\(\int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx\) [255]

Optimal result

Integrand size = 37, antiderivative size = 615 \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\frac {68 i b^2 d^3 \left (1+c^2 x^2\right )}{9 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 b^2 d^3 x \left (1+c^2 x^2\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b^2 d^3 \left (1+c^2 x^2\right )^2}{27 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b^2 d^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {22 i b d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b c d^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b c^2 d^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {11 i d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {i c d^3 x^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {5 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:

68/9*I*b^2*d^3*(c^2*x^2+1)/c/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-3/4*b^2*d 
^3*x*(c^2*x^2+1)/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-2/27*I*b^2*d^3*(c^2*x 
^2+1)^2/c/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+3/4*b^2*d^3*(c^2*x^2+1)^(1/2 
)*arcsinh(c*x)/c/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-22/3*I*b*d^3*x*(c^2*x 
^2+1)^(1/2)*(a+b*arcsinh(c*x))/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+3/2*b*c 
*d^3*x^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/(d+I*c*d*x)^(1/2)/(f-I*c*f*x 
)^(1/2)+2/9*I*b*c^2*d^3*x^3*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/(d+I*c*d* 
x)^(1/2)/(f-I*c*f*x)^(1/2)+11/3*I*d^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/c/( 
d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-3/2*d^3*x*(c^2*x^2+1)*(a+b*arcsinh(c*x) 
)^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-1/3*I*c*d^3*x^2*(c^2*x^2+1)*(a+b*a 
rcsinh(c*x))^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+5/6*d^3*(c^2*x^2+1)^(1/ 
2)*(a+b*arcsinh(c*x))^3/b/c/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)
 

Mathematica [A] (verified)

Time = 13.27 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.18 \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\frac {-1620 i a b c d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}+792 i a^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+1620 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-324 a^2 c d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-72 i a^2 c^2 d^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+180 b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3+162 a b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))-4 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (3 \text {arcsinh}(c x))+6 b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (27 b \cosh (2 \text {arcsinh}(c x))+2 i \left (4 b c x \left (-33+c^2 x^2\right )+27 a (5+2 i c x) \sqrt {1+c^2 x^2}-3 a \cosh (3 \text {arcsinh}(c x))\right )\right )+540 a^2 d^{5/2} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )-81 b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))+18 b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 \left (30 a+45 i b \sqrt {1+c^2 x^2}-i b \cosh (3 \text {arcsinh}(c x))-9 b \sinh (2 \text {arcsinh}(c x))\right )+12 i a b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (3 \text {arcsinh}(c x))}{216 c f \sqrt {1+c^2 x^2}} \] Input:

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

Output:

((-1620*I)*a*b*c*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (792*I)*a^2*d 
^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (1620*I)*b^2*d^ 
2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 324*a^2*c*d^2*x* 
Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (72*I)*a^2*c^2*d^2 
*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 180*b^2*d^2*S 
qrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 + 162*a*b*d^2*Sqrt[d + I 
*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - (4*I)*b^2*d^2*Sqrt[d + I* 
c*d*x]*Sqrt[f - I*c*f*x]*Cosh[3*ArcSinh[c*x]] + 6*b*d^2*Sqrt[d + I*c*d*x]* 
Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(27*b*Cosh[2*ArcSinh[c*x]] + (2*I)*(4*b*c*x 
*(-33 + c^2*x^2) + 27*a*(5 + (2*I)*c*x)*Sqrt[1 + c^2*x^2] - 3*a*Cosh[3*Arc 
Sinh[c*x]])) + 540*a^2*d^(5/2)*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqr 
t[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] - 81*b^2*d^2*Sqrt[d + I* 
c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 18*b*d^2*Sqrt[d + I*c*d*x] 
*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(30*a + (45*I)*b*Sqrt[1 + c^2*x^2] - I*b 
*Cosh[3*ArcSinh[c*x]] - 9*b*Sinh[2*ArcSinh[c*x]]) + (12*I)*a*b*d^2*Sqrt[d 
+ I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]])/(216*c*f*Sqrt[1 + c^2*x 
^2])
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.51, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {6211, 27, 6258, 3042, 3798, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3798

