Optimal. Leaf size=136 \[ \frac {4 x^{m+1} \sqrt {c-a^2 c x^2} \, _2F_1(1,m+1;m+2;a x)}{(m+1) \sqrt {1-a^2 x^2}}-\frac {3 x^{m+1} \sqrt {c-a^2 c x^2}}{(m+1) \sqrt {1-a^2 x^2}}-\frac {a x^{m+2} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.21, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6153, 6150, 88, 64} \[ \frac {4 x^{m+1} \sqrt {c-a^2 c x^2} \, _2F_1(1,m+1;m+2;a x)}{(m+1) \sqrt {1-a^2 x^2}}-\frac {3 x^{m+1} \sqrt {c-a^2 c x^2}}{(m+1) \sqrt {1-a^2 x^2}}-\frac {a x^{m+2} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 64
Rule 88
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{3 \tanh ^{-1}(a x)} x^m \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {x^m (1+a x)^2}{1-a x} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (-3 x^m-a x^{1+m}+\frac {4 x^m}{1-a x}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {3 x^{1+m} \sqrt {c-a^2 c x^2}}{(1+m) \sqrt {1-a^2 x^2}}-\frac {a x^{2+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-a^2 x^2}}+\frac {\left (4 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^m}{1-a x} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {3 x^{1+m} \sqrt {c-a^2 c x^2}}{(1+m) \sqrt {1-a^2 x^2}}-\frac {a x^{2+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-a^2 x^2}}+\frac {4 x^{1+m} \sqrt {c-a^2 c x^2} \, _2F_1(1,1+m;2+m;a x)}{(1+m) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 74, normalized size = 0.54 \[ -\frac {x^{m+1} \sqrt {c-a^2 c x^2} (-4 (m+2) \, _2F_1(1,m+1;m+2;a x)+m (a x+3)+a x+6)}{(m+1) (m+2) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (a x + 1\right )} x^{m}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \]
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 = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right )^{3} x^{m} \sqrt {-a^{2} c \,x^{2}+c}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}^{3} x^{m}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^m\,\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{m} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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