Integrand size = 18, antiderivative size = 496 \[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {c^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{2 d^3}-\frac {\sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}+\frac {\sqrt {b} c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}+\frac {\sqrt {b} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d^3}+\frac {\sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}-\frac {\sqrt {b} c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}-\frac {\sqrt {b} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d^3} \] Output:
c^2*(d*x+c)*(a+b*arcsinh(d*x+c))^(1/2)/d^3+1/3*(d*x+c)^3*(a+b*arcsinh(d*x+ c))^(1/2)/d^3-1/2*c*(a+b*arcsinh(d*x+c))^(1/2)*cosh(2*arcsinh(d*x+c))/d^3- 1/16*b^(1/2)*exp(a/b)*Pi^(1/2)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))/d^3 +1/4*b^(1/2)*c^2*exp(a/b)*Pi^(1/2)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2)) /d^3+1/16*b^(1/2)*c*exp(2*a/b)*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*(a+b*arcsinh(d *x+c))^(1/2)/b^(1/2))/d^3+1/144*b^(1/2)*exp(3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^ (1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))/d^3+1/16*b^(1/2)*Pi^(1/2)*erfi(( a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))/d^3/exp(a/b)-1/4*b^(1/2)*c^2*Pi^(1/2)*e rfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))/d^3/exp(a/b)+1/16*b^(1/2)*c*2^(1/2 )*Pi^(1/2)*erfi(2^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))/d^3/exp(2*a/b) -1/144*b^(1/2)*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^ (1/2))/d^3/exp(3*a/b)
Time = 2.51 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.32 \[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {-36 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}+144 c^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-72 c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))+\sqrt {b} \sqrt {3 \pi } \cosh \left (\frac {3 a}{b}\right ) \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+9 \sqrt {b} \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-36 \sqrt {b} c^2 \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+9 \sqrt {b} c \sqrt {2 \pi } \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\sqrt {b} \sqrt {3 \pi } \cosh \left (\frac {3 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-9 \sqrt {b} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+36 \sqrt {b} c^2 \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+9 \sqrt {b} \left (-1+4 c^2\right ) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )-9 \sqrt {b} c \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )+9 \sqrt {b} c \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+\sqrt {b} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {3 a}{b}\right )+\sqrt {b} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {3 a}{b}\right )+12 \sqrt {a+b \text {arcsinh}(c+d x)} \sinh (3 \text {arcsinh}(c+d x))}{144 d^3} \] Input:
Integrate[x^2*Sqrt[a + b*ArcSinh[c + d*x]],x]
Output:
(-36*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]] + 144*c^2*(c + d*x)*Sqrt[a + b *ArcSinh[c + d*x]] - 72*c*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + Sqrt[b]*Sqrt[3*Pi]*Cosh[(3*a)/b]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] + 9*Sqrt[b]*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[ c + d*x]]/Sqrt[b]] - 36*Sqrt[b]*c^2*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*Arc Sinh[c + d*x]]/Sqrt[b]] + 9*Sqrt[b]*c*Sqrt[2*Pi]*Cosh[(2*a)/b]*Erfi[(Sqrt[ 2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - Sqrt[b]*Sqrt[3*Pi]*Cosh[(3*a)/ b]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - 9*Sqrt[b]*Sqrt[P i]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 36*Sqrt[b]*c^2*S qrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 9*Sqrt[b]*( -1 + 4*c^2)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]) - 9*Sqrt[b]*c*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*Sinh[(2*a)/b] + 9*Sqrt[b]*c*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[ a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + Sqrt[b ]*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*Sinh[(3*a )/b] + Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt [b]]*Sinh[(3*a)/b] + 12*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[3*ArcSinh[c + d* x]])/(144*d^3)
Time = 1.86 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6274, 27, 6245, 7267, 7292, 7293, 2009}
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.
