\(\int \frac {d+c^2 d x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\) [449]

Optimal result

Integrand size = 23, antiderivative size = 228 \[ \int \frac {d+c^2 d x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {3 d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {d e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {3 d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {d e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c} \] Output:

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

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.29 \[ \int \frac {d+c^2 d x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {d e^{-3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \left (-e^{\frac {3 a}{b}}-3 e^{\frac {3 a}{b}+2 \text {arcsinh}(c x)}-3 e^{\frac {3 a}{b}+4 \text {arcsinh}(c x)}-e^{\frac {3 a}{b}+6 \text {arcsinh}(c x)}+3 e^{\frac {4 a}{b}+3 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} e^{3 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+3 e^{\frac {2 a}{b}+3 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}+3 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{4 b c \sqrt {a+b \text {arcsinh}(c x)}} \] Input:

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

Output:

(d*(-E^((3*a)/b) - 3*E^((3*a)/b + 2*ArcSinh[c*x]) - 3*E^((3*a)/b + 4*ArcSi 
nh[c*x]) - E^((3*a)/b + 6*ArcSinh[c*x]) + 3*E^((4*a)/b + 3*ArcSinh[c*x])*S 
qrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + Sqrt[3]*E^(3*ArcS 
inh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c* 
x]))/b] + 3*E^((2*a)/b + 3*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*G 
amma[1/2, -((a + b*ArcSinh[c*x])/b)] + Sqrt[3]*E^((6*a)/b + 3*ArcSinh[c*x] 
)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b]))/(4*b*c 
*E^(3*(a/b + ArcSinh[c*x]))*Sqrt[a + b*ArcSinh[c*x]])
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, 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.

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

\(\Big \downarrow \) 6205

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

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-2*d*(1 + c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (6*d*(-1/8*(Sq 
rt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]) - (Sqrt[b]*E 
^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/8 + 
 (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*E^(a/b)) + ( 
Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(8*E^ 
((3*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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((c^2*d*x^2+d)/(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(d*(3*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*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**5 + 3*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*asin 
h(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) + 3*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 + 3*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)*a*b*c**3 - 6*int((s 
qrt(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)*a*b*c**4 - 6*int((sqrt(c**2*x**2 ...