Optimal. Leaf size=248 \[ -\frac {3 \sqrt {a x+\sqrt {-b+a x}}}{2 a}+\sqrt {-b+a x} \left (-\frac {2}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}}}{a}\right )-\frac {4 (-3+2 b) \text {ArcTan}\left (\frac {1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-5+4 b}}\right )}{a \sqrt {-5+4 b}}+\frac {(-19+4 b) \log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}\right )}{4 a}+\frac {2 \log \left (1-2 a x-\sqrt {a x+\sqrt {-b+a x}}+2 \sqrt {-b+a x} \sqrt {a x+\sqrt {-b+a x}}\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.48, antiderivative size = 442, normalized size of antiderivative = 1.78, number of steps
used = 25, number of rules used = 14, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6874, 648,
632, 212, 642, 626, 635, 1003, 996, 1033, 1090, 1, 1039, 1038} \begin {gather*} \frac {\sqrt {\sqrt {a x-b}+a x} \left (2 \sqrt {a x-b}+1\right )}{2 a}-\frac {2 \sqrt {a x-b}}{a}-\frac {2 \sqrt {\sqrt {a x-b}+a x}}{a}+\frac {\log \left (-\sqrt {a x-b}-a x+1\right )}{a}+\frac {2 \tanh ^{-1}\left (\sqrt {\sqrt {a x-b}+a x}\right )}{a}+\frac {2 (3-2 b) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{\sqrt {5-4 b}}\right )}{a \sqrt {5-4 b}}-\frac {(1-4 b) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{4 a}+\frac {2 (1-b) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{a}+\frac {\tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{a}-\frac {4 (1-b) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{\sqrt {5-4 b} \sqrt {\sqrt {a x-b}+a x}}\right )}{a \sqrt {5-4 b}}-\frac {2 \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{\sqrt {5-4 b} \sqrt {\sqrt {a x-b}+a x}}\right )}{a \sqrt {5-4 b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1
Rule 212
Rule 626
Rule 632
Rule 635
Rule 642
Rule 648
Rule 996
Rule 1003
Rule 1033
Rule 1038
Rule 1039
Rule 1090
Rule 6874
Rubi steps
\begin {align*} \int \frac {\sqrt {-b+a x}}{1+\sqrt {a x+\sqrt {-b+a x}}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^2}{1+\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=\frac {2 \text {Subst}\left (\int \left (-1+\frac {-1+b+x}{-1+b+x+x^2}+\sqrt {b+x+x^2}+\frac {(1-b) \sqrt {b+x+x^2}}{-1+b+x+x^2}-\frac {x \sqrt {b+x+x^2}}{-1+b+x+x^2}\right ) \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}+\frac {2 \text {Subst}\left (\int \frac {-1+b+x}{-1+b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {2 \text {Subst}\left (\int \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {2 \text {Subst}\left (\int \frac {x \sqrt {b+x+x^2}}{-1+b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {(2 (1-b)) \text {Subst}\left (\int \frac {\sqrt {b+x+x^2}}{-1+b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}-\frac {2 \sqrt {a x+\sqrt {-b+a x}}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{2 a}+\frac {\text {Subst}\left (\int \frac {1+2 x}{-1+b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} (-1+b)-\frac {x}{2}+\frac {x^2}{2}}{\left (-1+b+x+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {(1-4 b) \text {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{4 a}-\frac {(3-2 b) \text {Subst}\left (\int \frac {1}{-1+b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {(2 (1-b)) \text {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {(2 (1-b)) \text {Subst}\left (\int \frac {1}{\left (-1+b+x+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}-\frac {2 \sqrt {a x+\sqrt {-b+a x}}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{2 a}+\frac {\log \left (1-a x-\sqrt {-b+a x}\right )}{a}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {2 \text {Subst}\left (\int \frac {\frac {1-b}{2}+\frac {1}{2} (-1+b)-x}{\left (-1+b+x+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {(1-4 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 {(2 (3-2 b)) \text {Subst}\left (\int \frac {1}{5-4 b-x^2} \, dx,x,1+2 \sqrt {-b+a x}\right )}{a}+\frac {(4 (1-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 )}{a}-\frac {(4 (1-b)) \text {Subst}\left (\int \frac {1}{1-4 (-1+b)-x^2} \, dx,x,\frac {1+2 \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}-\frac {2 \sqrt {a x+\sqrt {-b+a x}}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{2 a}+\frac {2 (3-2 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b}}\right )}{a \sqrt {5-4 b}}-\frac {(1-4 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{4 a}+\frac {2 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {4 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{a \sqrt {5-4 b}}+\frac {\log \left (1-a x-\sqrt {-b+a x}\right )}{a}+\frac {2 \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 )}{a}-\frac {2 \text {Subst}\left (\int \frac {x}{\left (-1+b+x+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}-\frac {2 \sqrt {a x+\sqrt {-b+a x}}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{2 a}+\frac {2 (3-2 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b}}\right )}{a \sqrt {5-4 b}}+\frac {\tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {(1-4 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{4 a}+\frac {2 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {4 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{a \sqrt {5-4 b}}+\frac {\log \left (1-a x-\sqrt {-b+a x}\right )}{a}+\frac {\text {Subst}\left (\int \frac {1}{\left (-1+b+x+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {\text {Subst}\left (\int \frac {1+2 x}{\left (-1+b+x+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}-\frac {2 \sqrt {a x+\sqrt {-b+a x}}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{2 a}+\frac {2 (3-2 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b}}\right )}{a \sqrt {5-4 b}}+\frac {\tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {(1-4 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{4 a}+\frac {2 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {4 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{a \sqrt {5-4 b}}+\frac {\log \left (1-a x-\sqrt {-b+a x}\right )}{a}+\frac {2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a x}}\right )}{a}-\frac {2 \text {Subst}\left (\int \frac {1}{1-4 (-1+b)-x^2} \, dx,x,\frac {1+2 \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}-\frac {2 \sqrt {a x+\sqrt {-b+a x}}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{2 a}+\frac {2 \tanh ^{-1}\left (\sqrt {a x+\sqrt {-b+a x}}\right )}{a}+\frac {2 (3-2 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b}}\right )}{a \sqrt {5-4 b}}+\frac {\tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {(1-4 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{4 a}+\frac {2 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {2 \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{a \sqrt {5-4 b}}-\frac {4 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{a \sqrt {5-4 b}}+\frac {\log \left (1-a x-\sqrt {-b+a x}\right )}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.60, size = 216, normalized size = 0.87 \begin {gather*} \frac {-8 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}} \left (-3+2 \sqrt {-b+a x}\right )-\frac {16 (-3+2 b) \text {ArcTan}\left (\frac {1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-5+4 b}}\right )}{\sqrt {-5+4 b}}+(-19+4 b) \log \left (a \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}\right )\right )+8 \log \left (1-2 b+2 (b-a x)+\sqrt {a x+\sqrt {-b+a x}} \left (-1+2 \sqrt {-b+a x}\right )\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1550\) vs.
\(2(208)=416\).
time = 0.24, size = 1551, normalized size = 6.25
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1551\) |
default | \(\text {Expression too large to display}\) | \(1551\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x - b}}{\sqrt {a x + \sqrt {a x - b}} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.76, size = 395, normalized size = 1.59 \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 - 11\right )} \log \left ({\left | -2 \, \sqrt {a x - b} + 2 \, \sqrt {a x + \sqrt {a x - b}} - 1 \right |}\right )}{4 \, a} + \frac {2 \, {\left (2 \, b - 3\right )} \arctan \left (-\frac {2 \, \sqrt {a x - b} - 2 \, \sqrt {a x + \sqrt {a x - b}} + 3}{\sqrt {4 \, b - 5}}\right )}{a \sqrt {4 \, b - 5}} - \frac {2 \, {\left (2 \, b - 3\right )} \arctan \left (-\frac {2 \, \sqrt {a x - b} - 2 \, \sqrt {a x + \sqrt {a x - b}} - 1}{\sqrt {4 \, b - 5}}\right )}{a \sqrt {4 \, b - 5}} + \frac {2 \, {\left (2 \, b - 3\right )} \arctan \left (\frac {2 \, \sqrt {a x - b} + 1}{\sqrt {4 \, b - 5}}\right )}{a \sqrt {4 \, b - 5}} + \frac {\log \left (a x + \sqrt {a x - b} - 1\right )}{a} - \frac {\log \left ({\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{2} + b + 3 \, \sqrt {a x - b} - 3 \, \sqrt {a x + \sqrt {a x - b}} + 1\right )}{a} + \frac {\log \left ({\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{2} + b - \sqrt {a x - b} + \sqrt {a x + \sqrt {a x - b}} - 1\right )}{a} - \frac {2 \, \sqrt {a x - b}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a\,x-b}}{\sqrt {a\,x+\sqrt {a\,x-b}}+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________