\(\displaystyle \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx\) |
\(\Big \downarrow \) 6274 |
\(\displaystyle \frac {\int x^2 \sqrt {a+b \text {arcsinh}(c+d x)}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int d^2 x^2 \sqrt {a+b \text {arcsinh}(c+d x)}d(c+d x)}{d^3}\) |
\(\Big \downarrow \) 6245 |
\(\displaystyle \frac {\int d^2 x^2 \sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}d\text {arcsinh}(c+d x)}{d^3}\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle \frac {2 \int d^2 x^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))d\sqrt {a+b \text {arcsinh}(c+d x)}}{b d^3}\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \frac {2 \int d^2 x^2 (a+b \text {arcsinh}(c+d x)) \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )d\sqrt {a+b \text {arcsinh}(c+d x)}}{b d^3}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {2 \int \left ((a+b \text {arcsinh}(c+d x)) \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) c^2+(a+b \text {arcsinh}(c+d x)) \sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right ) c+(a+b \text {arcsinh}(c+d x)) \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )d\sqrt {a+b \text {arcsinh}(c+d x)}}{b d^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (\frac {1}{8} \sqrt {\pi } b^{3/2} c^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } b^{3/2} c^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\pi } b^{3/2} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{2}} b^{3/2} c e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{96} \sqrt {\frac {\pi }{3}} b^{3/2} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } b^{3/2} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{2}} b^{3/2} c e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{96} \sqrt {\frac {\pi }{3}} b^{3/2} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} b c^2 \sqrt {a+b \text {arcsinh}(c+d x)} \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {1}{6} b \sqrt {a+b \text {arcsinh}(c+d x)} \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {1}{4} b c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b d^3}\) |
Input:
Int[x^2*Sqrt[a + b*ArcSinh[c + d*x]],x]
Output:
(2*(-1/4*(b*c*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[(2*a)/b - (2*(a + b*ArcSin h[c + d*x]))/b]) - (b^(3/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d* x]]/Sqrt[b]])/32 + (b^(3/2)*c^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/8 + (b^(3/2)*c*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[ a + b*ArcSinh[c + d*x]])/Sqrt[b]])/16 + (b^(3/2)*E^((3*a)/b)*Sqrt[Pi/3]*Er f[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/96 + (b^(3/2)*Sqrt[Pi]* Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(32*E^(a/b)) - (b^(3/2)*c^2*Sq rt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*E^(a/b)) + (b^(3/2)* c*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*E^( (2*a)/b)) - (b^(3/2)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]] )/Sqrt[b]])/(96*E^((3*a)/b)) - (b*c^2*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[a/ b - (a + b*ArcSinh[c + d*x])/b])/2 - (b*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[ a/b - (a + b*ArcSinh[c + d*x])/b]^3)/6))/(b*d^3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[1/c^(m + 1) Subst[Int[(a + b*x)^n*Cosh[x]*(c*d + e*Sinh[ x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
\[\int x^{2} \sqrt {a +b \,\operatorname {arcsinh}\left (d x +c \right )}d x\]
Input:
int(x^2*(a+b*arcsinh(d*x+c))^(1/2),x)
Output:
int(x^2*(a+b*arcsinh(d*x+c))^(1/2),x)
Exception generated. \[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2*(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx \] Input:
integrate(x**2*(a+b*asinh(d*x+c))**(1/2),x)
Output:
Integral(x**2*sqrt(a + b*asinh(c + d*x)), x)
\[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int { \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*arcsinh(d*x + c) + a)*x^2, x)
\[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int { \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*arcsinh(d*x + c) + a)*x^2, x)
Timed out. \[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int x^2\,\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )} \,d x \] Input:
int(x^2*(a + b*asinh(c + d*x))^(1/2),x)
Output:
int(x^2*(a + b*asinh(c + d*x))^(1/2), x)
\[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int \sqrt {a +b \mathit {asinh} \left (d x +c \right )}\, x^{2}d x \] Input:
int(x^2*(a+b*asinh(d*x+c))^(1/2),x)
Output:
int(sqrt(asinh(c + d*x)*b + a)*x**2,x)