Optimal. Leaf size=113 \[ \frac {x^{1+m}}{4 c^2 (1-a x)^2}+\frac {(2-m) x^{1+m}}{4 c^2 (1-a x)}+\frac {x^{1+m} \, _2F_1(1,1+m;2+m;-a x)}{8 c^2 (1+m)}+\frac {\left (1-4 m+2 m^2\right ) x^{1+m} \, _2F_1(1,1+m;2+m;a x)}{8 c^2 (1+m)} \]
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Rubi [A]
time = 0.12, 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 = {6285, 105, 156,
162, 66} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 105
Rule 156
Rule 162
Rule 6285
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 = 0.05, size = 92, normalized size = 0.81 \begin {gather*} \frac {x^{1+m} \left (2 (1+m) (3-2 a x+m (-1+a x))+(-1+a x)^2 \, _2F_1(1,1+m;2+m;-a x)+\left (1-4 m+2 m^2\right ) (-1+a x)^2 \, _2F_1(1,1+m;2+m;a x)\right )}{8 c^2 (1+m) (-1+a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - \frac {\int \frac {x^{m}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx}{c^{2}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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