Optimal. Leaf size=116 \[ \frac {16 \sqrt {2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}-\frac {16 c^2 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{7/2}}{7 a c}-\frac {4 (c-a c x)^{5/2}}{5 a}-\frac {8 c (c-a c x)^{3/2}}{3 a} \]
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Rubi [A] time = 0.13, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6167, 6130, 21, 50, 63, 206} \[ -\frac {16 c^2 \sqrt {c-a c x}}{a}+\frac {16 \sqrt {2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}-\frac {2 (c-a c x)^{7/2}}{7 a c}-\frac {4 (c-a c x)^{5/2}}{5 a}-\frac {8 c (c-a c x)^{3/2}}{3 a} \]
Antiderivative was successfully verified.
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Rule 21
Rule 50
Rule 63
Rule 206
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=-\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 ) \operatorname {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 = 0.07, size = 80, normalized size = 0.69 \[ \frac {2 c^2 \left (\left (15 a^3 x^3-87 a^2 x^2+269 a x-1037\right ) \sqrt {c-a c x}+840 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{105 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 182, normalized size = 1.57 \[ \left [\frac {2 \, {\left (420 \, \sqrt {2} c^{\frac {5}{2}} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + {\left (15 \, a^{3} c^{2} x^{3} - 87 \, a^{2} c^{2} x^{2} + 269 \, a c^{2} x - 1037 \, c^{2}\right )} \sqrt {-a c x + c}\right )}}{105 \, a}, -\frac {2 \, {\left (840 \, \sqrt {2} \sqrt {-c} c^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - {\left (15 \, a^{3} c^{2} x^{3} - 87 \, a^{2} c^{2} x^{2} + 269 \, a c^{2} x - 1037 \, c^{2}\right )} \sqrt {-a c x + c}\right )}}{105 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 134, normalized size = 1.16 \[ -\frac {16 \, \sqrt {2} c^{3} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} + \frac {2 \, {\left (15 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{6} c^{6} - 42 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{6} c^{7} - 140 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{6} c^{8} - 840 \, \sqrt {-a c x + c} a^{6} c^{9}\right )}}{105 \, a^{7} c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 87, normalized size = 0.75 \[ -\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {7}{2}}}{7}+\frac {2 c \left (-a c x +c \right )^{\frac {5}{2}}}{5}+\frac {4 \left (-a c x +c \right )^{\frac {3}{2}} c^{2}}{3}+8 \sqrt {-a c x +c}\, c^{3}-8 c^{\frac {7}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 109, normalized size = 0.94 \[ -\frac {2 \, {\left (420 \, \sqrt {2} c^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 15 \, {\left (-a c x + c\right )}^{\frac {7}{2}} + 42 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 140 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 840 \, \sqrt {-a c x + c} c^{3}\right )}}{105 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 95, normalized size = 0.82 \[ -\frac {4\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a}-\frac {8\,c\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a}-\frac {16\,c^2\,\sqrt {c-a\,c\,x}}{a}-\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a\,c}-\frac {\sqrt {2}\,c^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,16{}\mathrm {i}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 51.78, size = 110, normalized size = 0.95 \[ - \frac {16 \sqrt {2} c^{3} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {16 c^{2} \sqrt {- a c x + c}}{a} - \frac {8 c \left (- a c x + c\right )^{\frac {3}{2}}}{3 a} - \frac {4 \left (- a c x + c\right )^{\frac {5}{2}}}{5 a} - \frac {2 \left (- a c x + c\right )^{\frac {7}{2}}}{7 a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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