\(\int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx\) [235]

Optimal result

Integrand size = 35, antiderivative size = 298 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\frac {i b \sqrt {1+c^2 x^2}}{6 c d f^2 (i+c x) \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 x (a+b \text {arcsinh}(c x))}{3 d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {(i-c x) (a+b \text {arcsinh}(c x))}{3 c d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}+\frac {i b \sqrt {1+c^2 x^2} \arctan (c x)}{6 c d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{3 c d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:

1/6*I*b*(c^2*x^2+1)^(1/2)/c/d/f^2/(I+c*x)/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1 
/2)+2/3*x*(a+b*arcsinh(c*x))/d/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-1/3 
*(I-c*x)*(a+b*arcsinh(c*x))/c/d/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)/(c 
^2*x^2+1)+1/6*I*b*(c^2*x^2+1)^(1/2)*arctan(c*x)/c/d/f^2/(d+I*c*d*x)^(1/2)/ 
(f-I*c*f*x)^(1/2)-1/3*b*(c^2*x^2+1)^(1/2)*ln(c^2*x^2+1)/c/d/f^2/(d+I*c*d*x 
)^(1/2)/(f-I*c*f*x)^(1/2)
 

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.68 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\frac {\sqrt {f-i c f x} \left (4 i a-8 a c x+8 i a c^2 x^2-2 b \sqrt {1+c^2 x^2}+4 i b \left (1+2 i c x+2 c^2 x^2\right ) \text {arcsinh}(c x)+5 b (1-i c x) \sqrt {1+c^2 x^2} \log (d (-1+i c x))+3 b \sqrt {1+c^2 x^2} \log (d+i c d x)-3 i b c x \sqrt {1+c^2 x^2} \log (d+i c d x)\right )}{12 c d f^3 (i+c x)^2 \sqrt {d+i c d x}} \] Input:

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

Output:

(Sqrt[f - I*c*f*x]*((4*I)*a - 8*a*c*x + (8*I)*a*c^2*x^2 - 2*b*Sqrt[1 + c^2 
*x^2] + (4*I)*b*(1 + (2*I)*c*x + 2*c^2*x^2)*ArcSinh[c*x] + 5*b*(1 - I*c*x) 
*Sqrt[1 + c^2*x^2]*Log[d*(-1 + I*c*x)] + 3*b*Sqrt[1 + c^2*x^2]*Log[d + I*c 
*d*x] - (3*I)*b*c*x*Sqrt[1 + c^2*x^2]*Log[d + I*c*d*x]))/(12*c*d*f^3*(I + 
c*x)^2*Sqrt[d + I*c*d*x])
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.55, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6211, 27, 6252, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6252

\(\displaystyle \frac {d \left (c^2 x^2+1\right )^{5/2} \left (-b c \int \left (\frac {2 x}{3 \left (c^2 x^2+1\right )}-\frac {i-c x}{3 c \left (c^2 x^2+1\right )^2}\right )dx+\frac {2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {(-c x+i) (a+b \text {arcsinh}(c x))}{3 c \left (c^2 x^2+1\right )^{3/2}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \left (c^2 x^2+1\right )^{5/2} \left (\frac {2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {(-c x+i) (a+b \text {arcsinh}(c x))}{3 c \left (c^2 x^2+1\right )^{3/2}}-b c \left (-\frac {i \arctan (c x)}{6 c^2}-\frac {1+i c x}{6 c^2 \left (c^2 x^2+1\right )}+\frac {\log \left (c^2 x^2+1\right )}{3 c^2}\right )\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

Input:

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

Output:

(d*(1 + c^2*x^2)^(5/2)*(-1/3*((I - c*x)*(a + b*ArcSinh[c*x]))/(c*(1 + c^2* 
x^2)^(3/2)) + (2*x*(a + b*ArcSinh[c*x]))/(3*Sqrt[1 + c^2*x^2]) - b*c*(-1/6 
*(1 + I*c*x)/(c^2*(1 + c^2*x^2)) - ((I/6)*ArcTan[c*x])/c^2 + Log[1 + c^2*x 
^2]/(3*c^2))))/((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 6252

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_) + (g_.)*(x_))^(m_.)*((d_) + ( 
e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p 
, x]}, Simp[(a + b*ArcSinh[c*x])   u, x] - Simp[b*c   Int[1/Sqrt[1 + c^2*x^ 
2]   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ 
[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])
 

Maple [A] (verified)

