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 = 0.08, antiderivative size = 125, normalized size of antiderivative = 1.00, 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 138
Rule 139
Rule 5802
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*}
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Mathematica [A] time = 0.23, 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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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} {\left (e x + d\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.84, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{m} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )\, dx \]
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 \[ b {\left (\frac {{\left (e x + d\right )} {\left (e x + d\right )}^{m} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{e {\left (m + 1\right )}} - \int \frac {{\left (c^{2} e x^{2} + c^{2} d x\right )} {\left (e x + d\right )}^{m}}{c^{2} e {\left (m + 1\right )} x^{2} - e {\left (m + 1\right )}}\,{d x} + \int \frac {{\left (c e x + c d\right )} {\left (e x + d\right )}^{m}}{c^{3} e {\left (m + 1\right )} x^{3} - c e {\left (m + 1\right )} x + {\left (c^{2} e {\left (m + 1\right )} x^{2} - e {\left (m + 1\right )}\right )} \sqrt {c x + 1} \sqrt {c x - 1}}\,{d x}\right )} + \frac {{\left (e x + d\right )}^{m + 1} a}{e {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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