Integrand size = 12, antiderivative size = 211 \[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=-\frac {4 x}{9 b^2}+\frac {2 a^2 x}{b^2}-\frac {a (a+b x)^2}{2 b^3}+\frac {2 (a+b x)^3}{27 b^3}+\frac {4 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{9 b^3}-\frac {2 a^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{b^3}+\frac {a (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{b^3}-\frac {2 (a+b x)^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{9 b^3}-\frac {a \text {arcsinh}(a+b x)^2}{2 b^3}+\frac {a^3 \text {arcsinh}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^2 \]
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Time = 0.26 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5859, 5828, 5838, 5783, 5798, 8, 5812, 30} \[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\frac {a^3 \text {arcsinh}(a+b x)^2}{3 b^3}-\frac {2 a^2 \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{b^3}+\frac {2 a^2 x}{b^2}-\frac {a \text {arcsinh}(a+b x)^2}{2 b^3}+\frac {a (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{b^3}-\frac {2 (a+b x)^2 \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{9 b^3}+\frac {4 \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{9 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^2-\frac {a (a+b x)^2}{2 b^3}+\frac {2 (a+b x)^3}{27 b^3}-\frac {4 x}{9 b^2} \]
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Rule 8
Rule 30
Rule 5783
Rule 5798
Rule 5812
Rule 5828
Rule 5838
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \text {arcsinh}(x)^2 \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{3} x^3 \text {arcsinh}(a+b x)^2-\frac {2}{3} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3 \text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = \frac {1}{3} x^3 \text {arcsinh}(a+b x)^2-\frac {2}{3} \text {Subst}\left (\int \left (-\frac {a^3 \text {arcsinh}(x)}{b^3 \sqrt {1+x^2}}+\frac {3 a^2 x \text {arcsinh}(x)}{b^3 \sqrt {1+x^2}}-\frac {3 a x^2 \text {arcsinh}(x)}{b^3 \sqrt {1+x^2}}+\frac {x^3 \text {arcsinh}(x)}{b^3 \sqrt {1+x^2}}\right ) \, dx,x,a+b x\right ) \\ & = \frac {1}{3} x^3 \text {arcsinh}(a+b x)^2-\frac {2 \text {Subst}\left (\int \frac {x^3 \text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{3 b^3}+\frac {(2 a) \text {Subst}\left (\int \frac {x^2 \text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b^3}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {x \text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b^3}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {\text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{3 b^3} \\ & = -\frac {2 a^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{b^3}+\frac {a (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{b^3}-\frac {2 (a+b x)^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{9 b^3}+\frac {a^3 \text {arcsinh}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^2+\frac {2 \text {Subst}\left (\int x^2 \, dx,x,a+b x\right )}{9 b^3}+\frac {4 \text {Subst}\left (\int \frac {x \text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{9 b^3}-\frac {a \text {Subst}(\int x \, dx,x,a+b x)}{b^3}-\frac {a \text {Subst}\left (\int \frac {\text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b^3}+\frac {\left (2 a^2\right ) \text {Subst}(\int 1 \, dx,x,a+b x)}{b^3} \\ & = \frac {2 a^2 x}{b^2}-\frac {a (a+b x)^2}{2 b^3}+\frac {2 (a+b x)^3}{27 b^3}+\frac {4 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{9 b^3}-\frac {2 a^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{b^3}+\frac {a (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{b^3}-\frac {2 (a+b x)^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{9 b^3}-\frac {a \text {arcsinh}(a+b x)^2}{2 b^3}+\frac {a^3 \text {arcsinh}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^2-\frac {4 \text {Subst}(\int 1 \, dx,x,a+b x)}{9 b^3} \\ & = -\frac {4 x}{9 b^2}+\frac {2 a^2 x}{b^2}-\frac {a (a+b x)^2}{2 b^3}+\frac {2 (a+b x)^3}{27 b^3}+\frac {4 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{9 b^3}-\frac {2 a^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{b^3}+\frac {a (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{b^3}-\frac {2 (a+b x)^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{9 b^3}-\frac {a \text {arcsinh}(a+b x)^2}{2 b^3}+\frac {a^3 \text {arcsinh}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)^2 \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.51 \[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\frac {b x \left (-24+66 a^2-15 a b x+4 b^2 x^2\right )-6 \sqrt {1+a^2+2 a b x+b^2 x^2} \left (-4+11 a^2-5 a b x+2 b^2 x^2\right ) \text {arcsinh}(a+b x)+9 \left (-3 a+2 a^3+2 b^3 x^3\right ) \text {arcsinh}(a+b x)^2}{54 b^3} \]
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Time = 0.17 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {\frac {\left (b x +a \right )^{3} \operatorname {arcsinh}\left (b x +a \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}}{9}-\frac {4 b x}{9}-\frac {4 a}{9}+\frac {2 \left (b x +a \right )^{3}}{27}-\frac {a \left (2 \operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )^{2}-2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )^{2}+\left (b x +a \right )^{2}+1\right )}{2}+a^{2} \left (\operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )-2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+2 b x +2 a \right )}{b^{3}}\) | \(190\) |
default | \(\frac {\frac {\left (b x +a \right )^{3} \operatorname {arcsinh}\left (b x +a \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}}{9}-\frac {4 b x}{9}-\frac {4 a}{9}+\frac {2 \left (b x +a \right )^{3}}{27}-\frac {a \left (2 \operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )^{2}-2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )^{2}+\left (b x +a \right )^{2}+1\right )}{2}+a^{2} \left (\operatorname {arcsinh}\left (b x +a \right )^{2} \left (b x +a \right )-2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+2 b x +2 a \right )}{b^{3}}\) | \(190\) |
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Time = 0.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.69 \[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\frac {4 \, b^{3} x^{3} - 15 \, a b^{2} x^{2} + 6 \, {\left (11 \, a^{2} - 4\right )} b x + 9 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} - 3 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} - 6 \, {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} - 4\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{54 \, b^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.15 \[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\begin {cases} \frac {a^{3} \operatorname {asinh}^{2}{\left (a + b x \right )}}{3 b^{3}} + \frac {11 a^{2} x}{9 b^{2}} - \frac {11 a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{9 b^{3}} - \frac {5 a x^{2}}{18 b} + \frac {5 a x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{9 b^{2}} - \frac {a \operatorname {asinh}^{2}{\left (a + b x \right )}}{2 b^{3}} + \frac {x^{3} \operatorname {asinh}^{2}{\left (a + b x \right )}}{3} + \frac {2 x^{3}}{27} - \frac {2 x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{9 b} - \frac {4 x}{9 b^{2}} + \frac {4 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{9 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {asinh}^{2}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \]
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\[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\int { x^{2} \operatorname {arsinh}\left (b x + a\right )^{2} \,d x } \]
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\[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\int { x^{2} \operatorname {arsinh}\left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int x^2 \text {arcsinh}(a+b x)^2 \, dx=\int x^2\,{\mathrm {asinh}\left (a+b\,x\right )}^2 \,d x \]
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