Optimal. Leaf size=51 \[ \frac {x^{1+m} \sqrt {1-a^2 x^2} \, _2F_1(1,1+m;2+m;a x)}{(1+m) \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6288, 6285, 66}
\begin {gather*} \frac {\sqrt {1-a^2 x^2} x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{(m+1) \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 6285
Rule 6288
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^m}{\sqrt {c-a^2 c x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{\tanh ^{-1}(a x)} x^m}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x^m}{1-a x} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {x^{1+m} \sqrt {1-a^2 x^2} \, _2F_1(1,1+m;2+m;a x)}{(1+m) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 51, normalized size = 1.00 \begin {gather*} \frac {x^{1+m} \sqrt {1-a^2 x^2} \, _2F_1(1,1+m;2+m;a x)}{(1+m) \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +1\right ) x^{m}}{\sqrt {-a^{2} x^{2}+1}\, \sqrt {-a^{2} c \,x^{2}+c}}\, dx\]
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 \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} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, 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 = -0.02 \begin {gather*} \int \frac {x^m\,\left (a\,x+1\right )}{\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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