Integrand size = 21, antiderivative size = 105 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x)) \, dx=\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2}}{32 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{16 d}-\frac {3 b e^3 \text {arcsinh}(c+d x)}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))}{4 d} \]
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Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5859, 12, 5776, 327, 221} \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x)) \, dx=\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))}{4 d}-\frac {3 b e^3 \text {arcsinh}(c+d x)}{32 d}-\frac {b e^3 \sqrt {(c+d x)^2+1} (c+d x)^3}{16 d}+\frac {3 b e^3 \sqrt {(c+d x)^2+1} (c+d x)}{32 d} \]
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Rule 12
Rule 221
Rule 327
Rule 5776
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^3 x^3 (a+b \text {arcsinh}(x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int x^3 (a+b \text {arcsinh}(x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))}{4 d}-\frac {\left (b e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d} \\ & = -\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{16 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))}{4 d}+\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{16 d} \\ & = \frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2}}{32 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{16 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{32 d} \\ & = \frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2}}{32 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{16 d}-\frac {3 b e^3 \text {arcsinh}(c+d x)}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))}{4 d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.79 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x)) \, dx=\frac {e^3 \left (3 b (c+d x) \sqrt {1+(c+d x)^2}-2 b (c+d x)^3 \sqrt {1+(c+d x)^2}-3 b \text {arcsinh}(c+d x)+8 (c+d x)^4 (a+b \text {arcsinh}(c+d x))\right )}{32 d} \]
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Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {e^{3} a \left (d x +c \right )^{4}}{4}+e^{3} b \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}\) | \(86\) |
default | \(\frac {\frac {e^{3} a \left (d x +c \right )^{4}}{4}+e^{3} b \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}\) | \(86\) |
parts | \(\frac {e^{3} a \left (d x +c \right )^{4}}{4 d}+\frac {e^{3} b \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}\) | \(88\) |
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Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (93) = 186\).
Time = 0.25 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.17 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x)) \, dx=\frac {8 \, a d^{4} e^{3} x^{4} + 32 \, a c d^{3} e^{3} x^{3} + 48 \, a c^{2} d^{2} e^{3} x^{2} + 32 \, a c^{3} d e^{3} x + {\left (8 \, b d^{4} e^{3} x^{4} + 32 \, b c d^{3} e^{3} x^{3} + 48 \, b c^{2} d^{2} e^{3} x^{2} + 32 \, b c^{3} d e^{3} x + {\left (8 \, b c^{4} - 3 \, b\right )} e^{3}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (2 \, b d^{3} e^{3} x^{3} + 6 \, b c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, b c^{2} - b\right )} d e^{3} x + {\left (2 \, b c^{3} - 3 \, b c\right )} e^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{32 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (94) = 188\).
Time = 0.28 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.75 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x)) \, dx=\begin {cases} a c^{3} e^{3} x + \frac {3 a c^{2} d e^{3} x^{2}}{2} + a c d^{2} e^{3} x^{3} + \frac {a d^{3} e^{3} x^{4}}{4} + \frac {b c^{4} e^{3} \operatorname {asinh}{\left (c + d x \right )}}{4 d} + b c^{3} e^{3} x \operatorname {asinh}{\left (c + d x \right )} - \frac {b c^{3} e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{16 d} + \frac {3 b c^{2} d e^{3} x^{2} \operatorname {asinh}{\left (c + d x \right )}}{2} - \frac {3 b c^{2} e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{16} + b c d^{2} e^{3} x^{3} \operatorname {asinh}{\left (c + d x \right )} - \frac {3 b c d e^{3} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{16} + \frac {3 b c e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{32 d} + \frac {b d^{3} e^{3} x^{4} \operatorname {asinh}{\left (c + d x \right )}}{4} - \frac {b d^{2} e^{3} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{16} + \frac {3 b e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{32} - \frac {3 b e^{3} \operatorname {asinh}{\left (c + d x \right )}}{32 d} & \text {for}\: d \neq 0 \\c^{3} e^{3} x \left (a + b \operatorname {asinh}{\left (c \right )}\right ) & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 790 vs. \(2 (93) = 186\).
