Optimal. Leaf size=57 \[ -\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac {2 (c-a c x)^{3/2}}{a^2 c}-\frac {4 \sqrt {c-a c x}}{a^2} \]
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Rubi [A] time = 0.09, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6130, 21, 77} \[ -\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac {2 (c-a c x)^{3/2}}{a^2 c}-\frac {4 \sqrt {c-a c x}}{a^2} \]
Antiderivative was successfully verified.
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Rule 21
Rule 77
Rule 6130
Rubi steps
\begin {align*} \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\\ &=c \int \frac {x (1+a x)}{\sqrt {c-a c x}} \, dx\\ &=c \int \left (\frac {2}{a \sqrt {c-a c x}}-\frac {3 \sqrt {c-a c x}}{a c}+\frac {(c-a c x)^{3/2}}{a c^2}\right ) \, dx\\ &=-\frac {4 \sqrt {c-a c x}}{a^2}+\frac {2 (c-a c x)^{3/2}}{a^2 c}-\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 31, normalized size = 0.54 \[ -\frac {2 \left (a^2 x^2+3 a x+6\right ) \sqrt {c-a c x}}{5 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 27, normalized size = 0.47 \[ -\frac {2 \, {\left (a^{2} x^{2} + 3 \, a x + 6\right )} \sqrt {-a c x + c}}{5 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 92, normalized size = 1.61 \[ \frac {2 \, {\left (\frac {5 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {-a c x + c} c\right )}}{a c} - \frac {3 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} - 10 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {-a c x + c} c^{2}}{a c^{2}}\right )}}{15 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 28, normalized size = 0.49 \[ -\frac {2 \sqrt {-a c x +c}\, \left (a^{2} x^{2}+3 a x +6\right )}{5 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 44, normalized size = 0.77 \[ -\frac {2 \, {\left ({\left (-a c x + c\right )}^{\frac {5}{2}} - 5 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 10 \, \sqrt {-a c x + c} c^{2}\right )}}{5 \, a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 46, normalized size = 0.81 \[ -\frac {2\,{\left (c-a\,c\,x\right )}^{5/2}-10\,c\,{\left (c-a\,c\,x\right )}^{3/2}+20\,c^2\,\sqrt {c-a\,c\,x}}{5\,a^2\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.41, size = 48, normalized size = 0.84 \[ \frac {2 \left (- 2 c^{2} \sqrt {- a c x + c} + c \left (- a c x + c\right )^{\frac {3}{2}} - \frac {\left (- a c x + c\right )^{\frac {5}{2}}}{5}\right )}{a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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