Optimal. Leaf size=56 \[ \frac {2 x \sqrt {1-a x} F_1\left (\frac {3}{2};\frac {n+1}{2},-\frac {n}{2};\frac {5}{2};a x,-a x\right )}{3 \sqrt {c-\frac {c}{a x}}} \]
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Rubi [A] time = 0.15, antiderivative size = 56, 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 = {6134, 6129, 133} \[ \frac {2 x \sqrt {1-a x} F_1\left (\frac {3}{2};\frac {n+1}{2},-\frac {n}{2};\frac {5}{2};a x,-a x\right )}{3 \sqrt {c-\frac {c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx &=\frac {\sqrt {1-a x} \int \frac {e^{n \tanh ^{-1}(a x)} \sqrt {x}}{\sqrt {1-a x}} \, dx}{\sqrt {c-\frac {c}{a x}} \sqrt {x}}\\ &=\frac {\sqrt {1-a x} \int \sqrt {x} (1-a x)^{-\frac {1}{2}-\frac {n}{2}} (1+a x)^{n/2} \, dx}{\sqrt {c-\frac {c}{a x}} \sqrt {x}}\\ &=\frac {2 x \sqrt {1-a x} F_1\left (\frac {3}{2};\frac {1+n}{2},-\frac {n}{2};\frac {5}{2};a x,-a x\right )}{3 \sqrt {c-\frac {c}{a x}}}\\ \end {align*}
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Mathematica [F] time = 180.01, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \sqrt {\frac {a c x - c}{a x}}}{a c x - c}, x\right ) \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {c - \frac {c}{a x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{\sqrt {c -\frac {c}{a x}}}\, dx \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {c - \frac {c}{a x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{\sqrt {c-\frac {c}{a\,x}}} \,d x \]
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 \[ \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{\sqrt {- c \left (-1 + \frac {1}{a x}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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