\(\int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx\) [262]

Optimal result

Integrand size = 37, antiderivative size = 719 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=-\frac {2 i b^2 d^2 \left (1+c^2 x^2\right )}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 i d^2 (a+b \text {arcsinh}(c x))^2}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {4 d^2 x (a+b \text {arcsinh}(c x))^2}{f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {4 d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{b c f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {16 i b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {8 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {8 b^2 d^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {8 b^2 d^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 b^2 d^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:

-2*I*b^2*d^2*(c^2*x^2+1)/c/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+2*I*b*d^2 
*x*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1 
/2)-4*I*d^2*(a+b*arcsinh(c*x))^2/c/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+4 
*d^2*x*(a+b*arcsinh(c*x))^2/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+4*d^2*(c 
^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2/c/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/ 
2)-I*d^2*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/c/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x 
)^(1/2)-d^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^3/b/c/f/(d+I*c*d*x)^(1/2) 
/(f-I*c*f*x)^(1/2)+16*I*b*d^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))*arctan( 
c*x+(c^2*x^2+1)^(1/2))/c/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-8*b*d^2*(c^ 
2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)/c/f/(d+I 
*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+8*b^2*d^2*(c^2*x^2+1)^(1/2)*polylog(2,-I*( 
c*x+(c^2*x^2+1)^(1/2)))/c/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-8*b^2*d^2* 
(c^2*x^2+1)^(1/2)*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/c/f/(d+I*c*d*x)^(1/ 
2)/(f-I*c*f*x)^(1/2)-4*b^2*d^2*(c^2*x^2+1)^(1/2)*polylog(2,-(c*x+(c^2*x^2+ 
1)^(1/2))^2)/c/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 16.39 (sec) , antiderivative size = 1084, normalized size of antiderivative = 1.51 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

((3*a^2*d*(5 - I*c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(f^2*(I + c*x)) 
 - (9*a^2*d^(3/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - 
 I*c*f*x]])/f^(3/2) - (b^2*d*(-I + c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x 
]*(-18*Pi*ArcSinh[c*x] - (6 - 6*I)*ArcSinh[c*x]^2 + I*ArcSinh[c*x]^3 - 12* 
(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcS 
inh[c*x]] + 12*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] - 24*Pi*Log[Cosh[ 
ArcSinh[c*x]/2]] - (24*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - ((12*I)*ArcSin 
h[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x] 
/2])))/(f^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/ 
2])^2) - (I*b^2*d*(-I + c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(((-6*I)* 
c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ((6 + 6*I)*ArcSinh[c*x]^2)/Sqrt[1 + 
c^2*x^2] + (2*ArcSinh[c*x]^3)/Sqrt[1 + c^2*x^2] + (3*I)*(2 + ArcSinh[c*x]^ 
2) + ((6*I)*(2*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + Pi*(3 
*ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] - 2*Log[-Cos[(Pi + (2*I)*ArcSinh 
[c*x])/4]] + 4*Log[Cosh[ArcSinh[c*x]/2]]) + (4*I)*PolyLog[2, (-I)/E^ArcSin 
h[c*x]]))/Sqrt[1 + c^2*x^2] - (12*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Sq 
rt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]))))/(f^2*(C 
osh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) + (3*a*b*d*Sqrt[d + I*c*d 
*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - I*Sinh[Arc 
Sinh[c*x]/2])) + 4*ArcSinh[c*x]*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSi...
 

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.44, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6211, 27, 6259, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6259

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^{3/2} \left (\frac {16 i b \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c}-\frac {i \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c}+\frac {4 x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {4 i (a+b \text {arcsinh}(c x))^2}{c \sqrt {c^2 x^2+1}}-\frac {(a+b \text {arcsinh}(c x))^3}{b c}+\frac {4 (a+b \text {arcsinh}(c x))^2}{c}-\frac {8 b \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{c}+2 i a b x+\frac {8 b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c}-\frac {8 b^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c}-\frac {4 b^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c}+2 i b^2 x \text {arcsinh}(c x)-\frac {2 i b^2 \sqrt {c^2 x^2+1}}{c}\right )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\)

Input:

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

Output:

