\(\int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{5/2}} \, dx\) [267]

Optimal result

Integrand size = 37, antiderivative size = 776 \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{5/2}} \, dx=\frac {2 i b^2 d^3 \left (1+c^2 x^2\right )}{c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {28 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {5 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {112 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {112 b^2 d^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {8 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {16 i b^2 d^3 \sqrt {1+c^2 x^2} \tan \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {28 i d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 i d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:

2*I*b^2*d^3*(c^2*x^2+1)/c/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-2*I*b*d^ 
3*x*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x) 
^(1/2)+28/3*d^3*(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)+I*d^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/c/f^2/(d+I*c 
*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+5/3*d^3*(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)+112/3*b*d^3*(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)-112/3*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^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+8/3*b*d^3*(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)+16/3*I*b^2*d^3*(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)+28/3*I*d 
^3*(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)-4/3*I*d^3*(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. \(2552\) vs. \(2(776)=1552\).

Time = 23.76 (sec) , antiderivative size = 2552, normalized size of antiderivative = 3.29 \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{5/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*((I*a^2*d^2)/f^3 + (((8*I)/3) 
*a^2*d^2)/(f^3*(I + c*x)^2) - (28*a^2*d^2)/(3*f^3*(I + c*x))))/c + (5*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)]])/(c*f^(5/2)) - ((I/3)*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] + I*Sinh[A 
rcSinh[c*x]/2])*(-(Cosh[(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*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]] + Log[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^2*Sqrt[I*((-I)*d + c*d*x)]*Sqr 
t[(-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*ArcSinh[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...
 

Rubi [A] (verified)

Time = 1.64 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.47, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6259

\(\displaystyle \frac {d^5 \left (c^2 x^2+1\right )^{5/2} \int \left (\frac {i c x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {12 i (a+b \text {arcsinh}(c x))^2}{(c x+i) \sqrt {c^2 x^2+1}}-\frac {8 (a+b \text {arcsinh}(c x))^2}{(c x+i)^2 \sqrt {c^2 x^2+1}}+\frac {5 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}\right )dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^5 \left (c^2 x^2+1\right )^{5/2} \left (\frac {i \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c}+\frac {5 (a+b \text {arcsinh}(c x))^3}{3 b c}+\frac {28 (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {112 b \log \left (1+i e^{-\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 c}+\frac {28 i \tan \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {8 b \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) (a+b \text {arcsinh}(c x))}{3 c}-\frac {4 i \tan \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) (a+b \text {arcsinh}(c x))^2}{3 c}-2 i a b x-\frac {112 b^2 \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )}{3 c}-2 i b^2 x \text {arcsinh}(c x)+\frac {16 i b^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c}+\frac {2 i b^2 \sqrt {c^2 x^2+1}}{c}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

Input:

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

Output:

(d^5*(1 + c^2*x^2)^(5/2)*((-2*I)*a*b*x + ((2*I)*b^2*Sqrt[1 + c^2*x^2])/c - 
 (2*I)*b^2*x*ArcSinh[c*x] + (28*(a + b*ArcSinh[c*x])^2)/(3*c) + (I*Sqrt[1 
+ c^2*x^2]*(a + b*ArcSinh[c*x])^2)/c + (5*(a + b*ArcSinh[c*x])^3)/(3*b*c) 
+ (112*b*(a + b*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]])/(3*c) - (112*b^2* 
PolyLog[2, (-I)/E^ArcSinh[c*x]])/(3*c) + (8*b*(a + b*ArcSinh[c*x])*Sec[Pi/ 
4 + (I/2)*ArcSinh[c*x]]^2)/(3*c) + (((16*I)/3)*b^2*Tan[Pi/4 + (I/2)*ArcSin 
h[c*x]])/c + (((28*I)/3)*(a + b*ArcSinh[c*x])^2*Tan[Pi/4 + (I/2)*ArcSinh[c 
*x]])/c - (((4*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 = 7.73 (sec) , antiderivative size = 1134, normalized size of antiderivative = 1.46

method	result	size
default	\(\text {Expression too large to display}\)	\(1134\)

Input:

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

Output:

5/3*d^2*(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/c/f^3+1/2*I*d^2*(arcsinh(x*c)^2*b^2+2*arcsinh(x*c)*a*b-2*b 
^2*arcsinh(x*c)+a^2-2*a*b+2*b^2)*(c^2*x^2+(c^2*x^2+1)^(1/2)*x*c+1)*(I*(x*c 
-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/(c^2*x^2+1)/c/f^3+1/2*I*d^2*(arcsinh(x*c 
)^2*b^2+2*arcsinh(x*c)*a*b+2*b^2*arcsinh(x*c)+a^2+2*a*b+2*b^2)*(c^2*x^2-(c 
^2*x^2+1)^(1/2)*x*c+1)*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/(c^2*x^2+1 
)/c/f^3-4/3*d^2*(518*a*b+4554*arcsinh(x*c)^2*b^2*c^2*x^2+1369*a^2+1369*arc 
sinh(x*c)^2*b^2+518*b^2*arcsinh(x*c)+888*b^2+882*a*b*c^4*x^4+1288*a*b*c^2* 
x^2-252*(c^2*x^2+1)^(1/2)*b^2*c^3*x^3-236*(c^2*x^2+1)^(1/2)*b^2*c*x+882*(c 
^2*x^2+1)^(1/2)*arcsinh(x*c)*b^2*c^3*x^3+442*(c^2*x^2+1)^(1/2)*arcsinh(x*c 
)*b^2*c*x+882*arcsinh(x*c)*b^2*c^4*x^4+1288*arcsinh(x*c)*b^2*c^2*x^2+132*I 
*(c^2*x^2+1)^(1/2)*b^2*c^2*x^2+308*I*a*b*c*x+420*I*a*b*c^3*x^3+882*(c^2*x^ 
2+1)^(1/2)*a*b*c^3*x^3+442*(c^2*x^2+1)^(1/2)*a*b*c*x+9108*arcsinh(x*c)*a*b 
*c^2*x^2+2738*arcsinh(x*c)*a*b-714*I*(c^2*x^2+1)^(1/2)*a*b*c^2*x^2+308*I*a 
rcsinh(x*c)*b^2*c*x+420*I*arcsinh(x*c)*b^2*c^3*x^3-370*I*(c^2*x^2+1)^(1/2) 
*arcsinh(x*c)*b^2+2016*b^2*c^4*x^4+2680*b^2*c^2*x^2+7938*arcsinh(x*c)*a*b* 
c^4*x^4+3969*arcsinh(x*c)^2*b^2*c^4*x^4+4554*a^2*c^2*x^2-714*I*arcsinh(x*c 
)*(c^2*x^2+1)^(1/2)*b^2*c^2*x^2+456*I*b^2*c^3*x^3+232*I*b^2*c*x-370*I*(c^2 
*x^2+1)^(1/2)*a*b+148*I*(c^2*x^2+1)^(1/2)*b^2+3969*a^2*c^4*x^4)*(7*x^3*c^3 
-9*I*c^2*x^2-7*x^2*c^2*(c^2*x^2+1)^(1/2)+3*x*c-5*I-7*(c^2*x^2+1)^(1/2))...
 

Fricas [F]

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

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

Output:

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

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*d**2*( - 30*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asin(sqrt( - c*i*x 
 + 1)/sqrt(2))*a**2*c*x - 30*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**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)*a*b*c**4*i*x - 6*sqrt(c*i*x + 1 
)*sqrt( - c*i*x + 1)*int((sqrt(c*i*x + 1)*asinh(c*x)*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)*a*b*c 
**3 + 12*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 + 12*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int((sqr 
t(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*s 
qrt(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 + 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int((sqrt(c*i*x + 1)*asinh 
(c*x)**2*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**4*i*x - 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 
 1)*int((sqrt(c*i*x + 1)*asinh(c*x)**2*x**2)/(sqrt( - c*i*x + 1)*c**2*x...