Optimal. Leaf size=80 \[ \frac {x^{m+1} \, _2F_1\left (\frac {3}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{c (m+1)}+\frac {a x^{m+2} \, _2F_1\left (\frac {3}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{c (m+2)} \]
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Rubi [A] time = 0.11, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6148, 808, 364} \[ \frac {x^{m+1} \, _2F_1\left (\frac {3}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{c (m+1)}+\frac {a x^{m+2} \, _2F_1\left (\frac {3}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{c (m+2)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 808
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^m}{c-a^2 c x^2} \, dx &=\frac {\int \frac {x^m (1+a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac {\int \frac {x^m}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}+\frac {a \int \frac {x^{1+m}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac {x^{1+m} \, _2F_1\left (\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{c (1+m)}+\frac {a x^{2+m} \, _2F_1\left (\frac {3}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{c (2+m)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 82, normalized size = 1.02 \[ \frac {\frac {x^{m+1} \, _2F_1\left (\frac {3}{2},\frac {m+1}{2};\frac {m+1}{2}+1;a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \, _2F_1\left (\frac {3}{2},\frac {m+2}{2};\frac {m+2}{2}+1;a^2 x^2\right )}{m+2}}{c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{m}}{a^{3} c x^{3} - a^{2} c x^{2} - a c x + c}, x\right ) \]
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 \[ \int -\frac {{\left (a x + 1\right )} x^{m}}{{\left (a^{2} c x^{2} - c\right )} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right ) x^{m}}{\sqrt {-a^{2} x^{2}+1}\, \left (-a^{2} c \,x^{2}+c \right )}\, dx \]
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 \[ -\int \frac {{\left (a x + 1\right )} x^{m}}{{\left (a^{2} c x^{2} - c\right )} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,\left (a\,x+1\right )}{\left (c-a^2\,c\,x^2\right )\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \frac {\int \frac {x^{m}}{- a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x x^{m}}{- a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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