Integrand size = 25, antiderivative size = 360 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{64 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}-\frac {3 b^{3/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}+\frac {3 b^{3/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {3 b^{3/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {3 b^{3/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{128 d} \]
[Out]
Time = 0.71 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5859, 12, 5777, 5812, 5783, 5780, 5556, 3389, 2211, 2236, 2235} \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=-\frac {3 \sqrt {\pi } b^{3/2} e^3 e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^3 e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {3 \sqrt {\pi } b^{3/2} e^3 e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^3 e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}-\frac {3 b e^3 \sqrt {(c+d x)^2+1} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}+\frac {9 b e^3 \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{64 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{32 d} \]
[In]
[Out]
Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5777
Rule 5780
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))^{3/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int x^3 (a+b \text {arcsinh}(x))^{3/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {x^4 \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d} \\ & = -\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}+\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{32 d}+\frac {\left (3 b^2 e^3\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{64 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{64 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{64 d}-\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {\sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{64 d}-\frac {\left (9 b^2 e^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{128 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{64 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 \sqrt {x}}-\frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{64 d}+\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{128 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{64 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{512 d}+\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{256 d}+\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{128 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{64 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}-\frac {\left (3 b 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 {arcsinh}(c+d x)\right )}{1024 d}+\frac {\left (3 b 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 {arcsinh}(c+d x)\right )}{1024 d}+\frac {\left (3 b 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 {arcsinh}(c+d x)\right )}{512 d}-\frac {\left (3 b 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 {arcsinh}(c+d x)\right )}{512 d}+\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{256 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{64 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{512 d}+\frac {\left (3 b e^3\right ) \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{512 d}+\frac {\left (3 b e^3\right ) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{256 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{256 d}+\frac {\left (9 b 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 {arcsinh}(c+d x)\right )}{512 d}-\frac {\left (9 b 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 {arcsinh}(c+d x)\right )}{512 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{64 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}-\frac {3 b^{3/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}+\frac {3 b^{3/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{512 d}+\frac {3 b^{3/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {3 b^{3/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{512 d}+\frac {\left (9 b e^3\right ) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{256 d}-\frac {\left (9 b e^3\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{256 d} \\ & = \frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{64 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}-\frac {3 b^{3/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}+\frac {3 b^{3/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {3 b^{3/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {3 b^{3/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{128 d} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.62 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\frac {b e^3 e^{-\frac {4 a}{b}} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-\sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {5}{2},-\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )+8 \sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {5}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \left (-8 \sqrt {2} \Gamma \left (\frac {5}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+e^{\frac {2 a}{b}} \Gamma \left (\frac {5}{2},\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )}{512 d \sqrt {-\frac {(a+b \text {arcsinh}(c+d x))^2}{b^2}}} \]
[In]
[Out]
\[\int \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {3}{2}}d x\]
[In]
[Out]
Exception generated. \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=e^{3} \left (\int a c^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int a d^{3} x^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b c^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 3 a c d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int 3 a c^{2} d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b d^{3} x^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 3 b c d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 3 b c^{2} d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx\right ) \]
[In]
[Out]
\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
[In]
[Out]