Optimal. Leaf size=119 \[ -\frac {4 \left (3 a x+3 b-2 c^2+5 c+15\right ) \sqrt {\sqrt {a x+b}+c}}{15 a}-\frac {4 (c+5) \sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}}{15 a}-\frac {4 \sqrt {-c-1} \tan ^{-1}\left (\frac {\sqrt {-c-1} \sqrt {\sqrt {a x+b}+c}}{c+1}\right )}{a} \]
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Rubi [A] time = 0.12, antiderivative size = 106, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {513, 446, 88, 50, 63, 206} \begin {gather*} -\frac {4 \left (\sqrt {a x+b}+c\right )^{5/2}}{5 a}-\frac {4 (1-c) \left (\sqrt {a x+b}+c\right )^{3/2}}{3 a}-\frac {4 \sqrt {\sqrt {a x+b}+c}}{a}+\frac {4 \sqrt {c+1} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a x+b}+c}}{\sqrt {c+1}}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 206
Rule 446
Rule 513
Rubi steps
\begin {align*} \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{1-\sqrt {b+a x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {c+\sqrt {x}} \sqrt {x}}{1-\sqrt {x}} \, dx,x,b+a x\right )}{a}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x^2 \sqrt {c+x}}{1-x} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=\frac {2 \operatorname {Subst}\left (\int \left ((-1+c) \sqrt {c+x}+\frac {\sqrt {c+x}}{1-x}-(c+x)^{3/2}\right ) \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=-\frac {4 (1-c) \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a}-\frac {4 \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a}+\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt {c+x}}{1-x} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=-\frac {4 \sqrt {c+\sqrt {b+a x}}}{a}-\frac {4 (1-c) \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a}-\frac {4 \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a}+\frac {(2 (1+c)) \operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=-\frac {4 \sqrt {c+\sqrt {b+a x}}}{a}-\frac {4 (1-c) \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a}-\frac {4 \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a}+\frac {(4 (1+c)) \operatorname {Subst}\left (\int \frac {1}{1+c-x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=-\frac {4 \sqrt {c+\sqrt {b+a x}}}{a}-\frac {4 (1-c) \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a}-\frac {4 \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a}+\frac {4 \sqrt {1+c} \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {b+a x}}}{\sqrt {1+c}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 96, normalized size = 0.81 \begin {gather*} \frac {60 \sqrt {c+1} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a x+b}+c}}{\sqrt {c+1}}\right )-4 \sqrt {\sqrt {a x+b}+c} \left (c \left (\sqrt {a x+b}+5\right )+5 \sqrt {a x+b}+3 a x+3 b-2 c^2+15\right )}{15 a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 108, normalized size = 0.91 \begin {gather*} \frac {4 \sqrt {c+\sqrt {b+a x}} \left (-15-5 c+2 c^2-5 \sqrt {b+a x}-c \sqrt {b+a x}-3 (b+a x)\right )}{15 a}-\frac {4 \sqrt {-1-c} \tan ^{-1}\left (\frac {\sqrt {-1-c} \sqrt {c+\sqrt {b+a x}}}{1+c}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 200, normalized size = 1.68 \begin {gather*} \left [\frac {2 \, {\left (2 \, {\left (2 \, c^{2} - 3 \, a x - \sqrt {a x + b} {\left (c + 5\right )} - 3 \, b - 5 \, c - 15\right )} \sqrt {c + \sqrt {a x + b}} + 15 \, \sqrt {c + 1} \log \left (\frac {a x + 2 \, {\left (\sqrt {a x + b} \sqrt {c + 1} + \sqrt {c + 1}\right )} \sqrt {c + \sqrt {a x + b}} + 2 \, \sqrt {a x + b} {\left (c + 1\right )} + b + 2 \, c + 1}{a x + b - 1}\right )\right )}}{15 \, a}, \frac {4 \, {\left ({\left (2 \, c^{2} - 3 \, a x - \sqrt {a x + b} {\left (c + 5\right )} - 3 \, b - 5 \, c - 15\right )} \sqrt {c + \sqrt {a x + b}} - 15 \, \sqrt {-c - 1} \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}} \sqrt {-c - 1}}{c + 1}\right )\right )}}{15 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 107, normalized size = 0.90 \begin {gather*} -\frac {4 \, {\left (c + 1\right )} \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-c - 1}}\right )}{a \sqrt {-c - 1}} - \frac {4 \, {\left (3 \, a^{4} {\left (c + \sqrt {a x + b}\right )}^{\frac {5}{2}} - 5 \, a^{4} {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} c + 5 \, a^{4} {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} + 15 \, a^{4} \sqrt {c + \sqrt {a x + b}}\right )}}{15 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 85, normalized size = 0.71
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {2 \left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}-\frac {2 c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+\frac {2 \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+2 \sqrt {c +\sqrt {a x +b}}-2 \sqrt {1+c}\, \arctanh \left (\frac {\sqrt {c +\sqrt {a x +b}}}{\sqrt {1+c}}\right )\right )}{a}\) | \(85\) |
default | \(-\frac {2 \left (\frac {2 \left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}-\frac {2 c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+\frac {2 \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+2 \sqrt {c +\sqrt {a x +b}}-2 \sqrt {1+c}\, \arctanh \left (\frac {\sqrt {c +\sqrt {a x +b}}}{\sqrt {1+c}}\right )\right )}{a}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.80, size = 95, normalized size = 0.80 \begin {gather*} -\frac {2 \, {\left (6 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {5}{2}} - 10 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} {\left (c - 1\right )} + 15 \, \sqrt {c + 1} \log \left (\frac {\sqrt {c + \sqrt {a x + b}} - \sqrt {c + 1}}{\sqrt {c + \sqrt {a x + b}} + \sqrt {c + 1}}\right ) + 30 \, \sqrt {c + \sqrt {a x + b}}\right )}}{15 \, a} \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 {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}}{\sqrt {b+a\,x}-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.58, size = 90, normalized size = 0.76 \begin {gather*} \frac {2 \left (\frac {2 \left (c - 1\right ) \left (c + \sqrt {a x + b}\right )^{\frac {3}{2}}}{3} - \frac {2 \left (c + \sqrt {a x + b}\right )^{\frac {5}{2}}}{5} - 2 \sqrt {c + \sqrt {a x + b}} - \frac {2 \left (c + 1\right ) \operatorname {atan}{\left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {- c - 1}} \right )}}{\sqrt {- c - 1}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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