\(\displaystyle \frac {d^3 \sqrt {c^2 x^2+1} \int \left (-i x^3 (a+b \text {arcsinh}(c x))^2 c^6-3 x^2 (a+b \text {arcsinh}(c x))^2 c^5+3 i x (a+b \text {arcsinh}(c x))^2 c^4+(a+b \text {arcsinh}(c x))^2 c^3\right )d\text {arcsinh}(c x)}{c^4 \sqrt {d+i c d x} \sqrt {f-i c f x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^3 \sqrt {c^2 x^2+1} \left (\frac {2}{9} i b c^6 x^3 (a+b \text {arcsinh}(c x))+\frac {3}{2} b c^5 x^2 (a+b \text {arcsinh}(c x))-\frac {22}{3} i b c^4 x (a+b \text {arcsinh}(c x))+\frac {5 c^3 (a+b \text {arcsinh}(c x))^3}{6 b}-\frac {1}{3} i c^5 x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-\frac {3}{2} c^4 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2+\frac {11}{3} i c^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} b^2 c^3 \text {arcsinh}(c x)-\frac {3}{4} b^2 c^4 x \sqrt {c^2 x^2+1}-\frac {2}{27} i b^2 c^3 \left (c^2 x^2+1\right )^{3/2}+\frac {68}{9} i b^2 c^3 \sqrt {c^2 x^2+1}\right )}{c^4 \sqrt {d+i c d x} \sqrt {f-i c f x}}\)

Input:

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

Output:

(d^3*Sqrt[1 + c^2*x^2]*(((68*I)/9)*b^2*c^3*Sqrt[1 + c^2*x^2] - (3*b^2*c^4* 
x*Sqrt[1 + c^2*x^2])/4 - ((2*I)/27)*b^2*c^3*(1 + c^2*x^2)^(3/2) + (3*b^2*c 
^3*ArcSinh[c*x])/4 - ((22*I)/3)*b*c^4*x*(a + b*ArcSinh[c*x]) + (3*b*c^5*x^ 
2*(a + b*ArcSinh[c*x]))/2 + ((2*I)/9)*b*c^6*x^3*(a + b*ArcSinh[c*x]) + ((1 
1*I)/3)*c^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2 - (3*c^4*x*Sqrt[1 + c 
^2*x^2]*(a + b*ArcSinh[c*x])^2)/2 - (I/3)*c^5*x^2*Sqrt[1 + c^2*x^2]*(a + b 
*ArcSinh[c*x])^2 + (5*c^3*(a + b*ArcSinh[c*x])^3)/(6*b)))/(c^4*Sqrt[d + I* 
c*d*x]*Sqrt[f - I*c*f*x])
 

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ 
m, 0] || NeQ[a^2 - b^2, 0])
 

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 6258

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/S 
qrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d])   Subst[I 
nt[(a + b*x)^n*(c*f + g*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b 
, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && GtQ[d, 0] && (Gt 
Q[m, 0] || IGtQ[n, 0])
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1476 vs. \(2 (511 ) = 1022\).

Time = 6.34 (sec) , antiderivative size = 1477, normalized size of antiderivative = 2.40

method	result	size
default	\(\text {Expression too large to display}\)	\(1477\)
parts	\(\text {Expression too large to display}\)	\(1477\)

Input:

int((d+I*c*d*x)^(5/2)*(a+b*arcsinh(x*c))^2/(f-I*c*f*x)^(1/2),x,method=_RET 
URNVERBOSE)
 

Output:

