\(\int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx\) [202]

Optimal result

Integrand size = 35, antiderivative size = 304 \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx=-\frac {i b d x \sqrt {d+i c d x} \sqrt {f-i c f x}}{3 \sqrt {1+c^2 x^2}}-\frac {b c d x^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{4 \sqrt {1+c^2 x^2}}-\frac {i b c^2 d x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 \sqrt {1+c^2 x^2}}+\frac {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))+\frac {i d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c}+\frac {d \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}} \] Output:

-1/3*I*b*d*x*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)-1/4*b*c 
*d*x^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)-1/9*I*b*c^2*d 
*x^3*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)+1/2*d*x*(d+I*c* 
d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))+1/3*I*d*(d+I*c*d*x)^(1/2)* 
(f-I*c*f*x)^(1/2)*(c^2*x^2+1)*(a+b*arcsinh(c*x))/c+1/4*d*(d+I*c*d*x)^(1/2) 
*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))^2/b/c/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 2.56 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.90 \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx=\frac {12 a d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (2 i+3 c x+2 i c^2 x^2\right )+36 a d^{3/2} \sqrt {f} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+\frac {9 b d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (2 \text {arcsinh}(c x)^2-\cosh (2 \text {arcsinh}(c x))+2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))\right )}{\sqrt {1+c^2 x^2}}-\frac {2 i b d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (9 c x-3 \text {arcsinh}(c x) \left (3 \sqrt {1+c^2 x^2}+\cosh (3 \text {arcsinh}(c x))\right )+\sinh (3 \text {arcsinh}(c x))\right )}{\sqrt {1+c^2 x^2}}}{72 c} \] Input:

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

Output:

(12*a*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(2*I + 3*c*x + (2*I)*c^2*x^2) 
+ 36*a*d^(3/2)*Sqrt[f]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqr 
t[f - I*c*f*x]] + (9*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(2*ArcSinh[c* 
x]^2 - Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]]))/Sqrt[1 
 + c^2*x^2] - ((2*I)*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(9*c*x - 3*Ar 
cSinh[c*x]*(3*Sqrt[1 + c^2*x^2] + Cosh[3*ArcSinh[c*x]]) + Sinh[3*ArcSinh[c 
*x]]))/Sqrt[1 + c^2*x^2])/(72*c)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.49, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6211, 27, 6253, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6253

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {i \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c}+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{9} i b c^2 x^3-\frac {1}{4} b c x^2-\frac {i b x}{3}\right )}{\sqrt {c^2 x^2+1}}\)

Input:

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

Output:

(d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*((-1/3*I)*b*x - (b*c*x^2)/4 - (I/9) 
*b*c^2*x^3 + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/2 + ((I/3)*(1 + c^ 
2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/c + (a + b*ArcSinh[c*x])^2/(4*b*c)))/Sq 
rt[1 + c^2*x^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 6253

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d 
_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a 
+ b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] 
 && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n 
, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] 
&& LtQ[p, -2]))
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (245 ) = 490\).

Time = 4.88 (sec) , antiderivative size = 760, normalized size of antiderivative = 2.50

method	result	size
default	\(\frac {i a \left (i c d x +d \right )^{\frac {3}{2}} \left (-i c f x +f \right )^{\frac {3}{2}}}{3 c f}+\frac {i a d \sqrt {i c d x +d}\, \left (-i c f x +f \right )^{\frac {3}{2}}}{2 c f}-\frac {i a d \sqrt {-i c f x +f}\, \sqrt {i c d x +d}}{2 c}+\frac {a \,d^{2} f \sqrt {\left (-i c f x +f \right ) \left (i c d x +d \right )}\, \ln \left (\frac {c^{2} d f x}{\sqrt {c^{2} d f}}+\sqrt {c^{2} d f \,x^{2}+d f}\right )}{2 \sqrt {-i c f x +f}\, \sqrt {i c d x +d}\, \sqrt {c^{2} d f}}+b \left (\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \operatorname {arcsinh}\left (x c \right )^{2} d}{4 \sqrt {c^{2} x^{2}+1}\, c}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{72 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right ) d}{8 \left (c^{2} x^{2}+1\right ) c}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right ) d}{8 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{72 \left (c^{2} x^{2}+1\right ) c}\right )\)	\(760\)
parts	\(\frac {i a \left (i c d x +d \right )^{\frac {3}{2}} \left (-i c f x +f \right )^{\frac {3}{2}}}{3 c f}+\frac {i a d \sqrt {i c d x +d}\, \left (-i c f x +f \right )^{\frac {3}{2}}}{2 c f}-\frac {i a d \sqrt {-i c f x +f}\, \sqrt {i c d x +d}}{2 c}+\frac {a \,d^{2} f \sqrt {\left (-i c f x +f \right ) \left (i c d x +d \right )}\, \ln \left (\frac {c^{2} d f x}{\sqrt {c^{2} d f}}+\sqrt {c^{2} d f \,x^{2}+d f}\right )}{2 \sqrt {-i c f x +f}\, \sqrt {i c d x +d}\, \sqrt {c^{2} d f}}+b \left (\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \operatorname {arcsinh}\left (x c \right )^{2} d}{4 \sqrt {c^{2} x^{2}+1}\, c}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{72 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right ) d}{8 \left (c^{2} x^{2}+1\right ) c}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right ) d}{8 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{72 \left (c^{2} x^{2}+1\right ) c}\right )\)	\(760\)

