Optimal. Leaf size=64 \[ \frac {2 c (a x+1) \sqrt {1-a^2 x^2} x^m (-a x)^{-m} \, _2F_1\left (\frac {3}{2},-m;\frac {5}{2};a x+1\right )}{3 a \sqrt {c-a c x}} \]
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Rubi [A] time = 0.12, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6128, 892, 67, 65} \[ \frac {2 c (a x+1) \sqrt {1-a^2 x^2} x^m (-a x)^{-m} \, _2F_1\left (\frac {3}{2},-m;\frac {5}{2};a x+1\right )}{3 a \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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Rule 65
Rule 67
Rule 892
Rule 6128
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx &=c \int \frac {x^m \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}} \, dx\\ &=\frac {\left (c \sqrt {1-a^2 x^2}\right ) \int x^m \sqrt {\frac {1}{c}+\frac {a x}{c}} \, dx}{\sqrt {\frac {1}{c}+\frac {a x}{c}} \sqrt {c-a c x}}\\ &=\frac {\left (c x^m (-a x)^{-m} \sqrt {1-a^2 x^2}\right ) \int (-a x)^m \sqrt {\frac {1}{c}+\frac {a x}{c}} \, dx}{\sqrt {\frac {1}{c}+\frac {a x}{c}} \sqrt {c-a c x}}\\ &=\frac {2 c x^m (-a x)^{-m} (1+a x) \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {3}{2},-m;\frac {5}{2};1+a x\right )}{3 a \sqrt {c-a c x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.72 \[ \frac {x^{m+1} \sqrt {c-a c x} \, _2F_1\left (-\frac {1}{2},m+1;m+2;-a x\right )}{(m+1) \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} x^{m}}{a x - 1}, x\right ) \]
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 \[ \int \frac {\sqrt {-a c x + c} {\left (a x + 1\right )} x^{m}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right ) x^{m} \sqrt {-a c x +c}}{\sqrt {-a^{2} x^{2}+1}}\, dx \]
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 \[ \int \frac {\sqrt {-a c x + c} {\left (a x + 1\right )} x^{m}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^m\,\sqrt {c-a\,c\,x}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{m} \sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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