Optimal. Leaf size=136 \[ \frac {2 (c-a c x)^{7/2}}{5 a c^2 \sqrt {1-a^2 x^2}}+\frac {8 (c-a c x)^{5/2}}{5 a c \sqrt {1-a^2 x^2}}+\frac {64 (c-a c x)^{3/2}}{5 a \sqrt {1-a^2 x^2}}-\frac {256 c \sqrt {c-a c x}}{5 a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6127, 657, 649} \[ \frac {2 (c-a c x)^{7/2}}{5 a c^2 \sqrt {1-a^2 x^2}}+\frac {8 (c-a c x)^{5/2}}{5 a c \sqrt {1-a^2 x^2}}+\frac {64 (c-a c x)^{3/2}}{5 a \sqrt {1-a^2 x^2}}-\frac {256 c \sqrt {c-a c x}}{5 a \sqrt {1-a^2 x^2}} \]
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)} (c-a c x)^{3/2} \, dx &=\frac {\int \frac {(c-a c x)^{9/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {2 (c-a c x)^{7/2}}{5 a c^2 \sqrt {1-a^2 x^2}}+\frac {12 \int \frac {(c-a c x)^{7/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{5 c^2}\\ &=\frac {8 (c-a c x)^{5/2}}{5 a c \sqrt {1-a^2 x^2}}+\frac {2 (c-a c x)^{7/2}}{5 a c^2 \sqrt {1-a^2 x^2}}+\frac {32 \int \frac {(c-a c x)^{5/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{5 c}\\ &=\frac {64 (c-a c x)^{3/2}}{5 a \sqrt {1-a^2 x^2}}+\frac {8 (c-a c x)^{5/2}}{5 a c \sqrt {1-a^2 x^2}}+\frac {2 (c-a c x)^{7/2}}{5 a c^2 \sqrt {1-a^2 x^2}}+\frac {128}{5} \int \frac {(c-a c x)^{3/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {256 c \sqrt {c-a c x}}{5 a \sqrt {1-a^2 x^2}}+\frac {64 (c-a c x)^{3/2}}{5 a \sqrt {1-a^2 x^2}}+\frac {8 (c-a c x)^{5/2}}{5 a c \sqrt {1-a^2 x^2}}+\frac {2 (c-a c x)^{7/2}}{5 a c^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 61, normalized size = 0.45 \[ -\frac {2 c^2 \sqrt {1-a x} \left (a^3 x^3-7 a^2 x^2+43 a x+91\right )}{5 a \sqrt {a x+1} \sqrt {c-a c x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.54, size = 62, normalized size = 0.46 \[ \frac {2 \, {\left (a^{3} c x^{3} - 7 \, a^{2} c x^{2} + 43 \, a c x + 91 \, c\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{5 \, {\left (a^{3} x^{2} - a\right )}} \]
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 = 62, normalized size = 0.46 \[ \frac {2 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (-a c x +c \right )^{\frac {3}{2}} \left (x^{3} a^{3}-7 a^{2} x^{2}+43 a x +91\right )}{5 \left (a x +1\right )^{2} \left (a x -1\right )^{3} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 61, normalized size = 0.45 \[ -\frac {2 \, {\left (a^{3} c^{\frac {3}{2}} x^{3} - 7 \, a^{2} c^{\frac {3}{2}} x^{2} + 43 \, a c^{\frac {3}{2}} x + 91 \, c^{\frac {3}{2}}\right )} \sqrt {a x + 1} {\left (a x - 1\right )}}{5 \, {\left (a^{3} x^{2} - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 99, normalized size = 0.73 \[ -\frac {\sqrt {c-a\,c\,x}\,\left (\frac {182\,c\,\sqrt {1-a^2\,x^2}}{5\,a^3}+\frac {2\,c\,x^3\,\sqrt {1-a^2\,x^2}}{5}-\frac {14\,c\,x^2\,\sqrt {1-a^2\,x^2}}{5\,a}+\frac {86\,c\,x\,\sqrt {1-a^2\,x^2}}{5\,a^2}\right )}{\frac {1}{a^2}-x^2} \]
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 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {3}{2}} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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