Integrand size = 23, antiderivative size = 136 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=-\frac {4}{9} b^2 e^2 x+\frac {2 b^2 e^2 (c+d x)^3}{27 d}+\frac {4 b e^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{9 d}-\frac {2 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{3 d} \]
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Time = 0.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5859, 12, 5776, 5812, 5798, 8, 30} \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{3 d}-\frac {2 b e^2 \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {4 b e^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {2 b^2 e^2 (c+d x)^3}{27 d}-\frac {4}{9} b^2 e^2 x \]
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Rule 8
Rule 12
Rule 30
Rule 5776
Rule 5798
Rule 5812
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^2 x^2 (a+b \text {arcsinh}(x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int x^2 (a+b \text {arcsinh}(x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {x^3 (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d} \\ & = -\frac {2 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{3 d}+\frac {\left (4 b e^2\right ) \text {Subst}\left (\int \frac {x (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{9 d}+\frac {\left (2 b^2 e^2\right ) \text {Subst}\left (\int x^2 \, dx,x,c+d x\right )}{9 d} \\ & = \frac {2 b^2 e^2 (c+d x)^3}{27 d}+\frac {4 b e^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{9 d}-\frac {2 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{3 d}-\frac {\left (4 b^2 e^2\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{9 d} \\ & = -\frac {4}{9} b^2 e^2 x+\frac {2 b^2 e^2 (c+d x)^3}{27 d}+\frac {4 b e^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{9 d}-\frac {2 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{3 d} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.08 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {e^2 \left (-12 b^2 (c+d x)+\left (9 a^2+2 b^2\right ) (c+d x)^3+6 a b \left (2-(c+d x)^2\right ) \sqrt {1+(c+d x)^2}+6 b \left (3 a (c+d x)^3+2 b \sqrt {1+(c+d x)^2}-b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+9 b^2 (c+d x)^3 \text {arcsinh}(c+d x)^2\right )}{27 d} \]
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Time = 0.41 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3}+e^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{9}-\frac {4 d x}{9}-\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(163\) |
default | \(\frac {\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3}+e^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{9}-\frac {4 d x}{9}-\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(163\) |
parts | \(\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{9}-\frac {4 d x}{9}-\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )}{d}+\frac {2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(168\) |
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Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (122) = 244\).
Time = 0.27 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.63 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c d^{2} e^{2} x^{2} + 3 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{2} - 4 \, b^{2}\right )} d e^{2} x + 9 \, {\left (b^{2} d^{3} e^{2} x^{3} + 3 \, b^{2} c d^{2} e^{2} x^{2} + 3 \, b^{2} c^{2} d e^{2} x + b^{2} c^{3} e^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 6 \, {\left (3 \, a b d^{3} e^{2} x^{3} + 9 \, a b c d^{2} e^{2} x^{2} + 9 \, a b c^{2} d e^{2} x + 3 \, a b c^{3} e^{2} - {\left (b^{2} d^{2} e^{2} x^{2} + 2 \, b^{2} c d e^{2} x + {\left (b^{2} c^{2} - 2 \, b^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 6 \, {\left (a b d^{2} e^{2} x^{2} + 2 \, a b c d e^{2} x + {\left (a b c^{2} - 2 \, a b\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{27 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (126) = 252\).
Time = 0.32 (sec) , antiderivative size = 610, normalized size of antiderivative = 4.49 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\begin {cases} a^{2} c^{2} e^{2} x + a^{2} c d e^{2} x^{2} + \frac {a^{2} d^{2} e^{2} x^{3}}{3} + \frac {2 a b c^{3} e^{2} \operatorname {asinh}{\left (c + d x \right )}}{3 d} + 2 a b c^{2} e^{2} x \operatorname {asinh}{\left (c + d x \right )} - \frac {2 a b c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9 d} + 2 a b c d e^{2} x^{2} \operatorname {asinh}{\left (c + d x \right )} - \frac {4 a b c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9} + \frac {2 a b d^{2} e^{2} x^{3} \operatorname {asinh}{\left (c + d x \right )}}{3} - \frac {2 a b d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9} + \frac {4 a b e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9 d} + \frac {b^{2} c^{3} e^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3 d} + b^{2} c^{2} e^{2} x \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 b^{2} c^{2} e^{2} x}{9} - \frac {2 b^{2} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{9 d} + b^{2} c d e^{2} x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 b^{2} c d e^{2} x^{2}}{9} - \frac {4 b^{2} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{9} + \frac {b^{2} d^{2} e^{2} x^{3} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3} + \frac {2 b^{2} d^{2} e^{2} x^{3}}{27} - \frac {2 b^{2} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{9} - \frac {4 b^{2} e^{2} x}{9} + \frac {4 b^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{9 d} & \text {for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \]
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\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
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\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \]
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