Integrand size = 25, antiderivative size = 835 \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=-\frac {1813 b^3 e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{1125 d}+\frac {119 b^3 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{1125 d}-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{1000 d}+\frac {14 b^2 e^4 (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{15 d}-\frac {7 b^2 e^4 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{7/2}}{5 d}+\frac {105 b^{7/2} e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {119 b^{7/2} e^4 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{18000 d}-\frac {21 b^{7/2} e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {21 b^{7/2} e^4 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {105 b^{7/2} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {119 b^{7/2} e^4 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{18000 d}-\frac {21 b^{7/2} e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {21 b^{7/2} e^4 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64000 d} \]
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Time = 2.13 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.00, number of steps used = 77, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {5859, 12, 5777, 5812, 5798, 5772, 5774, 3388, 2211, 2236, 2235, 5780, 5556} \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\frac {e^4 (a+b \text {arcsinh}(c+d x))^{7/2} (c+d x)^5}{5 d}+\frac {7 b^2 e^4 (a+b \text {arcsinh}(c+d x))^{3/2} (c+d x)^5}{100 d}-\frac {7 b e^4 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2} (c+d x)^4}{50 d}-\frac {21 b^3 e^4 \sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)} (c+d x)^4}{1000 d}-\frac {7 b^2 e^4 (a+b \text {arcsinh}(c+d x))^{3/2} (c+d x)^3}{45 d}+\frac {14 b e^4 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2} (c+d x)^2}{75 d}+\frac {119 b^3 e^4 \sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)} (c+d x)^2}{1125 d}+\frac {14 b^2 e^4 (a+b \text {arcsinh}(c+d x))^{3/2} (c+d x)}{15 d}-\frac {28 b e^4 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}}{75 d}+\frac {105 b^{7/2} e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {21 b^{7/2} e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64000 d}-\frac {119 b^{7/2} e^4 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{18000 d}+\frac {21 b^{7/2} e^4 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {105 b^{7/2} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {21 b^{7/2} e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64000 d}-\frac {119 b^{7/2} e^4 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{18000 d}+\frac {21 b^{7/2} e^4 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64000 d}-\frac {1813 b^3 e^4 \sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}{1125 d} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5556
Rule 5772
Rule 5774
Rule 5777
Rule 5780
Rule 5798
Rule 5812
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^4 x^4 (a+b \text {arcsinh}(x))^{7/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 \text {Subst}\left (\int x^4 (a+b \text {arcsinh}(x))^{7/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{7/2}}{5 d}-\frac {\left (7 b e^4\right ) \text {Subst}\left (\int \frac {x^5 (a+b \text {arcsinh}(x))^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{10 d} \\ & = -\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{7/2}}{5 d}+\frac {\left (14 b e^4\right ) \text {Subst}\left (\int \frac {x^3 (a+b \text {arcsinh}(x))^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{25 d}+\frac {\left (7 b^2 e^4\right ) \text {Subst}\left (\int x^4 (a+b \text {arcsinh}(x))^{3/2} \, dx,x,c+d x\right )}{20 d} \\ & = \frac {7 b^2 e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{3/2}}{100 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{7/2}}{5 d}-\frac {\left (28 b e^4\right ) \text {Subst}\left (\int \frac {x (a+b \text {arcsinh}(x))^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{75 d}-\frac {\left (7 b^2 e^4\right ) \text {Subst}\left (\int x^2 (a+b \text {arcsinh}(x))^{3/2} \, dx,x,c+d x\right )}{15 d}-\frac {\left (21 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^5 \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{200 d} \\ & = -\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{1000 d}-\frac {7 b^2 e^4 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{7/2}}{5 d}+\frac {\left (14 b^2 e^4\right ) \text {Subst}\left (\int (a+b \text {arcsinh}(x))^{3/2} \, dx,x,c+d x\right )}{15 d}+\frac {\left (21 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^3 \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{250 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^3 \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{30 d}+\frac {\left (21 b^4 e^4\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{2000 d} \\ & = \frac {119 b^3 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{1125 d}-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{1000 d}+\frac {14 b^2 e^4 (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{15 d}-\frac {7 b^2 e^4 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{7/2}}{5 d}+\frac {\left (21 b^3 e^4\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{2000 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{125 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{45 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{5 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{500 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{180 d} \\ & = -\frac {1813 b^3 e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{1125 d}+\frac {119 b^3 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{1125 d}-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{1000 d}+\frac {14 b^2 e^4 (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{15 d}-\frac {7 b^2 e^4 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{7/2}}{5 d}+\frac {\left (21 b^3 e^4\right ) \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 \sqrt {x}}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 \sqrt {x}}+\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{2000 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{500 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{180 d}+\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{250 d}+\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{90 d}+\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{10 d} \\ & = -\frac {1813 b^3 e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{1125 d}+\frac {119 b^3 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{1125 d}-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{1000 d}+\frac {14 b^2 e^4 (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{15 d}-\frac {7 b^2 e^4 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{7/2}}{5 d}+\frac {\left (21 b^3 e^4\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{32000 d}+\frac {\left (21 b^3 e^4\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{16000 d}-\frac {\left (63 b^3 e^4\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{32000 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}-\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{500 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{250 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}-\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{180 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{90 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{10 d} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.39 \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\frac {b^4 e^4 e^{-\frac {5 a}{b}} \left (-506250 e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {9}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )+81 \sqrt {5} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {9}{2},-\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )-3125 \sqrt {3} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {9}{2},-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )+506250 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {9}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+3125 \sqrt {3} e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {9}{2},\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-81 \sqrt {5} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {9}{2},\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{8100000 d \sqrt {a+b \text {arcsinh}(c+d x)}} \]
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\[\int \left (d e x +c e \right )^{4} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {7}{2}}d x\]
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Exception generated. \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
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\[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^4\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2} \,d x \]
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