Integrand size = 16, antiderivative size = 226 \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 x^2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3} \]
1/4*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c^3-1/ 4*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c^3/exp(a/b)-1/4 *exp(3*a/b)*erf(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2) /b^(3/2)/c^3+1/4*erfi(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi ^(1/2)/b^(3/2)/c^3/exp(3*a/b)-2*x^2*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh(c*x ))^(1/2)
Time = 0.30 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.28 \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {e^{-3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \left (-e^{\frac {3 a}{b}}+e^{\frac {3 a}{b}+2 \text {arcsinh}(c x)}+e^{\frac {3 a}{b}+4 \text {arcsinh}(c x)}-e^{\frac {3 a}{b}+6 \text {arcsinh}(c x)}-e^{\frac {4 a}{b}+3 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} e^{3 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-e^{\frac {2 a}{b}+3 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}+3 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{4 b c^3 \sqrt {a+b \text {arcsinh}(c x)}} \]
(-E^((3*a)/b) + E^((3*a)/b + 2*ArcSinh[c*x]) + E^((3*a)/b + 4*ArcSinh[c*x] ) - E^((3*a)/b + 6*ArcSinh[c*x]) - E^((4*a)/b + 3*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + Sqrt[3]*E^(3*ArcSinh[c*x]) *Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] - E^((2*a)/b + 3*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, - ((a + b*ArcSinh[c*x])/b)] + Sqrt[3]*E^((6*a)/b + 3*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b])/(4*b*c^3*E^(3*(a/b + ArcSinh[c*x]))*Sqrt[a + b*ArcSinh[c*x]])
Time = 0.48 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6193, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 6193 |
\(\displaystyle \frac {2 \int \left (\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\) |
(-2*x^2*Sqrt[1 + c^2*x^2])/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (2*((Sqrt[b]*E ^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/8 - (Sqrt[b]*E^((3* a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/8 - (Sqr t[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*E^(a/b)) + (Sqrt[ b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(8*E^((3*a )/b))))/(b^2*c^3)
3.2.48.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Si mp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 1]
\[\int \frac {x^{2}}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]
Exception generated. \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {x^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \]