\(\int \frac {x^2 (d+c^2 d x^2)^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\) [454]

Optimal result

Integrand size = 28, antiderivative size = 457 \[ \int \frac {x^2 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d^2 x^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {5 d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {d^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {3 d^2 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {d^2 e^{\frac {7 a}{b}} \sqrt {7 \pi } \text {erf}\left (\frac {\sqrt {7} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {3 d^2 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{-\frac {7 a}{b}} \sqrt {7 \pi } \text {erfi}\left (\frac {\sqrt {7} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3} \] Output:

-2*d^2*x^2*(c^2*x^2+1)^(5/2)/b/c/(a+b*arcsinh(c*x))^(1/2)+5/64*d^2*exp(a/b 
)*Pi^(1/2)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3-1/64*d^2*exp( 
3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3 
/2)/c^3-3/64*d^2*exp(5*a/b)*5^(1/2)*Pi^(1/2)*erf(5^(1/2)*(a+b*arcsinh(c*x) 
)^(1/2)/b^(1/2))/b^(3/2)/c^3-1/64*d^2*exp(7*a/b)*7^(1/2)*Pi^(1/2)*erf(7^(1 
/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3-5/64*d^2*Pi^(1/2)*erfi(( 
a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3/exp(a/b)+1/64*d^2*3^(1/2)*Pi^ 
(1/2)*erfi(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3/exp(3*a/b 
)+3/64*d^2*5^(1/2)*Pi^(1/2)*erfi(5^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2)) 
/b^(3/2)/c^3/exp(5*a/b)+1/64*d^2*7^(1/2)*Pi^(1/2)*erfi(7^(1/2)*(a+b*arcsin 
h(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3/exp(7*a/b)
 

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.26 \[ \int \frac {x^2 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {d^2 e^{-7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \left (e^{\frac {7 a}{b}}+3 e^{\frac {7 a}{b}+2 \text {arcsinh}(c x)}+e^{\frac {7 a}{b}+4 \text {arcsinh}(c x)}-5 e^{\frac {7 a}{b}+6 \text {arcsinh}(c x)}-5 e^{\frac {7 a}{b}+8 \text {arcsinh}(c x)}+e^{\frac {7 a}{b}+10 \text {arcsinh}(c x)}+3 e^{\frac {7 a}{b}+12 \text {arcsinh}(c x)}+e^{\frac {7 a}{b}+14 \text {arcsinh}(c x)}+5 e^{\frac {8 a}{b}+7 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )-\sqrt {7} e^{7 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )-3 \sqrt {5} e^{\frac {2 a}{b}+7 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\sqrt {3} e^{\frac {4 a}{b}+7 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+5 e^{\frac {6 a}{b}+7 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-\sqrt {3} e^{\frac {10 a}{b}+7 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-3 \sqrt {5} e^{\frac {12 a}{b}+7 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\sqrt {7} e^{7 \left (\frac {2 a}{b}+\text {arcsinh}(c x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{64 b c^3 \sqrt {a+b \text {arcsinh}(c x)}} \] Input:

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

Output:

-1/64*(d^2*(E^((7*a)/b) + 3*E^((7*a)/b + 2*ArcSinh[c*x]) + E^((7*a)/b + 4* 
ArcSinh[c*x]) - 5*E^((7*a)/b + 6*ArcSinh[c*x]) - 5*E^((7*a)/b + 8*ArcSinh[ 
c*x]) + E^((7*a)/b + 10*ArcSinh[c*x]) + 3*E^((7*a)/b + 12*ArcSinh[c*x]) + 
E^((7*a)/b + 14*ArcSinh[c*x]) + 5*E^((8*a)/b + 7*ArcSinh[c*x])*Sqrt[a/b + 
ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] - Sqrt[7]*E^(7*ArcSinh[c*x])* 
Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-7*(a + b*ArcSinh[c*x]))/b] - 
3*Sqrt[5]*E^((2*a)/b + 7*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gam 
ma[1/2, (-5*(a + b*ArcSinh[c*x]))/b] - Sqrt[3]*E^((4*a)/b + 7*ArcSinh[c*x] 
)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] 
+ 5*E^((6*a)/b + 7*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2 
, -((a + b*ArcSinh[c*x])/b)] - Sqrt[3]*E^((10*a)/b + 7*ArcSinh[c*x])*Sqrt[ 
a/b + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b] - 3*Sqrt[5]*E^( 
(12*a)/b + 7*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (5*(a + b*A 
rcSinh[c*x]))/b] - Sqrt[7]*E^(7*((2*a)/b + ArcSinh[c*x]))*Sqrt[a/b + ArcSi 
nh[c*x]]*Gamma[1/2, (7*(a + b*ArcSinh[c*x]))/b]))/(b*c^3*E^(7*(a/b + ArcSi 
nh[c*x]))*Sqrt[a + b*ArcSinh[c*x]])
 

