\(\int \frac {x (d+c^2 d x^2)^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\) [455]

Optimal result

Integrand size = 26, antiderivative size = 358 \[ \int \frac {x \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d^2 x \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 d^2 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 d^2 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{-\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2} \] Output:

-2*d^2*x*(c^2*x^2+1)^(5/2)/b/c/(a+b*arcsinh(c*x))^(1/2)+1/4*d^2*exp(4*a/b) 
*Pi^(1/2)*erf(2*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^2+5/32*d^2*exp 
(2*a/b)*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^( 
3/2)/c^2+1/32*d^2*exp(6*a/b)*6^(1/2)*Pi^(1/2)*erf(6^(1/2)*(a+b*arcsinh(c*x 
))^(1/2)/b^(1/2))/b^(3/2)/c^2+1/4*d^2*Pi^(1/2)*erfi(2*(a+b*arcsinh(c*x))^( 
1/2)/b^(1/2))/b^(3/2)/c^2/exp(4*a/b)+5/32*d^2*2^(1/2)*Pi^(1/2)*erfi(2^(1/2 
)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^2/exp(2*a/b)+1/32*d^2*6^(1/2 
)*Pi^(1/2)*erfi(6^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^2/exp( 
6*a/b)
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.98 \[ \int \frac {x \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {d^2 e^{-\frac {6 a}{b}} \left (-\sqrt {6} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )-8 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-5 \sqrt {2} e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+5 \sqrt {2} e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+8 e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+\sqrt {6} e^{\frac {12 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )+10 e^{\frac {6 a}{b}} \sinh (2 \text {arcsinh}(c x))+8 e^{\frac {6 a}{b}} \sinh (4 \text {arcsinh}(c x))+2 e^{\frac {6 a}{b}} \sinh (6 \text {arcsinh}(c x))\right )}{32 b c^2 \sqrt {a+b \text {arcsinh}(c x)}} \] Input:

Integrate[(x*(d + c^2*d*x^2)^2)/(a + b*ArcSinh[c*x])^(3/2),x]
 

Output:

-1/32*(d^2*(-(Sqrt[6]*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-6*(a + 
b*ArcSinh[c*x]))/b]) - 8*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma 
[1/2, (-4*(a + b*ArcSinh[c*x]))/b] - 5*Sqrt[2]*E^((4*a)/b)*Sqrt[-((a + b*A 
rcSinh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c*x]))/b] + 5*Sqrt[2]*E^((8 
*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (2*(a + b*ArcSinh[c*x]))/b] + 8 
*E^((10*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (4*(a + b*ArcSinh[c*x])) 
/b] + Sqrt[6]*E^((12*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (6*(a + b*A 
rcSinh[c*x]))/b] + 10*E^((6*a)/b)*Sinh[2*ArcSinh[c*x]] + 8*E^((6*a)/b)*Sin 
h[4*ArcSinh[c*x]] + 2*E^((6*a)/b)*Sinh[6*ArcSinh[c*x]]))/(b*c^2*E^((6*a)/b 
)*Sqrt[a + b*ArcSinh[c*x]])
 

Rubi [A] (verified)

Time = 2.25 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.57, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6229, 6206, 3042, 3793, 2009, 6234, 5971, 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.

\(\displaystyle \int \frac {x \left (c^2 d x^2+d\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\)

\(\Big \downarrow \) 6229

\(\displaystyle \frac {2 d^2 \int \frac {\left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b c}+\frac {12 c d^2 \int \frac {x^2 \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 d^2 x \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 6206

\(\displaystyle \frac {2 d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {12 c d^2 \int \frac {x^2 \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 d^2 x \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 d^2 \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )^4}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {12 c d^2 \int \frac {x^2 \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 d^2 x \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {2 d^2 \int \left (\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {3}{8 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {12 c d^2 \int \frac {x^2 \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 d^2 x \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {12 c d^2 \int \frac {x^2 \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}+\frac {2 d^2 \left (\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}-\frac {2 d^2 x \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {12 d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {2 d^2 \left (\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}-\frac {2 d^2 x \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {12 d^2 \int \left (\frac {\cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {1}{16 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {2 d^2 \left (\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}-\frac {2 d^2 x \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 d^2 \left (\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}+\frac {12 d^2 \left (\frac {1}{64} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \sqrt {b} e^{\frac {6 a}{b}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \sqrt {b} e^{-\frac {6 a}{b}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}-\frac {2 d^2 x \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

