Integrand size = 22, antiderivative size = 120 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {b d x \sqrt {1+c^2 x^2}}{24 c^3}-\frac {b d x^3 \sqrt {1+c^2 x^2}}{36 c}-\frac {1}{36} b c d x^5 \sqrt {1+c^2 x^2}-\frac {b d \text {arcsinh}(c x)}{24 c^4}+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x)) \]
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Time = 0.07 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {14, 5803, 12, 470, 327, 221} \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))-\frac {b d \text {arcsinh}(c x)}{24 c^4}-\frac {1}{36} b c d x^5 \sqrt {c^2 x^2+1}-\frac {b d x^3 \sqrt {c^2 x^2+1}}{36 c}+\frac {b d x \sqrt {c^2 x^2+1}}{24 c^3} \]
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Rule 12
Rule 14
Rule 221
Rule 327
Rule 470
Rule 5803
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x))-(b c) \int \frac {d x^4 \left (3+2 c^2 x^2\right )}{12 \sqrt {1+c^2 x^2}} \, dx \\ & = \frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x))-\frac {1}{12} (b c d) \int \frac {x^4 \left (3+2 c^2 x^2\right )}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {1}{36} b c d x^5 \sqrt {1+c^2 x^2}+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x))-\frac {1}{9} (b c d) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {b d x^3 \sqrt {1+c^2 x^2}}{36 c}-\frac {1}{36} b c d x^5 \sqrt {1+c^2 x^2}+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x))+\frac {(b d) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{12 c} \\ & = \frac {b d x \sqrt {1+c^2 x^2}}{24 c^3}-\frac {b d x^3 \sqrt {1+c^2 x^2}}{36 c}-\frac {1}{36} b c d x^5 \sqrt {1+c^2 x^2}+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x))-\frac {(b d) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{24 c^3} \\ & = \frac {b d x \sqrt {1+c^2 x^2}}{24 c^3}-\frac {b d x^3 \sqrt {1+c^2 x^2}}{36 c}-\frac {1}{36} b c d x^5 \sqrt {1+c^2 x^2}-\frac {b d \text {arcsinh}(c x)}{24 c^4}+\frac {1}{4} d x^4 (a+b \text {arcsinh}(c x))+\frac {1}{6} c^2 d x^6 (a+b \text {arcsinh}(c x)) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.73 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {d \left (6 a c^4 x^4 \left (3+2 c^2 x^2\right )+b c x \sqrt {1+c^2 x^2} \left (3-2 c^2 x^2-2 c^4 x^4\right )+3 b \left (-1+6 c^4 x^4+4 c^6 x^6\right ) \text {arcsinh}(c x)\right )}{72 c^4} \]
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Time = 0.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.91
method | result | size |
parts | \(d a \left (\frac {1}{6} c^{2} x^{6}+\frac {1}{4} x^{4}\right )+\frac {d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{36}+\frac {c x \sqrt {c^{2} x^{2}+1}}{24}-\frac {\operatorname {arcsinh}\left (c x \right )}{24}\right )}{c^{4}}\) | \(109\) |
derivativedivides | \(\frac {d a \left (\frac {1}{6} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{36}+\frac {c x \sqrt {c^{2} x^{2}+1}}{24}-\frac {\operatorname {arcsinh}\left (c x \right )}{24}\right )}{c^{4}}\) | \(113\) |
default | \(\frac {d a \left (\frac {1}{6} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{36}+\frac {c x \sqrt {c^{2} x^{2}+1}}{24}-\frac {\operatorname {arcsinh}\left (c x \right )}{24}\right )}{c^{4}}\) | \(113\) |
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Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.91 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {12 \, a c^{6} d x^{6} + 18 \, a c^{4} d x^{4} + 3 \, {\left (4 \, b c^{6} d x^{6} + 6 \, b c^{4} d x^{4} - b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (2 \, b c^{5} d x^{5} + 2 \, b c^{3} d x^{3} - 3 \, b c d x\right )} \sqrt {c^{2} x^{2} + 1}}{72 \, c^{4}} \]
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Time = 0.60 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.15 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{2} d x^{6}}{6} + \frac {a d x^{4}}{4} + \frac {b c^{2} d x^{6} \operatorname {asinh}{\left (c x \right )}}{6} - \frac {b c d x^{5} \sqrt {c^{2} x^{2} + 1}}{36} + \frac {b d x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {b d x^{3} \sqrt {c^{2} x^{2} + 1}}{36 c} + \frac {b d x \sqrt {c^{2} x^{2} + 1}}{24 c^{3}} - \frac {b d \operatorname {asinh}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d x^{4}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.38 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{6} \, a c^{2} d x^{6} + \frac {1}{4} \, a d x^{4} + \frac {1}{288} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} b c^{2} d + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b d \]
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Exception generated. \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right ) \,d x \]
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