\(\int \frac {(d+c^2 d x^2)^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\) [73]

Optimal result

Integrand size = 25, antiderivative size = 415 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {15 d^2 \sqrt {d+c^2 d x^2} \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c \sqrt {1+c^2 x^2}}-\frac {3 d^2 \sqrt {d+c^2 d x^2} \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{4 b^2 c \sqrt {1+c^2 x^2}}-\frac {3 d^2 \sqrt {d+c^2 d x^2} \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c \sqrt {1+c^2 x^2}}+\frac {15 d^2 \sqrt {d+c^2 d x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c \sqrt {1+c^2 x^2}}+\frac {3 d^2 \sqrt {d+c^2 d x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c \sqrt {1+c^2 x^2}}+\frac {3 d^2 \sqrt {d+c^2 d x^2} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c \sqrt {1+c^2 x^2}} \] Output:

-d^2*(c^2*x^2+1)^(5/2)*(c^2*d*x^2+d)^(1/2)/b/c/(a+b*arcsinh(c*x))-15/16*d^ 
2*(c^2*d*x^2+d)^(1/2)*Chi(2*(a+b*arcsinh(c*x))/b)*sinh(2*a/b)/b^2/c/(c^2*x 
^2+1)^(1/2)-3/4*d^2*(c^2*d*x^2+d)^(1/2)*Chi(4*(a+b*arcsinh(c*x))/b)*sinh(4 
*a/b)/b^2/c/(c^2*x^2+1)^(1/2)-3/16*d^2*(c^2*d*x^2+d)^(1/2)*Chi(6*(a+b*arcs 
inh(c*x))/b)*sinh(6*a/b)/b^2/c/(c^2*x^2+1)^(1/2)+15/16*d^2*(c^2*d*x^2+d)^( 
1/2)*cosh(2*a/b)*Shi(2*(a+b*arcsinh(c*x))/b)/b^2/c/(c^2*x^2+1)^(1/2)+3/4*d 
^2*(c^2*d*x^2+d)^(1/2)*cosh(4*a/b)*Shi(4*(a+b*arcsinh(c*x))/b)/b^2/c/(c^2* 
x^2+1)^(1/2)+3/16*d^2*(c^2*d*x^2+d)^(1/2)*cosh(6*a/b)*Shi(6*(a+b*arcsinh(c 
*x))/b)/b^2/c/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.82 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {d^2 \sqrt {d+c^2 d x^2} \left (16 b+48 b c^2 x^2+48 b c^4 x^4+16 b c^6 x^6+15 (a+b \text {arcsinh}(c x)) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+12 (a+b \text {arcsinh}(c x)) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+3 a \text {Chi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+3 b \text {arcsinh}(c x) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )-15 a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-15 b \text {arcsinh}(c x) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-12 a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-12 b \text {arcsinh}(c x) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-3 a \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-3 b \text {arcsinh}(c x) \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )\right )}{16 b^2 c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \] Input:

Integrate[(d + c^2*d*x^2)^(5/2)/(a + b*ArcSinh[c*x])^2,x]
 

Output:

-1/16*(d^2*Sqrt[d + c^2*d*x^2]*(16*b + 48*b*c^2*x^2 + 48*b*c^4*x^4 + 16*b* 
c^6*x^6 + 15*(a + b*ArcSinh[c*x])*CoshIntegral[2*(a/b + ArcSinh[c*x])]*Sin 
h[(2*a)/b] + 12*(a + b*ArcSinh[c*x])*CoshIntegral[4*(a/b + ArcSinh[c*x])]* 
Sinh[(4*a)/b] + 3*a*CoshIntegral[6*(a/b + ArcSinh[c*x])]*Sinh[(6*a)/b] + 3 
*b*ArcSinh[c*x]*CoshIntegral[6*(a/b + ArcSinh[c*x])]*Sinh[(6*a)/b] - 15*a* 
Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - 15*b*ArcSinh[c*x]*Cos 
h[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - 12*a*Cosh[(4*a)/b]*SinhI 
ntegral[4*(a/b + ArcSinh[c*x])] - 12*b*ArcSinh[c*x]*Cosh[(4*a)/b]*SinhInte 
gral[4*(a/b + ArcSinh[c*x])] - 3*a*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + Arc 
Sinh[c*x])] - 3*b*ArcSinh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh 
[c*x])]))/(b^2*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.57, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6205, 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.

