b}+\frac {\text {Subst}\left (\int \frac {b^2-\frac {3 x}{2}}{\sqrt {-b^2+x+b x^2}} \, dx,x,\sqrt {b+a x}\right )}{a b} \\ & = -\frac {3 \sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}}{2 a b^2}+\frac {\sqrt {b+a x} \sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}}{a b}+\frac {\left (3+4 b^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2+x+b x^2}} \, dx,x,\sqrt {b+a x}\right )}{4 a b^2} \\ & = -\frac {3 \sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}}{2 a b^2}+\frac {\sqrt {b+a x} \sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}}{a b}+\frac {\left (3+4 b^3\right ) \text {Subst}\left (\int \frac {1}{4 b-x^2} \, dx,x,\frac {1+2 b \sqrt {b+a x}}{\sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}}\right )}{2 a b^2} \\ & = -\frac {3 \sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}}{2 a b^2}+\frac {\sqrt {b+a x} \sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}}{a b}+\frac {\left (3+4 b^3\right ) \text {arctanh}\left (\frac {1+2 b \sqrt {b+a x}}{2 \sqrt {b} \sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}}\right )}{4 a b^{5/2}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\frac {2 \sqrt {b} \sqrt {a b x+\sqrt {b+a x}} \left (-3+2 b \sqrt {b+a x}\right )-\left (3+4 b^3\right ) \log \left (a b^2 \left (1+2 b \sqrt {b+a x}-2 \sqrt {b} \sqrt {a b x+\sqrt {b+a x}}\right )\right )}{4 a b^{5/2}} \]
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Time = 0.16 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.36
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {a x +b}\, \sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}}{b}-\frac {3 \left (\frac {\sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}}{b}-\frac {\ln \left (\frac {\frac {1}{2}+b \sqrt {a x +b}}{\sqrt {b}}+\sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}\right )}{2 b^{\frac {3}{2}}}\right )}{2 b}+\sqrt {b}\, \ln \left (\frac {\frac {1}{2}+b \sqrt {a x +b}}{\sqrt {b}}+\sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}\right )}{a}\) | \(161\) |
default | \(\frac {\frac {\sqrt {a x +b}\, \sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}}{b}-\frac {3 \left (\frac {\sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}}{b}-\frac {\ln \left (\frac {\frac {1}{2}+b \sqrt {a x +b}}{\sqrt {b}}+\sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}\right )}{2 b^{\frac {3}{2}}}\right )}{2 b}+\sqrt {b}\, \ln \left (\frac {\frac {1}{2}+b \sqrt {a x +b}}{\sqrt {b}}+\sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}\right )}{a}\) | \(161\) |
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Timed out. \[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (99) = 198\).
Time = 0.64 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.92 \[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\begin {cases} \frac {2 \left (\begin {cases} \left (\frac {b}{2} + \frac {3}{8 b^{2}}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {- b^{2} + b \left (a x + b\right ) + \sqrt {a x + b}} + 2 b \sqrt {a x + b} + 1 \right )}}{\sqrt {b}} & \text {for}\: b^{2} + \frac {1}{4 b} \neq 0 \\\frac {\left (\sqrt {a x + b} + \frac {1}{2 b}\right ) \log {\left (\sqrt {a x + b} + \frac {1}{2 b} \right )}}{\sqrt {b \left (\sqrt {a x + b} + \frac {1}{2 b}\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \left (\frac {\sqrt {a x + b}}{2 b} - \frac {3}{4 b^{2}}\right ) \sqrt {- b^{2} + b \left (a x + b\right ) + \sqrt {a x + b}} & \text {for}\: b \neq 0 \\2 b^{4} \sqrt {- b^{2} + \sqrt {a x + b}} + \frac {4 b^{2} \left (- b^{2} + \sqrt {a x + b}\right )^{\frac {3}{2}}}{3} + \frac {2 \left (- b^{2} + \sqrt {a x + b}\right )^{\frac {5}{2}}}{5} & \text {otherwise} \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\\sqrt [4]{b} x & \text {otherwise} \end {cases} \]
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\[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\int { \frac {\sqrt {a x + b}}{\sqrt {a b x + \sqrt {a x + b}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\int \frac {\sqrt {b+a\,x}}{\sqrt {\sqrt {b+a\,x}+a\,b\,x}} \,d x \]
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