Optimal. Leaf size=87 \[ \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {Int}\left (\frac {(d+e x)^{m+1}}{x \sqrt {1-c^2 x^2}},x\right )}{e (m+1)}+\frac {(d+e x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{e (m+1)} \]
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Rubi [A] time = 0.05, 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 = {} \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{e (1+m)}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(d+e x)^{1+m}}{x \sqrt {1-c^2 x^2}} \, dx}{e (1+m)}\\ \end {align*}
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Mathematica [A] time = 1.96, size = 0, normalized size = 0.00 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \operatorname {arsech}\left (c x\right ) + a\right )} {\left (e x + d\right )}^{m}, x\right ) \]
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 \[ \int {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.10, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{m} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )\, dx \]
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 \[ b {\left (\frac {{\left (e x + d\right )} {\left (e x + d\right )}^{m} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right ) - {\left (e x + d\right )} {\left (e x + d\right )}^{m} \log \relax (x)}{e {\left (m + 1\right )}} - \int \frac {{\left (c^{2} e {\left (m + 1\right )} x^{3} \log \relax (c) - {\left (e {\left (m + 1\right )} \log \relax (c) - e\right )} x + d\right )} {\left (e x + d\right )}^{m}}{c^{2} e {\left (m + 1\right )} x^{3} - e {\left (m + 1\right )} x}\,{d x} + \int \frac {{\left (c^{2} e x^{2} + c^{2} d x\right )} {\left (e x + d\right )}^{m}}{c^{2} e {\left (m + 1\right )} x^{2} + {\left (c^{2} e {\left (m + 1\right )} x^{2} - e {\left (m + 1\right )}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - e {\left (m + 1\right )}}\,{d x}\right )} + \frac {{\left (e x + d\right )}^{m + 1} a}{e {\left (m + 1\right )}} \]
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 \[ \int \left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^m \,d x \]
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 \[ \int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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