Optimal. Leaf size=80 \[ \frac {x^{1+m} \, _2F_1\left (\frac {7}{2},\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{c^3 (1+m)}+\frac {a x^{2+m} \, _2F_1\left (\frac {7}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{c^3 (2+m)} \]
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Rubi [A]
time = 0.08, 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 = {6283, 822, 371}
\begin {gather*} \frac {x^{m+1} \, _2F_1\left (\frac {7}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{c^3 (m+1)}+\frac {a x^{m+2} \, _2F_1\left (\frac {7}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{c^3 (m+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 822
Rule 6283
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^m}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x^m (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac {\int \frac {x^m}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}+\frac {a \int \frac {x^{1+m}}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac {x^{1+m} \, _2F_1\left (\frac {7}{2},\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{c^3 (1+m)}+\frac {a x^{2+m} \, _2F_1\left (\frac {7}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{c^3 (2+m)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 82, normalized size = 1.02 \begin {gather*} \frac {\frac {x^{1+m} \, _2F_1\left (\frac {7}{2},\frac {1+m}{2};1+\frac {1+m}{2};a^2 x^2\right )}{1+m}+\frac {a x^{2+m} \, _2F_1\left (\frac {7}{2},\frac {2+m}{2};1+\frac {2+m}{2};a^2 x^2\right )}{2+m}}{c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +1\right ) x^{m}}{\sqrt {-a^{2} x^{2}+1}\, \left (-a^{2} c \,x^{2}+c \right )^{3}}\, dx\]
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^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x x^{m}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \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 )}{{\left (c-a^2\,c\,x^2\right )}^3\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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