\(\int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx\) [512]

Optimal result

Integrand size = 28, antiderivative size = 235 \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-2 (3+n)} e^{-\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-2 (3+n)} e^{\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}} \]

[Out]

-1/8*(a+b*arcsinh(c*x))^(1+n)*(c^2*d*x^2+d)^(1/2)/b/c^3/(1+n)/(c^2*x^2+1)^(1/2)+(a+b*arcsinh(c*x))^n*GAMMA(1+n
,-4*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/(2^(6+2*n))/c^3/exp(4*a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*x^2+
1)^(1/2)-exp(4*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,4*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/(2^(6+2*n))/c^3
/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5819, 5556, 3388, 2212} \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^{n+1}}{8 b c^3 (n+1) \sqrt {c^2 x^2+1}}+\frac {2^{-2 (n+3)} e^{-\frac {4 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {2^{-2 (n+3)} e^{\frac {4 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}} \]

[In]

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

[Out]

-1/8*(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(b*c^3*(1 + n)*Sqrt[1 + c^2*x^2]) + (Sqrt[d + c^2*d*x^
2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(3 + n))*c^3*E^((4*a)/b)*Sqrt[1 + c
^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) - (E^((4*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n,
(4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(3 + n))*c^3*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 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 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 ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^3 \sqrt {1+c^2 x^2}} \\ & = \frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int \left (-\frac {x^n}{8}+\frac {1}{8} x^n \cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int x^n \cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int e^{-i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b c^3 \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int e^{i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {4^{-3-n} e^{-\frac {4 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 {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {4^{-3-n} e^{\frac {4 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 {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.73 \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {8 (a+b \text {arcsinh}(c x))}{b (1+n)}+4^{-n} e^{-\frac {4 a}{b}} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^{-n} \left (\left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{64 c^3 \sqrt {d \left (1+c^2 x^2\right )}} \]

[In]

Integrate[x^2*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*((-8*(a + b*ArcSinh[c*x]))/(b*(1 + n)) + ((a/b + ArcSinh[c*x])^n*G
amma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b] - E^((8*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*Ar
cSinh[c*x]))/b])/(4^n*E^((4*a)/b)*(-((a + b*ArcSinh[c*x])^2/b^2))^n)))/(64*c^3*Sqrt[d*(1 + c^2*x^2)])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate(x^2*(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^2, x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(x^2*(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^2, x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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