\(\int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{5/2}} \, dx\) [268]

Optimal result

Integrand size = 37, antiderivative size = 608 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{5/2}} \, dx=\frac {8 d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {32 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {32 b^2 d^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {4 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {8 i b^2 d^2 \sqrt {1+c^2 x^2} \tan \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {8 i d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:

8/3*d^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2/c/f^2/(d+I*c*d*x)^(1/2)/(f- 
I*c*f*x)^(1/2)+1/3*d^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^3/b/c/f^2/(d+I 
*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+32/3*b*d^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh( 
c*x))*ln(1+I/(c*x+(c^2*x^2+1)^(1/2)))/c/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^ 
(1/2)-32/3*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^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+4/3*b*d^2*(c^2*x^2+1)^(1/2)*(a+ 
b*arcsinh(c*x))*sec(1/4*Pi+1/2*I*arcsinh(c*x))^2/c/f^2/(d+I*c*d*x)^(1/2)/( 
f-I*c*f*x)^(1/2)+8/3*I*b^2*d^2*(c^2*x^2+1)^(1/2)*tan(1/4*Pi+1/2*I*arcsinh( 
c*x))/c/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+8/3*I*d^2*(c^2*x^2+1)^(1/2 
)*(a+b*arcsinh(c*x))^2*tan(1/4*Pi+1/2*I*arcsinh(c*x))/c/f^2/(d+I*c*d*x)^(1 
/2)/(f-I*c*f*x)^(1/2)-2/3*I*d^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2*sec 
(1/4*Pi+1/2*I*arcsinh(c*x))^2*tan(1/4*Pi+1/2*I*arcsinh(c*x))/c/f^2/(d+I*c* 
d*x)^(1/2)/(f-I*c*f*x)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1617\) vs. \(2(608)=1216\).

Time = 16.46 (sec) , antiderivative size = 1617, normalized size of antiderivative = 2.66 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{5/2}} \, dx =\text {Too large to display} \] Input:

Integrate[((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(5/2) 
,x]
 

Output:

(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*((((4*I)/3)*a^2*d)/(f^3*(I + 
c*x)^2) - (8*a^2*d)/(3*f^3*(I + c*x))))/c + (a^2*d^(3/2)*Log[c*d*f*x + Sqr 
t[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(c*f^(5/2)) - ( 
(I/3)*a*b*d*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*( 
1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(-(Cosh[(3* 
ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sq 
rt[1 + c^2*x^2]])) + Cosh[ArcSinh[c*x]/2]*(4*I + 3*ArcSinh[c*x] - 6*ArcTan 
[Coth[ArcSinh[c*x]/2]] + (3*I)*Log[Sqrt[1 + c^2*x^2]]) + 2*(Sqrt[1 + c^2*x 
^2]*(I*ArcSinh[c*x] + (2*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Log[Sqrt[1 + c^ 
2*x^2]]) + 2*(1 + I*ArcSinh[c*x] + (2*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Lo 
g[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(c*f^3*(1 + I*c*x)*Sqrt[-((( 
-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x] 
/2])^4) + (a*b*d*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-( 
d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(Cosh[ 
(3*ArcSinh[c*x])/2]*((14*I - 3*ArcSinh[c*x])*ArcSinh[c*x] + (28*I)*ArcTan[ 
Tanh[ArcSinh[c*x]/2]] - 14*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]* 
(8 + (6*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 - (84*I)*ArcTan[Tanh[ArcSinh[c* 
x]/2]] + 42*Log[Sqrt[1 + c^2*x^2]]) - (2*I)*(4 + (4*I)*ArcSinh[c*x] + 6*Ar 
cSinh[c*x]^2 - (56*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 28*Log[Sqrt[1 + c^2*x 
^2]] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(14*I + 3*ArcSinh[c*x]) - (28*I)...
 

Rubi [A] (verified)

Time = 2.06 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.48, 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.

\(\displaystyle \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{5/2}} \, dx\)

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6259

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^4 \left (c^2 x^2+1\right )^{5/2} \left (\frac {(a+b \text {arcsinh}(c x))^3}{3 b c}+\frac {8 (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {32 b \log \left (1+i e^{-\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 c}+\frac {8 i \tan \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {4 b \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) (a+b \text {arcsinh}(c x))}{3 c}-\frac {2 i \tan \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) (a+b \text {arcsinh}(c x))^2}{3 c}-\frac {32 b^2 \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )}{3 c}+\frac {8 i b^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

Input:

Int[((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(5/2),x]
 

Output:

