Integrand size = 24, antiderivative size = 236 \[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}+\frac {d e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2} \]
[Out]
Time = 0.49 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5814, 5791, 3393, 3388, 2211, 2236, 2235, 5819, 5556} \[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {\sqrt {\pi } d e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} d e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}+\frac {\sqrt {\pi } d e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} d e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}-\frac {2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5556
Rule 5791
Rule 5814
Rule 5819
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {(2 d) \int \frac {\sqrt {1+c^2 x^2}}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b c}+\frac {(8 c d) \int \frac {x^2 \sqrt {1+c^2 x^2}}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b} \\ & = -\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {(2 d) \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2}+\frac {(8 d) \text {Subst}\left (\int \frac {\cosh ^2\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 x)\right )}{b^2 c^2} \\ & = -\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {(2 d) \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 {arcsinh}(c x)\right )}{b^2 c^2}+\frac {(8 d) \text {Subst}\left (\int \left (-\frac {1}{8 \sqrt {x}}+\frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2} \\ & = -\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2}+\frac {d \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2} \\ & = -\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d \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 x)\right )}{2 b^2 c^2}+\frac {d \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 x)\right )}{2 b^2 c^2}+\frac {d \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 x)\right )}{2 b^2 c^2}+\frac {d \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 x)\right )}{2 b^2 c^2} \\ & = -\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}+\frac {d \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}+\frac {d \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}+\frac {d \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2} \\ & = -\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}+\frac {d e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.96 \[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {d e^{-\frac {4 a}{b}} \left (\sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+\sqrt {2} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-e^{\frac {4 a}{b}} \left (\sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+2 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))\right )\right )}{4 b c^2 \sqrt {a+b \text {arcsinh}(c x)}} \]
[In]
[Out]
\[\int \frac {x \left (c^{2} d \,x^{2}+d \right )}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]
[In]
[Out]
Exception generated. \[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=d \left (\int \frac {x}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {c^{2} x^{3}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \]
[In]
[Out]
\[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Timed out. \[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {x\,\left (d\,c^2\,x^2+d\right )}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
[In]
[Out]