\(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\)

\(\Big \downarrow \) 6205

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

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

Input:

Int[(d + c^2*d*x^2)^(5/2)/(a + b*ArcSinh[c*x])^2,x]
 

Output:

-((Sqrt[1 + c^2*x^2]*(d + c^2*d*x^2)^(5/2))/(b*c*(a + b*ArcSinh[c*x]))) + 
(6*d^2*Sqrt[d + c^2*d*x^2]*((-5*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]*S 
inh[(2*a)/b])/32 - (CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b]*Sinh[(4*a)/b] 
)/8 - (CoshIntegral[(6*(a + b*ArcSinh[c*x]))/b]*Sinh[(6*a)/b])/32 + (5*Cos 
h[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/32 + (Cosh[(4*a)/b]*S 
inhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/8 + (Cosh[(6*a)/b]*SinhIntegral[( 
6*(a + b*ArcSinh[c*x]))/b])/32))/(b^2*c*Sqrt[1 + c^2*x^2])
 

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 6205

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[Simp[Sqrt[1 + c^2*x^2]*(d + e*x^2)^p]*((a + b*ArcSinh[c*x] 
)^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Simp[(d + e*x 
^2)^p/(1 + c^2*x^2)^p]   Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x]) 
^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && LtQ[n, 
 -1]
 

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 [A] (verified)

Time = 1.73 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.47

method	result	size
default	\(-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (32 b \,c^{7} x^{7}+32 \sqrt {c^{2} x^{2}+1}\, b \,c^{6} x^{6}+96 b \,c^{5} x^{5}+96 \sqrt {c^{2} x^{2}+1}\, b \,c^{4} x^{4}+96 b \,c^{3} x^{3}+96 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}+12 \,\operatorname {arcsinh}\left (x c \right ) b \,\operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arcsinh}\left (x c \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arcsinh}\left (x c \right )+4 a}{b}}+3 \,\operatorname {arcsinh}\left (x c \right ) b \,\operatorname {expIntegral}_{1}\left (-6 \,\operatorname {arcsinh}\left (x c \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arcsinh}\left (x c \right )+6 a}{b}}+15 \,\operatorname {arcsinh}\left (x c \right ) b \,\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arcsinh}\left (x c \right )+2 a}{b}}-12 \,{\mathrm e}^{\frac {b \,\operatorname {arcsinh}\left (x c \right )+4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arcsinh}\left (x c \right )+\frac {4 a}{b}\right ) b \,\operatorname {arcsinh}\left (x c \right )-15 \,\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arcsinh}\left (x c \right )+2 a}{b}} b \,\operatorname {arcsinh}\left (x c \right )-3 \,\operatorname {expIntegral}_{1}\left (6 \,\operatorname {arcsinh}\left (x c \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arcsinh}\left (x c \right )+6 a}{b}} b \,\operatorname {arcsinh}\left (x c \right )+12 a \,\operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arcsinh}\left (x c \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arcsinh}\left (x c \right )+4 a}{b}}+3 a \,\operatorname {expIntegral}_{1}\left (-6 \,\operatorname {arcsinh}\left (x c \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arcsinh}\left (x c \right )+6 a}{b}}+15 a \,\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arcsinh}\left (x c \right )+2 a}{b}}-12 \,{\mathrm e}^{\frac {b \,\operatorname {arcsinh}\left (x c \right )+4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arcsinh}\left (x c \right )+\frac {4 a}{b}\right ) a -15 \,\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arcsinh}\left (x c \right )+2 a}{b}} a -3 \,\operatorname {expIntegral}_{1}\left (6 \,\operatorname {arcsinh}\left (x c \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arcsinh}\left (x c \right )+6 a}{b}} a +32 b x c +32 \sqrt {c^{2} x^{2}+1}\, b \right ) d^{2}}{32 \left (c^{2} x^{2}+1\right ) c \,b^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}\)	\(609\)

Input:

int((c^2*d*x^2+d)^(5/2)/(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/32*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*x*c+1)*(32*b*c^7*x^ 
7+32*(c^2*x^2+1)^(1/2)*b*c^6*x^6+96*b*c^5*x^5+96*(c^2*x^2+1)^(1/2)*b*c^4*x 
^4+96*b*c^3*x^3+96*(c^2*x^2+1)^(1/2)*b*c^2*x^2+12*arcsinh(x*c)*b*Ei(1,-4*a 
rcsinh(x*c)-4*a/b)*exp(-(-b*arcsinh(x*c)+4*a)/b)+3*arcsinh(x*c)*b*Ei(1,-6* 
arcsinh(x*c)-6*a/b)*exp(-(-b*arcsinh(x*c)+6*a)/b)+15*arcsinh(x*c)*b*Ei(1,- 
2*arcsinh(x*c)-2*a/b)*exp(-(-b*arcsinh(x*c)+2*a)/b)-12*exp((b*arcsinh(x*c) 
+4*a)/b)*Ei(1,4*arcsinh(x*c)+4*a/b)*b*arcsinh(x*c)-15*Ei(1,2*arcsinh(x*c)+ 
2*a/b)*exp((b*arcsinh(x*c)+2*a)/b)*b*arcsinh(x*c)-3*Ei(1,6*arcsinh(x*c)+6* 
a/b)*exp((b*arcsinh(x*c)+6*a)/b)*b*arcsinh(x*c)+12*a*Ei(1,-4*arcsinh(x*c)- 
4*a/b)*exp(-(-b*arcsinh(x*c)+4*a)/b)+3*a*Ei(1,-6*arcsinh(x*c)-6*a/b)*exp(- 
(-b*arcsinh(x*c)+6*a)/b)+15*a*Ei(1,-2*arcsinh(x*c)-2*a/b)*exp(-(-b*arcsinh 
(x*c)+2*a)/b)-12*exp((b*arcsinh(x*c)+4*a)/b)*Ei(1,4*arcsinh(x*c)+4*a/b)*a- 
15*Ei(1,2*arcsinh(x*c)+2*a/b)*exp((b*arcsinh(x*c)+2*a)/b)*a-3*Ei(1,6*arcsi 
nh(x*c)+6*a/b)*exp((b*arcsinh(x*c)+6*a)/b)*a+32*b*x*c+32*(c^2*x^2+1)^(1/2) 
*b)*d^2/(c^2*x^2+1)/c/b^2/(a+b*arcsinh(x*c))
 

