\(\int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx\) [243]

Optimal result

Integrand size = 37, antiderivative size = 817 \[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {16 i b^2 d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}{75 c}+\frac {15}{64} b^2 d^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x}+\frac {8 i b^2 d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}{225 c}+\frac {1}{32} b^2 d^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )+\frac {2 i b^2 d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )^2}{125 c}-\frac {9 b^2 d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)}{64 c \sqrt {1+c^2 x^2}}-\frac {2 i b d^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{5 \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 f x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}-\frac {4 i b c^2 d^2 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {2 i b c^4 d^2 f x^5 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}-\frac {b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{8 c}+\frac {3}{8} d^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {i d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{5 c}+\frac {d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{8 b c \sqrt {1+c^2 x^2}} \] Output:

16/75*I*b^2*d^2*f*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/c+15/64*b^2*d^2*f*x* 
(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)+8/225*I*b^2*d^2*f*(d+I*c*d*x)^(1/2)*(f 
-I*c*f*x)^(1/2)*(c^2*x^2+1)/c+1/32*b^2*d^2*f*x*(d+I*c*d*x)^(1/2)*(f-I*c*f* 
x)^(1/2)*(c^2*x^2+1)+2/125*I*b^2*d^2*f*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2) 
*(c^2*x^2+1)^2/c-9/64*b^2*d^2*f*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*arcsin 
h(c*x)/c/(c^2*x^2+1)^(1/2)-2/5*I*b*d^2*f*x*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^( 
1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)-3/8*b*c*d^2*f*x^2*(d+I*c*d*x)^(1 
/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)-4/15*I*b*c^2*d^ 
2*f*x^3*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1) 
^(1/2)-2/25*I*b*c^4*d^2*f*x^5*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arc 
sinh(c*x))/(c^2*x^2+1)^(1/2)-1/8*b*d^2*f*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/ 
2)*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c+3/8*d^2*f*x*(d+I*c*d*x)^(1/2)*(f 
-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))^2+1/4*d^2*f*x*(d+I*c*d*x)^(1/2)*(f-I*c* 
f*x)^(1/2)*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2+1/5*I*d^2*f*(d+I*c*d*x)^(1/2)* 
(f-I*c*f*x)^(1/2)*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/c+1/8*d^2*f*(d+I*c*d* 
x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))^3/b/c/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 3.89 (sec) , antiderivative size = 1084, normalized size of antiderivative = 1.33 \[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx =\text {Too large to display} \] Input:

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

Output:

((-72000*I)*a*b*c*d^2*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (57600*I)* 
a^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (72000*I 
)*b^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 180000 
*a^2*c*d^2*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (11 
5200*I)*a^2*c^2*d^2*f*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2 
*x^2] + 72000*a^2*c^3*d^2*f*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 
 + c^2*x^2] + (57600*I)*a^2*c^4*d^2*f*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f 
*x]*Sqrt[1 + c^2*x^2] + 36000*b^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x 
]*ArcSinh[c*x]^3 - 72000*a*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cos 
h[2*ArcSinh[c*x]] + (4000*I)*b^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] 
*Cosh[3*ArcSinh[c*x]] - 4500*a*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] 
*Cosh[4*ArcSinh[c*x]] + (288*I)*b^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f 
*x]*Cosh[5*ArcSinh[c*x]] + 108000*a^2*d^(5/2)*f^(3/2)*Sqrt[1 + c^2*x^2]*Lo 
g[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 36000*b 
^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] - (12000 
*I)*a*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] + 1 
125*b^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[4*ArcSinh[c*x]] + 1 
800*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(60*a + (20 
*I)*b*Sqrt[1 + c^2*x^2] + (10*I)*b*Cosh[3*ArcSinh[c*x]] + (2*I)*b*Cosh[5*A 
rcSinh[c*x]] + 40*b*Sinh[2*ArcSinh[c*x]] + 5*b*Sinh[4*ArcSinh[c*x]]) - ...
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 383, 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, 6253, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6253