(d^4*(1 + c^2*x^2)^(5/2)*((8*(a + b*ArcSinh[c*x])^2)/(3*c) + (a + b*ArcSin 
h[c*x])^3/(3*b*c) + (32*b*(a + b*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]])/ 
(3*c) - (32*b^2*PolyLog[2, (-I)/E^ArcSinh[c*x]])/(3*c) + (4*b*(a + b*ArcSi 
nh[c*x])*Sec[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/(3*c) + (((8*I)/3)*b^2*Tan[Pi/4 
 + (I/2)*ArcSinh[c*x]])/c + (((8*I)/3)*(a + b*ArcSinh[c*x])^2*Tan[Pi/4 + ( 
I/2)*ArcSinh[c*x]])/c - (((2*I)/3)*(a + b*ArcSinh[c*x])^2*Sec[Pi/4 + (I/2) 
*ArcSinh[c*x]]^2*Tan[Pi/4 + (I/2)*ArcSinh[c*x]])/c))/((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 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 = 6.26 (sec) , antiderivative size = 837, normalized size of antiderivative = 1.38

method	result	size
default	\(\frac {d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{3} \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{3 \sqrt {c^{2} x^{2}+1}\, b \,f^{3} c}-\frac {4 d \left (20 a b +105 \operatorname {arcsinh}\left (x c \right )^{2} b^{2} c^{2} x^{2}+25 a^{2}+25 \operatorname {arcsinh}\left (x c \right )^{2} b^{2}+20 b^{2} \operatorname {arcsinh}\left (x c \right )+30 b^{2}+48 a b \,c^{4} x^{4}+52 a b \,c^{2} x^{2}-24 \sqrt {c^{2} x^{2}+1}\, b^{2} c^{3} x^{3}-20 \sqrt {c^{2} x^{2}+1}\, b^{2} c x +48 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) b^{2} c^{3} x^{3}+10 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) b^{2} c x +48 \,\operatorname {arcsinh}\left (x c \right ) b^{2} c^{4} x^{4}+52 \,\operatorname {arcsinh}\left (x c \right ) b^{2} c^{2} x^{2}+48 \sqrt {c^{2} x^{2}+1}\, a b \,c^{3} x^{3}+10 \sqrt {c^{2} x^{2}+1}\, a b c x +210 \,\operatorname {arcsinh}\left (x c \right ) a b \,c^{2} x^{2}+50 \,\operatorname {arcsinh}\left (x c \right ) a b -36 i \sqrt {c^{2} x^{2}+1}\, a b \,c^{2} x^{2}-36 i \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, b^{2} c^{2} x^{2}+42 i b^{2} c^{3} x^{3}+10 i b^{2} c x -10 i \sqrt {c^{2} x^{2}+1}\, a b -10 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) b^{2}+120 b^{2} c^{4} x^{4}+118 b^{2} c^{2} x^{2}+288 \,\operatorname {arcsinh}\left (x c \right ) a b \,c^{4} x^{4}+6 i \sqrt {c^{2} x^{2}+1}\, b^{2} c^{2} x^{2}+36 i a b \,c^{3} x^{3}+36 i \operatorname {arcsinh}\left (x c \right ) b^{2} c^{3} x^{3}+144 \operatorname {arcsinh}\left (x c \right )^{2} b^{2} c^{4} x^{4}+105 a^{2} c^{2} x^{2}+20 i a b c x +20 i \operatorname {arcsinh}\left (x c \right ) b^{2} c x +144 a^{2} c^{4} x^{4}+10 i \sqrt {c^{2} x^{2}+1}\, b^{2}\right ) \left (2 x^{3} c^{3}-3 i c^{2} x^{2}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}-i-2 \sqrt {c^{2} x^{2}+1}\right ) \sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}}{3 f^{3} \left (144 c^{4} x^{4}+105 c^{2} x^{2}+25\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {16 d \left (b \operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) b -2 a \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )+2 a \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) b \right ) b \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{3 \sqrt {c^{2} x^{2}+1}\, f^{3} c}\)	\(837\)

Input:

int((d+I*c*d*x)^(3/2)*(a+b*arcsinh(x*c))^2/(f-I*c*f*x)^(5/2),x,method=_RET 
URNVERBOSE)
 

Output:

