Optimal. Leaf size=306 \[ 4 \sqrt {c+\sqrt {b+a x}}+\frac {2 \left (\sqrt {2} a^2+2 \sqrt {2} b+\sqrt {2} a \sqrt {a^2+4 b}+\sqrt {2} a c+\sqrt {2} \sqrt {a^2+4 b} c\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a-\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a-\sqrt {a^2+4 b}-2 c}}+\frac {2 \left (-\sqrt {2} a^2-2 \sqrt {2} b+\sqrt {2} a \sqrt {a^2+4 b}-\sqrt {2} a c+\sqrt {2} \sqrt {a^2+4 b} c\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a+\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a+\sqrt {a^2+4 b}-2 c}} \]
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Rubi [A]
time = 1.81, antiderivative size = 256, normalized size of antiderivative = 0.84, number of steps
used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {838, 840, 1180,
212} \begin {gather*} \frac {2 \sqrt {2} \left (-a \left (\sqrt {a^2+4 b}-c\right )-c \sqrt {a^2+4 b}+a^2+2 b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {-\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-\sqrt {a^2+4 b}+a+2 c}}-\frac {2 \sqrt {2} \left (a \left (\sqrt {a^2+4 b}+c\right )+c \sqrt {a^2+4 b}+a^2+2 b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {\sqrt {a^2+4 b}+a+2 c}}+4 \sqrt {\sqrt {a x+b}+c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 838
Rule 840
Rule 1180
Rubi steps
\begin {align*} \int \frac {\sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx &=-\left (2 \text {Subst}\left (\int \frac {x \sqrt {c+x}}{b+a x-x^2} \, dx,x,\sqrt {b+a x}\right )\right )\\ &=4 \sqrt {c+\sqrt {b+a x}}+2 \text {Subst}\left (\int \frac {-b-(a+c) x}{\sqrt {c+x} \left (b+a x-x^2\right )} \, dx,x,\sqrt {b+a x}\right )\\ &=4 \sqrt {c+\sqrt {b+a x}}+4 \text {Subst}\left (\int \frac {-b-(-a-c) c+(-a-c) x^2}{b-a c-c^2+(a+2 c) x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=4 \sqrt {c+\sqrt {b+a x}}+\frac {\left (2 \left (a^2+2 b-a \left (\sqrt {a^2+4 b}-c\right )-\sqrt {a^2+4 b} c\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {a^2+4 b}+\frac {1}{2} (a+2 c)-x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {a^2+4 b}}-\frac {\left (2 \left (a^2+2 b+\sqrt {a^2+4 b} c+a \left (\sqrt {a^2+4 b}+c\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {a^2+4 b}+\frac {1}{2} (a+2 c)-x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {a^2+4 b}}\\ &=4 \sqrt {c+\sqrt {b+a x}}+\frac {2 \sqrt {2} \left (a^2+2 b-a \left (\sqrt {a^2+4 b}-c\right )-\sqrt {a^2+4 b} c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {a-\sqrt {a^2+4 b}+2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {a-\sqrt {a^2+4 b}+2 c}}-\frac {2 \sqrt {2} \left (a^2+2 b+\sqrt {a^2+4 b} c+a \left (\sqrt {a^2+4 b}+c\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {a+\sqrt {a^2+4 b}+2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {a+\sqrt {a^2+4 b}+2 c}}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 265, normalized size = 0.87 \begin {gather*} 4 \sqrt {c+\sqrt {b+a x}}+\frac {2 \sqrt {2} \left (a^2+2 b+a \sqrt {a^2+4 b}+a c+\sqrt {a^2+4 b} c\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a-\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a-\sqrt {a^2+4 b}-2 c}}+\frac {2 \sqrt {2} \left (-a^2-2 b+a \sqrt {a^2+4 b}-a c+\sqrt {a^2+4 b} c\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a+\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a+\sqrt {a^2+4 b}-2 c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 212, normalized size = 0.69
method | result | size |
derivativedivides | \(4 \sqrt {c +\sqrt {a x +b}}+\frac {4 \left (a \sqrt {a^{2}+4 b}+c \sqrt {a^{2}+4 b}-a^{2}-a c -2 b \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}+\frac {4 \left (a \sqrt {a^{2}+4 b}+c \sqrt {a^{2}+4 b}+a^{2}+a c +2 b \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\) | \(212\) |
default | \(4 \sqrt {c +\sqrt {a x +b}}+\frac {4 \left (a \sqrt {a^{2}+4 b}+c \sqrt {a^{2}+4 b}-a^{2}-a c -2 b \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}+\frac {4 \left (a \sqrt {a^{2}+4 b}+c \sqrt {a^{2}+4 b}+a^{2}+a c +2 b \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\) | \(212\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1106 vs.
