Integrand size = 10, antiderivative size = 131 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\frac {7 a x^2 \sqrt {1+(a+b x)^2}}{48 b^2}-\frac {x^3 \sqrt {1+(a+b x)^2}}{16 b}-\frac {\left (4 a \left (16-19 a^2\right )-\left (9-26 a^2\right ) (a+b x)\right ) \sqrt {1+(a+b x)^2}}{96 b^4}-\frac {\left (3-24 a^2+8 a^4\right ) \text {arcsinh}(a+b x)}{32 b^4}+\frac {1}{4} x^4 \text {arcsinh}(a+b x) \]
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Time = 0.13 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5859, 5828, 757, 847, 794, 221} \[ \int x^3 \text {arcsinh}(a+b x) \, dx=-\frac {\left (4 a \left (16-19 a^2\right )-\left (9-26 a^2\right ) (a+b x)\right ) \sqrt {(a+b x)^2+1}}{96 b^4}-\frac {\left (8 a^4-24 a^2+3\right ) \text {arcsinh}(a+b x)}{32 b^4}+\frac {1}{4} x^4 \text {arcsinh}(a+b x)+\frac {7 a x^2 \sqrt {(a+b x)^2+1}}{48 b^2}-\frac {x^3 \sqrt {(a+b x)^2+1}}{16 b} \]
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Rule 221
Rule 757
Rule 794
Rule 847
Rule 5828
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^3 \text {arcsinh}(x) \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{4} x^4 \text {arcsinh}(a+b x)-\frac {1}{4} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^4}{\sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {x^3 \sqrt {1+(a+b x)^2}}{16 b}+\frac {1}{4} x^4 \text {arcsinh}(a+b x)-\frac {1}{16} \text {Subst}\left (\int \frac {\left (-\frac {3-4 a^2}{b^2}-\frac {7 a x}{b^2}\right ) \left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = \frac {7 a x^2 \sqrt {1+(a+b x)^2}}{48 b^2}-\frac {x^3 \sqrt {1+(a+b x)^2}}{16 b}+\frac {1}{4} x^4 \text {arcsinh}(a+b x)-\frac {1}{48} \text {Subst}\left (\int \frac {\left (\frac {a \left (23-12 a^2\right )}{b^3}-\frac {\left (9-26 a^2\right ) x}{b^3}\right ) \left (-\frac {a}{b}+\frac {x}{b}\right )}{\sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = \frac {7 a x^2 \sqrt {1+(a+b x)^2}}{48 b^2}-\frac {x^3 \sqrt {1+(a+b x)^2}}{16 b}-\frac {\left (4 a \left (16-19 a^2\right )-\left (9-26 a^2\right ) (a+b x)\right ) \sqrt {1+(a+b x)^2}}{96 b^4}+\frac {1}{4} x^4 \text {arcsinh}(a+b x)-\frac {\left (3-24 a^2+8 a^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{32 b^4} \\ & = \frac {7 a x^2 \sqrt {1+(a+b x)^2}}{48 b^2}-\frac {x^3 \sqrt {1+(a+b x)^2}}{16 b}-\frac {\left (4 a \left (16-19 a^2\right )-\left (9-26 a^2\right ) (a+b x)\right ) \sqrt {1+(a+b x)^2}}{96 b^4}-\frac {\left (3-24 a^2+8 a^4\right ) \text {arcsinh}(a+b x)}{32 b^4}+\frac {1}{4} x^4 \text {arcsinh}(a+b x) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.73 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\frac {\sqrt {1+a^2+2 a b x+b^2 x^2} \left (50 a^3+9 b x-26 a^2 b x-6 b^3 x^3+a \left (-55+14 b^2 x^2\right )\right )-3 \left (3-24 a^2+8 a^4-8 b^4 x^4\right ) \text {arcsinh}(a+b x)}{96 b^4} \]
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Time = 0.05 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.53
method | result | size |
derivativedivides | \(\frac {-\operatorname {arcsinh}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arcsinh}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\left (b x +a \right )^{3} \sqrt {1+\left (b x +a \right )^{2}}}{16}+\frac {3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (b x +a \right )}{32}+a^{3} \sqrt {1+\left (b x +a \right )^{2}}-\frac {3 a^{2} \left (\frac {\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{2}-\frac {\operatorname {arcsinh}\left (b x +a \right )}{2}\right )}{2}+a \left (\frac {\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{3}-\frac {2 \sqrt {1+\left (b x +a \right )^{2}}}{3}\right )}{b^{4}}\) | \(200\) |
default | \(\frac {-\operatorname {arcsinh}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arcsinh}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\left (b x +a \right )^{3} \sqrt {1+\left (b x +a \right )^{2}}}{16}+\frac {3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (b x +a \right )}{32}+a^{3} \sqrt {1+\left (b x +a \right )^{2}}-\frac {3 a^{2} \left (\frac {\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{2}-\frac {\operatorname {arcsinh}\left (b x +a \right )}{2}\right )}{2}+a \left (\frac {\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{3}-\frac {2 \sqrt {1+\left (b x +a \right )^{2}}}{3}\right )}{b^{4}}\) | \(200\) |
