\(\int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{5/2}} \, dx\) [231]

Optimal result

Integrand size = 35, antiderivative size = 483 \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{5/2}} \, dx=-\frac {i b d^3 x \sqrt {1+c^2 x^2}}{f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {8 i b d^3 \sqrt {1+c^2 x^2}}{3 c f^2 (i+c x) \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {5 b d^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2}{2 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {20 i d^3 (1+i c x) (a+b \text {arcsinh}(c x))}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i d^3 (1+i c x)^4 (a+b \text {arcsinh}(c x))}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}+\frac {5 i d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {5 d^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {28 b d^3 \sqrt {1+c^2 x^2} \log (i+c x)}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:

-I*b*d^3*x*(c^2*x^2+1)^(1/2)/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+8/3*I 
*b*d^3*(c^2*x^2+1)^(1/2)/c/f^2/(I+c*x)/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2) 
-5/2*b*d^3*(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2/c/f^2/(d+I*c*d*x)^(1/2)/(f-I*c 
*f*x)^(1/2)+20/3*I*d^3*(1+I*c*x)*(a+b*arcsinh(c*x))/c/f^2/(d+I*c*d*x)^(1/2 
)/(f-I*c*f*x)^(1/2)-2/3*I*d^3*(1+I*c*x)^4*(a+b*arcsinh(c*x))/c/f^2/(d+I*c* 
d*x)^(1/2)/(f-I*c*f*x)^(1/2)/(c^2*x^2+1)+5/3*I*d^3*(c^2*x^2+1)*(a+b*arcsin 
h(c*x))/c/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+5*d^3*(c^2*x^2+1)^(1/2)* 
arcsinh(c*x)*(a+b*arcsinh(c*x))/c/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+ 
28/3*b*d^3*(c^2*x^2+1)^(1/2)*ln(I+c*x)/c/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x) 
^(1/2)
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1083\) vs. \(2(483)=966\).

Time = 12.48 (sec) , antiderivative size = 1083, normalized size of antiderivative = 2.24 \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(((4*I)*a*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-23 + (34*I)*c*x + 3*c^ 
2*x^2))/(f^3*(I + c*x)^2) + (60*a*d^(5/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sq 
rt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/f^(5/2) - ((2*I)*b*d^2*Sqrt[d + I*c*d* 
x]*Sqrt[f - I*c*f*x]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(-(Co 
sh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] + (I 
/2)*Log[1 + c^2*x^2])) + Cosh[ArcSinh[c*x]/2]*(4*I + 3*ArcSinh[c*x] - 6*Ar 
cTan[Coth[ArcSinh[c*x]/2]] + ((3*I)/2)*Log[1 + c^2*x^2]) + 2*(2 + (2*I)*Ar 
cSinh[c*x] + (4*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Log[1 + c^2*x^2] + (Sqrt 
[1 + c^2*x^2]*((2*I)*ArcSinh[c*x] + (4*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + L 
og[1 + c^2*x^2]))/2)*Sinh[ArcSinh[c*x]/2]))/(f^3*(1 + I*c*x)*(Cosh[ArcSinh 
[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4) + (2*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f 
- I*c*f*x]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(Cosh[(3*ArcSin 
h[c*x])/2]*((14*I - 3*ArcSinh[c*x])*ArcSinh[c*x] + (28*I)*ArcTan[Tanh[ArcS 
inh[c*x]/2]] - 7*Log[1 + c^2*x^2]) + Cosh[ArcSinh[c*x]/2]*(8 + (6*I)*ArcSi 
nh[c*x] + 9*ArcSinh[c*x]^2 - (84*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 21*Log[ 
1 + c^2*x^2]) - (2*I)*(4 + (4*I)*ArcSinh[c*x] + 6*ArcSinh[c*x]^2 - (56*I)* 
ArcTan[Tanh[ArcSinh[c*x]/2]] + 14*Log[1 + c^2*x^2] + Sqrt[1 + c^2*x^2]*(Ar 
cSinh[c*x]*(14*I + 3*ArcSinh[c*x]) - (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 
 7*Log[1 + c^2*x^2]))*Sinh[ArcSinh[c*x]/2]))/(f^3*(1 + I*c*x)*(Cosh[ArcSin 
h[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4) - (I*b*d^2*Sqrt[d + I*c*d*x]*Sqr...
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.47, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6211, 27, 6252, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6252

