Optimal. Leaf size=103 \[ \frac {2 (c-a c x)^{5/2}}{3 a c^2 \sqrt {1-a^2 x^2}}+\frac {16 (c-a c x)^{3/2}}{3 a c \sqrt {1-a^2 x^2}}-\frac {64 \sqrt {c-a c x}}{3 a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.10, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6127, 657, 649} \[ \frac {2 (c-a c x)^{5/2}}{3 a c^2 \sqrt {1-a^2 x^2}}+\frac {16 (c-a c x)^{3/2}}{3 a c \sqrt {1-a^2 x^2}}-\frac {64 \sqrt {c-a c x}}{3 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 649
Rule 657
Rule 6127
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \sqrt {c-a c x} \, dx &=\frac {\int \frac {(c-a c x)^{7/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {2 (c-a c x)^{5/2}}{3 a c^2 \sqrt {1-a^2 x^2}}+\frac {8 \int \frac {(c-a c x)^{5/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=\frac {16 (c-a c x)^{3/2}}{3 a c \sqrt {1-a^2 x^2}}+\frac {2 (c-a c x)^{5/2}}{3 a c^2 \sqrt {1-a^2 x^2}}+\frac {32 \int \frac {(c-a c x)^{3/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {64 \sqrt {c-a c x}}{3 a \sqrt {1-a^2 x^2}}+\frac {16 (c-a c x)^{3/2}}{3 a c \sqrt {1-a^2 x^2}}+\frac {2 (c-a c x)^{5/2}}{3 a c^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.50 \[ \frac {2 c \sqrt {1-a x} \left (a^2 x^2-10 a x-23\right )}{3 a \sqrt {a x+1} \sqrt {c-a c x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 49, normalized size = 0.48 \[ -\frac {2 \, {\left (a^{2} x^{2} - 10 \, a x - 23\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{3 \, {\left (a^{3} x^{2} - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 54, normalized size = 0.52 \[ \frac {2 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \sqrt {-a c x +c}\, \left (a^{2} x^{2}-10 a x -23\right )}{3 \left (a x +1\right )^{2} \left (a x -1\right )^{2} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 50, normalized size = 0.49 \[ \frac {2 \, {\left (a^{2} \sqrt {c} x^{2} - 10 \, a \sqrt {c} x - 23 \, \sqrt {c}\right )} \sqrt {a x + 1} {\left (a x - 1\right )}}{3 \, {\left (a^{3} x^{2} - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 40, normalized size = 0.39 \[ -\frac {2\,\sqrt {c-a\,c\,x}\,\left (-a^2\,x^2+10\,a\,x+23\right )}{3\,a\,\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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