Fricas [F]

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

integrate((c^2*d*x^2+d)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
 

Output:

integral((c^4*d^2*x^4 + 2*c^2*d^2*x^2 + d^2)*sqrt(c^2*d*x^2 + d)/(b^2*arcs 
inh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2), x)
 

Sympy [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \] Input:

integrate((c**2*d*x**2+d)**(5/2)/(a+b*asinh(c*x))**2,x)
 

Output:

Integral((d*(c**2*x**2 + 1))**(5/2)/(a + b*asinh(c*x))**2, x)
 

Maxima [F]

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

integrate((c^2*d*x^2+d)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
 

Output:

-((c^6*d^(5/2)*x^6 + 3*c^4*d^(5/2)*x^4 + 3*c^2*d^(5/2)*x^2 + d^(5/2))*(c^2 
*x^2 + 1) + (c^7*d^(5/2)*x^7 + 3*c^5*d^(5/2)*x^5 + 3*c^3*d^(5/2)*x^3 + c*d 
^(5/2)*x)*sqrt(c^2*x^2 + 1))/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1)*a*b*c^2*x + 
a*b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log(c*x + sqrt 
(c^2*x^2 + 1))) + integrate(((6*c^6*d^(5/2)*x^6 + 11*c^4*d^(5/2)*x^4 + 4*c 
^2*d^(5/2)*x^2 - d^(5/2))*(c^2*x^2 + 1)^(3/2) + 6*(2*c^7*d^(5/2)*x^7 + 5*c 
^5*d^(5/2)*x^5 + 4*c^3*d^(5/2)*x^3 + c*d^(5/2)*x)*(c^2*x^2 + 1) + (6*c^8*d 
^(5/2)*x^8 + 19*c^6*d^(5/2)*x^6 + 21*c^4*d^(5/2)*x^4 + 9*c^2*d^(5/2)*x^2 + 
 d^(5/2))*sqrt(c^2*x^2 + 1))/(a*b*c^4*x^4 + (c^2*x^2 + 1)*a*b*c^2*x^2 + 2* 
a*b*c^2*x^2 + a*b + (b^2*c^4*x^4 + (c^2*x^2 + 1)*b^2*c^2*x^2 + 2*b^2*c^2*x 
^2 + b^2 + 2*(b^2*c^3*x^3 + b^2*c*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2 
*x^2 + 1)) + 2*(a*b*c^3*x^3 + a*b*c*x)*sqrt(c^2*x^2 + 1)), x)
 

Giac [F]

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

integrate((c^2*d*x^2+d)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="giac")
 

Output:

integrate((c^2*d*x^2 + d)^(5/2)/(b*arcsinh(c*x) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {{\left (d\,c^2\,x^2+d\right )}^{5/2}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \] Input:

int((d + c^2*d*x^2)^(5/2)/(a + b*asinh(c*x))^2,x)
 

Output:

int((d + c^2*d*x^2)^(5/2)/(a + b*asinh(c*x))^2, x)
 

Reduce [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\sqrt {d}\, d^{2} \left (\int \frac {\sqrt {c^{2} x^{2}+1}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x +\left (\int \frac {\sqrt {c^{2} x^{2}+1}\, x^{4}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}+2 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, x^{2}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}\right ) \] Input:

int((c^2*d*x^2+d)^(5/2)/(a+b*asinh(c*x))^2,x)
 

Output:

sqrt(d)*d**2*(int(sqrt(c**2*x**2 + 1)/(asinh(c*x)**2*b**2 + 2*asinh(c*x)*a 
*b + a**2),x) + int((sqrt(c**2*x**2 + 1)*x**4)/(asinh(c*x)**2*b**2 + 2*asi 
nh(c*x)*a*b + a**2),x)*c**4 + 2*int((sqrt(c**2*x**2 + 1)*x**2)/(asinh(c*x) 
**2*b**2 + 2*asinh(c*x)*a*b + a**2),x)*c**2)