Time = 6.99 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.47

method	result	size
default	\(a \left (\frac {i}{c d f \sqrt {i c d x +d}\, \left (-i c f x +f \right )^{\frac {3}{2}}}+\frac {-\frac {2 i \sqrt {i c d x +d}}{3 d c f \left (-i c f x +f \right )^{\frac {3}{2}}}-\frac {2 i \sqrt {i c d x +d}}{3 c d \,f^{2} \sqrt {-i c f x +f}}}{d}\right )+\frac {b \left (4 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-5 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) x^{4} c^{4}-3 \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right ) x^{4} c^{4}+4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+i x c +8 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-10 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) x^{2} c^{2}-6 \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right ) x^{2} c^{2}+6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )+i x^{3} c^{3}+4 \,\operatorname {arcsinh}\left (x c \right )-5 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )-3 \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )+1\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \sqrt {c^{2} x^{2}+1}}{6 f^{3} d^{2} c \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )}\)	\(438\)
parts	\(a \left (\frac {i}{c d f \sqrt {i c d x +d}\, \left (-i c f x +f \right )^{\frac {3}{2}}}+\frac {-\frac {2 i \sqrt {i c d x +d}}{3 d c f \left (-i c f x +f \right )^{\frac {3}{2}}}-\frac {2 i \sqrt {i c d x +d}}{3 c d \,f^{2} \sqrt {-i c f x +f}}}{d}\right )+\frac {b \left (4 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-5 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) x^{4} c^{4}-3 \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right ) x^{4} c^{4}+4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+i x c +8 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-10 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) x^{2} c^{2}-6 \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right ) x^{2} c^{2}+6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )+i x^{3} c^{3}+4 \,\operatorname {arcsinh}\left (x c \right )-5 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )-3 \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )+1\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \sqrt {c^{2} x^{2}+1}}{6 f^{3} d^{2} c \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )}\)	\(438\)

Input:

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

Output:

a*(I/c/d/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(3/2)+2/d*(-1/3*I/d/c/f/(f-I*c*f* 
x)^(3/2)*(d+I*c*d*x)^(1/2)-1/3*I/c/d/f^2/(f-I*c*f*x)^(1/2)*(d+I*c*d*x)^(1/ 
2)))+1/6*b*(4*arcsinh(x*c)*c^4*x^4-5*ln(x*c+(c^2*x^2+1)^(1/2)+I)*x^4*c^4-3 
*ln(x*c+(c^2*x^2+1)^(1/2)-I)*x^4*c^4+4*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^3* 
c^3+I*x*c+8*arcsinh(x*c)*c^2*x^2-10*ln(x*c+(c^2*x^2+1)^(1/2)+I)*x^2*c^2-6* 
ln(x*c+(c^2*x^2+1)^(1/2)-I)*x^2*c^2+6*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x*c+c 
^2*x^2-2*I*(c^2*x^2+1)^(1/2)*arcsinh(x*c)+I*x^3*c^3+4*arcsinh(x*c)-5*ln(x* 
c+(c^2*x^2+1)^(1/2)+I)-3*ln(x*c+(c^2*x^2+1)^(1/2)-I)+1)*(I*(x*c-I)*d)^(1/2 
)*(-I*(I+x*c)*f)^(1/2)*(c^2*x^2+1)^(1/2)/f^3/d^2/c/(c^6*x^6+3*c^4*x^4+3*c^ 
2*x^2+1)
 

Fricas [F]

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

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

Output:

