Integrand size = 14, antiderivative size = 183 \[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=-\frac {2 x \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {8 x^2}{3 b^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^2}+\frac {2 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^2} \]
-2/3*exp(2*a/b)*erf(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^( 1/2)/b^(5/2)/c^2+2/3*erfi(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*2^(1/2 )*Pi^(1/2)/b^(5/2)/c^2/exp(2*a/b)-2/3*x*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh (c*x))^(3/2)-4/3/b^2/c^2/(a+b*arcsinh(c*x))^(1/2)-8/3*x^2/b^2/(a+b*arcsinh (c*x))^(1/2)
Time = 0.55 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.09 \[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\frac {e^{-2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \left (-4 \sqrt {2} b e^{2 \text {arcsinh}(c x)} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+e^{\frac {2 a}{b}} \left (-4 a+b-4 a e^{4 \text {arcsinh}(c x)}-b e^{4 \text {arcsinh}(c x)}-4 b \left (1+e^{4 \text {arcsinh}(c x)}\right ) \text {arcsinh}(c x)+4 \sqrt {2} e^{2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x)) \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{6 b^2 c^2 (a+b \text {arcsinh}(c x))^{3/2}} \]
(-4*Sqrt[2]*b*E^(2*ArcSinh[c*x])*(-((a + b*ArcSinh[c*x])/b))^(3/2)*Gamma[1 /2, (-2*(a + b*ArcSinh[c*x]))/b] + E^((2*a)/b)*(-4*a + b - 4*a*E^(4*ArcSin h[c*x]) - b*E^(4*ArcSinh[c*x]) - 4*b*(1 + E^(4*ArcSinh[c*x]))*ArcSinh[c*x] + 4*Sqrt[2]*E^(2*(a/b + ArcSinh[c*x]))*Sqrt[a/b + ArcSinh[c*x]]*(a + b*Ar cSinh[c*x])*Gamma[1/2, (2*(a + b*ArcSinh[c*x]))/b]))/(6*b^2*c^2*E^(2*(a/b + ArcSinh[c*x]))*(a + b*ArcSinh[c*x])^(3/2))
Result contains complex when optimal does not.
Time = 1.31 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.13, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6194, 6198, 6233, 6195, 25, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634}
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.
| \(\displaystyle \int \frac {x}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {2 \int \frac {1}{\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}dx}{3 b c}+\frac {4 c \int \frac {x^2}{\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}dx}{3 b}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {4 c \int \frac {x^2}{\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}dx}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {4 c \left (\frac {4 \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b c}-\frac {2 x^2}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle \frac {4 c \left (\frac {4 \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {2 x^2}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 c \left (-\frac {4 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {2 x^2}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {4 c \left (-\frac {4 \int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 \sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {2 x^2}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 c \left (-\frac {2 \int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {2 x^2}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 c \left (-\frac {2 x^2}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 \int -\frac {i \sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^3}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {4 c \left (-\frac {2 x^2}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 i \int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^3}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {4 c \left (-\frac {2 x^2}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 i \left (\frac {1}{2} i \int \frac {e^{-2 \text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i \int \frac {e^{2 \text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))\right )}{b^2 c^3}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {4 c \left (-\frac {2 x^2}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 i \left (i \int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-i \int e^{\frac {2 (a+b \text {arcsinh}(c x))}{b}-\frac {2 a}{b}}d\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^3}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {4 c \left (-\frac {2 x^2}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 i \left (i \int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {4 c \left (-\frac {2 x^2}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\) |
(-2*x*Sqrt[1 + c^2*x^2])/(3*b*c*(a + b*ArcSinh[c*x])^(3/2)) - 4/(3*b^2*c^2 *Sqrt[a + b*ArcSinh[c*x]]) + (4*c*((-2*x^2)/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + ((2*I)*((I/2)*Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*Ar cSinh[c*x]])/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b *ArcSinh[c*x]])/Sqrt[b]])/E^((2*a)/b)))/(b^2*c^3)))/(3*b)
3.2.52.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
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] + (- Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n + 1)/ Sqrt[1 + c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*A rcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c ^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Simp[f*(m/(b*c* (n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e , c^2*d] && LtQ[n, -1]
\[\int \frac {x}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\int \frac {x}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\int { \frac {x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\int { \frac {x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{5/2}} \,d x \]