Integrand size = 25, antiderivative size = 469 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=-\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {15 b^{5/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {15 b^{5/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d} \]
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Time = 1.49 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5996, 12, 5884, 5939, 5893, 5953, 3393, 3388, 2211, 2236, 2235} \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=-\frac {15 \sqrt {\pi } b^{5/2} e^3 e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^3 e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {15 \sqrt {\pi } b^{5/2} e^3 e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^3 e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {5 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {15 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5884
Rule 5893
Rule 5939
Rule 5953
Rule 5996
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^3 x^3 (a+b \text {arccosh}(x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int x^3 (a+b \text {arccosh}(x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (5 b e^3\right ) \text {Subst}\left (\int \frac {x^4 (a+b \text {arccosh}(x))^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{8 d} \\ & = -\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b e^3\right ) \text {Subst}\left (\int \frac {x^2 (a+b \text {arccosh}(x))^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{32 d}+\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int x^3 \sqrt {a+b \text {arccosh}(x)} \, dx,x,c+d x\right )}{64 d} \\ & = \frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b e^3\right ) \text {Subst}\left (\int \frac {(a+b \text {arccosh}(x))^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{64 d}+\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int x \sqrt {a+b \text {arccosh}(x)} \, dx,x,c+d x\right )}{128 d}-\frac {\left (15 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {-1+x} \sqrt {1+x} \sqrt {a+b \text {arccosh}(x)}} \, dx,x,c+d x\right )}{512 d} \\ & = \frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cosh ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{512 d}-\frac {\left (45 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} \sqrt {a+b \text {arccosh}(x)}} \, dx,x,c+d x\right )}{512 d} \\ & = \frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 \sqrt {x}}+\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \text {arccosh}(c+d x)\right )}{512 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{512 d} \\ & = -\frac {45 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{4096 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{1024 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \text {arccosh}(c+d x)\right )}{512 d} \\ & = -\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{8192 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{8192 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2048 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2048 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{1024 d} \\ & = -\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c+d x)}\right )}{4096 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c+d x)}\right )}{4096 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c+d x)}\right )}{1024 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c+d x)}\right )}{1024 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2048 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2048 d} \\ & = -\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {15 b^{5/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {15 b^{5/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c+d x)}\right )}{1024 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c+d x)}\right )}{1024 d} \\ & = -\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {15 b^{5/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {15 b^{5/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(968\) vs. \(2(469)=938\).
Time = 8.60 (sec) , antiderivative size = 968, normalized size of antiderivative = 2.06 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=e^3 \left (\frac {a^2 e^{-\frac {4 a}{b}} \sqrt {a+b \text {arccosh}(c+d x)} \left (\sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )+4 \sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \left (4 \sqrt {2} \Gamma \left (\frac {3}{2},\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )\right )\right )}{128 d \sqrt {-\frac {(a+b \text {arccosh}(c+d x))^2}{b^2}}}+\frac {a \sqrt {b} \left ((8 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {4 a}{b}\right )-\sinh \left (\frac {4 a}{b}\right )\right )+(8 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {4 a}{b}\right )+\sinh \left (\frac {4 a}{b}\right )\right )+8 \left ((4 a+3 b) \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )+(4 a-3 b) \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+8 \sqrt {b} \sqrt {a+b \text {arccosh}(c+d x)} (4 \text {arccosh}(c+d x) \cosh (2 \text {arccosh}(c+d x))-3 \sinh (2 \text {arccosh}(c+d x)))\right )+8 \sqrt {b} \sqrt {a+b \text {arccosh}(c+d x)} (8 \text {arccosh}(c+d x) \cosh (4 \text {arccosh}(c+d x))-3 \sinh (4 \text {arccosh}(c+d x)))\right )}{1024 d}+\frac {-\sqrt {b} \left (64 a^2+48 a b+15 b^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {4 a}{b}\right )-\sinh \left (\frac {4 a}{b}\right )\right )-\sqrt {b} \left (64 a^2-48 a b+15 b^2\right ) \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {4 a}{b}\right )+\sinh \left (\frac {4 a}{b}\right )\right )-16 \left (\sqrt {b} \left (16 a^2+24 a b+15 b^2\right ) \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )+\sqrt {b} \left (16 a^2-24 a b+15 b^2\right ) \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )-8 b \sqrt {a+b \text {arccosh}(c+d x)} \left (b \left (15+16 \text {arccosh}(c+d x)^2\right ) \cosh (2 \text {arccosh}(c+d x))+4 (a-5 b \text {arccosh}(c+d x)) \sinh (2 \text {arccosh}(c+d x))\right )\right )+8 b \sqrt {a+b \text {arccosh}(c+d x)} \left (b \left (15+64 \text {arccosh}(c+d x)^2\right ) \cosh (4 \text {arccosh}(c+d x))+8 (a-5 b \text {arccosh}(c+d x)) \sinh (4 \text {arccosh}(c+d x))\right )}{16384 d}\right ) \]
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\[\int \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]
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Exception generated. \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
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\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
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