\(\int x (a+b \text {arcsinh}(c x))^{5/2} \, dx\) [143]

Optimal result

Integrand size = 14, antiderivative size = 223 \[ \int x (a+b \text {arcsinh}(c x))^{5/2} \, dx=\frac {15 b^2 \sqrt {a+b \text {arcsinh}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{5/2}-\frac {15 b^{5/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{256 c^2}-\frac {15 b^{5/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{256 c^2} \]

[Out]

1/4*(a+b*arcsinh(c*x))^(5/2)/c^2+1/2*x^2*(a+b*arcsinh(c*x))^(5/2)-15/512*b^(5/2)*exp(2*a/b)*erf(2^(1/2)*(a+b*a
rcsinh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/c^2-15/512*b^(5/2)*erfi(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))
*2^(1/2)*Pi^(1/2)/c^2/exp(2*a/b)-5/8*b*x*(a+b*arcsinh(c*x))^(3/2)*(c^2*x^2+1)^(1/2)/c+15/64*b^2*(a+b*arcsinh(c
*x))^(1/2)/c^2+15/32*b^2*x^2*(a+b*arcsinh(c*x))^(1/2)

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5777, 5812, 5783, 5819, 3393, 3388, 2211, 2236, 2235} \[ \int x (a+b \text {arcsinh}(c x))^{5/2} \, dx=-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{256 c^2}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{256 c^2}+\frac {15 b^2 \sqrt {a+b \text {arcsinh}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {5 b x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{5/2} \]

[In]

Int[x*(a + b*ArcSinh[c*x])^(5/2),x]

[Out]

(15*b^2*Sqrt[a + b*ArcSinh[c*x]])/(64*c^2) + (15*b^2*x^2*Sqrt[a + b*ArcSinh[c*x]])/32 - (5*b*x*Sqrt[1 + c^2*x^
2]*(a + b*ArcSinh[c*x])^(3/2))/(8*c) + (a + b*ArcSinh[c*x])^(5/2)/(4*c^2) + (x^2*(a + b*ArcSinh[c*x])^(5/2))/2
- (15*b^(5/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(256*c^2) - (15*b^(5/2)
*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(256*c^2*E^((2*a)/b))

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 5777

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcSinh[c*x])^n/(
m + 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /;
FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[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]

Rule 5812

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Dist[f^2*((m - 1)/(c^2*
(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)
))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0
]

Rule 5819

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{5/2}-\frac {1}{4} (5 b c) \int \frac {x^2 (a+b \text {arcsinh}(c x))^{3/2}}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {5 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{8 c}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{5/2}+\frac {1}{16} \left (15 b^2\right ) \int x \sqrt {a+b \text {arcsinh}(c x)} \, dx+\frac {(5 b) \int \frac {(a+b \text {arcsinh}(c x))^{3/2}}{\sqrt {1+c^2 x^2}} \, dx}{8 c} \\ & = \frac {15}{32} b^2 x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{5/2}-\frac {1}{64} \left (15 b^3 c\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx \\ & = \frac {15}{32} b^2 x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{5/2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {\sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 c^2} \\ & = \frac {15}{32} b^2 x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{5/2}+\frac {\left (15 b^2\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 c^2} \\ & = \frac {15 b^2 \sqrt {a+b \text {arcsinh}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{5/2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{128 c^2} \\ & = \frac {15 b^2 \sqrt {a+b \text {arcsinh}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{5/2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{256 c^2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{256 c^2} \\ & = \frac {15 b^2 \sqrt {a+b \text {arcsinh}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{5/2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{128 c^2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{128 c^2} \\ & = \frac {15 b^2 \sqrt {a+b \text {arcsinh}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{5/2}-\frac {15 b^{5/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{256 c^2}-\frac {15 b^{5/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{256 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.52 \[ \int x (a+b \text {arcsinh}(c x))^{5/2} \, dx=\frac {e^{-\frac {2 a}{b}} \left (-b^3 \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {7}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+b^3 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {7}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{32 \sqrt {2} c^2 \sqrt {a+b \text {arcsinh}(c x)}} \]

[In]

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

[Out]

(-(b^3*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[7/2, (-2*(a + b*ArcSinh[c*x]))/b]) + b^3*E^((4*a)/b)*Sqrt[a/b + A
rcSinh[c*x]]*Gamma[7/2, (2*(a + b*ArcSinh[c*x]))/b])/(32*Sqrt[2]*c^2*E^((2*a)/b)*Sqrt[a + b*ArcSinh[c*x]])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x*(a + b*asinh(c*x))**(5/2), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int x (a+b \text {arcsinh}(c x))^{5/2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

int(x*(a + b*asinh(c*x))^(5/2),x)

[Out]

int(x*(a + b*asinh(c*x))^(5/2), x)