\(\int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx\) [513]

Optimal result

Integrand size = 26, antiderivative size = 355 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {3^{-1-n} e^{-\frac {3 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {e^{-\frac {a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {e^{a/b} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-1-n} e^{\frac {3 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}} \]

[Out]

1/8*3^(-1-n)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-3*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^2/exp(3*a/b)/(((-a-
b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)+1/8*(a+b*arcsinh(c*x))^n*GAMMA(1+n,(-a-b*arcsinh(c*x))/b)*(c^2*d*x^2+d
)^(1/2)/c^2/exp(a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)+1/8*exp(a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n
,(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^2/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)+1/8*3^(-1-n)*exp(3
*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,3*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^2/(((a+b*arcsinh(c*x))/b)^n
)/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5819, 5556, 3389, 2212} \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {3^{-n-1} e^{-\frac {3 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{8 c^2 \sqrt {c^2 x^2+1}}+\frac {e^{-\frac {a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 c^2 \sqrt {c^2 x^2+1}}+\frac {e^{a/b} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 c^2 \sqrt {c^2 x^2+1}}+\frac {3^{-n-1} e^{\frac {3 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{8 c^2 \sqrt {c^2 x^2+1}} \]

[In]

Int[x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n,x]

[Out]

(3^(-1 - n)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b])/(8*c^2*E^((3
*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1
+ n, -((a + b*ArcSinh[c*x])/b)])/(8*c^2*E^(a/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (E^(a/b)*Sq
rt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b*ArcSinh[c*x])/b])/(8*c^2*Sqrt[1 + c^2*x^2]*((a +
b*ArcSinh[c*x])/b)^n) + (3^(-1 - n)*E^((3*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (3*(a
+ b*ArcSinh[c*x]))/b])/(8*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n)

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3389

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

Rule 5556

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]

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 {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int x^n \cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^2 \sqrt {1+c^2 x^2}} \\ & = -\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int \left (\frac {1}{4} x^n \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )+\frac {1}{4} x^n \sinh \left (\frac {a}{b}-\frac {x}{b}\right )\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^2 \sqrt {1+c^2 x^2}} \\ & = -\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int x^n \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b c^2 \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int x^n \sinh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b c^2 \sqrt {1+c^2 x^2}} \\ & = -\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b c^2 \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b c^2 \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b c^2 \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b c^2 \sqrt {1+c^2 x^2}} \\ & = \frac {3^{-1-n} e^{-\frac {3 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {e^{-\frac {a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {e^{a/b} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-1-n} e^{\frac {3 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.65 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {d e^{-\frac {3 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (3 e^{\frac {4 a}{b}} \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^{-n} \Gamma \left (1+n,\frac {a}{b}+\text {arcsinh}(c x)\right )+\left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \left (3^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+3 e^{\frac {2 a}{b}} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c x)}{b}\right )+3^{-n} e^{\frac {6 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{2 n} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{24 c^2 \sqrt {d \left (1+c^2 x^2\right )}} \]

[In]

Integrate[x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n,x]

[Out]

(d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*((3*E^((4*a)/b)*Gamma[1 + n, a/b + ArcSinh[c*x]])/(a/b + ArcSinh[c
*x])^n + (Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b]/3^n + 3*E^((2*a)/b)*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)
] + (E^((6*a)/b)*(-((a + b*ArcSinh[c*x])/b))^(2*n)*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/(3^n*(-((a + b*Ar
cSinh[c*x])^2/b^2))^n))/(-((a + b*ArcSinh[c*x])/b))^n))/(24*c^2*E^((3*a)/b)*Sqrt[d*(1 + c^2*x^2)])

Maple [F]

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

[In]

int(x*(a+b*arcsinh(c*x))^n*(c^2*d*x^2+d)^(1/2),x)

[Out]

int(x*(a+b*arcsinh(c*x))^n*(c^2*d*x^2+d)^(1/2),x)

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^n*x, x)

Sympy [F]

\[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\int x \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{n}\, dx \]

[In]

integrate(x*(a+b*asinh(c*x))**n*(c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(x*sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))**n, x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^n*x, x)

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

int(x*(a + b*asinh(c*x))^n*(d + c^2*d*x^2)^(1/2),x)

[Out]

int(x*(a + b*asinh(c*x))^n*(d + c^2*d*x^2)^(1/2), x)