1/3*I*a^2/c/f*(d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(1/2)+5/6*I*a^2*d/c/f*(d+I*c*d 
*x)^(3/2)*(f-I*c*f*x)^(1/2)+5/2*I*a^2*d^2/c/f*(d+I*c*d*x)^(1/2)*(f-I*c*f*x 
)^(1/2)+5/2*a^2*d^3*((f-I*c*f*x)*(d+I*c*d*x))^(1/2)/(d+I*c*d*x)^(1/2)/(f-I 
*c*f*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*(5/6*d^2*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)/f/c-1/216*I*d^2*(9*arcsinh(x*c)^2-6*arcsinh(x*c)+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)*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/f/c/(c^2*x^2+1)-3/16*d^2*(2*ar 
csinh(x*c)^2-2*arcsinh(x*c)+1)*(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))*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/f/c/(c^2*x^2 
+1)+15/8*I*d^2*(arcsinh(x*c)^2-2*arcsinh(x*c)+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)/f/c/(c^2*x^2+1)+15/8*I*d 
^2*(arcsinh(x*c)^2+2*arcsinh(x*c)+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)/f/c/(c^2*x^2+1)-3/16*d^2*(2*arcsinh( 
x*c)^2+2*arcsinh(x*c)+1)*(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))*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/f/c/(c^2*x^2+1)-1/ 
216*I*d^2*(9*arcsinh(x*c)^2+6*arcsinh(x*c)+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)*(I*(x*c-I)*d)^(1/2)*(-I*(I 
+x*c)*f)^(1/2)/f/c/(c^2*x^2+1))+2*a*b*(5/4*d^2*arcsinh(x*c)^2*(-I*(I+x*c)* 
f)^(1/2)*(I*(x*c-I)*d)^(1/2)/(c^2*x^2+1)^(1/2)/c/f-1/72*I*d^2*(-1+3*arc...
 

Fricas [F]

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

integrate((d+I*c*d*x)^(5/2)*(a+b*arcsinh(c*x))^2/(f-I*c*f*x)^(1/2),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)* 
sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f))/(c*f*x + I*f), x)
 

Sympy [F(-1)]

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

integrate((d+I*c*d*x)**(5/2)*(a+b*asinh(c*x))**2/(f-I*c*f*x)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((d+I*c*d*x)^(5/2)*(a+b*arcsinh(c*x))^2/(f-I*c*f*x)^(1/2),x, algo 
rithm="maxima")
 

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\frac {\sqrt {d}\, d^{2} \left (30 \mathit {asin} \left (\frac {\sqrt {-c i x +1}}{\sqrt {2}}\right ) a^{2} i -2 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c^{2} i \,x^{2}-9 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c x +22 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} i -12 \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {-c i x +1}}d x \right ) a b \,c^{3}+24 \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right ) x}{\sqrt {-c i x +1}}d x \right ) a b \,c^{2} i +12 \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )}{\sqrt {-c i x +1}}d x \right ) a b c -6 \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )^{2} x^{2}}{\sqrt {-c i x +1}}d x \right ) b^{2} c^{3}+12 \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )^{2} x}{\sqrt {-c i x +1}}d x \right ) b^{2} c^{2} i +6 \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )^{2}}{\sqrt {-c i x +1}}d x \right ) b^{2} c \right )}{6 \sqrt {f}\, c} \] Input:

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

Output:

(sqrt(d)*d**2*(30*asin(sqrt( - c*i*x + 1)/sqrt(2))*a**2*i - 2*sqrt(c*i*x + 
 1)*sqrt( - c*i*x + 1)*a**2*c**2*i*x**2 - 9*sqrt(c*i*x + 1)*sqrt( - c*i*x 
+ 1)*a**2*c*x + 22*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*a**2*i - 12*int((sqr 
t(c*i*x + 1)*asinh(c*x)*x**2)/sqrt( - c*i*x + 1),x)*a*b*c**3 + 24*int((sqr 
t(c*i*x + 1)*asinh(c*x)*x)/sqrt( - c*i*x + 1),x)*a*b*c**2*i + 12*int((sqrt 
(c*i*x + 1)*asinh(c*x))/sqrt( - c*i*x + 1),x)*a*b*c - 6*int((sqrt(c*i*x + 
1)*asinh(c*x)**2*x**2)/sqrt( - c*i*x + 1),x)*b**2*c**3 + 12*int((sqrt(c*i* 
x + 1)*asinh(c*x)**2*x)/sqrt( - c*i*x + 1),x)*b**2*c**2*i + 6*int((sqrt(c* 
i*x + 1)*asinh(c*x)**2)/sqrt( - c*i*x + 1),x)*b**2*c))/(6*sqrt(f)*c)