Time = 0.21 (sec) , antiderivative size = 790, normalized size of antiderivative = 7.52 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x)) \, dx=\frac {1}{4} \, a d^{3} e^{3} x^{4} + a c d^{2} e^{3} x^{3} + \frac {3}{2} \, a c^{2} d e^{3} x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} x}{d^{2}} - \frac {{\left (c^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} c}{d^{3}}\right )}\right )} b c^{2} d e^{3} + \frac {1}{6} \, {\left (6 \, x^{3} \operatorname {arsinh}\left (d x + c\right ) - d {\left (\frac {2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} x^{2}}{d^{2}} - \frac {15 \, c^{3} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{4}} - \frac {5 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} c x}{d^{3}} + \frac {9 \, {\left (c^{2} + 1\right )} c \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{4}} + \frac {15 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} c^{2}}{d^{4}} - \frac {4 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (c^{2} + 1\right )}}{d^{4}}\right )}\right )} b c d^{2} e^{3} + \frac {1}{96} \, {\left (24 \, x^{4} \operatorname {arsinh}\left (d x + c\right ) - {\left (\frac {6 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} x^{3}}{d^{2}} - \frac {14 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} c x^{2}}{d^{3}} + \frac {105 \, c^{4} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{5}} + \frac {35 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} c^{2} x}{d^{4}} - \frac {90 \, {\left (c^{2} + 1\right )} c^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{5}} - \frac {105 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} c^{3}}{d^{5}} - \frac {9 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (c^{2} + 1\right )} x}{d^{4}} + \frac {9 \, {\left (c^{2} + 1\right )}^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{5}} + \frac {55 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (c^{2} + 1\right )} c}{d^{5}}\right )} d\right )} b d^{3} e^{3} + a c^{3} e^{3} x + \frac {{\left ({\left (d x + c\right )} \operatorname {arsinh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} + 1}\right )} b c^{3} e^{3}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (93) = 186\).
Time = 0.74 (sec) , antiderivative size = 613, normalized size of antiderivative = 5.84 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x)) \, dx=\frac {1}{4} \, a d^{3} e^{3} x^{4} + a c d^{2} e^{3} x^{3} + \frac {3}{2} \, a c^{2} d e^{3} x^{2} - {\left (d {\left (\frac {c \log \left (-c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} {\left | d \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )\right )} b c^{3} e^{3} + \frac {3}{4} \, {\left (2 \, x^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (\frac {x}{d^{2}} - \frac {3 \, c}{d^{3}}\right )} - \frac {{\left (2 \, c^{2} - 1\right )} \log \left (-c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} {\left | d \right |}\right )}{d^{2} {\left | d \right |}}\right )} d\right )} b c^{2} d e^{3} + \frac {1}{6} \, {\left (6 \, x^{3} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (x {\left (\frac {2 \, x}{d^{2}} - \frac {5 \, c}{d^{3}}\right )} + \frac {11 \, c^{2} d - 4 \, d}{d^{5}}\right )} + \frac {3 \, {\left (2 \, c^{3} - 3 \, c\right )} \log \left (-c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} {\left | d \right |}\right )}{d^{3} {\left | d \right |}}\right )} d\right )} b c d^{2} e^{3} + \frac {1}{96} \, {\left (24 \, x^{4} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (2 \, x {\left (\frac {3 \, x}{d^{2}} - \frac {7 \, c}{d^{3}}\right )} + \frac {26 \, c^{2} d^{3} - 9 \, d^{3}}{d^{7}}\right )} x - \frac {5 \, {\left (10 \, c^{3} d^{2} - 11 \, c d^{2}\right )}}{d^{7}}\right )} - \frac {3 \, {\left (8 \, c^{4} - 24 \, c^{2} + 3\right )} \log \left (-c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} {\left | d \right |}\right )}{d^{4} {\left | d \right |}}\right )} d\right )} b d^{3} e^{3} + a c^{3} e^{3} x \]
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Timed out. \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x)) \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right ) \,d x \]
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