\(\displaystyle \frac {d^5 \left (c^2 x^2+1\right )^{5/2} \left (-b c \int \left (-\frac {2 i (i c x+1)^4}{3 c \left (c^2 x^2+1\right )^2}+\frac {20 i (i c x+1)}{3 c \left (c^2 x^2+1\right )}+\frac {5 \text {arcsinh}(c x)}{c \sqrt {c^2 x^2+1}}+\frac {5 i}{3 c}\right )dx-\frac {2 i (1+i c x)^4 (a+b \text {arcsinh}(c x))}{3 c \left (c^2 x^2+1\right )^{3/2}}+\frac {20 i (1+i c x) (a+b \text {arcsinh}(c x))}{3 c \sqrt {c^2 x^2+1}}+\frac {5 i \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c}+\frac {5 \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{c}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^5 \left (c^2 x^2+1\right )^{5/2} \left (-\frac {2 i (1+i c x)^4 (a+b \text {arcsinh}(c x))}{3 c \left (c^2 x^2+1\right )^{3/2}}+\frac {20 i (1+i c x) (a+b \text {arcsinh}(c x))}{3 c \sqrt {c^2 x^2+1}}+\frac {5 i \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c}+\frac {5 \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{c}-b c \left (\frac {5 \text {arcsinh}(c x)^2}{2 c^2}-\frac {8 i}{3 c^2 (c x+i)}-\frac {28 \log (c x+i)}{3 c^2}+\frac {i x}{c}\right )\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

Input:

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

Output:

(d^5*(1 + c^2*x^2)^(5/2)*((((-2*I)/3)*(1 + I*c*x)^4*(a + b*ArcSinh[c*x]))/ 
(c*(1 + c^2*x^2)^(3/2)) + (((20*I)/3)*(1 + I*c*x)*(a + b*ArcSinh[c*x]))/(c 
*Sqrt[1 + c^2*x^2]) + (((5*I)/3)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c 
 + (5*ArcSinh[c*x]*(a + b*ArcSinh[c*x]))/c - b*c*((I*x)/c - ((8*I)/3)/(c^2 
*(I + c*x)) + (5*ArcSinh[c*x]^2)/(2*c^2) - (28*Log[I + c*x])/(3*c^2))))/(( 
d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 6211

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_ 
) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 + c^2*x 
^2)^q)   Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x], 
x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^ 
2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
 

rule 6252

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_) + (g_.)*(x_))^(m_.)*((d_) + ( 
e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p 
, x]}, Simp[(a + b*ArcSinh[c*x])   u, x] - Simp[b*c   Int[1/Sqrt[1 + c^2*x^ 
2]   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ 
[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])
 

Maple [A] (verified)

Time = 10.80 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {d^{2} \left (56 a +16 b +4 i b \,c^{3} x^{3}+10 i b x c +46 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) b +84 i \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}-56 a \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-24 \sqrt {c^{2} x^{2}+1}\, a c x +30 \operatorname {arcsinh}\left (x c \right )^{2} b \,c^{2} x^{2}+60 \,\operatorname {arcsinh}\left (x c \right ) a \,c^{2} x^{2}-112 \,\operatorname {arcsinh}\left (x c \right ) b \,c^{2} x^{2}-24 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) b c x +16 b \,c^{2} x^{2}-56 b \,c^{3} x^{3} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}+224 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) b \,c^{2} x^{2}+15 \operatorname {arcsinh}\left (x c \right )^{2} b \,c^{4} x^{4}+30 \,\operatorname {arcsinh}\left (x c \right ) a \,c^{4} x^{4}-56 \,\operatorname {arcsinh}\left (x c \right ) b \,c^{4} x^{4}-56 b \,\operatorname {arcsinh}\left (x c \right )+112 a \,c^{2} x^{2}+46 i \sqrt {c^{2} x^{2}+1}\, a -6 i b \,c^{5} x^{5}+84 i \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+112 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) b \,c^{4} x^{4}+6 i \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, b \,c^{4} x^{4}+56 a \,c^{4} x^{4}+15 b \operatorname {arcsinh}\left (x c \right )^{2}+30 \,\operatorname {arcsinh}\left (x c \right ) a +6 i \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+112 b \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{6 \sqrt {c^{2} x^{2}+1}\, f^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) c}\)	\(512\)

Input:

int((d+I*c*d*x)^(5/2)*(a+b*arcsinh(x*c))/(f-I*c*f*x)^(5/2),x,method=_RETUR 
NVERBOSE)
 

