Optimal. Leaf size=97 \[ \frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac {2 (c-a c x)^{3/2}}{3 a^2 c}+\frac {4 \sqrt {c-a c x}}{a^2}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^2} \]
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Rubi [A] time = 0.17, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6167, 6130, 21, 80, 50, 63, 206} \[ \frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac {2 (c-a c x)^{3/2}}{3 a^2 c}+\frac {4 \sqrt {c-a c x}}{a^2}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 21
Rule 50
Rule 63
Rule 80
Rule 206
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} x \sqrt {c-a c x} \, dx\\ &=-\int \frac {x (1-a x) \sqrt {c-a c x}}{1+a x} \, dx\\ &=-\frac {\int \frac {x (c-a c x)^{3/2}}{1+a x} \, dx}{c}\\ &=\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac {\int \frac {(c-a c x)^{3/2}}{1+a x} \, dx}{a c}\\ &=\frac {2 (c-a c x)^{3/2}}{3 a^2 c}+\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac {2 \int \frac {\sqrt {c-a c x}}{1+a x} \, dx}{a}\\ &=\frac {4 \sqrt {c-a c x}}{a^2}+\frac {2 (c-a c x)^{3/2}}{3 a^2 c}+\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac {(4 c) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx}{a}\\ &=\frac {4 \sqrt {c-a c x}}{a^2}+\frac {2 (c-a c x)^{3/2}}{3 a^2 c}+\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}-\frac {8 \operatorname {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a^2}\\ &=\frac {4 \sqrt {c-a c x}}{a^2}+\frac {2 (c-a c x)^{3/2}}{3 a^2 c}+\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 70, normalized size = 0.72 \[ \frac {2 \left (3 a^2 x^2-11 a x+38\right ) \sqrt {c-a c x}-60 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{15 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 136, normalized size = 1.40 \[ \left [\frac {2 \, {\left (15 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + {\left (3 \, a^{2} x^{2} - 11 \, a x + 38\right )} \sqrt {-a c x + c}\right )}}{15 \, a^{2}}, \frac {2 \, {\left (30 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) + {\left (3 \, a^{2} x^{2} - 11 \, a x + 38\right )} \sqrt {-a c x + c}\right )}}{15 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 105, normalized size = 1.08 \[ \frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a^{2} \sqrt {-c}} + \frac {2 \, {\left (3 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{8} c^{8} + 5 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{8} c^{9} + 30 \, \sqrt {-a c x + c} a^{8} c^{10}\right )}}{15 \, a^{10} c^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 73, normalized size = 0.75 \[ \frac {\frac {2 \left (-a c x +c \right )^{\frac {5}{2}}}{5}+\frac {2 c \left (-a c x +c \right )^{\frac {3}{2}}}{3}+4 \sqrt {-a c x +c}\, c^{2}-4 c^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 95, normalized size = 0.98 \[ \frac {2 \, {\left (15 \, \sqrt {2} c^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 3 \, {\left (-a c x + c\right )}^{\frac {5}{2}} + 5 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 30 \, \sqrt {-a c x + c} c^{2}\right )}}{15 \, a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 80, normalized size = 0.82 \[ \frac {4\,\sqrt {c-a\,c\,x}}{a^2}+\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a^2\,c}+\frac {2\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a^2\,c^2}+\frac {\sqrt {2}\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i}}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.29, size = 92, normalized size = 0.95 \[ \frac {2 \left (\frac {2 \sqrt {2} c^{3} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 2 c^{2} \sqrt {- a c x + c} + \frac {c \left (- a c x + c\right )^{\frac {3}{2}}}{3} + \frac {\left (- a c x + c\right )^{\frac {5}{2}}}{5}\right )}{a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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