Optimal. Leaf size=97 \[ -\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac {2 (c-a c x)^{3/2}}{3 a^3 c}-\frac {4 \sqrt {c-a c x}}{a^3}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^3} \]
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Rubi [A] time = 0.25, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6167, 6130, 21, 88, 50, 63, 206} \[ -\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac {2 (c-a c x)^{3/2}}{3 a^3 c}-\frac {4 \sqrt {c-a c x}}{a^3}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^3} \]
Antiderivative was successfully verified.
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Rule 21
Rule 50
Rule 63
Rule 88
Rule 206
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx\\ &=-\int \frac {x^2 (1-a x) \sqrt {c-a c x}}{1+a x} \, dx\\ &=-\frac {\int \frac {x^2 (c-a c x)^{3/2}}{1+a x} \, dx}{c}\\ &=-\frac {\int \left (\frac {(c-a c x)^{3/2}}{a^2 (1+a x)}-\frac {(c-a c x)^{5/2}}{a^2 c}\right ) \, dx}{c}\\ &=-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac {\int \frac {(c-a c x)^{3/2}}{1+a x} \, dx}{a^2 c}\\ &=-\frac {2 (c-a c x)^{3/2}}{3 a^3 c}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac {2 \int \frac {\sqrt {c-a c x}}{1+a x} \, dx}{a^2}\\ &=-\frac {4 \sqrt {c-a c x}}{a^3}-\frac {2 (c-a c x)^{3/2}}{3 a^3 c}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac {(4 c) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx}{a^2}\\ &=-\frac {4 \sqrt {c-a c x}}{a^3}-\frac {2 (c-a c x)^{3/2}}{3 a^3 c}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac {8 \operatorname {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a^3}\\ &=-\frac {4 \sqrt {c-a c x}}{a^3}-\frac {2 (c-a c x)^{3/2}}{3 a^3 c}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 78, normalized size = 0.80 \[ \frac {2 \left (3 a^3 x^3-9 a^2 x^2+16 a x-52\right ) \sqrt {c-a c x}+84 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{21 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 153, normalized size = 1.58 \[ \left [\frac {2 \, {\left (21 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + {\left (3 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 16 \, a x - 52\right )} \sqrt {-a c x + c}\right )}}{21 \, a^{3}}, -\frac {2 \, {\left (42 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - {\left (3 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 16 \, a x - 52\right )} \sqrt {-a c x + c}\right )}}{21 \, a^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 105, normalized size = 1.08 \[ -\frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a^{3} \sqrt {-c}} + \frac {2 \, {\left (3 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{18} c^{18} - 7 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{18} c^{20} - 42 \, \sqrt {-a c x + c} a^{18} c^{21}\right )}}{21 \, a^{21} c^{21}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 75, normalized size = 0.77 \[ -\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {7}{2}}}{7}+\frac {\left (-a c x +c \right )^{\frac {3}{2}} c^{2}}{3}+2 \sqrt {-a c x +c}\, c^{3}-2 c^{\frac {7}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c^{3} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 97, normalized size = 1.00 \[ -\frac {2 \, {\left (21 \, \sqrt {2} c^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 3 \, {\left (-a c x + c\right )}^{\frac {7}{2}} + 7 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 42 \, \sqrt {-a c x + c} c^{3}\right )}}{21 \, a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 80, normalized size = 0.82 \[ -\frac {4\,\sqrt {c-a\,c\,x}}{a^3}-\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a^3\,c}-\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a^3\,c^3}-\frac {\sqrt {2}\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i}}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.25, size = 95, normalized size = 0.98 \[ - \frac {2 \left (\frac {2 \sqrt {2} c^{4} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 2 c^{3} \sqrt {- a c x + c} + \frac {c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3} + \frac {\left (- a c x + c\right )^{\frac {7}{2}}}{7}\right )}{a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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