Optimal. Leaf size=363 \[ -\frac {\sqrt {-b+a x} \sqrt {\frac {a x+\sqrt {-b+a x}}{\left (1+\sqrt {-b+a x}\right )^2}}}{2 a}+\frac {(-3-2 b+2 a x) \sqrt {\frac {a x+\sqrt {-b+a x}}{\left (1+\sqrt {-b+a x}\right )^2}}}{2 a}+\frac {(-3-4 b) \tanh ^{-1}\left (\frac {\sqrt {b}-\sqrt {\frac {a x+\sqrt {-b+a x}}{\left (1+\sqrt {-b+a x}\right )^2}}-\sqrt {-b+a x} \sqrt {\frac {a x+\sqrt {-b+a x}}{\left (1+\sqrt {-b+a x}\right )^2}}}{1+\sqrt {-b+a x}}\right )}{2 a}+\frac {2 \sqrt {b} \log \left (1+\sqrt {-b+a x}\right )}{a}-\frac {2 \sqrt {b} \log \left (1-2 b+2 \sqrt {b} \sqrt {\frac {a x+\sqrt {-b+a x}}{\left (1+\sqrt {-b+a x}\right )^2}}+\sqrt {-b+a x} \left (1+2 \sqrt {b} \sqrt {\frac {a x+\sqrt {-b+a x}}{\left (1+\sqrt {-b+a x}\right )^2}}\right )\right )}{a} \]
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Rubi [A]
time = 0.23, antiderivative size = 148, normalized size of antiderivative = 0.41, number of steps
used = 7, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {828, 857, 635,
212, 738} \begin {gather*} -\frac {\sqrt {\sqrt {a x-b}+a x} \left (3-2 \sqrt {a x-b}\right )}{2 a}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a x-b}-2 b+1}{2 \sqrt {b} \sqrt {\sqrt {a x-b}+a x}}\right )}{a}+\frac {(4 b+3) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 828
Rule 857
Rubi steps
\begin {align*} \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{1+\sqrt {-b+a x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x \sqrt {b+x+x^2}}{1+x} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {\left (3-2 \sqrt {-b+a x}\right ) \sqrt {a x+\sqrt {-b+a x}}}{2 a}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (-3+4 b)-\frac {1}{2} (3+4 b) x}{(1+x) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{2 a}\\ &=-\frac {\left (3-2 \sqrt {-b+a x}\right ) \sqrt {a x+\sqrt {-b+a x}}}{2 a}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {(3+4 b) \text {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{4 a}\\ &=-\frac {\left (3-2 \sqrt {-b+a x}\right ) \sqrt {a x+\sqrt {-b+a x}}}{2 a}+\frac {(4 b) \text {Subst}\left (\int \frac {1}{4 b-x^2} \, dx,x,\frac {-1+2 b-\sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}+\frac {(3+4 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 )}{2 a}\\ &=-\frac {\left (3-2 \sqrt {-b+a x}\right ) \sqrt {a x+\sqrt {-b+a x}}}{2 a}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {1-2 b+\sqrt {-b+a x}}{2 \sqrt {b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}+\frac {(3+4 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{4 a}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 137, normalized size = 0.38 \begin {gather*} -\frac {2 \left (3-2 \sqrt {-b+a x}\right ) \sqrt {a x+\sqrt {-b+a x}}+16 \sqrt {b} \tanh ^{-1}\left (\frac {1+\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}}{\sqrt {b}}\right )+(3+4 b) \log \left (a \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}\right )\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 214, normalized size = 0.59
method | result | size |
derivativedivides | \(\frac {\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{2}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{4}-2 \sqrt {\left (1+\sqrt {a x -b}\right )^{2}+b -\sqrt {a x -b}-1}+\ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (1+\sqrt {a x -b}\right )^{2}+b -\sqrt {a x -b}-1}\right )+2 \sqrt {b}\, \ln \left (\frac {2 b -\sqrt {a x -b}-1+2 \sqrt {b}\, \sqrt {\left (1+\sqrt {a x -b}\right )^{2}+b -\sqrt {a x -b}-1}}{1+\sqrt {a x -b}}\right )}{a}\) | \(214\) |
default | \(\frac {\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{2}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{4}-2 \sqrt {\left (1+\sqrt {a x -b}\right )^{2}+b -\sqrt {a x -b}-1}+\ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (1+\sqrt {a x -b}\right )^{2}+b -\sqrt {a x -b}-1}\right )+2 \sqrt {b}\, \ln \left (\frac {2 b -\sqrt {a x -b}-1+2 \sqrt {b}\, \sqrt {\left (1+\sqrt {a x -b}\right )^{2}+b -\sqrt {a x -b}-1}}{1+\sqrt {a x -b}}\right )}{a}\) | \(214\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x + \sqrt {a x - b}}}{\sqrt {a x - b} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 127, normalized size = 0.35 \begin {gather*} \frac {1}{2} \, \sqrt {a x + \sqrt {a x - b}} {\left (\frac {2 \, \sqrt {a x - b}}{a} - \frac {3}{a}\right )} - \frac {{\left (4 \, b + 3\right )} \log \left ({\left | -2 \, \sqrt {a x - b} + 2 \, \sqrt {a x + \sqrt {a x - b}} - 1 \right |}\right )}{4 \, a} - \frac {4 \, b \arctan \left (-\frac {\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}} + 1}{\sqrt {-b}}\right )}{a \sqrt {-b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a\,x+\sqrt {a\,x-b}}}{\sqrt {a\,x-b}+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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