Optimal. Leaf size=81 \[ -\frac {2^{\frac {n}{2}+1} \sqrt {c-a c x} (1-a x)^{-n/2} \, _2F_1\left (\frac {1-n}{2},-\frac {n}{2};\frac {3-n}{2};\frac {1}{2} (1-a x)\right )}{a c (1-n)} \]
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Rubi [A] time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6130, 23, 69} \[ -\frac {2^{\frac {n}{2}+1} \sqrt {c-a c x} (1-a x)^{-n/2} \, _2F_1\left (\frac {1-n}{2},-\frac {n}{2};\frac {3-n}{2};\frac {1}{2} (1-a x)\right )}{a c (1-n)} \]
Antiderivative was successfully verified.
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Rule 23
Rule 69
Rule 6130
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx &=\int \frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{\sqrt {c-a c x}} \, dx\\ &=\left ((1-a x)^{-n/2} (c-a c x)^{n/2}\right ) \int (1+a x)^{n/2} (c-a c x)^{-\frac {1}{2}-\frac {n}{2}} \, dx\\ &=-\frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} \sqrt {c-a c x} \, _2F_1\left (\frac {1-n}{2},-\frac {n}{2};\frac {3-n}{2};\frac {1}{2} (1-a x)\right )}{a c (1-n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 78, normalized size = 0.96 \[ \frac {2^{\frac {n}{2}+1} \sqrt {c-a c x} (1-a x)^{-n/2} \, _2F_1\left (\frac {1}{2}-\frac {n}{2},-\frac {n}{2};\frac {3}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )}{a c (n-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a c x + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c x - c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a c x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{\sqrt {-a c x +c}}\, dx \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a c x + c}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{\sqrt {c-a\,c\,x}} \,d x \]
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 \[ \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{\sqrt {- c \left (a x - 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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