Integrand size = 10, antiderivative size = 152 \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\frac {15 \sqrt {\text {arcsinh}(a x)}}{64 a^2}+\frac {15}{32} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{8 a}+\frac {\text {arcsinh}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{256 a^2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{256 a^2} \]
[Out]
Time = 0.22 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5777, 5812, 5783, 5819, 3393, 3388, 2211, 2235, 2236} \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=-\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{256 a^2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{256 a^2}-\frac {5 x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{8 a}+\frac {\text {arcsinh}(a x)^{5/2}}{4 a^2}+\frac {15 \sqrt {\text {arcsinh}(a x)}}{64 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}+\frac {15}{32} x^2 \sqrt {\text {arcsinh}(a x)} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5777
Rule 5783
Rule 5812
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {1}{4} (5 a) \int \frac {x^2 \text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{8 a}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}+\frac {15}{16} \int x \sqrt {\text {arcsinh}(a x)} \, dx+\frac {5 \int \frac {\text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx}{8 a} \\ & = \frac {15}{32} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{8 a}+\frac {\text {arcsinh}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {1}{64} (15 a) \int \frac {x^2}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx \\ & = \frac {15}{32} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{8 a}+\frac {\text {arcsinh}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^2} \\ & = \frac {15}{32} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{8 a}+\frac {\text {arcsinh}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}+\frac {15 \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{64 a^2} \\ & = \frac {15 \sqrt {\text {arcsinh}(a x)}}{64 a^2}+\frac {15}{32} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{8 a}+\frac {\text {arcsinh}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{128 a^2} \\ & = \frac {15 \sqrt {\text {arcsinh}(a x)}}{64 a^2}+\frac {15}{32} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{8 a}+\frac {\text {arcsinh}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{256 a^2}-\frac {15 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{256 a^2} \\ & = \frac {15 \sqrt {\text {arcsinh}(a x)}}{64 a^2}+\frac {15}{32} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{8 a}+\frac {\text {arcsinh}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{128 a^2}-\frac {15 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{128 a^2} \\ & = \frac {15 \sqrt {\text {arcsinh}(a x)}}{64 a^2}+\frac {15}{32} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{8 a}+\frac {\text {arcsinh}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{256 a^2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{256 a^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.34 \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\frac {\frac {\sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-2 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\Gamma \left (\frac {7}{2},2 \text {arcsinh}(a x)\right )}{32 \sqrt {2} a^2} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (-128 \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+160 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, a x -120 \sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}-64 \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }-60 \sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }+15 \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )+15 \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )\right )}{512 \sqrt {\pi }\, a^{2}}\) | \(136\) |
[In]
[Out]
Exception generated. \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\int x \operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}\, dx \]
[In]
[Out]
\[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Timed out. \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\int x\,{\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \]
[In]
[Out]