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (-\frac {2}{25} i b c^4 x^5 (a+b \text {arcsinh}(c x))-\frac {4}{15} i b c^2 x^3 (a+b \text {arcsinh}(c x))+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{8} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2+\frac {i \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c}-\frac {b \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{8 c}-\frac {3}{8} b c x^2 (a+b \text {arcsinh}(c x))-\frac {2}{5} i b x (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^3}{8 b c}-\frac {9 b^2 \text {arcsinh}(c x)}{64 c}+\frac {1}{32} b^2 x \left (c^2 x^2+1\right )^{3/2}+\frac {15}{64} b^2 x \sqrt {c^2 x^2+1}+\frac {2 i b^2 \left (c^2 x^2+1\right )^{5/2}}{125 c}+\frac {8 i b^2 \left (c^2 x^2+1\right )^{3/2}}{225 c}+\frac {16 i b^2 \sqrt {c^2 x^2+1}}{75 c}\right )}{\left (c^2 x^2+1\right )^{3/2}}\)

Input:

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

Output:

(d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*((((16*I)/75)*b^2*Sqrt[1 + c^2* 
x^2])/c + (15*b^2*x*Sqrt[1 + c^2*x^2])/64 + (((8*I)/225)*b^2*(1 + c^2*x^2) 
^(3/2))/c + (b^2*x*(1 + c^2*x^2)^(3/2))/32 + (((2*I)/125)*b^2*(1 + c^2*x^2 
)^(5/2))/c - (9*b^2*ArcSinh[c*x])/(64*c) - ((2*I)/5)*b*x*(a + b*ArcSinh[c* 
x]) - (3*b*c*x^2*(a + b*ArcSinh[c*x]))/8 - ((4*I)/15)*b*c^2*x^3*(a + b*Arc 
Sinh[c*x]) - ((2*I)/25)*b*c^4*x^5*(a + b*ArcSinh[c*x]) - (b*(1 + c^2*x^2)^ 
2*(a + b*ArcSinh[c*x]))/(8*c) + (3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x] 
)^2)/8 + (x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/4 + ((I/5)*(1 + c^ 
2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/c + (a + b*ArcSinh[c*x])^3/(8*b*c)))/ 
(1 + c^2*x^2)^(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 6253

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2568 vs. \(2 (676 ) = 1352\).

Time = 8.14 (sec) , antiderivative size = 2569, normalized size of antiderivative = 3.14

method	result	size
default	\(\text {Expression too large to display}\)	\(2569\)
parts	\(\text {Expression too large to display}\)	\(2569\)

Input:

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

Output:

1/5*I*a^2/c/f*(d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(5/2)+1/4*I*a^2*d/c/f*(d+I*c*d 
*x)^(3/2)*(f-I*c*f*x)^(5/2)+1/4*I*a^2*d^2/c/f*(d+I*c*d*x)^(1/2)*(f-I*c*f*x 
)^(5/2)-1/8*I*a^2*d^2/c*(f-I*c*f*x)^(3/2)*(d+I*c*d*x)^(1/2)-3/8*I*a^2*d^2* 
f/c*(f-I*c*f*x)^(1/2)*(d+I*c*d*x)^(1/2)+3/8*a^2*d^3*f^2*((f-I*c*f*x)*(d+I* 
c*d*x))^(1/2)/(f-I*c*f*x)^(1/2)/(d+I*c*d*x)^(1/2)*ln(c^2*d*f*x/(c^2*d*f)^( 
1/2)+(c^2*d*f*x^2+d*f)^(1/2))/(c^2*d*f)^(1/2)+b^2*(1/8*(I*(x*c-I)*d)^(1/2) 
*(-I*(I+x*c)*f)^(1/2)/(c^2*x^2+1)^(1/2)/c*arcsinh(x*c)^3*f*d^2+1/4000*I*(I 
*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(16*c^6*x^6+16*(c^2*x^2+1)^(1/2)*x^ 
5*c^5+28*c^4*x^4+20*(c^2*x^2+1)^(1/2)*c^3*x^3+13*c^2*x^2+5*(c^2*x^2+1)^(1/ 
2)*x*c+1)*(25*arcsinh(x*c)^2-10*arcsinh(x*c)+2)*f*d^2/(c^2*x^2+1)/c+1/512* 
(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(8*x^5*c^5+8*x^4*c^4*(c^2*x^2+1)^ 
(1/2)+12*x^3*c^3+8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c+(c^2*x^2+1)^(1/2))*(8*a 
rcsinh(x*c)^2-4*arcsinh(x*c)+1)*f*d^2/(c^2*x^2+1)/c+1/288*I*(I*(x*c-I)*d)^ 
(1/2)*(-I*(I+x*c)*f)^(1/2)*(4*c^4*x^4+4*(c^2*x^2+1)^(1/2)*c^3*x^3+5*c^2*x^ 
2+3*(c^2*x^2+1)^(1/2)*x*c+1)*(9*arcsinh(x*c)^2-6*arcsinh(x*c)+2)*f*d^2/(c^ 
2*x^2+1)/c+1/16*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(2*x^3*c^3+2*x^2* 
c^2*(c^2*x^2+1)^(1/2)+2*x*c+(c^2*x^2+1)^(1/2))*(2*arcsinh(x*c)^2-2*arcsinh 
(x*c)+1)*f*d^2/(c^2*x^2+1)/c+1/16*I*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/ 
2)*(c^2*x^2+(c^2*x^2+1)^(1/2)*x*c+1)*(arcsinh(x*c)^2-2*arcsinh(x*c)+2)*f*d 
^2/(c^2*x^2+1)/c+1/16*I*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(c^2*x...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad 
Argument TypeError: Bad Argument TypeError: Bad Argument TypeDone
 

Mupad [F(-1)]

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

\[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {f}\, \sqrt {d}\, d^{2} f \left (30 \mathit {asin} \left (\frac {\sqrt {-c i x +1}}{\sqrt {2}}\right ) a^{2} i +8 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c^{4} i \,x^{4}+10 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c^{3} x^{3}+16 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c^{2} i \,x^{2}+25 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c x +8 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} i +80 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) a b \,c^{4} i +80 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+80 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right ) x d x \right ) a b \,c^{2} i +80 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )d x \right ) a b c +40 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4} i +40 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+40 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )^{2} x d x \right ) b^{2} c^{2} i +40 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{40 c} \] Input:

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

Output:

(sqrt(f)*sqrt(d)*d**2*f*(30*asin(sqrt( - c*i*x + 1)/sqrt(2))*a**2*i + 8*sq 
rt(c*i*x + 1)*sqrt( - c*i*x + 1)*a**2*c**4*i*x**4 + 10*sqrt(c*i*x + 1)*sqr 
t( - c*i*x + 1)*a**2*c**3*x**3 + 16*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*a** 
2*c**2*i*x**2 + 25*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*a**2*c*x + 8*sqrt(c* 
i*x + 1)*sqrt( - c*i*x + 1)*a**2*i + 80*int(sqrt(c*i*x + 1)*sqrt( - c*i*x 
+ 1)*asinh(c*x)*x**3,x)*a*b*c**4*i + 80*int(sqrt(c*i*x + 1)*sqrt( - c*i*x 
+ 1)*asinh(c*x)*x**2,x)*a*b*c**3 + 80*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 
1)*asinh(c*x)*x,x)*a*b*c**2*i + 80*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)* 
asinh(c*x),x)*a*b*c + 40*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh(c*x) 
**2*x**3,x)*b**2*c**4*i + 40*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh( 
c*x)**2*x**2,x)*b**2*c**3 + 40*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asin 
h(c*x)**2*x,x)*b**2*c**2*i + 40*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asi 
nh(c*x)**2,x)*b**2*c))/(40*c)