Optimal. Leaf size=172 \[ \frac {c (2 m+3) \sqrt {1-a^2 x^2} x^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{(m+1) (m+2) \sqrt {c-a^2 c x^2}}-\frac {2 a c \sqrt {1-a^2 x^2} x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{(m+2) \sqrt {c-a^2 c x^2}}-\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{m+2} \]
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Rubi [A] time = 0.28, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6152, 1809, 808, 365, 364} \[ \frac {c (2 m+3) \sqrt {1-a^2 x^2} x^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{(m+1) (m+2) \sqrt {c-a^2 c x^2}}-\frac {2 a c \sqrt {1-a^2 x^2} x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{(m+2) \sqrt {c-a^2 c x^2}}-\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{m+2} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 808
Rule 1809
Rule 6152
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx &=c \int \frac {x^m (1-a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-\frac {\int \frac {x^m \left (-a^2 c (3+2 m)+2 a^3 c (2+m) x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{a^2 (2+m)}\\ &=-\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-(2 a c) \int \frac {x^{1+m}}{\sqrt {c-a^2 c x^2}} \, dx+\frac {(c (3+2 m)) \int \frac {x^m}{\sqrt {c-a^2 c x^2}} \, dx}{2+m}\\ &=-\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-\frac {\left (2 a c \sqrt {1-a^2 x^2}\right ) \int \frac {x^{1+m}}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-a^2 c x^2}}+\frac {\left (c (3+2 m) \sqrt {1-a^2 x^2}\right ) \int \frac {x^m}{\sqrt {1-a^2 x^2}} \, dx}{(2+m) \sqrt {c-a^2 c x^2}}\\ &=-\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}+\frac {c (3+2 m) x^{1+m} \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{(1+m) (2+m) \sqrt {c-a^2 c x^2}}-\frac {2 a c x^{2+m} \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{(2+m) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 111, normalized size = 0.65 \[ \frac {x^{m+1} \left (\frac {2 \sqrt {c-a c x} F_1\left (m+1;\frac {1}{2},-\frac {1}{2};m+2;-a x,a x\right )}{\sqrt {1-a x}}-\frac {\sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{\sqrt {1-a^2 x^2}}\right )}{m+1} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x - 1\right )} x^{m}}{a x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {x^{m} \sqrt {-a^{2} c \,x^{2}+c}\, \left (-a^{2} x^{2}+1\right )}{\left (a x +1\right )^{2}}\, dx \]
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 \[ -\int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 1\right )} x^{m}}{{\left (a x + 1\right )}^{2}}\,{d x} \]
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 \[ -\int \frac {x^m\,\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]
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 \[ - \int \left (- \frac {x^{m} \sqrt {- a^{2} c x^{2} + c}}{a x + 1}\right )\, dx - \int \frac {a x x^{m} \sqrt {- a^{2} c x^{2} + c}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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