Optimal. Leaf size=407 \[ \frac {2 \left (\sqrt {2} a^3+\sqrt {2} a c \sqrt {a^2+4 b}+\sqrt {2} a^2 \sqrt {a^2+4 b}+\sqrt {2} b \sqrt {a^2+4 b}+\sqrt {2} a^2 c+3 \sqrt {2} a b+2 \sqrt {2} b c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {-\sqrt {a^2+4 b}-a-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-\sqrt {a^2+4 b}-a-2 c}}+\frac {2 \left (-\sqrt {2} a^3+\sqrt {2} a c \sqrt {a^2+4 b}+\sqrt {2} a^2 \sqrt {a^2+4 b}+\sqrt {2} b \sqrt {a^2+4 b}-\sqrt {2} a^2 c-3 \sqrt {2} a b-2 \sqrt {2} b c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {\sqrt {a^2+4 b}-a-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {\sqrt {a^2+4 b}-a-2 c}}+\frac {4}{3} (3 a+c) \sqrt {\sqrt {a x+b}+c}+\frac {4}{3} \sqrt {a x+b} \sqrt {\sqrt {a x+b}+c} \]
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Rubi [A] time = 4.65, antiderivative size = 255, normalized size of antiderivative = 0.63, number of steps used = 7, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {897, 1287, 1166, 206} \begin {gather*} -\frac {2 \sqrt {2} \left (a^2-\frac {a^3+a^2 c+3 a b+2 b c}{\sqrt {a^2+4 b}}+a c+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 {-\sqrt {a^2+4 b}+a+2 c}}-\frac {2 \sqrt {2} \left (a^2+\frac {a^3+a^2 c+3 a b+2 b c}{\sqrt {a^2+4 b}}+a c+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 {\sqrt {a^2+4 b}+a+2 c}}+\frac {4}{3} \left (\sqrt {a x+b}+c\right )^{3/2}+4 a \sqrt {\sqrt {a x+b}+c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 897
Rule 1166
Rule 1287
Rubi steps
\begin {align*} \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {x^2 \sqrt {c+x}}{b+a x-x^2} \, dx,x,\sqrt {b+a x}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {x^2 \left (-c+x^2\right )^2}{b-a c-c^2+(a+2 c) x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \left (-a-x^2+\frac {a (b-c (a+c))+\left (a^2+b+a c\right ) x^2}{b-a c-c^2+(a+2 c) x^2-x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\right )\\ &=4 a \sqrt {c+\sqrt {b+a x}}+\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-4 \operatorname {Subst}\left (\int \frac {a (b-c (a+c))+\left (a^2+b+a c\right ) x^2}{b-a c-c^2+(a+2 c) x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=4 a \sqrt {c+\sqrt {b+a x}}+\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-\left (2 \left (a^2+b+a c-\frac {a^3+3 a b+a^2 c+2 b c}{\sqrt {a^2+4 b}}\right )\right ) \operatorname {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 )-\left (2 \left (a^2+b+a c+\frac {a^3+3 a b+a^2 c+2 b c}{\sqrt {a^2+4 b}}\right )\right ) \operatorname {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 )\\ &=4 a \sqrt {c+\sqrt {b+a x}}+\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-\frac {2 \sqrt {2} \left (a^2+b+a c-\frac {a^3+3 a b+a^2 c+2 b c}{\sqrt {a^2+4 b}}\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-\sqrt {a^2+4 b}+2 c}}-\frac {2 \sqrt {2} \left (a^2+b+a c+\frac {a^3+3 a b+a^2 c+2 b c}{\sqrt {a^2+4 b}}\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+\sqrt {a^2+4 b}+2 c}}\\ \end {align*}
