\(\int \frac {(c+a^2 c x^2)^{5/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx\) [495]

Optimal result

Integrand size = 23, antiderivative size = 396 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {5 c^2 \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}}{8 a \sqrt {1+a^2 x^2}}+\frac {3 c^2 \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {c^2 \sqrt {\frac {\pi }{6}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {6} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {3 c^2 \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {c^2 \sqrt {\frac {\pi }{6}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {6} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}} \]

[Out]

1/384*c^2*erf(6^(1/2)*arcsinh(a*x)^(1/2))*6^(1/2)*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+1/384*c^2*e
rfi(6^(1/2)*arcsinh(a*x)^(1/2))*6^(1/2)*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+15/128*c^2*erf(2^(1/2
)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+15/128*c^2*erfi(2^(1/2)*arcsinh
(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+3/64*c^2*erf(2*arcsinh(a*x)^(1/2))*Pi^(1
/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+3/64*c^2*erfi(2*arcsinh(a*x)^(1/2))*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a
/(a^2*x^2+1)^(1/2)+5/8*c^2*(a^2*c*x^2+c)^(1/2)*arcsinh(a*x)^(1/2)/a/(a^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5791, 3393, 3388, 2211, 2235, 2236} \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {3 \sqrt {\pi } c^2 \sqrt {a^2 c x^2+c} \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {a^2 x^2+1}}+\frac {15 \sqrt {\frac {\pi }{2}} c^2 \sqrt {a^2 c x^2+c} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {\frac {\pi }{6}} c^2 \sqrt {a^2 c x^2+c} \text {erf}\left (\sqrt {6} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {a^2 x^2+1}}+\frac {3 \sqrt {\pi } c^2 \sqrt {a^2 c x^2+c} \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {a^2 x^2+1}}+\frac {15 \sqrt {\frac {\pi }{2}} c^2 \sqrt {a^2 c x^2+c} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {\frac {\pi }{6}} c^2 \sqrt {a^2 c x^2+c} \text {erfi}\left (\sqrt {6} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {a^2 x^2+1}}+\frac {5 c^2 \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}}{8 a \sqrt {a^2 x^2+1}} \]

[In]

Int[(c + a^2*c*x^2)^(5/2)/Sqrt[ArcSinh[a*x]],x]

[Out]

(5*c^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/(8*a*Sqrt[1 + a^2*x^2]) + (3*c^2*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*E
rf[2*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2]) + (15*c^2*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[A
rcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2]) + (c^2*Sqrt[Pi/6]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[6]*Sqrt[ArcSinh[a*x]]])
/(64*a*Sqrt[1 + a^2*x^2]) + (3*c^2*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2
*x^2]) + (15*c^2*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2]) + (
c^2*Sqrt[Pi/6]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[6]*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2])

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 2235

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]

Rule 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5791

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*c))*Simp[(d
+ e*x^2)^p/(1 + c^2*x^2)^p], Subst[Int[x^n*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; Free
Q[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^6(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a \sqrt {1+a^2 x^2}} \\ & = \frac {\left (c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {5}{16 \sqrt {x}}+\frac {15 \cosh (2 x)}{32 \sqrt {x}}+\frac {3 \cosh (4 x)}{16 \sqrt {x}}+\frac {\cosh (6 x)}{32 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a \sqrt {1+a^2 x^2}} \\ & = \frac {5 c^2 \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}}{8 a \sqrt {1+a^2 x^2}}+\frac {\left (c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (6 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{32 a \sqrt {1+a^2 x^2}} \\ & = \frac {5 c^2 \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}}{8 a \sqrt {1+a^2 x^2}}+\frac {\left (c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-6 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{6 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{64 a \sqrt {1+a^2 x^2}} \\ & = \frac {5 c^2 \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}}{8 a \sqrt {1+a^2 x^2}}+\frac {\left (c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-6 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {\left (c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{6 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}} \\ & = \frac {5 c^2 \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}}{8 a \sqrt {1+a^2 x^2}}+\frac {3 c^2 \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {c^2 \sqrt {\frac {\pi }{6}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {6} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {3 c^2 \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {c^2 \sqrt {\frac {\pi }{6}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {6} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.50 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {c^2 \sqrt {c+a^2 c x^2} \left (240 \text {arcsinh}(a x)+\sqrt {6} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-6 \text {arcsinh}(a x)\right )+18 \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arcsinh}(a x)\right )+45 \sqrt {2} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )-45 \sqrt {2} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )-18 \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},4 \text {arcsinh}(a x)\right )-\sqrt {6} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},6 \text {arcsinh}(a x)\right )\right )}{384 a \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \]

[In]

Integrate[(c + a^2*c*x^2)^(5/2)/Sqrt[ArcSinh[a*x]],x]

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*(240*ArcSinh[a*x] + Sqrt[6]*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -6*ArcSinh[a*x]] + 18*Sqrt
[-ArcSinh[a*x]]*Gamma[1/2, -4*ArcSinh[a*x]] + 45*Sqrt[2]*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -2*ArcSinh[a*x]] - 45*
Sqrt[2]*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 2*ArcSinh[a*x]] - 18*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 4*ArcSinh[a*x]] - Sqr
t[6]*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 6*ArcSinh[a*x]]))/(384*a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])

Maple [F]

\[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{\sqrt {\operatorname {arcsinh}\left (a x \right )}}d x\]

[In]

int((a^2*c*x^2+c)^(5/2)/arcsinh(a*x)^(1/2),x)

[Out]

int((a^2*c*x^2+c)^(5/2)/arcsinh(a*x)^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a^2*c*x^2+c)^(5/2)/arcsinh(a*x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\text {Timed out} \]

[In]

integrate((a**2*c*x**2+c)**(5/2)/asinh(a*x)**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{\sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \]

[In]

integrate((a^2*c*x^2+c)^(5/2)/arcsinh(a*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)/sqrt(arcsinh(a*x)), x)

Giac [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{\sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \]

[In]

integrate((a^2*c*x^2+c)^(5/2)/arcsinh(a*x)^(1/2),x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)/sqrt(arcsinh(a*x)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^{5/2}}{\sqrt {\mathrm {asinh}\left (a\,x\right )}} \,d x \]

[In]

int((c + a^2*c*x^2)^(5/2)/asinh(a*x)^(1/2),x)

[Out]

int((c + a^2*c*x^2)^(5/2)/asinh(a*x)^(1/2), x)