Optimal. Leaf size=113 \[ \frac {\left (2 m^2-4 m+1\right ) x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{8 c^2 (m+1)}+\frac {x^{m+1} \, _2F_1(1,m+1;m+2;-a x)}{8 c^2 (m+1)}+\frac {(2-m) x^{m+1}}{4 c^2 (1-a x)}+\frac {x^{m+1}}{4 c^2 (1-a x)^2} \]
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Rubi [A] time = 0.18, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6150, 103, 151, 156, 64} \[ \frac {\left (2 m^2-4 m+1\right ) x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{8 c^2 (m+1)}+\frac {x^{m+1} \, _2F_1(1,m+1;m+2;-a x)}{8 c^2 (m+1)}+\frac {(2-m) x^{m+1}}{4 c^2 (1-a x)}+\frac {x^{m+1}}{4 c^2 (1-a x)^2} \]
Antiderivative was successfully verified.
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Rule 64
Rule 103
Rule 151
Rule 156
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {x^m}{(1-a x)^3 (1+a x)} \, dx}{c^2}\\ &=\frac {x^{1+m}}{4 c^2 (1-a x)^2}-\frac {\int \frac {x^m \left (-a (3-m)-a^2 (1-m) x\right )}{(1-a x)^2 (1+a x)} \, dx}{4 a c^2}\\ &=\frac {x^{1+m}}{4 c^2 (1-a x)^2}+\frac {(2-m) x^{1+m}}{4 c^2 (1-a x)}+\frac {\int \frac {x^m \left (2 a^2 (1-m)^2-2 a^3 (2-m) m x\right )}{(1-a x) (1+a x)} \, dx}{8 a^2 c^2}\\ &=\frac {x^{1+m}}{4 c^2 (1-a x)^2}+\frac {(2-m) x^{1+m}}{4 c^2 (1-a x)}+\frac {\int \frac {x^m}{1+a x} \, dx}{8 c^2}+\frac {\left (1-4 m+2 m^2\right ) \int \frac {x^m}{1-a x} \, dx}{8 c^2}\\ &=\frac {x^{1+m}}{4 c^2 (1-a x)^2}+\frac {(2-m) x^{1+m}}{4 c^2 (1-a x)}+\frac {x^{1+m} \, _2F_1(1,1+m;2+m;-a x)}{8 c^2 (1+m)}+\frac {\left (1-4 m+2 m^2\right ) x^{1+m} \, _2F_1(1,1+m;2+m;a x)}{8 c^2 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 92, normalized size = 0.81 \[ \frac {x^{m+1} \left (\left (2 m^2-4 m+1\right ) (a x-1)^2 \, _2F_1(1,m+1;m+2;a x)+(a x-1)^2 \, _2F_1(1,m+1;m+2;-a x)+2 (m+1) (m (a x-1)-2 a x+3)\right )}{8 c^2 (m+1) (a x-1)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x^{m}}{a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x - c^{2}}, x\right ) \]
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 \[ \int -\frac {{\left (a x + 1\right )}^{2} x^{m}}{{\left (a^{2} c x^{2} - c\right )}^{2} {\left (a^{2} x^{2} - 1\right )}}\,{d x} \]
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 \[ \int \frac {\left (a x +1\right )^{2} x^{m}}{\left (-a^{2} x^{2}+1\right ) \left (-a^{2} c \,x^{2}+c \right )^{2}}\, dx \]
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 \[ -\int \frac {{\left (a x + 1\right )}^{2} x^{m}}{{\left (a^{2} c x^{2} - c\right )}^{2} {\left (a^{2} x^{2} - 1\right )}}\,{d x} \]
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 \[ -\int \frac {x^m\,{\left (a\,x+1\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^2\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \frac {\int \frac {x^{m}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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