Optimal. Leaf size=116 \[ \frac {16 c^2 \sqrt {c-a c x}}{a}+\frac {8 c (c-a c x)^{3/2}}{3 a}+\frac {4 (c-a c x)^{5/2}}{5 a}+\frac {2 (c-a c x)^{7/2}}{7 a c}-\frac {16 \sqrt {2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
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Rubi [A]
time = 0.07, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {16 \sqrt {2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}+\frac {16 c^2 \sqrt {c-a c x}}{a}+\frac {2 (c-a c x)^{7/2}}{7 a c}+\frac {4 (c-a c x)^{5/2}}{5 a}+\frac {8 c (c-a c x)^{3/2}}{3 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)^{5/2} \, dx &=\int \frac {(1-a x) (c-a c x)^{5/2}}{1+a x} \, dx\\ &=\frac {\int \frac {(c-a c x)^{7/2}}{1+a x} \, dx}{c}\\ &=\frac {2 (c-a c x)^{7/2}}{7 a c}+2 \int \frac {(c-a c x)^{5/2}}{1+a x} \, dx\\ &=\frac {4 (c-a c x)^{5/2}}{5 a}+\frac {2 (c-a c x)^{7/2}}{7 a c}+(4 c) \int \frac {(c-a c x)^{3/2}}{1+a x} \, dx\\ &=\frac {8 c (c-a c x)^{3/2}}{3 a}+\frac {4 (c-a c x)^{5/2}}{5 a}+\frac {2 (c-a c x)^{7/2}}{7 a c}+\left (8 c^2\right ) \int \frac {\sqrt {c-a c x}}{1+a x} \, dx\\ &=\frac {16 c^2 \sqrt {c-a c x}}{a}+\frac {8 c (c-a c x)^{3/2}}{3 a}+\frac {4 (c-a c x)^{5/2}}{5 a}+\frac {2 (c-a c x)^{7/2}}{7 a c}+\left (16 c^3\right ) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=\frac {16 c^2 \sqrt {c-a c x}}{a}+\frac {8 c (c-a c x)^{3/2}}{3 a}+\frac {4 (c-a c x)^{5/2}}{5 a}+\frac {2 (c-a c x)^{7/2}}{7 a c}-\frac {\left (32 c^2\right ) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a}\\ &=\frac {16 c^2 \sqrt {c-a c x}}{a}+\frac {8 c (c-a c x)^{3/2}}{3 a}+\frac {4 (c-a c x)^{5/2}}{5 a}+\frac {2 (c-a c x)^{7/2}}{7 a c}-\frac {16 \sqrt {2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 80, normalized size = 0.69 \begin {gather*} -\frac {2 c^2 \left (\sqrt {c-a c x} \left (-1037+269 a x-87 a^2 x^2+15 a^3 x^3\right )+840 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{105 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.84, size = 87, normalized size = 0.75
method | result | size |
risch | \(\frac {2 \left (15 a^{3} x^{3}-87 a^{2} x^{2}+269 a x -1037\right ) \left (a x -1\right ) c^{3}}{105 a \sqrt {-c \left (a x -1\right )}}-\frac {16 c^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a}\) | \(76\) |
derivativedivides | \(\frac {\frac {2 \left (-c x a +c \right )^{\frac {7}{2}}}{7}+\frac {4 c \left (-c x a +c \right )^{\frac {5}{2}}}{5}+\frac {8 c^{2} \left (-c x a +c \right )^{\frac {3}{2}}}{3}+16 c^{3} \sqrt {-c x a +c}-16 c^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a c}\) | \(87\) |
default | \(\frac {\frac {2 \left (-c x a +c \right )^{\frac {7}{2}}}{7}+\frac {4 c \left (-c x a +c \right )^{\frac {5}{2}}}{5}+\frac {8 c^{2} \left (-c x a +c \right )^{\frac {3}{2}}}{3}+16 c^{3} \sqrt {-c x a +c}-16 c^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a c}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 109, normalized size = 0.94 \begin {gather*} \frac {2 \, {\left (420 \, \sqrt {2} c^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 15 \, {\left (-a c x + c\right )}^{\frac {7}{2}} + 42 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 140 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 840 \, \sqrt {-a c x + c} c^{3}\right )}}{105 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 183, normalized size = 1.58 \begin {gather*} \left [\frac {2 \, {\left (420 \, \sqrt {2} c^{\frac {5}{2}} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) - {\left (15 \, a^{3} c^{2} x^{3} - 87 \, a^{2} c^{2} x^{2} + 269 \, a c^{2} x - 1037 \, c^{2}\right )} \sqrt {-a c x + c}\right )}}{105 \, a}, \frac {2 \, {\left (840 \, \sqrt {2} \sqrt {-c} c^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - {\left (15 \, a^{3} c^{2} x^{3} - 87 \, a^{2} c^{2} x^{2} + 269 \, a c^{2} x - 1037 \, c^{2}\right )} \sqrt {-a c x + c}\right )}}{105 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 44.70, size = 109, normalized size = 0.94 \begin {gather*} \frac {16 \sqrt {2} c^{3} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{a \sqrt {- c}} + \frac {16 c^{2} \sqrt {- a c x + c}}{a} + \frac {8 c \left (- a c x + c\right )^{\frac {3}{2}}}{3 a} + \frac {4 \left (- a c x + c\right )^{\frac {5}{2}}}{5 a} + \frac {2 \left (- a c x + c\right )^{\frac {7}{2}}}{7 a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 134, normalized size = 1.16 \begin {gather*} \frac {16 \, \sqrt {2} c^{3} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {2 \, {\left (15 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{6} c^{6} - 42 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{6} c^{7} - 140 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{6} c^{8} - 840 \, \sqrt {-a c x + c} a^{6} c^{9}\right )}}{105 \, a^{7} c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 95, normalized size = 0.82 \begin {gather*} \frac {4\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a}+\frac {8\,c\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a}+\frac {16\,c^2\,\sqrt {c-a\,c\,x}}{a}+\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a\,c}+\frac {\sqrt {2}\,c^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,16{}\mathrm {i}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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