Rubi [A] (verified)

Time = 2.05 (sec) , antiderivative size = 718, normalized size of antiderivative = 1.57, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6229, 6234, 25, 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.

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

\(\Big \downarrow \) 6229

\(\displaystyle \frac {4 d^2 \int \frac {x \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b c}+\frac {14 c d^2 \int \frac {x^3 \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 d^2 x^2 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {14 d^2 \int -\frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^3}+\frac {4 d^2 \int -\frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {2 d^2 x^2 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {14 d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {4 d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {2 d^2 x^2 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {14 d^2 \int \left (\frac {\sinh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {3 \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {4 d^2 \int \left (\frac {\sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {2 d^2 x^2 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 d^2 \left (-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}+\frac {14 d^2 \left (\frac {3}{128} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{128} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{128} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{128} \sqrt {\frac {\pi }{7}} \sqrt {b} e^{\frac {7 a}{b}} \text {erf}\left (\frac {\sqrt {7} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {3}{128} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{128} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{128} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{128} \sqrt {\frac {\pi }{7}} \sqrt {b} e^{-\frac {7 a}{b}} \text {erfi}\left (\frac {\sqrt {7} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 d^2 x^2 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

Input:

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

Output:

(-2*d^2*x^2*(1 + c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (4*d^2*( 
-1/16*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]) - ( 
Sqrt[b]*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt 
[b]])/32 - (Sqrt[b]*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh 
[c*x]])/Sqrt[b]])/32 + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqr 
t[b]])/(16*E^(a/b)) + (Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh 
[c*x]])/Sqrt[b]])/(32*E^((3*a)/b)) + (Sqrt[b]*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqr 
t[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*E^((5*a)/b))))/(b^2*c^3) + (14*d^2*(( 
3*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/128 + (S 
qrt[b]*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[ 
b]])/128 - (Sqrt[b]*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh 
[c*x]])/Sqrt[b]])/128 - (Sqrt[b]*E^((7*a)/b)*Sqrt[Pi/7]*Erf[(Sqrt[7]*Sqrt[ 
a + b*ArcSinh[c*x]])/Sqrt[b]])/128 - (3*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*A 
rcSinh[c*x]]/Sqrt[b]])/(128*E^(a/b)) - (Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*S 
qrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(128*E^((3*a)/b)) + (Sqrt[b]*Sqrt[Pi/5] 
*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(128*E^((5*a)/b)) + (Sq 
rt[b]*Sqrt[Pi/7]*Erfi[(Sqrt[7]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(128*E^ 
((7*a)/b))))/(b^2*c^3)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate(x^2*(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)
 

Sympy [F]

\[ \int \frac {x^2 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=d^{2} \left (\int \frac {x^{2}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {2 c^{2} x^{4}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {c^{4} x^{6}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \] Input:

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

Output:

d**2*(Integral(x**2/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a + b*asinh(c*x))*a 
sinh(c*x)), x) + Integral(2*c**2*x**4/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a 
 + b*asinh(c*x))*asinh(c*x)), x) + Integral(c**4*x**6/(a*sqrt(a + b*asinh( 
c*x)) + b*sqrt(a + b*asinh(c*x))*asinh(c*x)), x))
 

Maxima [F]

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

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 \left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=d^{2} \left (\left (\int \frac {\sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x^{6}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}+2 \left (\int \frac {\sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x^{4}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {\sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x^{2}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:

int(x^2*(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**6)/(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**4)/(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**2 
)/(asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b + a**2),x))