Optimal. Leaf size=39 \[ a x-\frac {b \sqrt {(c+d x)^2+1}}{d}+\frac {b (c+d x) \sinh ^{-1}(c+d x)}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5863, 5653, 261} \[ a x-\frac {b \sqrt {(c+d x)^2+1}}{d}+\frac {b (c+d x) \sinh ^{-1}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5653
Rule 5863
Rubi steps
\begin {align*} \int \left (a+b \sinh ^{-1}(c+d x)\right ) \, dx &=a x+b \int \sinh ^{-1}(c+d x) \, dx\\ &=a x+\frac {b \operatorname {Subst}\left (\int \sinh ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=a x+\frac {b (c+d x) \sinh ^{-1}(c+d x)}{d}-\frac {b \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}\\ &=a x-\frac {b \sqrt {1+(c+d x)^2}}{d}+\frac {b (c+d x) \sinh ^{-1}(c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 1.28 \[ a x-\frac {b \left (\sqrt {c^2+2 c d x+d^2 x^2+1}-c \sinh ^{-1}(c+d x)\right )}{d}+b x \sinh ^{-1}(c+d x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 65, normalized size = 1.67 \[ \frac {a d x + {\left (b d x + b c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} b}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 99, normalized size = 2.54 \[ -{\left (d {\left (\frac {c \log \left (-c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} {\left | d \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {{\left (d x + c\right )}^{2} + 1}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.92 \[ a x +\frac {b \left (\left (d x +c \right ) \arcsinh \left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 35, normalized size = 0.90 \[ a x + \frac {{\left ({\left (d x + c\right )} \operatorname {arsinh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} + 1}\right )} b}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 85, normalized size = 2.18 \[ a\,x+b\,x\,\mathrm {asinh}\left (c+d\,x\right )-\frac {b\,\sqrt {c^2+2\,c\,d\,x+d^2\,x^2+1}}{d}+\frac {b\,c\,d^2\,\ln \left (\sqrt {c^2+2\,c\,d\,x+d^2\,x^2+1}+\frac {x\,d^2+c\,d}{\sqrt {d^2}}\right )}{{\left (d^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 51, normalized size = 1.31 \[ a x + b \left (\begin {cases} \frac {c \operatorname {asinh}{\left (c + d x \right )}}{d} + x \operatorname {asinh}{\left (c + d x \right )} - \frac {\sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} & \text {for}\: d \neq 0 \\x \operatorname {asinh}{\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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