Optimal. Leaf size=51 \[ \frac {\sqrt {1-a^2 x^2} x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{(m+1) \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.18, 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 = {6153, 6150, 64} \[ \frac {\sqrt {1-a^2 x^2} x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{(m+1) \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 64
Rule 6150
Rule 6153
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 = 0.03, size = 51, normalized size = 1.00 \[ \frac {\sqrt {1-a^2 x^2} x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{(m+1) \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} x^{m}}{a^{3} c x^{3} - a^{2} c x^{2} - a c x + c}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )} x^{m}}{\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, 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 \[ \int \frac {{\left (a x + 1\right )} x^{m}}{\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^m\,\left (a\,x+1\right )}{\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{m} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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