Integrand size = 14, antiderivative size = 97 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=-\frac {3 b d \sqrt {1+c^2 x^2}}{4 c}-\frac {b (d+e x) \sqrt {1+c^2 x^2}}{4 c}-\frac {b \left (2 d^2-\frac {e^2}{c^2}\right ) \text {arcsinh}(c x)}{4 e}+\frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e} \]
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Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5828, 757, 655, 221} \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e}-\frac {b \text {arcsinh}(c x) \left (2 d^2-\frac {e^2}{c^2}\right )}{4 e}-\frac {b \sqrt {c^2 x^2+1} (d+e x)}{4 c}-\frac {3 b d \sqrt {c^2 x^2+1}}{4 c} \]
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Rule 221
Rule 655
Rule 757
Rule 5828
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e}-\frac {(b c) \int \frac {(d+e x)^2}{\sqrt {1+c^2 x^2}} \, dx}{2 e} \\ & = -\frac {b (d+e x) \sqrt {1+c^2 x^2}}{4 c}+\frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e}-\frac {b \int \frac {2 c^2 d^2-e^2+3 c^2 d e x}{\sqrt {1+c^2 x^2}} \, dx}{4 c e} \\ & = -\frac {3 b d \sqrt {1+c^2 x^2}}{4 c}-\frac {b (d+e x) \sqrt {1+c^2 x^2}}{4 c}+\frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e}-\frac {1}{4} \left (b \left (\frac {2 c d^2}{e}-\frac {e}{c}\right )\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {3 b d \sqrt {1+c^2 x^2}}{4 c}-\frac {b (d+e x) \sqrt {1+c^2 x^2}}{4 c}-\frac {b \left (2 d^2-\frac {e^2}{c^2}\right ) \text {arcsinh}(c x)}{4 e}+\frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=a d x+\frac {1}{2} a e x^2-\frac {b d \sqrt {1+c^2 x^2}}{c}-\frac {b e x \sqrt {1+c^2 x^2}}{4 c}+\frac {b e \text {arcsinh}(c x)}{4 c^2}+b d x \text {arcsinh}(c x)+\frac {1}{2} b e x^2 \text {arcsinh}(c x) \]
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Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87
method | result | size |
parts | \(a \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b \left (\frac {c \,\operatorname {arcsinh}\left (c x \right ) x^{2} e}{2}+\operatorname {arcsinh}\left (c x \right ) d c x -\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+2 d c \sqrt {c^{2} x^{2}+1}}{2 c}\right )}{c}\) | \(84\) |
derivativedivides | \(\frac {\frac {a \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{2}-d c \sqrt {c^{2} x^{2}+1}\right )}{c}}{c}\) | \(96\) |
default | \(\frac {\frac {a \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{2}-d c \sqrt {c^{2} x^{2}+1}\right )}{c}}{c}\) | \(96\) |
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Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {2 \, a c^{2} e x^{2} + 4 \, a c^{2} d x + {\left (2 \, b c^{2} e x^{2} + 4 \, b c^{2} d x + b e\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c e x + 4 \, b c d\right )} \sqrt {c^{2} x^{2} + 1}}{4 \, c^{2}} \]
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Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} a d x + \frac {a e x^{2}}{2} + b d x \operatorname {asinh}{\left (c x \right )} + \frac {b e x^{2} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {b e x \sqrt {c^{2} x^{2} + 1}}{4 c} + \frac {b e \operatorname {asinh}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{2} \, a e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b e + a d x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d}{c} \]
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Time = 0.34 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{2} \, a e x^{2} + {\left (x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{c}\right )} b d + \frac {1}{4} \, {\left (2 \, x^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} + \frac {\log \left (-x {\left | c \right |} + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2} {\left | c \right |}}\right )}\right )} b e + a d x \]
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Time = 2.76 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {a\,x\,\left (2\,d+e\,x\right )}{2}-\frac {b\,d\,\left (\sqrt {c^2\,x^2+1}-c\,x\,\mathrm {asinh}\left (c\,x\right )\right )}{c}-\frac {b\,e\,x\,\sqrt {c^2\,x^2+1}}{4\,c}+b\,e\,x\,\mathrm {asinh}\left (c\,x\right )\,\left (\frac {x}{2}+\frac {1}{4\,c^2\,x}\right ) \]
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