Integrand size = 20, antiderivative size = 349 \[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 e x^2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {e e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3} \]
-d*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c+1/4*e *exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c^3+d*erf i((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c/exp(a/b)-1/4*e*erfi ((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c^3/exp(a/b)-1/4*e*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*e*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*d*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh(c*x))^( 1/2)-2*e*x^2*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh(c*x))^(1/2)
Time = 0.95 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.87 \[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {e^{-3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \left (\left (4 c^2 d-e\right ) e^{\frac {4 a}{b}+3 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} e e^{3 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\left (4 c^2 d-e\right ) e^{\frac {2 a}{b}+3 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+e^{\frac {3 a}{b}} \left (-\left (\left (1+e^{2 \text {arcsinh}(c x)}\right ) \left (4 c^2 d e^{2 \text {arcsinh}(c x)}+e \left (-1+e^{2 \text {arcsinh}(c x)}\right )^2\right )\right )+\sqrt {3} e e^{3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{4 b c^3 \sqrt {a+b \text {arcsinh}(c x)}} \]
((4*c^2*d - e)*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*E^(3*ArcSinh[c*x])*Sqrt[-((a + b*Arc Sinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] + (4*c^2*d - e)*E^( (2*a)/b + 3*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] + E^((3*a)/b)*(-((1 + E^(2*ArcSinh[c*x]))*(4*c^2*d*E ^(2*ArcSinh[c*x]) + e*(-1 + E^(2*ArcSinh[c*x]))^2)) + Sqrt[3]*e*E^(3*(a/b + 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 = 0.93 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6208, 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.
\(\displaystyle \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 6208 |
\(\displaystyle \int \left (\frac {d}{(a+b \text {arcsinh}(c x))^{3/2}}+\frac {e x^2}{(a+b \text {arcsinh}(c x))^{3/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {\pi } e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {\sqrt {3 \pi } e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {\sqrt {\pi } e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {\sqrt {3 \pi } e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {\sqrt {\pi } d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {\sqrt {\pi } d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 d \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 e x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\) |
(-2*d*Sqrt[1 + c^2*x^2])/(b*c*Sqrt[a + b*ArcSinh[c*x]]) - (2*e*x^2*Sqrt[1 + c^2*x^2])/(b*c*Sqrt[a + b*ArcSinh[c*x]]) - (d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[ a + b*ArcSinh[c*x]]/Sqrt[b]])/(b^(3/2)*c) + (e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c^3) - (e*E^((3*a)/b)*Sqrt[3*Pi]*E rf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^3) + (d*Sqrt[ Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(b^(3/2)*c*E^(a/b)) - (e*Sqrt[ Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c^3*E^(a/b)) + (e*S qrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^3 *E^((3*a)/b))
3.7.43.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
\[\int \frac {e \,x^{2}+d}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]
Exception generated. \[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {d + e x^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {e x^{2} + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {e x^{2} + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {e\,x^2+d}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \]