\(\int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx\) [261]

Optimal result

Integrand size = 37, antiderivative size = 947 \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=-\frac {8 i b^2 d^3 \left (1+c^2 x^2\right )}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b^2 d^3 x \left (1+c^2 x^2\right )}{4 f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b^2 d^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {8 i b d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b c d^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {8 i d^3 (a+b \text {arcsinh}(c x))^2}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {8 d^3 x (a+b \text {arcsinh}(c x))^2}{f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {8 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 i d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {5 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{2 b c f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {32 i b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {16 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {16 b^2 d^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {16 b^2 d^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {8 b^2 d^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:

8*I*b*d^3*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)+1/4*b^2*d^3*x*(c^2*x^2+1)/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/ 
2)-1/4*b^2*d^3*(c^2*x^2+1)^(1/2)*arcsinh(c*x)/c/f/(d+I*c*d*x)^(1/2)/(f-I*c 
*f*x)^(1/2)-8*I*d^3*(a+b*arcsinh(c*x))^2/c/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x) 
^(1/2)-1/2*b*c*d^3*x^2*(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)-8*I*b^2*d^3*(c^2*x^2+1)/c/f/(d+I*c*d*x)^(1/2)/(f-I 
*c*f*x)^(1/2)+8*d^3*x*(a+b*arcsinh(c*x))^2/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x) 
^(1/2)+8*d^3*(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)-4*I*d^3*(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)+1/2*d^3*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/f/(d+I 
*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-5/2*d^3*(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)+32*I*b*d^3*(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)-16*b*d^3*(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)+16*b^2*d^3*(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)-16*b^2*d^3*(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^3*(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 [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2143\) vs. \(2(947)=1894\).

Time = 24.23 (sec) , antiderivative size = 2143, normalized size of antiderivative = 2.26 \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(((-4*I)*a^2*d^2)/f^2 + (a^2* 
c*d^2*x)/(2*f^2) + (8*a^2*d^2)/(f^2*(I + c*x))))/c - (15*a^2*d^(5/2)*Log[c 
*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(2* 
c*f^(3/2)) - ((4*I)*a*b*d^2*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]*(-(c*x) + 2*ArcSinh[c 
*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - I*ArcSinh[c*x]^2 + 4*ArcTan[Coth[Ar 
cSinh[c*x]/2]] - (2*I)*Log[Sqrt[1 + c^2*x^2]]) - ((-I)*c*x - (2*I)*ArcSinh 
[c*x] + I*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]^2 + (4*I)*ArcTan[C 
oth[ArcSinh[c*x]/2]] + 2*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2]))/(c 
*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSi 
nh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) - (a*b*d^2*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]* 
(8*ArcTan[Tanh[ArcSinh[c*x]/2]] + I*(ArcSinh[c*x]*(4*I + ArcSinh[c*x]) + 4 
*Log[Sqrt[1 + c^2*x^2]])) + (ArcSinh[c*x]*(-4*I + ArcSinh[c*x]) - (8*I)*Ar 
cTan[Tanh[ArcSinh[c*x]/2]] + 4*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2 
]))/(c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(I*Co 
sh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])) - (b^2*d^2*(-I + c*x)*Sqrt[I*( 
(-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(-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^ArcSinh[...
 

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 395, normalized size of antiderivative = 0.42, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6259

\(\displaystyle \frac {d^4 \left (c^2 x^2+1\right )^{3/2} \int \left (\frac {c^2 x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {4 i c x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {7 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {8 i (i-c x) (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{3/2}}\right )dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^4 \left (c^2 x^2+1\right )^{3/2} \left (\frac {32 i b \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c}+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-\frac {4 i \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c}+\frac {8 x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {8 i (a+b \text {arcsinh}(c x))^2}{c \sqrt {c^2 x^2+1}}-\frac {1}{2} b c x^2 (a+b \text {arcsinh}(c x))-\frac {5 (a+b \text {arcsinh}(c x))^3}{2 b c}+\frac {8 (a+b \text {arcsinh}(c x))^2}{c}-\frac {16 b \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{c}+8 i a b x+\frac {16 b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c}-\frac {16 b^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c}-\frac {8 b^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c}+8 i b^2 x \text {arcsinh}(c x)-\frac {b^2 \text {arcsinh}(c x)}{4 c}+\frac {1}{4} b^2 x \sqrt {c^2 x^2+1}-\frac {8 i b^2 \sqrt {c^2 x^2+1}}{c}\right )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\)

Input:

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

Output:

(d^4*(1 + c^2*x^2)^(3/2)*((8*I)*a*b*x - ((8*I)*b^2*Sqrt[1 + c^2*x^2])/c + 
(b^2*x*Sqrt[1 + c^2*x^2])/4 - (b^2*ArcSinh[c*x])/(4*c) + (8*I)*b^2*x*ArcSi 
nh[c*x] - (b*c*x^2*(a + b*ArcSinh[c*x]))/2 + (8*(a + b*ArcSinh[c*x])^2)/c 
- ((8*I)*(a + b*ArcSinh[c*x])^2)/(c*Sqrt[1 + c^2*x^2]) + (8*x*(a + b*ArcSi 
nh[c*x])^2)/Sqrt[1 + c^2*x^2] - ((4*I)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c* 
x])^2)/c + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/2 - (5*(a + b*ArcS 
inh[c*x])^3)/(2*b*c) + ((32*I)*b*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x 
]])/c - (16*b*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/c + (16*b^ 
2*PolyLog[2, (-I)*E^ArcSinh[c*x]])/c - (16*b^2*PolyLog[2, I*E^ArcSinh[c*x] 
])/c - (8*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 = 5.91 (sec) , antiderivative size = 953, normalized size of antiderivative = 1.01

method	result	size
default	\(-\frac {d^{2} \left (32 a^{2} b +a \,b^{2}+10 a^{3}-128 \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right ) a \,b^{2}+32 a^{2} b \,c^{2} x^{2}+3 a \,b^{2} c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, b^{3} c x +30 \,\operatorname {arcsinh}\left (x c \right ) a^{2} b \,c^{2} x^{2}-128 \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right ) a \,b^{2} c^{2} x^{2}-34 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2} b^{3} c x +64 \,\operatorname {arcsinh}\left (x c \right ) a \,b^{2} c^{2} x^{2}-2 \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, b^{3} c^{3} x^{3}+30 \operatorname {arcsinh}\left (x c \right )^{2} a \,b^{2} c^{2} x^{2}+30 a \,b^{2} \operatorname {arcsinh}\left (x c \right )^{2}-68 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) a \,b^{2} c x -4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, a \,b^{2} c^{3} x^{3}+2 a \,b^{2} c^{4} x^{4}+16 i \sqrt {c^{2} x^{2}+1}\, a^{2} b \,c^{2} x^{2}+16 i \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, b^{3} c^{2} x^{2}-34 \sqrt {c^{2} x^{2}+1}\, a^{2} b c x +10 b^{3} \operatorname {arcsinh}\left (x c \right )^{3}-32 b^{3} \operatorname {arcsinh}\left (x c \right )^{2}+b^{3} \operatorname {arcsinh}\left (x c \right )+48 i \sqrt {c^{2} x^{2}+1}\, a^{2} b +48 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2} b^{3}+30 a^{2} b \,\operatorname {arcsinh}\left (x c \right )+64 a \,b^{2} \operatorname {arcsinh}\left (x c \right )-\sqrt {c^{2} x^{2}+1}\, b^{3} c^{3} x^{3}-32 \operatorname {arcsinh}\left (x c \right )^{2} b^{3} c^{2} x^{2}+3 \,\operatorname {arcsinh}\left (x c \right ) b^{3} c^{2} x^{2}+2 \,\operatorname {arcsinh}\left (x c \right ) b^{3} c^{4} x^{4}+10 \operatorname {arcsinh}\left (x c \right )^{3} b^{3} c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, a^{2} b \,c^{3} x^{3}+10 a^{3} c^{2} x^{2}+32 i \sqrt {c^{2} x^{2}+1}\, b^{3} c^{2} x^{2}-32 i a \,b^{2} c^{3} x^{3}-32 i a \,b^{2} c x -32 i \operatorname {arcsinh}\left (x c \right ) b^{3} c^{3} x^{3}+96 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) a \,b^{2}-32 i \operatorname {arcsinh}\left (x c \right ) b^{3} c x +128 \operatorname {polylog}\left (2, i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) b^{3} c^{2} x^{2}+128 \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) b^{3} \operatorname {arcsinh}\left (x c \right )+128 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) a \,b^{2}+128 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) b^{3} c^{2} x^{2}+128 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) a \,b^{2} c^{2} x^{2}+128 \operatorname {polylog}\left (2, i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) b^{3}+32 i \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, a \,b^{2} c^{2} x^{2}+32 i \sqrt {c^{2} x^{2}+1}\, b^{3}\right ) \sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}\, \sqrt {c^{2} x^{2}+1}}{4 b \,f^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) c}\)	\(953\)

Input:

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

Output:

-1/4*d^2*(32*a^2*b+a*b^2+10*a^3-128*ln(x*c+(c^2*x^2+1)^(1/2))*a*b^2+32*a^2 
*b*c^2*x^2+3*a*b^2*c^2*x^2-(c^2*x^2+1)^(1/2)*b^3*c*x+128*ln(1-I*(x*c+(c^2* 
x^2+1)^(1/2)))*b^3*arcsinh(x*c)+128*ln(x*c+(c^2*x^2+1)^(1/2)+I)*a*b^2+30*a 
rcsinh(x*c)*a^2*b*c^2*x^2+128*polylog(2,I*(x*c+(c^2*x^2+1)^(1/2)))*b^3*c^2 
*x^2+48*I*(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*b^3+32*I*(c^2*x^2+1)^(1/2)*b^3* 
c^2*x^2-32*I*a*b^2*c^3*x^3-32*I*a*b^2*c*x-128*ln(x*c+(c^2*x^2+1)^(1/2))*a* 
b^2*c^2*x^2-34*(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*b^3*c*x+64*arcsinh(x*c)*a* 
b^2*c^2*x^2-2*arcsinh(x*c)^2*(c^2*x^2+1)^(1/2)*b^3*c^3*x^3+30*arcsinh(x*c) 
^2*a*b^2*c^2*x^2+30*a*b^2*arcsinh(x*c)^2-68*(c^2*x^2+1)^(1/2)*arcsinh(x*c) 
*a*b^2*c*x-4*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*a*b^2*c^3*x^3+2*a*b^2*c^4*x^4+ 
128*polylog(2,I*(x*c+(c^2*x^2+1)^(1/2)))*b^3-34*(c^2*x^2+1)^(1/2)*a^2*b*c* 
x+16*I*(c^2*x^2+1)^(1/2)*a^2*b*c^2*x^2+10*b^3*arcsinh(x*c)^3-32*b^3*arcsin 
h(x*c)^2+b^3*arcsinh(x*c)+30*a^2*b*arcsinh(x*c)+64*a*b^2*arcsinh(x*c)+32*I 
*(c^2*x^2+1)^(1/2)*b^3-(c^2*x^2+1)^(1/2)*b^3*c^3*x^3-32*arcsinh(x*c)^2*b^3 
*c^2*x^2+3*arcsinh(x*c)*b^3*c^2*x^2+2*arcsinh(x*c)*b^3*c^4*x^4+10*arcsinh( 
x*c)^3*b^3*c^2*x^2-2*(c^2*x^2+1)^(1/2)*a^2*b*c^3*x^3+48*I*(c^2*x^2+1)^(1/2 
)*a^2*b+16*I*(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*b^3*c^2*x^2+10*a^3*c^2*x^2+9 
6*I*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*a*b^2-32*I*arcsinh(x*c)*b^3*c^3*x^3-32* 
I*arcsinh(x*c)*b^3*c*x+128*arcsinh(x*c)*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))*b^ 
3*c^2*x^2+128*ln(x*c+(c^2*x^2+1)^(1/2)+I)*a*b^2*c^2*x^2+32*I*(c^2*x^2+1...
 

Fricas [F]

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

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

Output:

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

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/2*(c^2*d^3*x^3/(sqrt(c^2*d*f*x^2 + d*f)*f) - 8*I*c*d^3*x^2/(sqrt(c^2*d*f 
*x^2 + d*f)*f) + 17*d^3*x/(sqrt(c^2*d*f*x^2 + d*f)*f) - 15*d^3*arcsinh(c*x 
)/(sqrt(d*f)*c*f) - 24*I*d^3/(sqrt(c^2*d*f*x^2 + d*f)*c*f))*a^2 + integrat 
e((I*c*d*x + d)^(5/2)*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)^(5/2)*a*b*log(c*x + sqrt(c^2*x^2 + 1))/(-I*c*f*x + f 
)^(3/2), x)
 

Giac [F(-2)]

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

integrate((d+I*c*d*x)^(5/2)*(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:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\frac {\sqrt {d}\, d^{2} \left (-30 \sqrt {-c i x +1}\, \mathit {asin} \left (\frac {\sqrt {-c i x +1}}{\sqrt {2}}\right ) a^{2} i +4 \sqrt {-c i x +1}\, \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {-c i x +1}\, c i x -\sqrt {-c i x +1}}d x \right ) a b \,c^{3}-8 \sqrt {-c i x +1}\, \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right ) x}{\sqrt {-c i x +1}\, c i x -\sqrt {-c i x +1}}d x \right ) a b \,c^{2} i -4 \sqrt {-c i x +1}\, \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )}{\sqrt {-c i x +1}\, c i x -\sqrt {-c i x +1}}d x \right ) a b c +2 \sqrt {-c i x +1}\, \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )^{2} x^{2}}{\sqrt {-c i x +1}\, c i x -\sqrt {-c i x +1}}d x \right ) b^{2} c^{3}-4 \sqrt {-c i x +1}\, \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )^{2} x}{\sqrt {-c i x +1}\, c i x -\sqrt {-c i x +1}}d x \right ) b^{2} c^{2} i -2 \sqrt {-c i x +1}\, \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )^{2}}{\sqrt {-c i x +1}\, c i x -\sqrt {-c i x +1}}d x \right ) b^{2} c -\sqrt {c i x +1}\, a^{2} c^{2} i \,x^{2}-7 \sqrt {c i x +1}\, a^{2} c x -24 \sqrt {c i x +1}\, a^{2} i \right )}{2 \sqrt {f}\, \sqrt {-c i x +1}\, c f} \] Input:

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

Output:

(sqrt(d)*d**2*( - 30*sqrt( - c*i*x + 1)*asin(sqrt( - c*i*x + 1)/sqrt(2))*a 
**2*i + 4*sqrt( - c*i*x + 1)*int((sqrt(c*i*x + 1)*asinh(c*x)*x**2)/(sqrt( 
- c*i*x + 1)*c*i*x - sqrt( - c*i*x + 1)),x)*a*b*c**3 - 8*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 - 4*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 + 2*sqrt( 
- c*i*x + 1)*int((sqrt(c*i*x + 1)*asinh(c*x)**2*x**2)/(sqrt( - c*i*x + 1)* 
c*i*x - sqrt( - c*i*x + 1)),x)*b**2*c**3 - 4*sqrt( - c*i*x + 1)*int((sqrt( 
c*i*x + 1)*asinh(c*x)**2*x)/(sqrt( - c*i*x + 1)*c*i*x - sqrt( - c*i*x + 1) 
),x)*b**2*c**2*i - 2*sqrt( - c*i*x + 1)*int((sqrt(c*i*x + 1)*asinh(c*x)**2 
)/(sqrt( - c*i*x + 1)*c*i*x - sqrt( - c*i*x + 1)),x)*b**2*c - sqrt(c*i*x + 
 1)*a**2*c**2*i*x**2 - 7*sqrt(c*i*x + 1)*a**2*c*x - 24*sqrt(c*i*x + 1)*a** 
2*i))/(2*sqrt(f)*sqrt( - c*i*x + 1)*c*f)