(d^3*(1 + c^2*x^2)^(3/2)*((2*I)*a*b*x - ((2*I)*b^2*Sqrt[1 + c^2*x^2])/c + 
(2*I)*b^2*x*ArcSinh[c*x] + (4*(a + b*ArcSinh[c*x])^2)/c - ((4*I)*(a + b*Ar 
cSinh[c*x])^2)/(c*Sqrt[1 + c^2*x^2]) + (4*x*(a + b*ArcSinh[c*x])^2)/Sqrt[1 
 + c^2*x^2] - (I*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/c - (a + b*ArcS 
inh[c*x])^3/(b*c) + ((16*I)*b*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]]) 
/c - (8*b*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/c + (8*b^2*Pol 
yLog[2, (-I)*E^ArcSinh[c*x]])/c - (8*b^2*PolyLog[2, I*E^ArcSinh[c*x]])/c - 
 (4*b^2*PolyLog[2, -E^(2*ArcSinh[c*x])])/c))/((d + I*c*d*x)^(3/2)*(f - I*c 
*f*x)^(3/2))
 

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 6259

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

Maple [A] (verified)

Time = 7.32 (sec) , antiderivative size = 832, normalized size of antiderivative = 1.16

method	result	size
default	\(-\frac {i d \left (4 i \sqrt {c^{2} x^{2}+1}\, a^{2} b c x -8 i \operatorname {arcsinh}\left (x c \right ) a \,b^{2} c^{2} x^{2}-8 i \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) a \,b^{2} c^{2} x^{2}+4 i \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, b^{3} c x -2 a \,b^{2} c x +\sqrt {c^{2} x^{2}+1}\, a^{2} b \,c^{2} x^{2}-16 \arctan \left (x c +\sqrt {c^{2} x^{2}+1}\right ) a \,b^{2} c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, b^{3} c^{2} x^{2}+10 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) a \,b^{2}-2 \,\operatorname {arcsinh}\left (x c \right ) b^{3} c x -2 \,\operatorname {arcsinh}\left (x c \right ) b^{3} c^{3} x^{3}+2 \sqrt {c^{2} x^{2}+1}\, b^{3} c^{2} x^{2}-2 a \,b^{2} c^{3} x^{3}-3 i a \,b^{2} \operatorname {arcsinh}\left (x c \right )^{2}-4 i a^{2} b +5 \sqrt {c^{2} x^{2}+1}\, a^{2} b +2 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, a \,b^{2} c^{2} x^{2}-16 \arctan \left (x c +\sqrt {c^{2} x^{2}+1}\right ) a \,b^{2}+2 \sqrt {c^{2} x^{2}+1}\, b^{3}-4 i a^{2} b \,c^{2} x^{2}+4 i \operatorname {arcsinh}\left (x c \right )^{2} b^{3} c^{2} x^{2}-16 i \operatorname {polylog}\left (2, i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) b^{3} c^{2} x^{2}-i a^{3} c^{2} x^{2}-i a^{3}-i b^{3} \operatorname {arcsinh}\left (x c \right )^{3}+4 i b^{3} \operatorname {arcsinh}\left (x c \right )^{2}-16 i \operatorname {polylog}\left (2, i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) b^{3}+8 i \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, a \,b^{2} c x -16 i \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) \operatorname {arcsinh}\left (x c \right ) b^{3} c^{2} x^{2}-3 i \operatorname {arcsinh}\left (x c \right )^{2} a \,b^{2} c^{2} x^{2}+16 i \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right ) a \,b^{2} c^{2} x^{2}-3 i \operatorname {arcsinh}\left (x c \right ) a^{2} b \,c^{2} x^{2}-16 i \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) b^{3} \operatorname {arcsinh}\left (x c \right )+16 i \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right ) a \,b^{2}-8 i \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) a \,b^{2}-3 i a^{2} b \,\operatorname {arcsinh}\left (x c \right )-8 i a \,b^{2} \operatorname {arcsinh}\left (x c \right )+5 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2} b^{3}-i \operatorname {arcsinh}\left (x c \right )^{3} b^{3} c^{2} x^{2}\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \sqrt {c^{2} x^{2}+1}}{b \,f^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) c}\)	\(832\)

Input:

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

Output:

