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 = 0.0227585, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 5863
Rule 5653
Rule 261
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*}
Mathematica [A] time = 0.0281951, 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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Maple [A] time = 0.002, size = 36, normalized size = 0.9 \begin{align*} ax+{\frac{b}{d} \left ( \left ( dx+c \right ){\it Arcsinh} \left ( dx+c \right ) -\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11765, size = 47, normalized size = 1.21 \begin{align*} 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42392, size = 154, normalized size = 3.95 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.183411, size = 51, normalized size = 1.31 \begin{align*} 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}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.39628, size = 134, normalized size = 3.44 \begin{align*} -{\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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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