\(\int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx\) [237]

Optimal result

Integrand size = 37, antiderivative size = 680 \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {8 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c}+\frac {15}{64} b^2 d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {1}{32} b^2 c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}+\frac {4 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}{27 c}-\frac {15 b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)}{64 c \sqrt {1+c^2 x^2}}-\frac {4 i b d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}-\frac {4 i b c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {1}{4} c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {2 i d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {5 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{24 b c \sqrt {1+c^2 x^2}} \] Output:

8/9*I*b^2*d^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/c+15/64*b^2*d^2*x*(d+I*c 
*d*x)^(1/2)*(f-I*c*f*x)^(1/2)-1/32*b^2*c^2*d^2*x^3*(d+I*c*d*x)^(1/2)*(f-I* 
c*f*x)^(1/2)+4/27*I*b^2*d^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(c^2*x^2+1 
)/c-15/64*b^2*d^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*arcsinh(c*x)/c/(c^2* 
x^2+1)^(1/2)-4/3*I*b*d^2*x*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsin 
h(c*x))/(c^2*x^2+1)^(1/2)-3/8*b*c*d^2*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)-4/9*I*b*c^2*d^2*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)+1/8*b*c^3*d^2* 
x^4*(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*d^2*x*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))^2-1/4* 
c^2*d^2*x^3*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))^2+2/3*I 
*d^2*(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/ 
c+5/24*d^2*(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 = 3.23 (sec) , antiderivative size = 890, normalized size of antiderivative = 1.31 \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {-6912 i a b c d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}+4608 i a^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+6912 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+2592 a^2 c d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+4608 i a^2 c^2 d^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-1728 a^2 c^3 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+1440 b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3-1728 a b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))+256 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (3 \text {arcsinh}(c x))+108 a b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (4 \text {arcsinh}(c x))+4320 a^2 d^{5/2} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+864 b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))-768 i a b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (3 \text {arcsinh}(c x))-27 b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (4 \text {arcsinh}(c x))+12 b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (-576 i b c x+576 i a \sqrt {1+c^2 x^2}-144 b \cosh (2 \text {arcsinh}(c x))+192 i a \cosh (3 \text {arcsinh}(c x))+9 b \cosh (4 \text {arcsinh}(c x))+288 a \sinh (2 \text {arcsinh}(c x))-64 i b \sinh (3 \text {arcsinh}(c x))-36 a \sinh (4 \text {arcsinh}(c x))\right )+72 b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 \left (60 a+48 i b \sqrt {1+c^2 x^2}+16 i b \cosh (3 \text {arcsinh}(c x))+24 b \sinh (2 \text {arcsinh}(c x))-3 b \sinh (4 \text {arcsinh}(c x))\right )}{6912 c \sqrt {1+c^2 x^2}} \] Input:

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

Output:

((-6912*I)*a*b*c*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (4608*I)*a^2* 
d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (6912*I)*b^2*d 
^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 2592*a^2*c*d^2* 
x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (4608*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] - 1728*a^2* 
c^3*d^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 1440*b 
^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 1728*a*b*d^2*S 
qrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + (256*I)*b^2*d^2* 
Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[3*ArcSinh[c*x]] + 108*a*b*d^2*Sqr 
t[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 4320*a^2*d^(5/2)*S 
qrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*S 
qrt[f - I*c*f*x]] + 864*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2 
*ArcSinh[c*x]] - (768*I)*a*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[ 
3*ArcSinh[c*x]] - 27*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[4*Ar 
cSinh[c*x]] + 12*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(( 
-576*I)*b*c*x + (576*I)*a*Sqrt[1 + c^2*x^2] - 144*b*Cosh[2*ArcSinh[c*x]] + 
 (192*I)*a*Cosh[3*ArcSinh[c*x]] + 9*b*Cosh[4*ArcSinh[c*x]] + 288*a*Sinh[2* 
ArcSinh[c*x]] - (64*I)*b*Sinh[3*ArcSinh[c*x]] - 36*a*Sinh[4*ArcSinh[c*x]]) 
 + 72*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(60*a + (48 
*I)*b*Sqrt[1 + c^2*x^2] + (16*I)*b*Cosh[3*ArcSinh[c*x]] + 24*b*Sinh[2*A...
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.50, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6253

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\frac {1}{8} b c^3 x^4 (a+b \text {arcsinh}(c x))-\frac {4}{9} i b c^2 x^3 (a+b \text {arcsinh}(c x))+\frac {3}{8} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2+\frac {2 i \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c}-\frac {1}{4} c^2 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-\frac {3}{8} b c x^2 (a+b \text {arcsinh}(c x))-\frac {4}{3} i b x (a+b \text {arcsinh}(c x))+\frac {5 (a+b \text {arcsinh}(c x))^3}{24 b c}-\frac {15 b^2 \text {arcsinh}(c x)}{64 c}+\frac {15}{64} b^2 x \sqrt {c^2 x^2+1}+\frac {4 i b^2 \left (c^2 x^2+1\right )^{3/2}}{27 c}+\frac {8 i b^2 \sqrt {c^2 x^2+1}}{9 c}-\frac {1}{32} b^2 c^2 x^3 \sqrt {c^2 x^2+1}\right )}{\sqrt {c^2 x^2+1}}\)

