Optimal. Leaf size=101 \[ -\frac {2 \sqrt {1-a^2 x^2} (c-a c x)^{3/2}}{5 a^2 c}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a c x}}{5 a^2}-\frac {8 c \sqrt {1-a^2 x^2}}{5 a^2 \sqrt {c-a c x}} \]
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Rubi [A] time = 0.12, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6128, 795, 657, 649} \[ -\frac {2 \sqrt {1-a^2 x^2} (c-a c x)^{3/2}}{5 a^2 c}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a c x}}{5 a^2}-\frac {8 c \sqrt {1-a^2 x^2}}{5 a^2 \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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Rule 649
Rule 657
Rule 795
Rule 6128
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} x \sqrt {c-a c x} \, dx &=\frac {\int \frac {x (c-a c x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {2 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{5 a^2 c}-\frac {3 \int \frac {(c-a c x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{5 a c}\\ &=-\frac {2 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{5 a^2}-\frac {2 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{5 a^2 c}-\frac {4 \int \frac {\sqrt {c-a c x}}{\sqrt {1-a^2 x^2}} \, dx}{5 a}\\ &=-\frac {8 c \sqrt {1-a^2 x^2}}{5 a^2 \sqrt {c-a c x}}-\frac {2 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{5 a^2}-\frac {2 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{5 a^2 c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 0.46 \[ -\frac {2 c \sqrt {1-a^2 x^2} \left (a^2 x^2-3 a x+6\right )}{5 a^2 \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 49, normalized size = 0.49 \[ \frac {2 \, {\left (a^{2} x^{2} - 3 \, a x + 6\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{5 \, {\left (a^{3} x - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 65, normalized size = 0.64 \[ \frac {8 \, \sqrt {2} {\left | c \right |}}{5 \, a^{2} \sqrt {c}} - \frac {4 \, \sqrt {a c x + c} {\left | c \right |}}{a^{2} c} - \frac {2 \, {\left ({\left (a c x + c\right )}^{\frac {5}{2}} {\left | c \right |} - 5 \, {\left (a c x + c\right )}^{\frac {3}{2}} c {\left | c \right |}\right )}}{5 \, a^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 47, normalized size = 0.47 \[ \frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-a c x +c}\, \left (a^{2} x^{2}-3 a x +6\right )}{5 \left (a x -1\right ) a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 50, normalized size = 0.50 \[ -\frac {2 \, {\left (a^{2} \sqrt {c} x^{2} - 3 \, a \sqrt {c} x + 6 \, \sqrt {c}\right )} \sqrt {a x + 1} {\left (a x - 1\right )}}{5 \, {\left (a^{3} x - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 75, normalized size = 0.74 \[ \frac {\sqrt {c-a\,c\,x}\,\left (\frac {12\,\sqrt {1-a^2\,x^2}}{5\,a^3}-\frac {6\,x\,\sqrt {1-a^2\,x^2}}{5\,a^2}+\frac {2\,x^2\,\sqrt {1-a^2\,x^2}}{5\,a}\right )}{x-\frac {1}{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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