Integrand size = 18, antiderivative size = 153 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=2 b^2 d x-\frac {4 b^2 e x}{9 c^2}+\frac {2}{27} b^2 e x^3-\frac {2 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {4 b e \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3}-\frac {2 b e x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c}+d x (a+b \text {arcsinh}(c x))^2+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^2 \]
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Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5793, 5772, 5798, 8, 5776, 5812, 30} \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {2 b d \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c}-\frac {2 b e x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{9 c}+\frac {4 b e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{9 c^3}+d x (a+b \text {arcsinh}(c x))^2+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^2-\frac {4 b^2 e x}{9 c^2}+2 b^2 d x+\frac {2}{27} b^2 e x^3 \]
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Rule 8
Rule 30
Rule 5772
Rule 5776
Rule 5793
Rule 5798
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \int \left (d (a+b \text {arcsinh}(c x))^2+e x^2 (a+b \text {arcsinh}(c x))^2\right ) \, dx \\ & = d \int (a+b \text {arcsinh}(c x))^2 \, dx+e \int x^2 (a+b \text {arcsinh}(c x))^2 \, dx \\ & = d x (a+b \text {arcsinh}(c x))^2+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^2-(2 b c d) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{3} (2 b c e) \int \frac {x^3 (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 {2 b e x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c}+d x (a+b \text {arcsinh}(c x))^2+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^2+\left (2 b^2 d\right ) \int 1 \, dx+\frac {1}{9} \left (2 b^2 e\right ) \int x^2 \, dx+\frac {(4 b e) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{9 c} \\ & = 2 b^2 d x+\frac {2}{27} b^2 e x^3-\frac {2 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {4 b e \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3}-\frac {2 b e x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c}+d x (a+b \text {arcsinh}(c x))^2+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^2-\frac {\left (4 b^2 e\right ) \int 1 \, dx}{9 c^2} \\ & = 2 b^2 d x-\frac {4 b^2 e x}{9 c^2}+\frac {2}{27} b^2 e x^3-\frac {2 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {4 b e \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3}-\frac {2 b e x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c}+d x (a+b \text {arcsinh}(c x))^2+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^2 \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.07 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {9 a^2 c^3 x \left (3 d+e x^2\right )-6 a b \sqrt {1+c^2 x^2} \left (-2 e+c^2 \left (9 d+e x^2\right )\right )+2 b^2 c x \left (-6 e+c^2 \left (27 d+e x^2\right )\right )-6 b \left (-3 a c^3 x \left (3 d+e x^2\right )+b \sqrt {1+c^2 x^2} \left (-2 e+c^2 \left (9 d+e x^2\right )\right )\right ) \text {arcsinh}(c x)+9 b^2 c^3 x \left (3 d+e x^2\right ) \text {arcsinh}(c x)^2}{27 c^3} \]
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Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.41
method | result | size |
parts | \(a^{2} \left (\frac {1}{3} x^{3} e +d x \right )+\frac {b^{2} \left (\frac {e \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{27 c^{2}}+d \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\frac {c \,\operatorname {arcsinh}\left (c x \right ) x^{3} e}{3}+\operatorname {arcsinh}\left (c x \right ) d c x -\frac {e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d \,c^{2} \sqrt {c^{2} x^{2}+1}}{3 c^{2}}\right )}{c}\) | \(215\) |
derivativedivides | \(\frac {\frac {a^{2} \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (d \,c^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+\frac {e \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-d \,c^{2} \sqrt {c^{2} x^{2}+1}\right )}{c^{2}}}{c}\) | \(227\) |
default | \(\frac {\frac {a^{2} \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (d \,c^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+\frac {e \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-d \,c^{2} \sqrt {c^{2} x^{2}+1}\right )}{c^{2}}}{c}\) | \(227\) |
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Time = 0.28 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.37 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} e x^{3} + 9 \, {\left (b^{2} c^{3} e x^{3} + 3 \, b^{2} c^{3} d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 3 \, {\left (9 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{3} d - 4 \, b^{2} c e\right )} x + 6 \, {\left (3 \, a b c^{3} e x^{3} + 9 \, a b c^{3} d x - {\left (b^{2} c^{2} e x^{2} + 9 \, b^{2} c^{2} d - 2 \, b^{2} e\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (a b c^{2} e x^{2} + 9 \, a b c^{2} d - 2 \, a b e\right )} \sqrt {c^{2} x^{2} + 1}}{27 \, c^{3}} \]
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Time = 0.31 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.82 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} a^{2} d x + \frac {a^{2} e x^{3}}{3} + 2 a b d x \operatorname {asinh}{\left (c x \right )} + \frac {2 a b e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {2 a b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {2 a b e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {4 a b e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d x + \frac {b^{2} e x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {2 b^{2} e x^{3}}{27} - \frac {2 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {2 b^{2} e x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c} - \frac {4 b^{2} e x}{9 c^{2}} + \frac {4 b^{2} e \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\a^{2} \left (d x + \frac {e x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.42 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{3} \, b^{2} e x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} e x^{3} + b^{2} d x \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b e - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} e + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d}{c} \]
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Exception generated. \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right ) \,d x \]
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