\(\int \frac {(d+c^2 d x^2)^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\) [456]

Optimal result

Integrand size = 25, antiderivative size = 346 \[ \int \frac {\left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {5 d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 d^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}-\frac {d^2 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}+\frac {5 d^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {d^2 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c} \] Output:

-2*d^2*(c^2*x^2+1)^(5/2)/b/c/(a+b*arcsinh(c*x))^(1/2)-5/8*d^2*exp(a/b)*Pi^ 
(1/2)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c-5/16*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-1 
/16*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+5/8*d^2*Pi^(1/2)*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))/ 
b^(3/2)/c/exp(a/b)+5/16*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/exp(3*a/b)+1/16*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/exp(5*a/b)
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.27 \[ \int \frac {\left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {d^2 e^{-5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \left (-e^{\frac {5 a}{b}}-5 e^{\frac {5 a}{b}+2 \text {arcsinh}(c x)}-10 e^{\frac {5 a}{b}+4 \text {arcsinh}(c x)}-10 e^{\frac {5 a}{b}+6 \text {arcsinh}(c x)}-5 e^{\frac {5 a}{b}+8 \text {arcsinh}(c x)}-e^{\frac {5 a}{b}+10 \text {arcsinh}(c x)}+10 e^{\frac {6 a}{b}+5 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {5} e^{5 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+5 \sqrt {3} e^{\frac {2 a}{b}+5 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+10 e^{\frac {4 a}{b}+5 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+5 \sqrt {3} e^{\frac {8 a}{b}+5 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\sqrt {5} e^{5 \left (\frac {2 a}{b}+\text {arcsinh}(c x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{16 b c \sqrt {a+b \text {arcsinh}(c x)}} \] Input:

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

Output:

(d^2*(-E^((5*a)/b) - 5*E^((5*a)/b + 2*ArcSinh[c*x]) - 10*E^((5*a)/b + 4*Ar 
cSinh[c*x]) - 10*E^((5*a)/b + 6*ArcSinh[c*x]) - 5*E^((5*a)/b + 8*ArcSinh[c 
*x]) - E^((5*a)/b + 10*ArcSinh[c*x]) + 10*E^((6*a)/b + 5*ArcSinh[c*x])*Sqr 
t[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + Sqrt[5]*E^(5*ArcSin 
h[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-5*(a + b*ArcSinh[c*x] 
))/b] + 5*Sqrt[3]*E^((2*a)/b + 5*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x]) 
/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] + 10*E^((4*a)/b + 5*ArcSinh[c 
*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] 
 + 5*Sqrt[3]*E^((8*a)/b + 5*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1 
/2, (3*(a + b*ArcSinh[c*x]))/b] + Sqrt[5]*E^(5*((2*a)/b + ArcSinh[c*x]))*S 
qrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (5*(a + b*ArcSinh[c*x]))/b]))/(16*b*c*E 
^(5*(a/b + ArcSinh[c*x]))*Sqrt[a + b*ArcSinh[c*x]])
 

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.94, 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.

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

\(\Big \downarrow \) 6205

\(\displaystyle \frac {10 c d^2 \int \frac {x \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 d^2 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {10 d^2 \int -\frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c}-\frac {2 d^2 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {10 d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c}-\frac {2 d^2 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {10 d^2 \int \left (\frac {\sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c}-\frac {2 d^2 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {10 d^2 \left (-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c}-\frac {2 d^2 \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

Input:

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

Output:

(-2*d^2*(1 + c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (10*d^2*(-1/ 
16*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]) - (Sqr 
t[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]]/Sqrt[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]*Sqrt[a 
 + b*ArcSinh[c*x]])/Sqrt[b]])/(32*E^((5*a)/b))))/(b^2*c)
 

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 [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((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]

\[ \int \frac {\left (d+c^2 d x^2\right )^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=d^{2} \left (\int \frac {2 c^{2} x^{2}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {c^{4} x^{4}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {1}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \] Input:

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

Output:

d**2*(Integral(2*c**2*x**2/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a + b*asinh( 
c*x))*asinh(c*x)), x) + Integral(c**4*x**4/(a*sqrt(a + b*asinh(c*x)) + b*s 
qrt(a + b*asinh(c*x))*asinh(c*x)), x) + Integral(1/(a*sqrt(a + b*asinh(c*x 
)) + b*sqrt(a + b*asinh(c*x))*asinh(c*x)), x))
 

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(d**2*(2*asinh(c*x)*int((sqrt(asinh(c*x)*b + a)*x**6)/(asinh(c*x)**2*b**2* 
c**2*x**2 + asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b*c**2*x**2 + 2*asinh(c*x) 
*a*b + a**2*c**2*x**2 + a**2),x)*b**2*c**7 + 7*asinh(c*x)*int((sqrt(asinh( 
c*x)*b + a)*x**4)/(asinh(c*x)**2*b**2*c**2*x**2 + asinh(c*x)**2*b**2 + 2*a 
sinh(c*x)*a*b*c**2*x**2 + 2*asinh(c*x)*a*b + a**2*c**2*x**2 + a**2),x)*b** 
2*c**5 + 5*asinh(c*x)*int((sqrt(asinh(c*x)*b + a)*x**2)/(asinh(c*x)**2*b** 
2*c**2*x**2 + asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b*c**2*x**2 + 2*asinh(c* 
x)*a*b + a**2*c**2*x**2 + a**2),x)*b**2*c**3 - 6*asinh(c*x)*int((sqrt(c**2 
*x**2 + 1)*sqrt(asinh(c*x)*b + a)*asinh(c*x)*x**3)/(asinh(c*x)**2*b**2*c** 
2*x**2 + asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b*c**2*x**2 + 2*asinh(c*x)*a* 
b + a**2*c**2*x**2 + a**2),x)*b**2*c**4 - 6*asinh(c*x)*int((sqrt(c**2*x**2 
 + 1)*sqrt(asinh(c*x)*b + a)*x**3)/(asinh(c*x)**2*b**2*c**2*x**2 + asinh(c 
*x)**2*b**2 + 2*asinh(c*x)*a*b*c**2*x**2 + 2*asinh(c*x)*a*b + a**2*c**2*x* 
*2 + a**2),x)*a*b*c**4 + 2*sqrt(c**2*x**2 + 1)*sqrt(asinh(c*x)*b + a)*c**2 
*x**2 - 4*sqrt(c**2*x**2 + 1)*sqrt(asinh(c*x)*b + a) + 2*int((sqrt(asinh(c 
*x)*b + a)*x**6)/(asinh(c*x)**2*b**2*c**2*x**2 + asinh(c*x)**2*b**2 + 2*as 
inh(c*x)*a*b*c**2*x**2 + 2*asinh(c*x)*a*b + a**2*c**2*x**2 + a**2),x)*a*b* 
c**7 + 7*int((sqrt(asinh(c*x)*b + a)*x**4)/(asinh(c*x)**2*b**2*c**2*x**2 + 
 asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b*c**2*x**2 + 2*asinh(c*x)*a*b + a**2 
*c**2*x**2 + a**2),x)*a*b*c**5 + 5*int((sqrt(asinh(c*x)*b + a)*x**2)/(a...