Integrand size = 23, antiderivative size = 269 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=-\frac {15 b^2 e \sqrt {a+b \text {arccosh}(c+d x)}}{64 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{8 d}-\frac {e (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}-\frac {15 b^{5/2} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {15 b^{5/2} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{256 d} \]
[Out]
Time = 0.75 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5996, 12, 5884, 5939, 5893, 5953, 3393, 3388, 2211, 2236, 2235} \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{256 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{32 d}-\frac {15 b^2 e \sqrt {a+b \text {arccosh}(c+d x)}}{64 d}-\frac {5 b e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}}{8 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}-\frac {e (a+b \text {arccosh}(c+d x))^{5/2}}{4 d} \]
[In]
[Out]
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 x (a+b \text {arccosh}(x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int x (a+b \text {arccosh}(x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}-\frac {(5 b e) \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 )}{4 d} \\ & = -\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{8 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}-\frac {(5 b e) \text {Subst}\left (\int \frac {(a+b \text {arccosh}(x))^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{8 d}+\frac {\left (15 b^2 e\right ) \text {Subst}\left (\int x \sqrt {a+b \text {arccosh}(x)} \, dx,x,c+d x\right )}{16 d} \\ & = \frac {15 b^2 e (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{8 d}-\frac {e (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}-\frac {\left (15 b^3 e\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 )}{64 d} \\ & = \frac {15 b^2 e (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{8 d}-\frac {e (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}-\frac {\left (15 b^2 e\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 )}{64 d} \\ & = \frac {15 b^2 e (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{8 d}-\frac {e (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}-\frac {\left (15 b^2 e\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 )}{64 d} \\ & = -\frac {15 b^2 e \sqrt {a+b \text {arccosh}(c+d x)}}{64 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{8 d}-\frac {e (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}-\frac {\left (15 b^2 e\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 )}{128 d} \\ & = -\frac {15 b^2 e \sqrt {a+b \text {arccosh}(c+d x)}}{64 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{8 d}-\frac {e (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}-\frac {\left (15 b^2 e\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 )}{256 d}-\frac {\left (15 b^2 e\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 )}{256 d} \\ & = -\frac {15 b^2 e \sqrt {a+b \text {arccosh}(c+d x)}}{64 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{8 d}-\frac {e (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}-\frac {\left (15 b^2 e\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 )}{128 d}-\frac {\left (15 b^2 e\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 )}{128 d} \\ & = -\frac {15 b^2 e \sqrt {a+b \text {arccosh}(c+d x)}}{64 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{8 d}-\frac {e (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}-\frac {15 b^{5/2} e 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 )}{256 d}-\frac {15 b^{5/2} e 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 )}{256 d} \\ \end{align*}
Time = 1.28 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.85 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\frac {e \left (-15 b^{5/2} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )-15 b^{5/2} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+8 \sqrt {a+b \text {arccosh}(c+d x)} \left (\left (16 a^2+15 b^2\right ) \cosh (2 \text {arccosh}(c+d x))+16 b^2 \text {arccosh}(c+d x)^2 \cosh (2 \text {arccosh}(c+d x))-20 a b \sinh (2 \text {arccosh}(c+d x))+4 b \text {arccosh}(c+d x) (8 a \cosh (2 \text {arccosh}(c+d x))-5 b \sinh (2 \text {arccosh}(c+d x)))\right )\right )}{512 d} \]
[In]
[Out]
\[\int \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]
[In]
[Out]
Exception generated. \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
[In]
[Out]
\[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
[In]
[Out]