Input:

Int[(x*(d + c^2*d*x^2)^2)/(a + b*ArcSinh[c*x])^(3/2),x]
 

Output:

(-2*d^2*x*(1 + c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (2*d^2*((3 
*Sqrt[a + b*ArcSinh[c*x]])/4 + (Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a 
 + b*ArcSinh[c*x]])/Sqrt[b]])/32 + (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sq 
rt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/4 + (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sq 
rt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*E^((4*a)/b)) + (Sqrt[b]*Sqrt[Pi/2]*E 
rfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*E^((2*a)/b))))/(b^2*c^ 
2) + (12*d^2*(-1/8*Sqrt[a + b*ArcSinh[c*x]] + (Sqrt[b]*E^((4*a)/b)*Sqrt[Pi 
]*Erf[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/64 - (Sqrt[b]*E^((2*a)/b)*Sqr 
t[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/64 + (Sqrt[b]*E^( 
(6*a)/b)*Sqrt[Pi/6]*Erf[(Sqrt[6]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/64 + 
(Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*E^((4*a) 
/b)) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b] 
])/(64*E^((2*a)/b)) + (Sqrt[b]*Sqrt[Pi/6]*Erfi[(Sqrt[6]*Sqrt[a + b*ArcSinh 
[c*x]])/Sqrt[b]])/(64*E^((6*a)/b))))/(b^2*c^2)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[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 5971

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 6206

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

rule 6229

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Sqrt[1 + c^2*x^2]*(d + e*x^2)^p 
*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Simp[f*(m/(b*c*(n + 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] - Simp[c*((m + 2*p + 1)/(b*f* 
(n + 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}, x] && EqQ[e, c^2*d] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1 
, 0] && IGtQ[m, -3]
 

rule 6234

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> Simp[(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]
 

Maple [F]

\[\int \frac {x \left (c^{2} d \,x^{2}+d \right )^{2}}{\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{\frac {3}{2}}}d x\]

Input:

int(x*(c^2*d*x^2+d)^2/(a+b*arcsinh(x*c))^(3/2),x)
 

Output:

int(x*(c^2*d*x^2+d)^2/(a+b*arcsinh(x*c))^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x*(c^2*d*x^2+d)^2/(a+b*arcsinh(c*x))^(3/2),x, algorithm="fricas" 
)
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \frac {x \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=d^{2} \left (\int \frac {x}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {2 c^{2} x^{3}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {c^{4} x^{5}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \] Input:

integrate(x*(c**2*d*x**2+d)**2/(a+b*asinh(c*x))**(3/2),x)
 

Output:

d**2*(Integral(x/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a + b*asinh(c*x))*asin 
h(c*x)), x) + Integral(2*c**2*x**3/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a + 
b*asinh(c*x))*asinh(c*x)), x) + Integral(c**4*x**5/(a*sqrt(a + b*asinh(c*x 
)) + b*sqrt(a + b*asinh(c*x))*asinh(c*x)), x))
 

Maxima [F]

\[ \int \frac {x \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{2} x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x*(c^2*d*x^2+d)^2/(a+b*arcsinh(c*x))^(3/2),x, algorithm="maxima" 
)
 

Output:

integrate((c^2*d*x^2 + d)^2*x/(b*arcsinh(c*x) + a)^(3/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x*(c^2*d*x^2+d)^2/(a+b*arcsinh(c*x))^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {x\,{\left (d\,c^2\,x^2+d\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:

int((x*(d + c^2*d*x^2)^2)/(a + b*asinh(c*x))^(3/2),x)
 

Output:

int((x*(d + c^2*d*x^2)^2)/(a + b*asinh(c*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {x \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=d^{2} \left (\left (\int \frac {\sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x^{5}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}+2 \left (\int \frac {\sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x^{3}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {\sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:

int(x*(c^2*d*x^2+d)^2/(a+b*asinh(c*x))^(3/2),x)
 

Output:

d**2*(int((sqrt(asinh(c*x)*b + a)*x**5)/(asinh(c*x)**2*b**2 + 2*asinh(c*x) 
*a*b + a**2),x)*c**4 + 2*int((sqrt(asinh(c*x)*b + a)*x**3)/(asinh(c*x)**2* 
b**2 + 2*asinh(c*x)*a*b + a**2),x)*c**2 + int((sqrt(asinh(c*x)*b + a)*x)/( 
asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b + a**2),x))