\(2 (242) = 484\).
time = 0.80, size = 1106, normalized size = 3.61 \begin {gather*} -\sqrt {2} \sqrt {\frac {a^{3} + 3 \, a b + {\left (a^{2} + 2 \, b\right )} c + {\left (a^{2} + 4 \, b\right )} \sqrt {\frac {a^{4} + a^{2} c^{2} + 2 \, a^{2} b + b^{2} + 2 \, {\left (a^{3} + a b\right )} c}{a^{2} + 4 \, b}}}{a^{2} + 4 \, b}} \log \left (8 \, \sqrt {2} {\left (a^{4} + 5 \, a^{2} b + 4 \, b^{2} + {\left (a^{3} + 4 \, a b\right )} c - {\left (a^{3} + 4 \, a b\right )} \sqrt {\frac {a^{4} + a^{2} c^{2} + 2 \, a^{2} b + b^{2} + 2 \, {\left (a^{3} + a b\right )} c}{a^{2} + 4 \, b}}\right )} \sqrt {\frac {a^{3} + 3 \, a b + {\left (a^{2} + 2 \, b\right )} c + {\left (a^{2} + 4 \, b\right )} \sqrt {\frac {a^{4} + a^{2} c^{2} + 2 \, a^{2} b + b^{2} + 2 \, {\left (a^{3} + a b\right )} c}{a^{2} + 4 \, b}}}{a^{2} + 4 \, b}} + 32 \, {\left (a^{2} b + a b c + b^{2}\right )} \sqrt {c + \sqrt {a x + b}}\right ) + \sqrt {2} \sqrt {\frac {a^{3} + 3 \, a b + {\left (a^{2} + 2 \, b\right )} c + {\left (a^{2} + 4 \, b\right )} \sqrt {\frac {a^{4} + a^{2} c^{2} + 2 \, a^{2} b + b^{2} + 2 \, {\left (a^{3} + a b\right )} c}{a^{2} + 4 \, b}}}{a^{2} + 4 \, b}} \log \left (-8 \, \sqrt {2} {\left (a^{4} + 5 \, a^{2} b + 4 \, b^{2} + {\left (a^{3} + 4 \, a b\right )} c - {\left (a^{3} + 4 \, a b\right )} \sqrt {\frac {a^{4} + a^{2} c^{2} + 2 \, a^{2} b + b^{2} + 2 \, {\left (a^{3} + a b\right )} c}{a^{2} + 4 \, b}}\right )} \sqrt {\frac {a^{3} + 3 \, a b + {\left (a^{2} + 2 \, b\right )} c + {\left (a^{2} + 4 \, b\right )} \sqrt {\frac {a^{4} + a^{2} c^{2} + 2 \, a^{2} b + b^{2} + 2 \, {\left (a^{3} + a b\right )} c}{a^{2} + 4 \, b}}}{a^{2} + 4 \, b}} + 32 \, {\left (a^{2} b + a b c + b^{2}\right )} \sqrt {c + \sqrt {a x + b}}\right ) - \sqrt {2} \sqrt {\frac {a^{3} + 3 \, a b + {\left (a^{2} + 2 \, b\right )} c - {\left (a^{2} + 4 \, b\right )} \sqrt {\frac {a^{4} + a^{2} c^{2} + 2 \, a^{2} b + b^{2} + 2 \, {\left (a^{3} + a b\right )} c}{a^{2} + 4 \, b}}}{a^{2} + 4 \, b}} \log \left (8 \, \sqrt {2} {\left (a^{4} + 5 \, a^{2} b + 4 \, b^{2} + {\left (a^{3} + 4 \, a b\right )} c + {\left (a^{3} + 4 \, a b\right )} \sqrt {\frac {a^{4} + a^{2} c^{2} + 2 \, a^{2} b + b^{2} + 2 \, {\left (a^{3} + a b\right )} c}{a^{2} + 4 \, b}}\right )} \sqrt {\frac {a^{3} + 3 \, a b + {\left (a^{2} + 2 \, b\right )} c - {\left (a^{2} + 4 \, b\right )} \sqrt {\frac {a^{4} + a^{2} c^{2} + 2 \, a^{2} b + b^{2} + 2 \, {\left (a^{3} + a b\right )} c}{a^{2} + 4 \, b}}}{a^{2} + 4 \, b}} + 32 \, {\left (a^{2} b + a b c + b^{2}\right )} \sqrt {c + \sqrt {a x + b}}\right ) + \sqrt {2} \sqrt {\frac {a^{3} + 3 \, a b + {\left (a^{2} + 2 \, b\right )} c - {\left (a^{2} + 4 \, b\right )} \sqrt {\frac {a^{4} + a^{2} c^{2} + 2 \, a^{2} b + b^{2} + 2 \, {\left (a^{3} + a b\right )} c}{a^{2} + 4 \, b}}}{a^{2} + 4 \, b}} \log \left (-8 \, \sqrt {2} {\left (a^{4} + 5 \, a^{2} b + 4 \, b^{2} + {\left (a^{3} + 4 \, a b\right )} c + {\left (a^{3} + 4 \, a b\right )} \sqrt {\frac {a^{4} + a^{2} c^{2} + 2 \, a^{2} b + b^{2} + 2 \, {\left (a^{3} + a b\right )} c}{a^{2} + 4 \, b}}\right )} \sqrt {\frac {a^{3} + 3 \, a b + {\left (a^{2} + 2 \, b\right )} c - {\left (a^{2} + 4 \, b\right )} \sqrt {\frac {a^{4} + a^{2} c^{2} + 2 \, a^{2} b + b^{2} + 2 \, {\left (a^{3} + a b\right )} c}{a^{2} + 4 \, b}}}{a^{2} + 4 \, b}} + 32 \, {\left (a^{2} b + a b c + b^{2}\right )} \sqrt {c + \sqrt {a x + b}}\right ) + 4 \, \sqrt {c + \sqrt {a x + b}} \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 {c + \sqrt {a x + b}}}{x - \sqrt {a x + b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 210, normalized size = 0.69 \begin {gather*} -\frac {4 \, \sqrt {a^{2} + 4 \, b} \sqrt {-2 \, a - 4 \, c + 2 \, \sqrt {a^{2} + 4 \, b}} b \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-\frac {1}{2} \, a - c + \frac {1}{2} \, \sqrt {{\left (a + 2 \, c\right )}^{2} - 4 \, a c - 4 \, c^{2} + 4 \, b}}}\right )}{a^{3} + 4 \, a b + {\left (a^{2} + 4 \, b\right )}^{\frac {3}{2}}} + \frac {4 \, \sqrt {a^{2} + 4 \, b} \sqrt {-2 \, a - 4 \, c - 2 \, \sqrt {a^{2} + 4 \, b}} b \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-\frac {1}{2} \, a - c - \frac {1}{2} \, \sqrt {{\left (a + 2 \, c\right )}^{2} - 4 \, a c - 4 \, c^{2} + 4 \, b}}}\right )}{a^{3} + 4 \, a b - {\left (a^{2} + 4 \, b\right )}^{\frac {3}{2}}} + 4 \, \sqrt {c + \sqrt {a x + 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 {c+\sqrt {b+a\,x}}}{x-\sqrt {b+a\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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