Integrand size = 10, antiderivative size = 203 \[ \int x \text {arcsinh}(a+b x)^3 \, dx=\frac {6 a \sqrt {1+(a+b x)^2}}{b^2}-\frac {3 (a+b x) \sqrt {1+(a+b x)^2}}{8 b^2}+\frac {3 \text {arcsinh}(a+b x)}{8 b^2}-\frac {6 a (a+b x) \text {arcsinh}(a+b x)}{b^2}+\frac {3 (a+b x)^2 \text {arcsinh}(a+b x)}{4 b^2}+\frac {3 a \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b^2}-\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{4 b^2}+\frac {\text {arcsinh}(a+b x)^3}{4 b^2}-\frac {a^2 \text {arcsinh}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {arcsinh}(a+b x)^3 \]
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Time = 0.21 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5859, 5828, 5843, 3398, 3377, 2718, 3392, 30, 2715, 8} \[ \int x \text {arcsinh}(a+b x)^3 \, dx=-\frac {a^2 \text {arcsinh}(a+b x)^3}{2 b^2}+\frac {\text {arcsinh}(a+b x)^3}{4 b^2}-\frac {3 (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2}{4 b^2}+\frac {3 a \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2}{b^2}+\frac {3 (a+b x)^2 \text {arcsinh}(a+b x)}{4 b^2}-\frac {6 a (a+b x) \text {arcsinh}(a+b x)}{b^2}+\frac {3 \text {arcsinh}(a+b x)}{8 b^2}+\frac {1}{2} x^2 \text {arcsinh}(a+b x)^3-\frac {3 (a+b x) \sqrt {(a+b x)^2+1}}{8 b^2}+\frac {6 a \sqrt {(a+b x)^2+1}}{b^2} \]
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Rule 8
Rule 30
Rule 2715
Rule 2718
Rule 3377
Rule 3392
Rule 3398
Rule 5828
Rule 5843
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right ) \text {arcsinh}(x)^3 \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{2} x^2 \text {arcsinh}(a+b x)^3-\frac {3}{2} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \text {arcsinh}(x)^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = \frac {1}{2} x^2 \text {arcsinh}(a+b x)^3-\frac {3}{2} \text {Subst}\left (\int x^2 \left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^2 \, dx,x,\text {arcsinh}(a+b x)\right ) \\ & = \frac {1}{2} x^2 \text {arcsinh}(a+b x)^3-\frac {3}{2} \text {Subst}\left (\int \left (\frac {a^2 x^2}{b^2}-\frac {2 a x^2 \sinh (x)}{b^2}+\frac {x^2 \sinh ^2(x)}{b^2}\right ) \, dx,x,\text {arcsinh}(a+b x)\right ) \\ & = -\frac {a^2 \text {arcsinh}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {arcsinh}(a+b x)^3-\frac {3 \text {Subst}\left (\int x^2 \sinh ^2(x) \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^2}+\frac {(3 a) \text {Subst}\left (\int x^2 \sinh (x) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^2} \\ & = \frac {3 (a+b x)^2 \text {arcsinh}(a+b x)}{4 b^2}+\frac {3 a \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b^2}-\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{4 b^2}-\frac {a^2 \text {arcsinh}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {arcsinh}(a+b x)^3+\frac {3 \text {Subst}\left (\int x^2 \, dx,x,\text {arcsinh}(a+b x)\right )}{4 b^2}-\frac {3 \text {Subst}\left (\int \sinh ^2(x) \, dx,x,\text {arcsinh}(a+b x)\right )}{4 b^2}-\frac {(6 a) \text {Subst}(\int x \cosh (x) \, dx,x,\text {arcsinh}(a+b x))}{b^2} \\ & = -\frac {3 (a+b x) \sqrt {1+(a+b x)^2}}{8 b^2}-\frac {6 a (a+b x) \text {arcsinh}(a+b x)}{b^2}+\frac {3 (a+b x)^2 \text {arcsinh}(a+b x)}{4 b^2}+\frac {3 a \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b^2}-\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{4 b^2}+\frac {\text {arcsinh}(a+b x)^3}{4 b^2}-\frac {a^2 \text {arcsinh}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {arcsinh}(a+b x)^3+\frac {3 \text {Subst}(\int 1 \, dx,x,\text {arcsinh}(a+b x))}{8 b^2}+\frac {(6 a) \text {Subst}(\int \sinh (x) \, dx,x,\text {arcsinh}(a+b x))}{b^2} \\ & = \frac {6 a \sqrt {1+(a+b x)^2}}{b^2}-\frac {3 (a+b x) \sqrt {1+(a+b x)^2}}{8 b^2}+\frac {3 \text {arcsinh}(a+b x)}{8 b^2}-\frac {6 a (a+b x) \text {arcsinh}(a+b x)}{b^2}+\frac {3 (a+b x)^2 \text {arcsinh}(a+b x)}{4 