Integrand size = 10, antiderivative size = 90 \[ \int x^2 \text {arcsinh}(a+b x) \, dx=-\frac {x^2 \sqrt {1+(a+b x)^2}}{9 b}+\frac {\left (4-11 a^2+5 a b x\right ) \sqrt {1+(a+b x)^2}}{18 b^3}-\frac {a \left (3-2 a^2\right ) \text {arcsinh}(a+b x)}{6 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x) \]
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Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5859, 5828, 757, 794, 221} \[ \int x^2 \text {arcsinh}(a+b x) \, dx=-\frac {a \left (3-2 a^2\right ) \text {arcsinh}(a+b x)}{6 b^3}+\frac {\left (-11 a^2+5 a b x+4\right ) \sqrt {(a+b x)^2+1}}{18 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)-\frac {x^2 \sqrt {(a+b x)^2+1}}{9 b} \]
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Rule 221
Rule 757
Rule 794
Rule 5828
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) \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{3} x^3 \text {arcsinh}(a+b x)-\frac {1}{3} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3}{\sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {x^2 \sqrt {1+(a+b x)^2}}{9 b}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)-\frac {1}{9} \text {Subst}\left (\int \frac {\left (-\frac {2-3 a^2}{b^2}-\frac {5 a x}{b^2}\right ) \left (-\frac {a}{b}+\frac {x}{b}\right )}{\sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {x^2 \sqrt {1+(a+b x)^2}}{9 b}+\frac {\left (4-11 a^2+5 a b x\right ) \sqrt {1+(a+b x)^2}}{18 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x)-\frac {\left (a \left (3-2 a^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{6 b^3} \\ & = -\frac {x^2 \sqrt {1+(a+b x)^2}}{9 b}+\frac {\left (4-11 a^2+5 a b x\right ) \sqrt {1+(a+b x)^2}}{18 b^3}-\frac {a \left (3-2 a^2\right ) \text {arcsinh}(a+b x)}{6 b^3}+\frac {1}{3} x^3 \text {arcsinh}(a+b x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.82 \[ \int x^2 \text {arcsinh}(a+b x) \, dx=\frac {\left (4-11 a^2+5 a b x-2 b^2 x^2\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}+\left (-9 a+6 a^3+6 b^3 x^3\right ) \text {arcsinh}(a+b x)}{18 b^3} \]
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Time = 0.02 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.44
method | result | size |
derivativedivides | \(\frac {\operatorname {arcsinh}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}-\frac {\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (b x +a \right )^{2}}}{9}-a^{2} \sqrt {1+\left (b x +a \right )^{2}}+a \left (\frac {\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{2}-\frac {\operatorname {arcsinh}\left (b x +a \right )}{2}\right )}{b^{3}}\) | \(130\) |
default | \(\frac {\operatorname {arcsinh}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}-\frac {\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (b x +a \right )^{2}}}{9}-a^{2} \sqrt {1+\left (b x +a \right )^{2}}+a \left (\frac {\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{2}-\frac {\operatorname {arcsinh}\left (b x +a \right )}{2}\right )}{b^{3}}\) | \(130\) |
parts | \(\frac {x^{3} \operatorname {arcsinh}\left (b x +a \right )}{3}-\frac {b \left (\frac {x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{3 b^{2}}-\frac {5 a \left (\frac {x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}-\frac {3 a \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{2 b}-\frac {\left (a^{2}+1\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{3 b}-\frac {2 \left (a^{2}+1\right ) \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{3 b^{2}}\right )}{3}\) | \(285\) |
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Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.01 \[ \int x^2 \text {arcsinh}(a+b x) \, dx=\frac {3 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} - 3 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} - 4\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{18 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (83) = 166\).
Time = 0.25 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.89 \[ \int x^2 \text {arcsinh}(a+b x) \, dx=\begin {cases} \frac {a^{3} \operatorname {asinh}{\left (a + b x \right )}}{3 b^{3}} - \frac {11 a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{18 b^{3}} + \frac {5 a x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{18 b^{2}} - \frac {a \operatorname {asinh}{\left (a + b x \right )}}{2 b^{3}} + \frac {x^{3} \operatorname {asinh}{\left (a + b x \right )}}{3} - \frac {x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{9 b} + \frac {2 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{9 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {asinh}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (78) = 156\).
Time = 0.18 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.33 \[ \int x^2 \text {arcsinh}(a+b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {arsinh}\left (b x + a\right ) - \frac {1}{18} \, b {\left (\frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x^{2}}{b^{2}} - \frac {15 \, a^{3} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{4}} - \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{b^{3}} + \frac {9 \, {\left (a^{2} + 1\right )} a \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{4}} + \frac {15 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{b^{4}} - \frac {4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}}{b^{4}}\right )} \]
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Time = 0.43 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46 \[ \int x^2 \text {arcsinh}(a+b x) \, dx=\frac {1}{3} \, x^{3} \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right ) - \frac {1}{18} \, {\left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (x {\left (\frac {2 \, x}{b^{2}} - \frac {5 \, a}{b^{3}}\right )} + \frac {11 \, a^{2} b - 4 \, b}{b^{5}}\right )} + \frac {3 \, {\left (2 \, a^{3} - 3 \, a\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |}\right )}{b^{3} {\left | b \right |}}\right )} b \]
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Timed out. \[ \int x^2 \text {arcsinh}(a+b x) \, dx=\int x^2\,\mathrm {asinh}\left (a+b\,x\right ) \,d x \]
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