Optimal. Leaf size=116 \[ \frac {x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}-\frac {b c x^{2+m} \, _2F_1\left (\frac {3}{2},\frac {2+m}{2};\frac {4+m}{2};-c^2 x^2\right )}{2 d^2 (2+m)}+\frac {(1-m) \text {Int}\left (\frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2},x\right )}{2 d} \]
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Rubi [A]
time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^2} \, dx &=\frac {x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}-\frac {(b c) \int \frac {x^{1+m}}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {(1-m) \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx}{2 d}\\ &=\frac {x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}-\frac {b c x^{2+m} \, _2F_1\left (\frac {3}{2},\frac {2+m}{2};\frac {4+m}{2};-c^2 x^2\right )}{2 d^2 (2+m)}+\frac {(1-m) \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx}{2 d}\\ \end {align*}
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Mathematica [A]
time = 3.89, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (a +b \arcsinh \left (c x \right )\right )}{\left (c^{2} d \,x^{2}+d \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a x^{m}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{m} \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^m\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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