-1/24*(4*sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*b*c*x - 8* 
(2*b*c^2*x^2 + 2*I*b*c*x + b)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(c*x 
 + sqrt(c^2*x^2 + 1)) - 3*(c^4*d^2*f^3*x^3 + I*c^3*d^2*f^3*x^2 + c^2*d^2*f 
^3*x + I*c*d^2*f^3)*sqrt(b^2/(c^2*d^3*f^5))*log(-(I*sqrt(c^2*x^2 + 1)*sqrt 
(I*c*d*x + d)*sqrt(-I*c*f*x + f)*c*d*f^2*x*sqrt(b^2/(c^2*d^3*f^5)) + I*b*c 
^2*x^3 + I*b*x)/(b*c^3*x^3 - I*b*c^2*x^2 + b*c*x - I*b)) + 5*(c^4*d^2*f^3* 
x^3 + I*c^3*d^2*f^3*x^2 + c^2*d^2*f^3*x + I*c*d^2*f^3)*sqrt(b^2/(c^2*d^3*f 
^5))*log(-(I*sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*c*d*f^ 
2*x*sqrt(b^2/(c^2*d^3*f^5)) - I*b*c^2*x^3 - I*b*x)/(b*c^3*x^3 + I*b*c^2*x^ 
2 + b*c*x + I*b)) + 3*(c^4*d^2*f^3*x^3 + I*c^3*d^2*f^3*x^2 + c^2*d^2*f^3*x 
 + I*c*d^2*f^3)*sqrt(b^2/(c^2*d^3*f^5))*log(-(-I*sqrt(c^2*x^2 + 1)*sqrt(I* 
c*d*x + d)*sqrt(-I*c*f*x + f)*c*d*f^2*x*sqrt(b^2/(c^2*d^3*f^5)) + I*b*c^2* 
x^3 + I*b*x)/(b*c^3*x^3 - I*b*c^2*x^2 + b*c*x - I*b)) - 5*(c^4*d^2*f^3*x^3 
 + I*c^3*d^2*f^3*x^2 + c^2*d^2*f^3*x + I*c*d^2*f^3)*sqrt(b^2/(c^2*d^3*f^5) 
)*log(-(-I*sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*c*d*f^2* 
x*sqrt(b^2/(c^2*d^3*f^5)) - I*b*c^2*x^3 - I*b*x)/(b*c^3*x^3 + I*b*c^2*x^2 
+ b*c*x + I*b)) + 8*(c^4*d^2*f^3*x^3 + I*c^3*d^2*f^3*x^2 + c^2*d^2*f^3*x + 
 I*c*d^2*f^3)*sqrt(b^2/(c^2*d^3*f^5))*log((sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x 
+ d)*sqrt(-I*c*f*x + f)*c*d*f^2*x*sqrt(b^2/(c^2*d^3*f^5)) + b*c^2*x^3 + b* 
x)/(b*c^2*x^2 + b)) - 8*(c^4*d^2*f^3*x^3 + I*c^3*d^2*f^3*x^2 + c^2*d^2*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\frac {1}{12} \, b c {\left (\frac {2 i \, \sqrt {d} \sqrt {f}}{c^{3} d^{2} f^{3} x + i \, c^{2} d^{2} f^{3}} - \frac {5 \, \log \left (c x + i\right )}{c^{2} d^{\frac {3}{2}} f^{\frac {5}{2}}} - \frac {3 \, \log \left (c x - i\right )}{c^{2} d^{\frac {3}{2}} f^{\frac {5}{2}}}\right )} - \frac {1}{3} \, b {\left (\frac {3 i}{-3 i \, \sqrt {c^{2} d f x^{2} + d f} c^{2} d f^{2} x + 3 \, \sqrt {c^{2} d f x^{2} + d f} c d f^{2}} - \frac {2 \, x}{\sqrt {c^{2} d f x^{2} + d f} d f^{2}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {1}{3} \, a {\left (\frac {3 i}{-3 i \, \sqrt {c^{2} d f x^{2} + d f} c^{2} d f^{2} x + 3 \, \sqrt {c^{2} d f x^{2} + d f} c d f^{2}} - \frac {2 \, x}{\sqrt {c^{2} d f x^{2} + d f} d f^{2}}\right )} \] Input:

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

Output:

1/12*b*c*(2*I*sqrt(d)*sqrt(f)/(c^3*d^2*f^3*x + I*c^2*d^2*f^3) - 5*log(c*x 
+ I)/(c^2*d^(3/2)*f^(5/2)) - 3*log(c*x - I)/(c^2*d^(3/2)*f^(5/2))) - 1/3*b 
*(3*I/(-3*I*sqrt(c^2*d*f*x^2 + d*f)*c^2*d*f^2*x + 3*sqrt(c^2*d*f*x^2 + d*f 
)*c*d*f^2) - 2*x/(sqrt(c^2*d*f*x^2 + d*f)*d*f^2))*arcsinh(c*x) - 1/3*a*(3* 
I/(-3*I*sqrt(c^2*d*f*x^2 + d*f)*c^2*d*f^2*x + 3*sqrt(c^2*d*f*x^2 + d*f)*c* 
d*f^2) - 2*x/(sqrt(c^2*d*f*x^2 + d*f)*d*f^2))
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Warning, need to choose a branch fo 
r the root of a polynomial with parameters. This might be wrong.The choice 
 was done
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int(asinh(c*x)/(sqrt(c*i*x + 1)*s 
qrt( - c*i*x + 1)*c**3*i*x**3 - sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c**2*x* 
*2 + sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c*i*x - sqrt(c*i*x + 1)*sqrt( - c* 
i*x + 1)),x)*b*c**3*x**2 - 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int(asinh( 
c*x)/(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c**3*i*x**3 - sqrt(c*i*x + 1)*sqr 
t( - c*i*x + 1)*c**2*x**2 + sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c*i*x - sqr 
t(c*i*x + 1)*sqrt( - c*i*x + 1)),x)*b*c + 2*a*c**3*x**3 + 3*a*c*x - a*i)/( 
3*sqrt(f)*sqrt(d)*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c*d*f**2*(c**2*x**2 + 
 1))