Output:

1/6*d^2*(56*a+16*b-56*a*c^3*x^3*(c^2*x^2+1)^(1/2)-24*(c^2*x^2+1)^(1/2)*a*c 
*x+30*arcsinh(x*c)^2*b*c^2*x^2+60*arcsinh(x*c)*a*c^2*x^2-112*arcsinh(x*c)* 
b*c^2*x^2-24*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*b*c*x+16*b*c^2*x^2-56*b*c^3*x^ 
3*arcsinh(x*c)*(c^2*x^2+1)^(1/2)+224*ln(x*c+(c^2*x^2+1)^(1/2)+I)*b*c^2*x^2 
+112*ln(x*c+(c^2*x^2+1)^(1/2)+I)*b*c^4*x^4+46*I*(c^2*x^2+1)^(1/2)*arcsinh( 
x*c)*b+15*arcsinh(x*c)^2*b*c^4*x^4+30*arcsinh(x*c)*a*c^4*x^4-56*arcsinh(x* 
c)*b*c^4*x^4-56*b*arcsinh(x*c)+112*a*c^2*x^2+4*I*b*c^3*x^3+10*I*b*x*c-6*I* 
b*c^5*x^5+56*a*c^4*x^4+84*I*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*b*c^2*x^2+6*I*( 
c^2*x^2+1)^(1/2)*arcsinh(x*c)*b*c^4*x^4+15*b*arcsinh(x*c)^2+30*arcsinh(x*c 
)*a+84*I*(c^2*x^2+1)^(1/2)*a*c^2*x^2+6*I*(c^2*x^2+1)^(1/2)*a*c^4*x^4+46*I* 
(c^2*x^2+1)^(1/2)*a+112*b*ln(x*c+(c^2*x^2+1)^(1/2)+I))*(I*(x*c-I)*d)^(1/2) 
*(-I*(I+x*c)*f)^(1/2)/(c^2*x^2+1)^(1/2)/f^3/(c^4*x^4+2*c^2*x^2+1)/c
 

Fricas [F]

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

integrate((d+I*c*d*x)^(5/2)*(a+b*arcsinh(c*x))/(f-I*c*f*x)^(5/2),x, algori 
thm="fricas")
 

Output:

integral(((I*b*c^2*d^2*x^2 + 2*b*c*d^2*x - I*b*d^2)*sqrt(I*c*d*x + d)*sqrt 
(-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1)) + (I*a*c^2*d^2*x^2 + 2*a*c*d^2 
*x - I*a*d^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f))/(c^3*f^3*x^3 + 3*I*c^2 
*f^3*x^2 - 3*c*f^3*x - I*f^3), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((d+I*c*d*x)^(5/2)*(a+b*arcsinh(c*x))/(f-I*c*f*x)^(5/2),x, algori 
thm="maxima")
 

Output:

-1/3*(-3*I*(c^2*d*f*x^2 + d*f)^(5/2)/(c^5*f^5*x^4 + 4*I*c^4*f^5*x^3 - 6*c^ 
3*f^5*x^2 - 4*I*c^2*f^5*x + c*f^5) + 15*I*(c^2*d*f*x^2 + d*f)^(3/2)*d/(3*I 
*c^4*f^4*x^3 - 9*c^3*f^4*x^2 - 9*I*c^2*f^4*x + 3*c*f^4) - 10*I*sqrt(c^2*d* 
f*x^2 + d*f)*d^2/(c^3*f^3*x^2 + 2*I*c^2*f^3*x - c*f^3) - 105*I*sqrt(c^2*d* 
f*x^2 + d*f)*d^2/(-3*I*c^2*f^3*x + 3*c*f^3) - 15*d^3*arcsinh(c*x)/(c*f^3*s 
qrt(d/f)))*a + b*integrate((I*c*d*x + d)^(5/2)*log(c*x + sqrt(c^2*x^2 + 1) 
)/(-I*c*f*x + f)^(5/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d+I*c*d*x)^(5/2)*(a+b*arcsinh(c*x))/(f-I*c*f*x)^(5/2),x, algori 
thm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{5/2}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}}{{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \] Input:

int(((a + b*asinh(c*x))*(d + c*d*x*1i)^(5/2))/(f - c*f*x*1i)^(5/2),x)
 

