Integrand size = 16, antiderivative size = 140 \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=2 b^2 d x+\frac {1}{4} b^2 e x^2-\frac {2 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}-\frac {b e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c}-\frac {d^2 (a+b \text {arcsinh}(c x))^2}{2 e}+\frac {e (a+b \text {arcsinh}(c x))^2}{4 c^2}+\frac {(d+e x)^2 (a+b \text {arcsinh}(c x))^2}{2 e} \]
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Time = 0.21 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5828, 5838, 5783, 5798, 8, 5812, 30} \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {2 b d \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c}-\frac {b e x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c}+\frac {e (a+b \text {arcsinh}(c x))^2}{4 c^2}-\frac {d^2 (a+b \text {arcsinh}(c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \text {arcsinh}(c x))^2}{2 e}+2 b^2 d x+\frac {1}{4} b^2 e x^2 \]
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Rule 8
Rule 30
Rule 5783
Rule 5798
Rule 5812
Rule 5828
Rule 5838
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))^2}{2 e}-\frac {(b c) \int \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{e} \\ & = \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))^2}{2 e}-\frac {(b c) \int \left (\frac {d^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {2 d e x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {e^2 x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}\right ) \, dx}{e} \\ & = \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))^2}{2 e}-(2 b c d) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx-\frac {\left (b c d^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{e}-(b c e) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {2 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}-\frac {b e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c}-\frac {d^2 (a+b \text {arcsinh}(c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \text {arcsinh}(c x))^2}{2 e}+\left (2 b^2 d\right ) \int 1 \, dx+\frac {1}{2} \left (b^2 e\right ) \int x \, dx+\frac {(b e) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 c} \\ & = 2 b^2 d x+\frac {1}{4} b^2 e x^2-\frac {2 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}-\frac {b e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c}-\frac {d^2 (a+b \text {arcsinh}(c x))^2}{2 e}+\frac {e (a+b \text {arcsinh}(c x))^2}{4 c^2}+\frac {(d+e x)^2 (a+b \text {arcsinh}(c x))^2}{2 e} \\ \end{align*}
Time = 1.15 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {c \left (2 a^2 c x (2 d+e x)+b^2 c x (8 d+e x)-2 a b (4 d+e x) \sqrt {1+c^2 x^2}\right )+2 b \left (-b c (4 d+e x) \sqrt {1+c^2 x^2}+a \left (e+4 c^2 d x+2 c^2 e x^2\right )\right ) \text {arcsinh}(c x)+b^2 \left (e+4 c^2 d x+2 c^2 e x^2\right ) \text {arcsinh}(c x)^2}{4 c^2} \]
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Time = 0.31 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.31
method | result | size |
parts | \(a^{2} \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b^{2} \left (\frac {e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4 c}+d \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\frac {c \,\operatorname {arcsinh}\left (c x \right ) x^{2} e}{2}+\operatorname {arcsinh}\left (c x \right ) d c x -\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+2 d c \sqrt {c^{2} x^{2}+1}}{2 c}\right )}{c}\) | \(183\) |
derivativedivides | \(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (\frac {e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}+d c \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{2}-d c \sqrt {c^{2} x^{2}+1}\right )}{c}}{c}\) | \(193\) |
default | \(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (\frac {e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}+d c \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{2}-d c \sqrt {c^{2} x^{2}+1}\right )}{c}}{c}\) | \(193\) |
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Time = 0.26 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.31 \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {{\left (2 \, a^{2} + b^{2}\right )} c^{2} e x^{2} + 4 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{2} d x + {\left (2 \, b^{2} c^{2} e x^{2} + 4 \, b^{2} c^{2} d x + b^{2} e\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, {\left (2 \, a b c^{2} e x^{2} + 4 \, a b c^{2} d x + a b e - {\left (b^{2} c e x + 4 \, b^{2} c d\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, {\left (a b c e x + 4 \, a b c d\right )} \sqrt {c^{2} x^{2} + 1}}{4 \, c^{2}} \]
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Time = 0.23 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.66 \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} a^{2} d x + \frac {a^{2} e x^{2}}{2} + 2 a b d x \operatorname {asinh}{\left (c x \right )} + a b e x^{2} \operatorname {asinh}{\left (c x \right )} - \frac {2 a b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {a b e x \sqrt {c^{2} x^{2} + 1}}{2 c} + \frac {a b e \operatorname {asinh}{\left (c x \right )}}{2 c^{2}} + b^{2} d x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d x + \frac {b^{2} e x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {b^{2} e x^{2}}{4} - \frac {2 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {b^{2} e x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2 c} + \frac {b^{2} e \operatorname {asinh}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a^{2} \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.56 \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{2} \, b^{2} e x^{2} \operatorname {arsinh}\left (c x\right )^{2} + b^{2} d x \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} e x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} a b e + \frac {1}{4} \, {\left (c^{2} {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{4}}\right )} - 2 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} \operatorname {arsinh}\left (c x\right )\right )} b^{2} e + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d}{c} \]
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Exception generated. \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right ) \,d x \]
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