Optimal. Leaf size=136 \[ \frac {\left (-48 b^2+56 b-11\right ) \log \left (2 \sqrt {a x-b}-2 \sqrt {\sqrt {a x-b}+a x}+1\right )}{64 a}+\frac {\sqrt {\sqrt {a x-b}+a x} (8 a x-12 b-33)}{96 a}+\frac {\sqrt {a x-b} (24 a x+36 b-53) \sqrt {\sqrt {a x-b}+a x}}{48 a} \]
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Rubi [A] time = 0.46, antiderivative size = 166, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {1661, 640, 612, 621, 206} \begin {gather*} \frac {\sqrt {a x-b} \left (\sqrt {a x-b}+a x\right )^{3/2}}{2 a}-\frac {5 \left (\sqrt {a x-b}+a x\right )^{3/2}}{12 a}-\frac {(11-12 b) \left (2 \sqrt {a x-b}+1\right ) \sqrt {\sqrt {a x-b}+a x}}{32 a}+\frac {(11-12 b) (1-4 b) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{64 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \frac {(-1+a x) \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \left (-1+b+x^2\right ) \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a}+\frac {\operatorname {Subst}\left (\int \left (-4+3 b-\frac {5 x}{2}\right ) \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{2 a}\\ &=-\frac {5 \left (a x+\sqrt {-b+a x}\right )^{3/2}}{12 a}+\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a}-\frac {(11-12 b) \operatorname {Subst}\left (\int \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{8 a}\\ &=-\frac {5 \left (a x+\sqrt {-b+a x}\right )^{3/2}}{12 a}+\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a}-\frac {(11-12 b) \sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{32 a}+\frac {((11-12 b) (1-4 b)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{64 a}\\ &=-\frac {5 \left (a x+\sqrt {-b+a x}\right )^{3/2}}{12 a}+\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a}-\frac {(11-12 b) \sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{32 a}+\frac {((11-12 b) (1-4 b)) \operatorname {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}\\ &=-\frac {5 \left (a x+\sqrt {-b+a x}\right )^{3/2}}{12 a}+\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a}-\frac {(11-12 b) \sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{32 a}+\frac {(11-12 b) (1-4 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{64 a}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 132, normalized size = 0.97 \begin {gather*} \frac {3 \left (48 b^2-56 b+11\right ) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )+2 \sqrt {\sqrt {a x-b}+a x} \left (12 b \left (6 \sqrt {a x-b}-1\right )+8 a \left (6 x \sqrt {a x-b}+x\right )-106 \sqrt {a x-b}-33\right )}{192 a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 139, normalized size = 1.02 \begin {gather*} \frac {\left (-48 b^2+56 b-11\right ) \log \left (a \left (-2 \sqrt {a x-b}-1\right )+2 a \sqrt {\sqrt {a x-b}+a x}\right )}{64 a}+\frac {\sqrt {\sqrt {a x-b}+a x} \left (48 (a x-b)^{3/2}+8 (a x-b)+120 b \sqrt {a x-b}-106 \sqrt {a x-b}-4 b-33\right )}{96 a} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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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giac [A] time = 1.28, size = 181, normalized size = 1.33 \begin {gather*} -\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 ) - 48 \, {\left (4 \, b - 1\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 )} + 96 \, \sqrt {a x + \sqrt {a x - b}} {\left (2 \, \sqrt {a x - b} + 1\right )}}{192 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 251, normalized size = 1.85 \begin {gather*} \frac {\sqrt {a x -b}\, \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{2 a}-\frac {5 \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{12 a}-\frac {11 \sqrt {a x -b}\, \sqrt {a x +\sqrt {a x -b}}}{16 a}-\frac {11 \sqrt {a x +\sqrt {a x -b}}}{32 a}-\frac {7 \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right ) b}{8 a}+\frac {11 \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{64 a}+\frac {3 b \sqrt {a x +\sqrt {a x -b}}\, \sqrt {a x -b}}{4 a}+\frac {3 b \sqrt {a x +\sqrt {a x -b}}}{8 a}+\frac {3 \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right ) b^{2}}{4 a} \end {gather*}
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*} \int \frac {\sqrt {a x + \sqrt {a x - b}} {\left (a x - 1\right )}}{\sqrt {a x - b}}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {\sqrt {a\,x+\sqrt {a\,x-b}}\,\left (a\,x-1\right )}{\sqrt {a\,x-b}} \,d x \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 {\left (a x - 1\right ) \sqrt {a x + \sqrt {a x - b}}}{\sqrt {a x - b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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