Input:

int((d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(x*c)),x,method=_RETUR 
NVERBOSE)
 

Output:

1/3*I*a/c/f*(d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(3/2)+1/2*I*a*d/c/f*(d+I*c*d*x)^ 
(1/2)*(f-I*c*f*x)^(3/2)-1/2*I*a*d/c*(f-I*c*f*x)^(1/2)*(d+I*c*d*x)^(1/2)+1/ 
2*a*d^2*f*((f-I*c*f*x)*(d+I*c*d*x))^(1/2)/(f-I*c*f*x)^(1/2)/(d+I*c*d*x)^(1 
/2)*ln(c^2*d*f*x/(c^2*d*f)^(1/2)+(c^2*d*f*x^2+d*f)^(1/2))/(c^2*d*f)^(1/2)+ 
b*(1/4*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/(c^2*x^2+1)^(1/2)/c*arcsin 
h(x*c)^2*d+1/72*I*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(4*c^4*x^4+4*(c 
^2*x^2+1)^(1/2)*c^3*x^3+5*c^2*x^2+3*(c^2*x^2+1)^(1/2)*x*c+1)*(-1+3*arcsinh 
(x*c))*d/(c^2*x^2+1)/c+1/16*(-I*(I+x*c)*f)^(1/2)*(I*(x*c-I)*d)^(1/2)*(2*x^ 
3*c^3+2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c+(c^2*x^2+1)^(1/2))*(-1+2*arcsinh(x 
*c))*d/(c^2*x^2+1)/c+1/8*I*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(c^2*x 
^2+(c^2*x^2+1)^(1/2)*x*c+1)*(arcsinh(x*c)-1)*d/(c^2*x^2+1)/c+1/8*I*(I*(x*c 
-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*x*c+1)*(arcsi 
nh(x*c)+1)*d/(c^2*x^2+1)/c+1/16*(-I*(I+x*c)*f)^(1/2)*(I*(x*c-I)*d)^(1/2)*( 
2*x^3*c^3-2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c-(c^2*x^2+1)^(1/2))*(1+2*arcsin 
h(x*c))*d/(c^2*x^2+1)/c+1/72*I*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(4 
*c^4*x^4-4*(c^2*x^2+1)^(1/2)*c^3*x^3+5*c^2*x^2-3*(c^2*x^2+1)^(1/2)*x*c+1)* 
(1+3*arcsinh(x*c))*d/(c^2*x^2+1)/c)
 

Fricas [F]

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

integrate((d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x)),x, algori 
thm="fricas")
 

Output:

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

Sympy [F]

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

integrate((d+I*c*d*x)**(3/2)*(f-I*c*f*x)**(1/2)*(a+b*asinh(c*x)),x)
 

Output:

Integral((I*d*(c*x - I))**(3/2)*sqrt(-I*f*(c*x + I))*(a + b*asinh(c*x)), x 
)
 

Maxima [F(-2)]

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

integrate((d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x)),x, algori 
thm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F(-2)]

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

integrate((d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x)),x, algori 
thm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad 
Argument TypeError: Bad Argument TypeDone
 

Mupad [F(-1)]

Timed out. \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}} \,d x \] Input:

int((a + b*asinh(c*x))*(d + c*d*x*1i)^(3/2)*(f - c*f*x*1i)^(1/2),x)
 

Output:

int((a + b*asinh(c*x))*(d + c*d*x*1i)^(3/2)*(f - c*f*x*1i)^(1/2), x)
 

Reduce [F]

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

int((d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(1/2)*(a+b*asinh(c*x)),x)
                                                                                    
                                                                                    
 

Output:

(sqrt(f)*sqrt(d)*d*(6*asin(sqrt( - c*i*x + 1)/sqrt(2))*a*i + 2*sqrt(c*i*x 
+ 1)*sqrt( - c*i*x + 1)*a*c**2*i*x**2 + 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 
1)*a*c*x + 2*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*a*i + 6*int(sqrt(c*i*x + 1 
)*sqrt( - c*i*x + 1)*asinh(c*x)*x,x)*b*c**2*i + 6*int(sqrt(c*i*x + 1)*sqrt 
( - c*i*x + 1)*asinh(c*x),x)*b*c))/(6*c)