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Mathematica [A] time = 4.45, size = 328, normalized size = 0.81 \begin {gather*} \frac {\sqrt {\sqrt {a x+b}+c} \left (-2 a \left (-3 \sqrt {a^2+4 b}+\sqrt {a x+b}+c\right )+2 a \left (3 \sqrt {a^2+4 b}+\sqrt {a x+b}+c\right )-\frac {3 \left (a \sqrt {a^2+4 b}-a^2-2 b\right ) \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {\sqrt {a x+b}+c}{-\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {\frac {\sqrt {a x+b}+c}{-2 \sqrt {a^2+4 b}+2 a+4 c}}}-\frac {3 \sqrt {2} \left (a \sqrt {a^2+4 b}+a^2+2 b\right ) \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {\sqrt {a x+b}+c}{\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {\frac {\sqrt {a x+b}+c}{\sqrt {a^2+4 b}+a+2 c}}}+4 c \sqrt {a^2+4 b}+4 \sqrt {a^2+4 b} \sqrt {a x+b}\right )}{3 \sqrt {a^2+4 b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.89, size = 388, normalized size = 0.95 \begin {gather*} \frac {4}{3} \sqrt {c+\sqrt {b+a x}} \left (3 a+c+\sqrt {b+a x}\right )+\frac {2 \left (\sqrt {2} a^3+3 \sqrt {2} a b+\sqrt {2} a^2 \sqrt {a^2+4 b}+\sqrt {2} b \sqrt {a^2+4 b}+\sqrt {2} a^2 c+2 \sqrt {2} b c+\sqrt {2} a \sqrt {a^2+4 b} c\right ) \tan ^{-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 \left (-\sqrt {2} a^3-3 \sqrt {2} a b+\sqrt {2} a^2 \sqrt {a^2+4 b}+\sqrt {2} b \sqrt {a^2+4 b}-\sqrt {2} a^2 c-2 \sqrt {2} b c+\sqrt {2} a \sqrt {a^2+4 b} c\right ) \tan ^{-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 {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 2014, normalized size = 4.95
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 259, normalized size = 0.64 \begin {gather*} \frac {4 \, \sqrt {a^{2} + 4 \, b} \sqrt {-2 \, a - 4 \, c + 2 \, \sqrt {a^{2} + 4 \, b}} b^{2} \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^{4} + 6 \, a^{2} b + 8 \, b^{2} + {\left (a^{3} + 4 \, a b\right )} \sqrt {a^{2} + 4 \, b}} - \frac {4 \, \sqrt {a^{2} + 4 \, b} \sqrt {-2 \, a - 4 \, c - 2 \, \sqrt {a^{2} + 4 \, b}} b^{2} \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^{4} + 6 \, a^{2} b + 8 \, b^{2} - {\left (a^{3} + 4 \, a b\right )} \sqrt {a^{2} + 4 \, b}} + 4 \, a \sqrt {c + \sqrt {a x + b}} + \frac {4}{3} \, {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 268, normalized size = 0.66
method | result | size |
derivativedivides | \(\frac {4 \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+4 a \sqrt {c +\sqrt {a x +b}}+\frac {4 \left (a^{2} \sqrt {a^{2}+4 b}+a c \sqrt {a^{2}+4 b}+a^{3}+a^{2} c +b \sqrt {a^{2}+4 b}+3 a b +2 b c \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}+\frac {4 \left (a^{2} \sqrt {a^{2}+4 b}+a c \sqrt {a^{2}+4 b}-a^{3}-a^{2} c +b \sqrt {a^{2}+4 b}-3 a b -2 b c \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\) | \(268\) |
default | \(\frac {4 \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+4 a \sqrt {c +\sqrt {a x +b}}+\frac {4 \left (a^{2} \sqrt {a^{2}+4 b}+a c \sqrt {a^{2}+4 b}+a^{3}+a^{2} c +b \sqrt {a^{2}+4 b}+3 a b +2 b c \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}+\frac {4 \left (a^{2} \sqrt {a^{2}+4 b}+a c \sqrt {a^{2}+4 b}-a^{3}-a^{2} c +b \sqrt {a^{2}+4 b}-3 a b -2 b c \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\) | \(268\) |