b^2}+\frac {3 a \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b^2}-\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{4 b^2}+\frac {\text {arcsinh}(a+b x)^3}{4 b^2}-\frac {a^2 \text {arcsinh}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {arcsinh}(a+b x)^3 \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.64 \[ \int x \text {arcsinh}(a+b x)^3 \, dx=\frac {3 (15 a-b x) \sqrt {1+a^2+2 a b x+b^2 x^2}+\left (3-42 a^2-36 a b x+6 b^2 x^2\right ) \text {arcsinh}(a+b x)+6 (3 a-b x) \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^2+\left (2-4 a^2+4 b^2 x^2\right ) \text {arcsinh}(a+b x)^3}{8 b^2} \]
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Time = 0.12 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {arcsinh}\left (b x +a \right )^{3} \left (1+\left (b x +a \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )}{4}-\frac {\operatorname {arcsinh}\left (b x +a \right )^{3}}{4}+\frac {3 \,\operatorname {arcsinh}\left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right )}{4}-\frac {3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (b x +a \right )}{8}-a \left (\operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )-3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}\right )}{b^{2}}\) | \(169\) |
default | \(\frac {\frac {\operatorname {arcsinh}\left (b x +a \right )^{3} \left (1+\left (b x +a \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )}{4}-\frac {\operatorname {arcsinh}\left (b x +a \right )^{3}}{4}+\frac {3 \,\operatorname {arcsinh}\left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right )}{4}-\frac {3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (b x +a \right )}{8}-a \left (\operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )-3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}\right )}{b^{2}}\) | \(169\) |
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Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.89 \[ \int x \text {arcsinh}(a+b x)^3 \, dx=\frac {2 \, {\left (2 \, b^{2} x^{2} - 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x - 3 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 3 \, {\left (2 \, b^{2} x^{2} - 12 \, a b x - 14 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x - 15 \, a\right )}}{8 \, b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.22 \[ \int x \text {arcsinh}(a+b x)^3 \, dx=\begin {cases} - \frac {a^{2} \operatorname {asinh}^{3}{\left (a + b x \right )}}{2 b^{2}} - \frac {21 a^{2} \operatorname {asinh}{\left (a + b x \right )}}{4 b^{2}} - \frac {9 a x \operatorname {asinh}{\left (a + b x \right )}}{2 b} + \frac {9 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4 b^{2}} + \frac {45 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{8 b^{2}} + \frac {x^{2} \operatorname {asinh}^{3}{\left (a + b x \right )}}{2} + \frac {3 x^{2} \operatorname {asinh}{\left (a + b x \right )}}{4} - \frac {3 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4 b} - \frac {3 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{8 b} + \frac {\operatorname {asinh}^{3}{\left (a + b x \right )}}{4 b^{2}} + \frac {3 \operatorname {asinh}{\left (a + b x \right )}}{8 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {asinh}^{3}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \]
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\[ \int x \text {arcsinh}(a+b x)^3 \, dx=\int { x \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]
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\[ \int x \text {arcsinh}(a+b x)^3 \, dx=\int { x \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]
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Timed out. \[ \int x \text {arcsinh}(a+b x)^3 \, dx=\int x\,{\mathrm {asinh}\left (a+b\,x\right )}^3 \,d x \]
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