Output:

int(((a + b*asinh(c*x))*(d + c*d*x*1i)^(5/2))/(f - c*f*x*1i)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{5/2}} \, dx=\frac {\sqrt {d}\, d^{2} \left (-30 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asin} \left (\frac {\sqrt {-c i x +1}}{\sqrt {2}}\right ) a c x -30 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asin} \left (\frac {\sqrt {-c i x +1}}{\sqrt {2}}\right ) a i +3 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {-c i x +1}\, c^{2} x^{2}+2 \sqrt {-c i x +1}\, c i x -\sqrt {-c i x +1}}d x \right ) b \,c^{4} i x -3 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {-c i x +1}\, c^{2} x^{2}+2 \sqrt {-c i x +1}\, c i x -\sqrt {-c i x +1}}d x \right ) b \,c^{3}+6 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right ) x}{\sqrt {-c i x +1}\, c^{2} x^{2}+2 \sqrt {-c i x +1}\, c i x -\sqrt {-c i x +1}}d x \right ) b \,c^{3} x +6 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right ) x}{\sqrt {-c i x +1}\, c^{2} x^{2}+2 \sqrt {-c i x +1}\, c i x -\sqrt {-c i x +1}}d x \right ) b \,c^{2} i -3 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )}{\sqrt {-c i x +1}\, c^{2} x^{2}+2 \sqrt {-c i x +1}\, c i x -\sqrt {-c i x +1}}d x \right ) b \,c^{2} i x +3 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )}{\sqrt {-c i x +1}\, c^{2} x^{2}+2 \sqrt {-c i x +1}\, c i x -\sqrt {-c i x +1}}d x \right ) b c -3 a \,c^{3} x^{3}-31 a \,c^{2} i \,x^{2}-11 a c x -23 a i \right )}{3 \sqrt {f}\, \sqrt {c i x +1}\, \sqrt {-c i x +1}\, c \,f^{2} \left (c i x -1\right )} \] Input:

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

Output:

(sqrt(d)*d**2*( - 30*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asin(sqrt( - c*i*x 
 + 1)/sqrt(2))*a*c*x - 30*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asin(sqrt( - 
c*i*x + 1)/sqrt(2))*a*i + 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int((sqrt(c 
*i*x + 1)*asinh(c*x)*x**2)/(sqrt( - c*i*x + 1)*c**2*x**2 + 2*sqrt( - c*i*x 
 + 1)*c*i*x - sqrt( - c*i*x + 1)),x)*b*c**4*i*x - 3*sqrt(c*i*x + 1)*sqrt( 
- c*i*x + 1)*int((sqrt(c*i*x + 1)*asinh(c*x)*x**2)/(sqrt( - c*i*x + 1)*c** 
2*x**2 + 2*sqrt( - c*i*x + 1)*c*i*x - sqrt( - c*i*x + 1)),x)*b*c**3 + 6*sq 
rt(c*i*x + 1)*sqrt( - c*i*x + 1)*int((sqrt(c*i*x + 1)*asinh(c*x)*x)/(sqrt( 
 - c*i*x + 1)*c**2*x**2 + 2*sqrt( - c*i*x + 1)*c*i*x - sqrt( - c*i*x + 1)) 
,x)*b*c**3*x + 6*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int((sqrt(c*i*x + 1)*a 
sinh(c*x)*x)/(sqrt( - c*i*x + 1)*c**2*x**2 + 2*sqrt( - c*i*x + 1)*c*i*x - 
sqrt( - c*i*x + 1)),x)*b*c**2*i - 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int 
((sqrt(c*i*x + 1)*asinh(c*x))/(sqrt( - c*i*x + 1)*c**2*x**2 + 2*sqrt( - c* 
i*x + 1)*c*i*x - sqrt( - c*i*x + 1)),x)*b*c**2*i*x + 3*sqrt(c*i*x + 1)*sqr 
t( - c*i*x + 1)*int((sqrt(c*i*x + 1)*asinh(c*x))/(sqrt( - c*i*x + 1)*c**2* 
x**2 + 2*sqrt( - c*i*x + 1)*c*i*x - sqrt( - c*i*x + 1)),x)*b*c - 3*a*c**3* 
x**3 - 31*a*c**2*i*x**2 - 11*a*c*x - 23*a*i))/(3*sqrt(f)*sqrt(c*i*x + 1)*s 
qrt( - c*i*x + 1)*c*f**2*(c*i*x - 1))