parts | \(\frac {x^{4} \operatorname {arcsinh}\left (b x +a \right )}{4}-\frac {b \left (\frac {x^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}-\frac {7 a \left (\frac {x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{3 b^{2}}-\frac {5 a \left (\frac {x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}-\frac {3 a \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{2 b}-\frac {\left (a^{2}+1\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{3 b}-\frac {2 \left (a^{2}+1\right ) \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{3 b^{2}}\right )}{4 b}-\frac {3 \left (a^{2}+1\right ) \left (\frac {x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}-\frac {3 a \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{2 b}-\frac {\left (a^{2}+1\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{4 b^{2}}\right )}{4}\) | \(483\) |
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Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.84 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\frac {3 \, {\left (8 \, b^{4} x^{4} - 8 \, a^{4} + 24 \, a^{2} - 3\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - {\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} - 50 \, a^{3} + {\left (26 \, a^{2} - 9\right )} b x + 55 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{96 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (117) = 234\).
Time = 0.32 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.95 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\begin {cases} - \frac {a^{4} \operatorname {asinh}{\left (a + b x \right )}}{4 b^{4}} + \frac {25 a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{48 b^{4}} - \frac {13 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{48 b^{3}} + \frac {3 a^{2} \operatorname {asinh}{\left (a + b x \right )}}{4 b^{4}} + \frac {7 a x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{48 b^{2}} - \frac {55 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{96 b^{4}} + \frac {x^{4} \operatorname {asinh}{\left (a + b x \right )}}{4} - \frac {x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{16 b} + \frac {3 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{32 b^{3}} - \frac {3 \operatorname {asinh}{\left (a + b x \right )}}{32 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {asinh}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (114) = 228\).
Time = 0.18 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.43 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {arsinh}\left (b x + a\right ) - \frac {1}{96} \, {\left (\frac {6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x^{3}}{b^{2}} - \frac {14 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x^{2}}{b^{3}} + \frac {105 \, a^{4} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{5}} + \frac {35 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} x}{b^{4}} - \frac {90 \, {\left (a^{2} + 1\right )} a^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{5}} - \frac {105 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3}}{b^{5}} - \frac {9 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} x}{b^{4}} + \frac {9 \, {\left (a^{2} + 1\right )}^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{5}} + \frac {55 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} a}{b^{5}}\right )} b \]
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Time = 0.57 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.24 \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\frac {1}{4} \, x^{4} \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right ) - \frac {1}{96} \, {\left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left ({\left (2 \, x {\left (\frac {3 \, x}{b^{2}} - \frac {7 \, a}{b^{3}}\right )} + \frac {26 \, a^{2} b^{3} - 9 \, b^{3}}{b^{7}}\right )} x - \frac {5 \, {\left (10 \, a^{3} b^{2} - 11 \, a b^{2}\right )}}{b^{7}}\right )} - \frac {3 \, {\left (8 \, a^{4} - 24 \, a^{2} + 3\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |}\right )}{b^{4} {\left | b \right |}}\right )} b \]
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Timed out. \[ \int x^3 \text {arcsinh}(a+b x) \, dx=\int x^3\,\mathrm {asinh}\left (a+b\,x\right ) \,d x \]
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