Optimal. Leaf size=116 \[ -\frac {\sqrt {c-\frac {c}{a x}} x^{1+m} \sqrt {1-a^2 x^2}}{(1+m) (1-a x)}+\frac {(3+4 m) \sqrt {c-\frac {c}{a x}} x^{1+m} \, _2F_1\left (\frac {1}{2},\frac {1}{2}+m;\frac {3}{2}+m;-a x\right )}{(1+m) (1+2 m) \sqrt {1-a x}} \]
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Rubi [A]
time = 0.20, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6269, 6263,
895, 862, 66} \begin {gather*} \frac {(4 m+3) x^{m+1} \sqrt {c-\frac {c}{a x}} \, _2F_1\left (\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};-a x\right )}{(m+1) (2 m+1) \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} x^{m+1} \sqrt {c-\frac {c}{a x}}}{(m+1) (1-a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 862
Rule 895
Rule 6263
Rule 6269
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int e^{-\tanh ^{-1}(a x)} x^{-\frac {1}{2}+m} \sqrt {1-a x} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{-\frac {1}{2}+m} (1-a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x^{1+m} \sqrt {1-a^2 x^2}}{(1+m) (1-a x)}+\frac {\left ((3+4 m) \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{-\frac {1}{2}+m} \sqrt {1-a x}}{\sqrt {1-a^2 x^2}} \, dx}{2 (1+m) \sqrt {1-a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x^{1+m} \sqrt {1-a^2 x^2}}{(1+m) (1-a x)}+\frac {\left ((3+4 m) \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{-\frac {1}{2}+m}}{\sqrt {1+a x}} \, dx}{2 (1+m) \sqrt {1-a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x^{1+m} \sqrt {1-a^2 x^2}}{(1+m) (1-a x)}+\frac {(3+4 m) \sqrt {c-\frac {c}{a x}} x^{1+m} \, _2F_1\left (\frac {1}{2},\frac {1}{2}+m;\frac {3}{2}+m;-a x\right )}{(1+m) (1+2 m) \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 87, normalized size = 0.75 \begin {gather*} -\frac {2 \sqrt {c-\frac {c}{a x}} x^{1+m} \left (-\left ((3+2 m) \sqrt {1+a x}\right )+a (3+4 m) x \, _2F_1\left (\frac {1}{2},\frac {3}{2}+m;\frac {5}{2}+m;-a x\right )\right )}{\left (3+8 m+4 m^2\right ) \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.85, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \sqrt {c -\frac {c}{a x}}\, \sqrt {-a^{2} x^{2}+1}}{a x +1}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{m} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^m\,\sqrt {c-\frac {c}{a\,x}}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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