3.5.67 \(\int \frac {x^3 (d+c^2 d x^2)^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\) [467]

3.5.67.1 Optimal result

Integrand size = 28, antiderivative size = 474 \[ \int \frac {x^3 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d^2 x^3 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {3 d^2 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {d^2 e^{\frac {8 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {2 \sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {d^2 e^{\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {3 d^2 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {d^2 e^{-\frac {8 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {2 \sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {d^2 e^{-\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4} \]

output

-3/64*d^2*exp(2*a/b)*erf(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*2^(1/2) 
*Pi^(1/2)/b^(3/2)/c^4+1/64*d^2*exp(8*a/b)*erf(2*2^(1/2)*(a+b*arcsinh(c*x)) 
^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/c^4-3/64*d^2*erfi(2^(1/2)*(a+b*ar 
csinh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/c^4/exp(2*a/b)+1/64*d^ 
2*erfi(2*2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2 
)/c^4/exp(8*a/b)-1/32*d^2*exp(4*a/b)*erf(2*(a+b*arcsinh(c*x))^(1/2)/b^(1/2 
))*Pi^(1/2)/b^(3/2)/c^4-1/32*d^2*erfi(2*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))* 
Pi^(1/2)/b^(3/2)/c^4/exp(4*a/b)+1/64*d^2*exp(6*a/b)*erf(6^(1/2)*(a+b*arcsi 
nh(c*x))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/b^(3/2)/c^4+1/64*d^2*erfi(6^(1/2) 
*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/b^(3/2)/c^4/exp(6*a/b) 
-2*d^2*x^3*(c^2*x^2+1)^(5/2)/b/c/(a+b*arcsinh(c*x))^(1/2)
 

3.5.67.2 Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 462, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {d^2 e^{-\frac {8 a}{b}} \left (\sqrt {2} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )+\sqrt {6} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )-2 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-3 \sqrt {2} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+3 \sqrt {2} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+2 e^{\frac {12 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-\sqrt {6} e^{\frac {14 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )-\sqrt {2} e^{\frac {16 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )+6 e^{\frac {8 a}{b}} \sinh (2 \text {arcsinh}(c x))+2 e^{\frac {8 a}{b}} \sinh (4 \text {arcsinh}(c x))-2 e^{\frac {8 a}{b}} \sinh (6 \text {arcsinh}(c x))-e^{\frac {8 a}{b}} \sinh (8 \text {arcsinh}(c x))\right )}{64 b c^4 \sqrt {a+b \text {arcsinh}(c x)}} \]

input

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

output

(d^2*(Sqrt[2]*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-8*(a + b*ArcSin 
h[c*x]))/b] + Sqrt[6]*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/ 
2, (-6*(a + b*ArcSinh[c*x]))/b] - 2*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x] 
)/b)]*Gamma[1/2, (-4*(a + b*ArcSinh[c*x]))/b] - 3*Sqrt[2]*E^((6*a)/b)*Sqrt 
[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c*x]))/b] + 3*Sq 
rt[2]*E^((10*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (2*(a + b*ArcSinh[c 
*x]))/b] + 2*E^((12*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (4*(a + b*Ar 
cSinh[c*x]))/b] - Sqrt[6]*E^((14*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, 
 (6*(a + b*ArcSinh[c*x]))/b] - Sqrt[2]*E^((16*a)/b)*Sqrt[a/b + ArcSinh[c*x 
]]*Gamma[1/2, (8*(a + b*ArcSinh[c*x]))/b] + 6*E^((8*a)/b)*Sinh[2*ArcSinh[c 
*x]] + 2*E^((8*a)/b)*Sinh[4*ArcSinh[c*x]] - 2*E^((8*a)/b)*Sinh[6*ArcSinh[c 
*x]] - E^((8*a)/b)*Sinh[8*ArcSinh[c*x]]))/(64*b*c^4*E^((8*a)/b)*Sqrt[a + b 
*ArcSinh[c*x]])
 

3.5.67.3 Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6229, 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.

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

\(\Big \downarrow \) 6229

\(\displaystyle \frac {6 d^2 \int \frac {x^2 \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b c}+\frac {16 c d^2 \int \frac {x^4 \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 d^2 x^3 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 5971

\(\displaystyle \frac {16 d^2 \int \left (\frac {\cosh \left (\frac {8 a}{b}-\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{128 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {3}{128 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^4}+\frac {6 d^2 \int \left (\frac {\cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {1}{16 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^4}-\frac {2 d^2 x^3 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {16 d^2 \left (-\frac {1}{128} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{512} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {8 a}{b}} \text {erf}\left (\frac {2 \sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{128} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{512} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {8 a}{b}} \text {erfi}\left (\frac {2 \sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {3}{64} \sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^4}+\frac {6 d^2 \left (\frac {1}{64} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \sqrt {b} e^{\frac {6 a}{b}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \sqrt {b} e^{-\frac {6 a}{b}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^4}-\frac {2 d^2 x^3 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

input

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

output

(-2*d^2*x^3*(1 + c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (16*d^2* 
((3*Sqrt[a + b*ArcSinh[c*x]])/64 - (Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sq 
rt[a + b*ArcSinh[c*x]])/Sqrt[b]])/128 + (Sqrt[b]*E^((8*a)/b)*Sqrt[Pi/2]*Er 
f[(2*Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/512 - (Sqrt[b]*Sqrt[Pi]*E 
rfi[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(128*E^((4*a)/b)) + (Sqrt[b]*Sq 
rt[Pi/2]*Erfi[(2*Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(512*E^((8*a) 
/b))))/(b^2*c^4) + (6*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)*Sqrt[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]])/Sq 
rt[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^4)
 

3.5.67.3.1 Defintions of rubi rules used

rule 2009

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

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 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]
 

3.5.67.4 Maple [F]

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

input

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

output

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

3.5.67.5 Fricas [F(-2)]

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

input

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

output

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

3.5.67.6 Sympy [F]

\[ \int \frac {x^3 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=d^{2} \left (\int \frac {x^{3}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {2 c^{2} x^{5}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {c^{4} x^{7}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \]

input

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

output

d**2*(Integral(x**3/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a + b*asinh(c*x))*a 
sinh(c*x)), x) + Integral(2*c**2*x**5/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a 
 + b*asinh(c*x))*asinh(c*x)), x) + Integral(c**4*x**7/(a*sqrt(a + b*asinh( 
c*x)) + b*sqrt(a + b*asinh(c*x))*asinh(c*x)), x))
 

3.5.67.7 Maxima [F]

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

input

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

output

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

3.5.67.8 Giac [F(-2)]

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

input

integrate(x^3*(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
 

3.5.67.9 Mupad [F(-1)]

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

input

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

output

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