Optimal. Leaf size=125 \[ \frac{(d+e x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{e (m+1)}-\frac{\sqrt{2} b \sqrt{c x-1} (c d+e) (d+e x)^m \left (\frac{c (d+e x)}{c d+e}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-c x),\frac{e (1-c x)}{c d+e}\right )}{c e (m+1)} \]
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Rubi [A] time = 0.0811056, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {5802, 139, 138} \[ \frac{(d+e x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{e (m+1)}-\frac{\sqrt{2} b \sqrt{c x-1} (c d+e) (d+e x)^m \left (\frac{c (d+e x)}{c d+e}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-c x),\frac{e (1-c x)}{c d+e}\right )}{c e (m+1)} \]
Antiderivative was successfully verified.
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Rule 5802
Rule 139
Rule 138
Rubi steps
\begin{align*} \int (d+e x)^m \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{(d+e x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{e (1+m)}-\frac{(b c) \int \frac{(d+e x)^{1+m}}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{e (1+m)}\\ &=\frac{(d+e x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{e (1+m)}-\frac{\left (b (c d+e) (d+e x)^m \left (\frac{c (d+e x)}{c d+e}\right )^{-m}\right ) \int \frac{\left (\frac{c d}{c d+e}+\frac{c e x}{c d+e}\right )^{1+m}}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{e (1+m)}\\ &=-\frac{\sqrt{2} b (c d+e) \sqrt{-1+c x} (d+e x)^m \left (\frac{c (d+e x)}{c d+e}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-1-m;\frac{3}{2};\frac{1}{2} (1-c x),\frac{e (1-c x)}{c d+e}\right )}{c e (1+m)}+\frac{(d+e x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.223621, size = 177, normalized size = 1.42 \[ \frac{(d+e x)^m \left (\frac{c (d+e x)}{c d+e}\right )^{-m} \left (c (d+e x) \left (a+b \cosh ^{-1}(c x)\right ) \left (\frac{c (d+e x)}{c d+e}\right )^m-2 b e \sqrt{2 c x-2} F_1\left (\frac{1}{2};-\frac{1}{2},-m;\frac{3}{2};\frac{1}{2}-\frac{c x}{2},\frac{e-c e x}{c d+e}\right )+b \sqrt{2 c x-2} (e-c d) F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2}-\frac{c x}{2},\frac{e-c e x}{c d+e}\right )\right )}{c e (m+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 3.666, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}{\left (e x + d\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acosh}{\left (c x \right )}\right ) \left (d + e x\right )^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}{\left (e x + d\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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