Optimal. Leaf size=114 \[ \frac {2 (4 m+5) (a x+1) x^m \sqrt {c-a c x} (-a x)^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};a x+1\right )}{a (2 m+3) \sqrt {1-a^2 x^2}}-\frac {2 c \sqrt {1-a^2 x^2} x^{m+1}}{(2 m+3) \sqrt {c-a c x}} \]
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Rubi [A] time = 0.18, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6128, 881, 892, 67, 65} \[ \frac {2 (4 m+5) (a x+1) x^m \sqrt {c-a c x} (-a x)^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};a x+1\right )}{a (2 m+3) \sqrt {1-a^2 x^2}}-\frac {2 c \sqrt {1-a^2 x^2} x^{m+1}}{(2 m+3) \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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Rule 65
Rule 67
Rule 881
Rule 892
Rule 6128
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx &=\frac {\int \frac {x^m (c-a c x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {2 c x^{1+m} \sqrt {1-a^2 x^2}}{(3+2 m) \sqrt {c-a c x}}+\frac {(5+4 m) \int \frac {x^m \sqrt {c-a c x}}{\sqrt {1-a^2 x^2}} \, dx}{3+2 m}\\ &=-\frac {2 c x^{1+m} \sqrt {1-a^2 x^2}}{(3+2 m) \sqrt {c-a c x}}+\frac {\left ((5+4 m) \sqrt {\frac {1}{c}+\frac {a x}{c}} \sqrt {c-a c x}\right ) \int \frac {x^m}{\sqrt {\frac {1}{c}+\frac {a x}{c}}} \, dx}{(3+2 m) \sqrt {1-a^2 x^2}}\\ &=-\frac {2 c x^{1+m} \sqrt {1-a^2 x^2}}{(3+2 m) \sqrt {c-a c x}}+\frac {\left ((5+4 m) x^m (-a x)^{-m} \sqrt {\frac {1}{c}+\frac {a x}{c}} \sqrt {c-a c x}\right ) \int \frac {(-a x)^m}{\sqrt {\frac {1}{c}+\frac {a x}{c}}} \, dx}{(3+2 m) \sqrt {1-a^2 x^2}}\\ &=-\frac {2 c x^{1+m} \sqrt {1-a^2 x^2}}{(3+2 m) \sqrt {c-a c x}}+\frac {2 (5+4 m) x^m (-a x)^{-m} (1+a x) \sqrt {c-a c x} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1+a x\right )}{a (3+2 m) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 0.68 \[ -\frac {c \sqrt {1-a x} x^{m+1} \left (2 (m+1) \sqrt {a x+1}-(4 m+5) \, _2F_1\left (\frac {1}{2},m+1;m+2;-a x\right )\right )}{(m+1) (2 m+3) \sqrt {c-a c x}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} x^{m}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \sqrt {-a c x +c}\, \sqrt {-a^{2} x^{2}+1}}{a x +1}\, dx \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} x^{m}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{a\,x+1} \,d x \]
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 \[ \int \frac {x^{m} \sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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