Optimal. Leaf size=105 \[ -\frac {c 2^{\frac {n+5}{2}} \sqrt {c-a^2 c x^2} (1-a x)^{\frac {5-n}{2}} \, _2F_1\left (\frac {1}{2} (-n-3),\frac {5-n}{2};\frac {7-n}{2};\frac {1}{2} (1-a x)\right )}{a (5-n) \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6143, 6140, 69} \[ -\frac {c 2^{\frac {n+5}{2}} \sqrt {c-a^2 c x^2} (1-a x)^{\frac {5-n}{2}} \, _2F_1\left (\frac {1}{2} (-n-3),\frac {5-n}{2};\frac {7-n}{2};\frac {1}{2} (1-a x)\right )}{a (5-n) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int e^{n \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^{\frac {3}{2}-\frac {n}{2}} (1+a x)^{\frac {3}{2}+\frac {n}{2}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {2^{\frac {5+n}{2}} c (1-a x)^{\frac {5-n}{2}} \sqrt {c-a^2 c x^2} \, _2F_1\left (\frac {1}{2} (-3-n),\frac {5-n}{2};\frac {7-n}{2};\frac {1}{2} (1-a x)\right )}{a (5-n) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 102, normalized size = 0.97 \[ \frac {c 2^{\frac {n+5}{2}} \sqrt {c-a^2 c x^2} (1-a x)^{\frac {5}{2}-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-\frac {3}{2},\frac {5}{2}-\frac {n}{2};\frac {7}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )}{a (n-5) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c x^{2} - c\right )} \sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]
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 \[ \int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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