\(\int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx\) [248]

Optimal result

Integrand size = 37, antiderivative size = 604 \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=-\frac {8 f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {8 i b^2 f^2 \sqrt {1+c^2 x^2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {8 i f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {4 b f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {32 b f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )}{3 c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {32 b^2 f^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:

-8/3*f^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2/c/d^2/(d+I*c*d*x)^(1/2)/(f 
-I*c*f*x)^(1/2)+1/3*f^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^3/b/c/d^2/(d+ 
I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-8/3*I*b^2*f^2*(c^2*x^2+1)^(1/2)*cot(1/4*P 
i+1/2*I*arcsinh(c*x))/c/d^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-8/3*I*f^2* 
(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2*cot(1/4*Pi+1/2*I*arcsinh(c*x))/c/d^ 
2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+4/3*b*f^2*(c^2*x^2+1)^(1/2)*(a+b*arc 
sinh(c*x))*csc(1/4*Pi+1/2*I*arcsinh(c*x))^2/c/d^2/(d+I*c*d*x)^(1/2)/(f-I*c 
*f*x)^(1/2)+2/3*I*f^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2*cot(1/4*Pi+1/ 
2*I*arcsinh(c*x))*csc(1/4*Pi+1/2*I*arcsinh(c*x))^2/c/d^2/(d+I*c*d*x)^(1/2) 
/(f-I*c*f*x)^(1/2)+32/3*b*f^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/d^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+32/3*b^ 
2*f^2*(c^2*x^2+1)^(1/2)*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/c/d^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. \(1609\) vs. \(2(604)=1208\).

Time = 15.78 (sec) , antiderivative size = 1609, normalized size of antiderivative = 2.66 \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*((((-4*I)/3)*a^2*f)/(d^3*(-I 
+ c*x)^2) - (8*a^2*f)/(3*d^3*(-I + c*x))))/c + (a^2*f^(3/2)*Log[c*d*f*x + 
Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(c*d^(5/2)) 
+ ((I/3)*a*b*f*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])*((-I)*Co 
sh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] - I* 
Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]*(4 + (3*I)*ArcSinh[c*x] - ( 
6*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 3*Log[Sqrt[1 + c^2*x^2]]) + 2*(Sqrt[1 
+ c^2*x^2]*(ArcSinh[c*x] + 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + 
 c^2*x^2]]) + 2*(I + ArcSinh[c*x] + 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log 
[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(c*d^3*(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*f*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 + (3*I)*ArcSinh[c*x])*ArcSinh[c*x] - 28*ArcTan[Tanh 
[ArcSinh[c*x]/2]] + (14*I)*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]* 
(84*ArcTan[Tanh[ArcSinh[c*x]/2]] - I*(8 - (6*I)*ArcSinh[c*x] + 9*ArcSinh[c 
*x]^2 + 42*Log[Sqrt[1 + c^2*x^2]])) + 2*(4 - (4*I)*ArcSinh[c*x] + 6*ArcSin 
h[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)*Ar...
 

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 287, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6259

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(f^4*(1 + c^2*x^2)^(5/2)*((-8*(a + b*ArcSinh[c*x])^2)/(3*c) + (a + b*ArcSi 
nh[c*x])^3/(3*b*c) - (((8*I)/3)*b^2*Cot[Pi/4 + (I/2)*ArcSinh[c*x]])/c - (( 
(8*I)/3)*(a + b*ArcSinh[c*x])^2*Cot[Pi/4 + (I/2)*ArcSinh[c*x]])/c + (4*b*( 
a + b*ArcSinh[c*x])*Csc[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/(3*c) + (((2*I)/3)*( 
a + b*ArcSinh[c*x])^2*Cot[Pi/4 + (I/2)*ArcSinh[c*x]]*Csc[Pi/4 + (I/2)*ArcS 
inh[c*x]]^2)/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)))/((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 = 4.16 (sec) , antiderivative size = 837, normalized size of antiderivative = 1.39

method	result	size
default	\(\frac {f \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{3} \sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}}{3 \sqrt {c^{2} x^{2}+1}\, b c \,d^{3}}-\frac {4 f \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}+144 \operatorname {arcsinh}\left (x c \right )^{2} b^{2} c^{4} x^{4}+105 a^{2} c^{2} x^{2}-6 i \sqrt {c^{2} x^{2}+1}\, b^{2} c^{2} x^{2}-20 i a b c x -36 i a b \,c^{3} x^{3}-20 i \operatorname {arcsinh}\left (x c \right ) b^{2} c x -36 i \operatorname {arcsinh}\left (x c \right ) b^{2} c^{3} x^{3}+144 a^{2} c^{4} x^{4}-10 i \sqrt {c^{2} x^{2}+1}\, b^{2}\right ) \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+3 i c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}+i\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{3 d^{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 f \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 ) f}\, \sqrt {i \left (x c -i\right ) d}}{3 \sqrt {c^{2} x^{2}+1}\, c \,d^{3}}\)	\(837\)

Input:

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

Output:

1/3*f*(a+b*arcsinh(x*c))^3*(-I*(I+x*c)*f)^(1/2)*(I*(x*c-I)*d)^(1/2)/(c^2*x 
^2+1)^(1/2)/b/c/d^3-4/3*f*(20*a*b+105*arcsinh(x*c)^2*b^2*c^2*x^2-6*I*(c^2* 
x^2+1)^(1/2)*b^2*c^2*x^2-20*I*a*b*c*x-36*I*a*b*c^3*x^3+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+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*ar 
csinh(x*c)*b^2*c^4*x^4+52*arcsinh(x*c)*b^2*c^2*x^2+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+ 
36*I*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*b^2*c^2*x^2+210*arcsinh(x*c)*a*b*c^2*x 
^2+50*arcsinh(x*c)*a*b-20*I*arcsinh(x*c)*b^2*c*x-42*I*b^2*c^3*x^3-10*I*b^2 
*c*x+10*I*(c^2*x^2+1)^(1/2)*a*b-10*I*(c^2*x^2+1)^(1/2)*b^2+120*b^2*c^4*x^4 
+118*b^2*c^2*x^2-36*I*arcsinh(x*c)*b^2*c^3*x^3+288*arcsinh(x*c)*a*b*c^4*x^ 
4+144*arcsinh(x*c)^2*b^2*c^4*x^4+10*I*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*b^2+1 
05*a^2*c^2*x^2+144*a^2*c^4*x^4)*(2*x^3*c^3-2*x^2*c^2*(c^2*x^2+1)^(1/2)+3*I 
*c^2*x^2-2*(c^2*x^2+1)^(1/2)+I)*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/d 
^3/(144*c^4*x^4+105*c^2*x^2+25)/(c^2*x^2+1)^2/c-16/3*f*(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*(I+x*c)*f)^(1/2)*(I*(x*c-I)*d)^(1/2)/(c^2*x^2+1)^(1/2)/c/d^3
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(f)*f*( - 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*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)*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))/(sq 
rt(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*sqrt(c*i*x + 1)*sqrt( - 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**3*x + 3*sqrt(c*i*x + 1)*sqrt( - 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((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*...