-I*d*(8*I*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*a*b^2*c*x-16*I*ln(1-I*(x*c+(c^2*x 
^2+1)^(1/2)))*b^3*arcsinh(x*c)-3*I*a*b^2*arcsinh(x*c)^2+16*I*ln(x*c+(c^2*x 
^2+1)^(1/2))*a*b^2-8*I*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*a*b^2-3*I*a^2*b*arc 
sinh(x*c)-8*I*a*b^2*arcsinh(x*c)-16*I*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))*arcs 
inh(x*c)*b^3*c^2*x^2-3*I*arcsinh(x*c)^2*a*b^2*c^2*x^2+16*I*ln(x*c+(c^2*x^2 
+1)^(1/2))*a*b^2*c^2*x^2-8*I*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*a*b^2*c^2*x^2 
+4*I*(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*b^3*c*x-3*I*arcsinh(x*c)*a^2*b*c^2*x 
^2-8*I*arcsinh(x*c)*a*b^2*c^2*x^2-2*a*b^2*c*x+(c^2*x^2+1)^(1/2)*a^2*b*c^2* 
x^2-16*arctan(x*c+(c^2*x^2+1)^(1/2))*a*b^2*c^2*x^2+arcsinh(x*c)^2*(c^2*x^2 
+1)^(1/2)*b^3*c^2*x^2-4*I*a^2*b*c^2*x^2+10*(c^2*x^2+1)^(1/2)*arcsinh(x*c)* 
a*b^2-2*arcsinh(x*c)*b^3*c*x-2*arcsinh(x*c)*b^3*c^3*x^3+2*(c^2*x^2+1)^(1/2 
)*b^3*c^2*x^2-I*b^3*arcsinh(x*c)^3+4*I*b^3*arcsinh(x*c)^2-16*I*polylog(2,I 
*(x*c+(c^2*x^2+1)^(1/2)))*b^3-2*a*b^2*c^3*x^3-I*a^3*c^2*x^2+4*I*(c^2*x^2+1 
)^(1/2)*a^2*b*c*x+5*(c^2*x^2+1)^(1/2)*a^2*b+2*arcsinh(x*c)*(c^2*x^2+1)^(1/ 
2)*a*b^2*c^2*x^2-16*arctan(x*c+(c^2*x^2+1)^(1/2))*a*b^2+2*(c^2*x^2+1)^(1/2 
)*b^3-4*I*a^2*b-I*arcsinh(x*c)^3*b^3*c^2*x^2+4*I*arcsinh(x*c)^2*b^3*c^2*x^ 
2-16*I*polylog(2,I*(x*c+(c^2*x^2+1)^(1/2)))*b^3*c^2*x^2-I*a^3+5*(c^2*x^2+1 
)^(1/2)*arcsinh(x*c)^2*b^3)*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(c^2* 
x^2+1)^(1/2)/b/f^2/(c^4*x^4+2*c^2*x^2+1)/c
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

a^2*(-I*(c^2*d*f*x^2 + d*f)^(3/2)/(c^3*f^3*x^2 + 2*I*c^2*f^3*x - c*f^3) - 
6*I*sqrt(c^2*d*f*x^2 + d*f)*d/(-I*c^2*f^2*x + c*f^2) - 3*d^2*arcsinh(c*x)/ 
(c*f^2*sqrt(d/f))) + integrate((I*c*d*x + d)^(3/2)*b^2*log(c*x + sqrt(c^2* 
x^2 + 1))^2/(-I*c*f*x + f)^(3/2) + 2*(I*c*d*x + d)^(3/2)*a*b*log(c*x + sqr 
t(c^2*x^2 + 1))/(-I*c*f*x + f)^(3/2), x)
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio 
n over extensionUnable to transpose Error: Bad Argument ValueDegree mismat 
ch inside
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*d*( - 6*sqrt( - c*i*x + 1)*asin(sqrt( - c*i*x + 1)/sqrt(2))*a**2* 
i - 2*sqrt( - c*i*x + 1)*int((sqrt(c*i*x + 1)*asinh(c*x)*x)/(sqrt( - c*i*x 
 + 1)*c*i*x - sqrt( - c*i*x + 1)),x)*a*b*c**2*i - 2*sqrt( - c*i*x + 1)*int 
((sqrt(c*i*x + 1)*asinh(c*x))/(sqrt( - c*i*x + 1)*c*i*x - sqrt( - c*i*x + 
1)),x)*a*b*c - sqrt( - c*i*x + 1)*int((sqrt(c*i*x + 1)*asinh(c*x)**2*x)/(s 
qrt( - c*i*x + 1)*c*i*x - sqrt( - c*i*x + 1)),x)*b**2*c**2*i - sqrt( - c*i 
*x + 1)*int((sqrt(c*i*x + 1)*asinh(c*x)**2)/(sqrt( - c*i*x + 1)*c*i*x - sq 
rt( - c*i*x + 1)),x)*b**2*c - sqrt(c*i*x + 1)*a**2*c*x - 5*sqrt(c*i*x + 1) 
*a**2*i))/(sqrt(f)*sqrt( - c*i*x + 1)*c*f)