Optimal. Leaf size=87 \[ -\frac {4 \sqrt {\sqrt {a x+b}+c} \left (\sqrt {a x+b}+c+3 d\right )}{3 a}-\frac {4 d \sqrt {-c-d} \tan ^{-1}\left (\frac {\sqrt {-c-d} \sqrt {\sqrt {a x+b}+c}}{c+d}\right )}{a} \]
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Rubi [A] time = 0.06, antiderivative size = 81, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {431, 376, 80, 50, 63, 206} \begin {gather*} -\frac {4 d \sqrt {\sqrt {a x+b}+c}}{a}+\frac {4 d \sqrt {c+d} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a x+b}+c}}{\sqrt {c+d}}\right )}{a}-\frac {4 \left (\sqrt {a x+b}+c\right )^{3/2}}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 376
Rule 431
Rubi steps
\begin {align*} \int \frac {\sqrt {c+\sqrt {b+a x}}}{d-\sqrt {b+a x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {c+\sqrt {x}}}{d-\sqrt {x}} \, dx,x,b+a x\right )}{a}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x \sqrt {c+x}}{d-x} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=-\frac {4 \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a}+\frac {(2 d) \operatorname {Subst}\left (\int \frac {\sqrt {c+x}}{d-x} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=-\frac {4 d \sqrt {c+\sqrt {b+a x}}}{a}-\frac {4 \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a}+\frac {(2 d (c+d)) \operatorname {Subst}\left (\int \frac {1}{(d-x) \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=-\frac {4 d \sqrt {c+\sqrt {b+a x}}}{a}-\frac {4 \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a}+\frac {(4 d (c+d)) \operatorname {Subst}\left (\int \frac {1}{c+d-x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=-\frac {4 d \sqrt {c+\sqrt {b+a x}}}{a}-\frac {4 \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a}+\frac {4 d \sqrt {c+d} \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {b+a x}}}{\sqrt {c+d}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 73, normalized size = 0.84 \begin {gather*} \frac {12 d \sqrt {c+d} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a x+b}+c}}{\sqrt {c+d}}\right )-4 \sqrt {\sqrt {a x+b}+c} \left (\sqrt {a x+b}+c+3 d\right )}{3 a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 87, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt {c+\sqrt {b+a x}} \left (c+3 d+\sqrt {b+a x}\right )}{3 a}-\frac {4 \sqrt {-c-d} d \tan ^{-1}\left (\frac {\sqrt {-c-d} \sqrt {c+\sqrt {b+a x}}}{c+d}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 183, normalized size = 2.10 \begin {gather*} \left [\frac {2 \, {\left (3 \, \sqrt {c + d} d \log \left (-\frac {2 \, c d + d^{2} + a x + 2 \, \sqrt {a x + b} {\left (c + d\right )} + 2 \, {\left (\sqrt {c + d} d + \sqrt {a x + b} \sqrt {c + d}\right )} \sqrt {c + \sqrt {a x + b}} + b}{d^{2} - a x - b}\right ) - 2 \, {\left (c + 3 \, d + \sqrt {a x + b}\right )} \sqrt {c + \sqrt {a x + b}}\right )}}{3 \, a}, -\frac {4 \, {\left (3 \, \sqrt {-c - d} d \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}} \sqrt {-c - d}}{c + d}\right ) + {\left (c + 3 \, d + \sqrt {a x + b}\right )} \sqrt {c + \sqrt {a x + b}}\right )}}{3 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 82, normalized size = 0.94 \begin {gather*} -\frac {4 \, {\left (c d + d^{2}\right )} \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-c - d}}\right )}{a \sqrt {-c - d}} - \frac {4 \, {\left (a^{2} {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} + 3 \, a^{2} \sqrt {c + \sqrt {a x + b}} d\right )}}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 60, normalized size = 0.69
method | result | size |
derivativedivides | \(\frac {-\frac {4 \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}-4 d \sqrt {c +\sqrt {a x +b}}+4 d \sqrt {c +d}\, \arctanh \left (\frac {\sqrt {c +\sqrt {a x +b}}}{\sqrt {c +d}}\right )}{a}\) | \(60\) |
default | \(-\frac {2 \left (\frac {2 \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+2 d \sqrt {c +\sqrt {a x +b}}-2 d \sqrt {c +d}\, \arctanh \left (\frac {\sqrt {c +\sqrt {a x +b}}}{\sqrt {c +d}}\right )\right )}{a}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 88, normalized size = 1.01 \begin {gather*} -\frac {2 \, {\left (2 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} + 6 \, \sqrt {c + \sqrt {a x + b}} d + \frac {3 \, {\left (c d + d^{2}\right )} \log \left (-\frac {\sqrt {c + d} - \sqrt {c + \sqrt {a x + b}}}{\sqrt {c + d} + \sqrt {c + \sqrt {a x + b}}}\right )}{\sqrt {c + d}}\right )}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 63, normalized size = 0.72 \begin {gather*} \frac {4\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\sqrt {b+a\,x}}}{\sqrt {c+d}}\right )\,\sqrt {c+d}}{a}-\frac {4\,d\,\sqrt {c+\sqrt {b+a\,x}}}{a}-\frac {4\,{\left (c+\sqrt {b+a\,x}\right )}^{3/2}}{3\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.54, size = 78, normalized size = 0.90 \begin {gather*} - \frac {4 d \sqrt {c + \sqrt {a x + b}}}{a} - \frac {4 d \left (c + d\right ) \operatorname {atan}{\left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {- c - d}} \right )}}{a \sqrt {- c - d}} - \frac {4 \left (c + \sqrt {a x + b}\right )^{\frac {3}{2}}}{3 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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