1/3*d*(a+b*arcsinh(x*c))^3*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/(c^2*x 
^2+1)^(1/2)/b/f^3/c-4/3*d*(20*a*b+105*arcsinh(x*c)^2*b^2*c^2*x^2+25*a^2+25 
*arcsinh(x*c)^2*b^2+20*b^2*arcsinh(x*c)+30*b^2+48*a*b*c^4*x^4+52*a*b*c^2*x 
^2-24*(c^2*x^2+1)^(1/2)*b^2*c^3*x^3-20*(c^2*x^2+1)^(1/2)*b^2*c*x-10*I*(c^2 
*x^2+1)^(1/2)*arcsinh(x*c)*b^2+48*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*b^2*c^3*x 
^3+10*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*b^2*c*x+48*arcsinh(x*c)*b^2*c^4*x^4+5 
2*arcsinh(x*c)*b^2*c^2*x^2-10*I*(c^2*x^2+1)^(1/2)*a*b+6*I*(c^2*x^2+1)^(1/2 
)*b^2*c^2*x^2+10*I*(c^2*x^2+1)^(1/2)*b^2+42*I*b^2*c^3*x^3+10*I*b^2*c*x-36* 
I*(c^2*x^2+1)^(1/2)*a*b*c^2*x^2+48*(c^2*x^2+1)^(1/2)*a*b*c^3*x^3+10*(c^2*x 
^2+1)^(1/2)*a*b*c*x+210*arcsinh(x*c)*a*b*c^2*x^2+20*I*a*b*c*x+36*I*a*b*c^3 
*x^3+50*arcsinh(x*c)*a*b+120*b^2*c^4*x^4+118*b^2*c^2*x^2+288*arcsinh(x*c)* 
a*b*c^4*x^4+144*arcsinh(x*c)^2*b^2*c^4*x^4+105*a^2*c^2*x^2-36*I*arcsinh(x* 
c)*(c^2*x^2+1)^(1/2)*b^2*c^2*x^2+20*I*arcsinh(x*c)*b^2*c*x+36*I*arcsinh(x* 
c)*b^2*c^3*x^3+144*a^2*c^4*x^4)*(2*x^3*c^3-3*I*c^2*x^2-2*x^2*c^2*(c^2*x^2+ 
1)^(1/2)-I-2*(c^2*x^2+1)^(1/2))*(-I*(I+x*c)*f)^(1/2)*(I*(x*c-I)*d)^(1/2)/f 
^3/(144*c^4*x^4+105*c^2*x^2+25)/(c^2*x^2+1)^2/c-16/3*d*(b*arcsinh(x*c)^2-2 
*arcsinh(x*c)*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))*b-2*a*ln(x*c+(c^2*x^2+1)^(1/ 
2)+I)+2*a*ln(x*c+(c^2*x^2+1)^(1/2))-2*polylog(2,I*(x*c+(c^2*x^2+1)^(1/2))) 
*b)*b*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/(c^2*x^2+1)^(1/2)/f^3/c
 

Fricas [F]

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

integrate((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))^2/(f-I*c*f*x)^(5/2),x, algo 
rithm="fricas")
 

Output:

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

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

integrate((d+I*c*d*x)**(3/2)*(a+b*asinh(c*x))**2/(f-I*c*f*x)**(5/2),x)
 

Output:

Integral((I*d*(c*x - I))**(3/2)*(a + b*asinh(c*x))**2/(-I*f*(c*x + I))**(5 
/2), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))^2/(f-I*c*f*x)^(5/2),x, algo 
rithm="maxima")
 

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))^2/(f-I*c*f*x)^(5/2),x, algo 
rithm="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. \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}}{{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \] Input:

int(((a + b*asinh(c*x))^2*(d + c*d*x*1i)^(3/2))/(f - c*f*x*1i)^(5/2),x)
 

Output:

int(((a + b*asinh(c*x))^2*(d + c*d*x*1i)^(3/2))/(f - c*f*x*1i)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{5/2}} \, dx =\text {Too large to display} \] Input:

int((d+I*c*d*x)^(3/2)*(a+b*asinh(c*x))^2/(f-I*c*f*x)^(5/2),x)
 

Output:

(sqrt(d)*d*( - 6*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asin(sqrt( - c*i*x + 1 
)/sqrt(2))*a**2*c*x - 6*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asin(sqrt( - c* 
i*x + 1)/sqrt(2))*a**2*i + 6*sqrt(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)*a*b*c**3*x + 6*sqrt(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)*a*b*c**2*i - 6*sqr 
t(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) 
*a*b*c**2*i*x + 6*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 - s 
qrt( - c*i*x + 1)),x)*a*b*c + 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int((sq 
rt(c*i*x + 1)*asinh(c*x)**2*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**2*c**3*x + 3*sqrt(c*i*x + 1)*s 
qrt( - c*i*x + 1)*int((sqrt(c*i*x + 1)*asinh(c*x)**2*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**2*c** 
2*i - 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int((sqrt(c*i*x + 1)*asinh(c*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**2*c**2*i*x + 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int((s 
qrt(c*i*x + 1)*asinh(c*x)**2)/(sqrt( - c*i*x + 1)*c**2*x**2 + 2*sqrt( -...