Integrand size = 23, antiderivative size = 349 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=-\frac {45 b^4 e^3 (c+d x)^2}{128 d}+\frac {3 b^4 e^3 (c+d x)^4}{128 d}+\frac {45 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{64 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{32 d}-\frac {45 b^2 e^3 (a+b \text {arcsinh}(c+d x))^2}{128 d}-\frac {9 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{16 d}+\frac {3 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2}{16 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{8 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{4 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^4}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4}{4 d} \]
[Out]
Time = 0.44 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5859, 12, 5776, 5812, 5783, 30} \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{32 d}+\frac {45 b^3 e^3 (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{64 d}+\frac {3 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2}{16 d}-\frac {45 b^2 e^3 (a+b \text {arcsinh}(c+d x))^2}{128 d}-\frac {9 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{16 d}-\frac {b e^3 (c+d x)^3 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{4 d}+\frac {3 b e^3 (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{8 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4}{4 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^4}{32 d}+\frac {3 b^4 e^3 (c+d x)^4}{128 d}-\frac {45 b^4 e^3 (c+d x)^2}{128 d} \]
[In]
[Out]
Rule 12
Rule 30
Rule 5776
Rule 5783
Rule 5812
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^3 x^3 (a+b \text {arcsinh}(x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int x^3 (a+b \text {arcsinh}(x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4}{4 d}-\frac {\left (b e^3\right ) \text {Subst}\left (\int \frac {x^4 (a+b \text {arcsinh}(x))^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{4 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4}{4 d}+\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {x^2 (a+b \text {arcsinh}(x))^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d}+\frac {\left (3 b^2 e^3\right ) \text {Subst}\left (\int x^3 (a+b \text {arcsinh}(x))^2 \, dx,x,c+d x\right )}{4 d} \\ & = \frac {3 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2}{16 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{8 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{4 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d}-\frac {\left (9 b^2 e^3\right ) \text {Subst}\left (\int x (a+b \text {arcsinh}(x))^2 \, dx,x,c+d x\right )}{8 d}-\frac {\left (3 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^4 (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d} \\ & = -\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{32 d}-\frac {9 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{16 d}+\frac {3 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2}{16 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{8 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{4 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^4}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4}{4 d}+\frac {\left (9 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^2 (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{32 d}+\frac {\left (9 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^2 (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d}+\frac {\left (3 b^4 e^3\right ) \text {Subst}\left (\int x^3 \, dx,x,c+d x\right )}{32 d} \\ & = \frac {3 b^4 e^3 (c+d x)^4}{128 d}+\frac {45 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{64 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{32 d}-\frac {9 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{16 d}+\frac {3 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2}{16 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{8 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{4 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^4}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4}{4 d}-\frac {\left (9 b^3 e^3\right ) \text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{64 d}-\frac {\left (9 b^3 e^3\right ) \text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{16 d}-\frac {\left (9 b^4 e^3\right ) \text {Subst}(\int x \, dx,x,c+d x)}{64 d}-\frac {\left (9 b^4 e^3\right ) \text {Subst}(\int x \, dx,x,c+d x)}{16 d} \\ & = -\frac {45 b^4 e^3 (c+d x)^2}{128 d}+\frac {3 b^4 e^3 (c+d x)^4}{128 d}+\frac {45 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{64 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{32 d}-\frac {45 b^2 e^3 (a+b \text {arcsinh}(c+d x))^2}{128 d}-\frac {9 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{16 d}+\frac {3 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2}{16 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{8 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{4 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^4}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4}{4 d} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.36 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {e^3 \left (-9 b^2 \left (8 a^2+5 b^2\right ) (c+d x)^2+\left (32 a^4+24 a^2 b^2+3 b^4\right ) (c+d x)^4+2 a b (c+d x) \sqrt {1+(c+d x)^2} \left (24 a^2+45 b^2-2 \left (8 a^2+3 b^2\right ) (c+d x)^2\right )-6 a b \left (8 a^2+15 b^2\right ) \text {arcsinh}(c+d x)+2 b (c+d x) \left (-72 a b^2 (c+d x)+64 a^3 (c+d x)^3+24 a b^2 (c+d x)^3+72 a^2 b \sqrt {1+(c+d x)^2}+45 b^3 \sqrt {1+(c+d x)^2}-48 a^2 b (c+d x)^2 \sqrt {1+(c+d x)^2}-6 b^3 (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+3 b^2 \left (-24 a^2-15 b^2-24 b^2 (c+d x)^2+64 a^2 (c+d x)^4+8 b^2 (c+d x)^4+48 a b (c+d x) \sqrt {1+(c+d x)^2}-32 a b (c+d x)^3 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^2+16 b^3 \left (-3 a+8 a (c+d x)^4+3 b (c+d x) \sqrt {1+(c+d x)^2}-2 b (c+d x)^3 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^3+4 b^4 \left (-3+8 (c+d x)^4\right ) \text {arcsinh}(c+d x)^4\right )}{128 d} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.64
method | result | size |
derivativedivides | \(\frac {\frac {e^{3} a^{4} \left (d x +c \right )^{4}}{4}+e^{3} b^{4} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{4}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {3 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{16}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}+\frac {45 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{64}+\frac {27 \operatorname {arcsinh}\left (d x +c \right )^{2}}{128}+\frac {3 \left (d x +c \right )^{4}}{128}-\frac {45 \left (d x +c \right )^{2}}{128}-\frac {45}{128}-\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{16}\right )+4 e^{3} a \,b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{32}-\frac {3 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{128}+\frac {45 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{256}+\frac {27 \,\operatorname {arcsinh}\left (d x +c \right )}{256}-\frac {9 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{32}\right )+6 e^{3} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{16}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )+4 e^{3} b \,a^{3} \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}\) | \(573\) |
default | \(\frac {\frac {e^{3} a^{4} \left (d x +c \right )^{4}}{4}+e^{3} b^{4} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{4}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {3 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{16}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}+\frac {45 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{64}+\frac {27 \operatorname {arcsinh}\left (d x +c \right )^{2}}{128}+\frac {3 \left (d x +c \right )^{4}}{128}-\frac {45 \left (d x +c \right )^{2}}{128}-\frac {45}{128}-\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{16}\right )+4 e^{3} a \,b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{32}-\frac {3 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{128}+\frac {45 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{256}+\frac {27 \,\operatorname {arcsinh}\left (d x +c \right )}{256}-\frac {9 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{32}\right )+6 e^{3} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{16}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )+4 e^{3} b \,a^{3} \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}\) | \(573\) |
parts | \(\frac {e^{3} a^{4} \left (d x +c \right )^{4}}{4 d}+\frac {e^{3} b^{4} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{4}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {3 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{16}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}+\frac {45 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{64}+\frac {27 \operatorname {arcsinh}\left (d x +c \right )^{2}}{128}+\frac {3 \left (d x +c \right )^{4}}{128}-\frac {45 \left (d x +c \right )^{2}}{128}-\frac {45}{128}-\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{16}\right )}{d}+\frac {4 e^{3} a \,b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{32}-\frac {3 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{128}+\frac {45 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{256}+\frac {27 \,\operatorname {arcsinh}\left (d x +c \right )}{256}-\frac {9 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}+\frac {6 e^{3} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{16}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )}{d}+\frac {4 e^{3} b \,a^{3} \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}\) | \(584\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1241 vs. \(2 (319) = 638\).
Time = 0.29 (sec) , antiderivative size = 1241, normalized size of antiderivative = 3.56 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=\text {Too large to display} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 2876 vs. \(2 (325) = 650\).
Time = 1.12 (sec) , antiderivative size = 2876, normalized size of antiderivative = 8.24 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
[In]
[Out]
\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
[In]
[Out]
Timed out. \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]
[In]
[Out]