Integrand size = 16, antiderivative size = 124 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=-\frac {b (d+e x)^2 \sqrt {1+c^2 x^2}}{9 c}-\frac {b \left (4 \left (4 c^2 d^2-e^2\right )+5 c^2 d e x\right ) \sqrt {1+c^2 x^2}}{18 c^3}-\frac {b d \left (2 d^2-\frac {3 e^2}{c^2}\right ) \text {arcsinh}(c x)}{6 e}+\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e} \]
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Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5828, 757, 794, 221} \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e}-\frac {b d \text {arcsinh}(c x) \left (2 d^2-\frac {3 e^2}{c^2}\right )}{6 e}-\frac {b \sqrt {c^2 x^2+1} (d+e x)^2}{9 c}-\frac {b \sqrt {c^2 x^2+1} \left (4 \left (4 c^2 d^2-e^2\right )+5 c^2 d e x\right )}{18 c^3} \]
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Rule 221
Rule 757
Rule 794
Rule 5828
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e}-\frac {(b c) \int \frac {(d+e x)^3}{\sqrt {1+c^2 x^2}} \, dx}{3 e} \\ & = -\frac {b (d+e x)^2 \sqrt {1+c^2 x^2}}{9 c}+\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e}-\frac {b \int \frac {(d+e x) \left (3 c^2 d^2-2 e^2+5 c^2 d e x\right )}{\sqrt {1+c^2 x^2}} \, dx}{9 c e} \\ & = -\frac {b (d+e x)^2 \sqrt {1+c^2 x^2}}{9 c}-\frac {b \left (4 \left (4 c^2 d^2-e^2\right )+5 c^2 d e x\right ) \sqrt {1+c^2 x^2}}{18 c^3}+\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e}-\frac {1}{6} \left (b d \left (\frac {2 c d^2}{e}-\frac {3 e}{c}\right )\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {b (d+e x)^2 \sqrt {1+c^2 x^2}}{9 c}-\frac {b \left (4 \left (4 c^2 d^2-e^2\right )+5 c^2 d e x\right ) \sqrt {1+c^2 x^2}}{18 c^3}-\frac {b d \left (2 d^2-\frac {3 e^2}{c^2}\right ) \text {arcsinh}(c x)}{6 e}+\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-b \sqrt {1+c^2 x^2} \left (-4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 b c \left (6 c^2 d^2 x+2 c^2 e^2 x^3+3 d \left (e+2 c^2 e x^2\right )\right ) \text {arcsinh}(c x)}{18 c^3} \]
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Time = 0.32 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.40
method | result | size |
parts | \(\frac {a \left (e x +d \right )^{3}}{3 e}+\frac {b \left (\frac {c \,e^{2} \operatorname {arcsinh}\left (c x \right ) x^{3}}{3}+c \,\operatorname {arcsinh}\left (c x \right ) d e \,x^{2}+\operatorname {arcsinh}\left (c x \right ) c x \,d^{2}+\frac {c \,\operatorname {arcsinh}\left (c x \right ) d^{3}}{3 e}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (c x \right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{3 c^{2} e}\right )}{c}\) | \(174\) |
derivativedivides | \(\frac {\frac {a \left (e c x +c d \right )^{3}}{3 c^{2} e}+\frac {b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arcsinh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arcsinh}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (c x \right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{3 e}\right )}{c^{2}}}{c}\) | \(189\) |
default | \(\frac {\frac {a \left (e c x +c d \right )^{3}}{3 c^{2} e}+\frac {b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arcsinh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arcsinh}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (c x \right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{3 e}\right )}{c^{2}}}{c}\) | \(189\) |
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Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.19 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {6 \, a c^{3} e^{2} x^{3} + 18 \, a c^{3} d e x^{2} + 18 \, a c^{3} d^{2} x + 3 \, {\left (2 \, b c^{3} e^{2} x^{3} + 6 \, b c^{3} d e x^{2} + 6 \, b c^{3} d^{2} x + 3 \, b c d e\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (2 \, b c^{2} e^{2} x^{2} + 9 \, b c^{2} d e x + 18 \, b c^{2} d^{2} - 4 \, b e^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{18 \, c^{3}} \]
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Time = 0.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.53 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {asinh}{\left (c x \right )} + b d e x^{2} \operatorname {asinh}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {b d^{2} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {b d e x \sqrt {c^{2} x^{2} + 1}}{2 c} - \frac {b e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {b d e \operatorname {asinh}{\left (c x \right )}}{2 c^{2}} + \frac {2 b e^{2} \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.21 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b d e + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{2}}{c} \]
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Exception generated. \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^2 \,d x \]
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