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 + b} \sqrt {c + \sqrt {a x + b}}}{x - \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 = -0.00 \begin {gather*} \int \frac {\sqrt {c+\sqrt {b+a\,x}}\,\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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sympy [B] time = 123.16, size = 1095, normalized size = 2.69 \begin {gather*} - 4 a^{2} c \operatorname {RootSum} {\left (t^{4} \left (16 a^{5} c - 16 a^{4} b + 16 a^{4} c^{2} + 128 a^{3} b c - 128 a^{2} b^{2} + 128 a^{2} b c^{2} + 256 a b^{2} c - 256 b^{3} + 256 b^{2} c^{2}\right ) + t^{2} \left (- 4 a^{3} - 8 a^{2} c - 16 a b - 32 b c\right ) + 1, \left (t \mapsto t \log {\left (8 t^{3} a^{4} c - 8 t^{3} a^{3} b + 24 t^{3} a^{3} c^{2} + 16 t^{3} a^{2} b c + 16 t^{3} a^{2} c^{3} - 32 t^{3} a b^{2} + 96 t^{3} a b c^{2} - 64 t^{3} b^{2} c + 64 t^{3} b c^{3} - 2 t a^{2} - 4 t a c - 4 t b - 4 t c^{2} + \sqrt {c + \sqrt {a x + b}} \right )} \right )\right )} + 4 a^{2} \operatorname {RootSum} {\left (t^{4} \left (16 a^{4} + 128 a^{2} b + 256 b^{2}\right ) + t^{2} \left (- 4 a^{3} - 8 a^{2} c - 16 a b - 32 b c\right ) + a c - b + c^{2}, \left (t \mapsto t \log {\left (- 16 t^{3} a^{2} - 64 t^{3} b + 2 t a + 4 t c + \sqrt {c + \sqrt {a x + b}} \right )} \right )\right )} + 4 a b \operatorname {RootSum} {\left (t^{4} \left (16 a^{5} c - 16 a^{4} b + 16 a^{4} c^{2} + 128 a^{3} b c - 128 a^{2} b^{2} + 128 a^{2} b c^{2} + 256 a b^{2} c - 256 b^{3} + 256 b^{2} c^{2}\right ) + t^{2} \left (- 4 a^{3} - 8 a^{2} c - 16 a b - 32 b c\right ) + 1, \left (t \mapsto t \log {\left (8 t^{3} a^{4} c - 8 t^{3} a^{3} b + 24 t^{3} a^{3} c^{2} + 16 t^{3} a^{2} b c + 16 t^{3} a^{2} c^{3} - 32 t^{3} a b^{2} + 96 t^{3} a b c^{2} - 64 t^{3} b^{2} c + 64 t^{3} b c^{3} - 2 t a^{2} - 4 t a c - 4 t b - 4 t c^{2} + \sqrt {c + \sqrt {a x + b}} \right )} \right )\right )} - 4 a c^{2} \operatorname {RootSum} {\left (t^{4} \left (16 a^{5} c - 16 a^{4} b + 16 a^{4} c^{2} + 128 a^{3} b c - 128 a^{2} b^{2} + 128 a^{2} b c^{2} + 256 a b^{2} c - 256 b^{3} + 256 b^{2} c^{2}\right ) + t^{2} \left (- 4 a^{3} - 8 a^{2} c - 16 a b - 32 b c\right ) + 1, \left (t \mapsto t \log {\left (8 t^{3} a^{4} c - 8 t^{3} a^{3} b + 24 t^{3} a^{3} c^{2} + 16 t^{3} a^{2} b c + 16 t^{3} a^{2} c^{3} - 32 t^{3} a b^{2} + 96 t^{3} a b c^{2} - 64 t^{3} b^{2} c + 64 t^{3} b c^{3} - 2 t a^{2} - 4 t a c - 4 t b - 4 t c^{2} + \sqrt {c + \sqrt {a x + b}} \right )} \right )\right )} + 4 a c \operatorname {RootSum} {\left (t^{4} \left (16 a^{4} + 128 a^{2} b + 256 b^{2}\right ) + t^{2} \left (- 4 a^{3} - 8 a^{2} c - 16 a b - 32 b c\right ) + a c - b + c^{2}, \left (t \mapsto t \log {\left (- 16 t^{3} a^{2} - 64 t^{3} b + 2 t a + 4 t c + \sqrt {c + \sqrt {a x + b}} \right )} \right )\right )} + 4 a \sqrt {c + \sqrt {a x + b}} + 4 b \operatorname {RootSum} {\left (t^{4} \left (16 a^{4} + 128 a^{2} b + 256 b^{2}\right ) + t^{2} \left (- 4 a^{3} - 8 a^{2} c - 16 a b - 32 b c\right ) + a c - b + c^{2}, \left (t \mapsto t \log {\left (- 16 t^{3} a^{2} - 64 t^{3} b + 2 t a + 4 t c + \sqrt {c + \sqrt {a x + b}} \right )} \right )\right )} + \frac {4 \left (c + \sqrt {a x + b}\right )^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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