Optimal. Leaf size=21 \[ \text {Int}\left (\frac {(d+e x)^m}{\left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.03, 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 = {} \[ \int \frac {(d+e x)^m}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=\int \frac {(d+e x)^m}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.78, size = 0, normalized size = 0.00 \[ \int \frac {(d+e x)^m}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{m}}{\left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx \]
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 \[ -\frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{m} + {\left (c^{3} x^{3} + c x\right )} {\left (e x + d\right )}^{m}}{a b c^{3} x^{2} + \sqrt {c^{2} x^{2} + 1} a b c^{2} x + a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )} + \int \frac {{\left (c^{3} e {\left (m + 1\right )} x^{3} + c^{3} d x^{2} + c e {\left (m - 1\right )} x - c d\right )} {\left (c^{2} x^{2} + 1\right )} {\left (e x + d\right )}^{m} + {\left (2 \, c^{4} e {\left (m + 1\right )} x^{4} + 2 \, c^{4} d x^{3} + c^{2} e {\left (3 \, m + 1\right )} x^{2} + c^{2} d x + e m\right )} \sqrt {c^{2} x^{2} + 1} {\left (e x + d\right )}^{m} + {\left (c^{5} e {\left (m + 1\right )} x^{5} + c^{5} d x^{4} + 2 \, c^{3} e {\left (m + 1\right )} x^{3} + 2 \, c^{3} d x^{2} + c e {\left (m + 1\right )} x + c d\right )} {\left (e x + d\right )}^{m}}{a b c^{5} e x^{5} + a b c^{5} d x^{4} + 2 \, a b c^{3} e x^{3} + 2 \, a b c^{3} d x^{2} + a b c e x + a b c d + {\left (a b c^{3} e x^{3} + a b c^{3} d x^{2}\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (b^{2} c^{5} e x^{5} + b^{2} c^{5} d x^{4} + 2 \, b^{2} c^{3} e x^{3} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c e x + b^{2} c d + {\left (b^{2} c^{3} e x^{3} + b^{2} c^{3} d x^{2}\right )} {\left (c^{2} x^{2} + 1\right )} + 2 \, {\left (b^{2} c^{4} e x^{4} + b^{2} c^{4} d x^{3} + b^{2} c^{2} e x^{2} + b^{2} c^{2} d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a b c^{4} e x^{4} + a b c^{4} d x^{3} + a b c^{2} e x^{2} + a b c^{2} d x\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (d+e\,x\right )}^m}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{m}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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