Optimal. Leaf size=95 \[ \frac {8 c \sqrt {c-a c x}}{a}+\frac {4 (c-a c x)^{3/2}}{3 a}+\frac {2 (c-a c x)^{5/2}}{5 a c}-\frac {8 \sqrt {2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
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Rubi [A]
time = 0.06, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6265, 21, 52,
65, 212} \begin {gather*} -\frac {8 \sqrt {2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}+\frac {2 (c-a c x)^{5/2}}{5 a c}+\frac {4 (c-a c x)^{3/2}}{3 a}+\frac {8 c \sqrt {c-a c x}}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 52
Rule 65
Rule 212
Rule 6265
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=\int \frac {(1-a x) (c-a c x)^{3/2}}{1+a x} \, dx\\ &=\frac {\int \frac {(c-a c x)^{5/2}}{1+a x} \, dx}{c}\\ &=\frac {2 (c-a c x)^{5/2}}{5 a c}+2 \int \frac {(c-a c x)^{3/2}}{1+a x} \, dx\\ &=\frac {4 (c-a c x)^{3/2}}{3 a}+\frac {2 (c-a c x)^{5/2}}{5 a c}+(4 c) \int \frac {\sqrt {c-a c x}}{1+a x} \, dx\\ &=\frac {8 c \sqrt {c-a c x}}{a}+\frac {4 (c-a c x)^{3/2}}{3 a}+\frac {2 (c-a c x)^{5/2}}{5 a c}+\left (8 c^2\right ) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=\frac {8 c \sqrt {c-a c x}}{a}+\frac {4 (c-a c x)^{3/2}}{3 a}+\frac {2 (c-a c x)^{5/2}}{5 a c}-\frac {(16 c) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a}\\ &=\frac {8 c \sqrt {c-a c x}}{a}+\frac {4 (c-a c x)^{3/2}}{3 a}+\frac {2 (c-a c x)^{5/2}}{5 a c}-\frac {8 \sqrt {2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 71, normalized size = 0.75 \begin {gather*} \frac {2 c \sqrt {c-a c x} \left (73-16 a x+3 a^2 x^2\right )-120 \sqrt {2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{15 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.96, size = 73, normalized size = 0.77
method | result | size |
risch | \(-\frac {2 \left (3 a^{2} x^{2}-16 a x +73\right ) \left (a x -1\right ) c^{2}}{15 a \sqrt {-c \left (a x -1\right )}}-\frac {8 c^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a}\) | \(68\) |
derivativedivides | \(\frac {\frac {2 \left (-c x a +c \right )^{\frac {5}{2}}}{5}+\frac {4 c \left (-c x a +c \right )^{\frac {3}{2}}}{3}+8 c^{2} \sqrt {-c x a +c}-8 c^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a c}\) | \(73\) |
default | \(\frac {\frac {2 \left (-c x a +c \right )^{\frac {5}{2}}}{5}+\frac {4 c \left (-c x a +c \right )^{\frac {3}{2}}}{3}+8 c^{2} \sqrt {-c x a +c}-8 c^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a c}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 95, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (30 \, \sqrt {2} c^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 3 \, {\left (-a c x + c\right )}^{\frac {5}{2}} + 10 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 60 \, \sqrt {-a c x + c} c^{2}\right )}}{15 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 145, normalized size = 1.53 \begin {gather*} \left [\frac {2 \, {\left (30 \, \sqrt {2} c^{\frac {3}{2}} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + {\left (3 \, a^{2} c x^{2} - 16 \, a c x + 73 \, c\right )} \sqrt {-a c x + c}\right )}}{15 \, a}, \frac {2 \, {\left (60 \, \sqrt {2} \sqrt {-c} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) + {\left (3 \, a^{2} c x^{2} - 16 \, a c x + 73 \, c\right )} \sqrt {-a c x + c}\right )}}{15 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 34.22, size = 90, normalized size = 0.95 \begin {gather*} \frac {8 \sqrt {2} c^{2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{a \sqrt {- c}} + \frac {8 c \sqrt {- a c x + c}}{a} + \frac {4 \left (- a c x + c\right )^{\frac {3}{2}}}{3 a} + \frac {2 \left (- a c x + c\right )^{\frac {5}{2}}}{5 a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 107, normalized size = 1.13 \begin {gather*} \frac {8 \, \sqrt {2} c^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} + \frac {2 \, {\left (3 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{4} c^{4} + 10 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{4} c^{5} + 60 \, \sqrt {-a c x + c} a^{4} c^{6}\right )}}{15 \, a^{5} c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 78, normalized size = 0.82 \begin {gather*} \frac {4\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a}+\frac {8\,c\,\sqrt {c-a\,c\,x}}{a}+\frac {2\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a\,c}+\frac {\sqrt {2}\,c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,8{}\mathrm {i}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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