Input:

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

Output:

(d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*((((8*I)/9)*b^2*Sqrt[1 + c^2*x^2] 
)/c + (15*b^2*x*Sqrt[1 + c^2*x^2])/64 - (b^2*c^2*x^3*Sqrt[1 + c^2*x^2])/32 
 + (((4*I)/27)*b^2*(1 + c^2*x^2)^(3/2))/c - (15*b^2*ArcSinh[c*x])/(64*c) - 
 ((4*I)/3)*b*x*(a + b*ArcSinh[c*x]) - (3*b*c*x^2*(a + b*ArcSinh[c*x]))/8 - 
 ((4*I)/9)*b*c^2*x^3*(a + b*ArcSinh[c*x]) + (b*c^3*x^4*(a + b*ArcSinh[c*x] 
))/8 + (3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/8 - (c^2*x^3*Sqrt[1 
+ c^2*x^2]*(a + b*ArcSinh[c*x])^2)/4 + (((2*I)/3)*(1 + c^2*x^2)^(3/2)*(a + 
 b*ArcSinh[c*x])^2)/c + (5*(a + b*ArcSinh[c*x])^3)/(24*b*c)))/Sqrt[1 + c^2 
*x^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. 1971 vs. \(2 (559 ) = 1118\).

Time = 8.14 (sec) , antiderivative size = 1972, normalized size of antiderivative = 2.90

method	result	size
default	\(\text {Expression too large to display}\)	\(1972\)
parts	\(\text {Expression too large to display}\)	\(1972\)

Input:

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

Output:

1/4*I*a^2/c/f*(d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(3/2)+5/12*I*a^2*d/c/f*(d+I*c* 
d*x)^(3/2)*(f-I*c*f*x)^(3/2)+5/8*I*a^2*d^2/c/f*(d+I*c*d*x)^(1/2)*(f-I*c*f* 
x)^(3/2)-5/8*I*a^2*d^2/c*(f-I*c*f*x)^(1/2)*(d+I*c*d*x)^(1/2)+5/8*a^2*d^3*f 
*((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*(5/24 
*(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*d^2-1/512*(-I*(I+x*c)*f)^(1/2)*(I*(x*c-I)*d)^(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*arcsinh(x*c)^2-4*arcsinh(x*c)+1)*d^2/(c^2*x^2+1)/c+1/108*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 
)*d^2/(c^2*x^2+1)/c+1/16*(-I*(I+x*c)*f)^(1/2)*(I*(x*c-I)*d)^(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)*d^2/(c^2*x^2+1)/c+1/4*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)*d^2/(c^2*x^2+1)/c+1/4*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)*d^2/(c^2*x^2 
+1)/c+1/16*(-I*(I+x*c)*f)^(1/2)*(I*(x*c-I)*d)^(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)*d^2/(c^2*x^2+1)/c+1/108*I*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*...
 

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

integrate((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(1/2)*(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:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

\[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {f}\, \sqrt {d}\, d^{2} \left (30 \mathit {asin} \left (\frac {\sqrt {-c i x +1}}{\sqrt {2}}\right ) a^{2} i -6 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c^{3} x^{3}+16 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c^{2} i \,x^{2}+9 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c x +16 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} i -48 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+96 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right ) x d x \right ) a b \,c^{2} i +48 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )d x \right ) a b c -24 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+48 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )^{2} x d x \right ) b^{2} c^{2} i +24 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{24 c} \] Input:

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

Output:

(sqrt(f)*sqrt(d)*d**2*(30*asin(sqrt( - c*i*x + 1)/sqrt(2))*a**2*i - 6*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 + 9*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*a**2*c 
*x + 16*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*a**2*i - 48*int(sqrt(c*i*x + 1) 
*sqrt( - c*i*x + 1)*asinh(c*x)*x**2,x)*a*b*c**3 + 96*int(sqrt(c*i*x + 1)*s 
qrt( - c*i*x + 1)*asinh(c*x)*x,x)*a*b*c**2*i + 48*int(sqrt(c*i*x + 1)*sqrt 
( - c*i*x + 1)*asinh(c*x),x)*a*b*c - 24*int(sqrt(c*i*x + 1)*sqrt( - c*i*x 
+ 1)*asinh(c*x)**2*x**2,x)*b**2*c**3 + 48*int(sqrt(c*i*x + 1)*sqrt( - c*i* 
x + 1)*asinh(c*x)**2*x,x)*b**2*c**2*i + 24*int(sqrt(c*i*x + 1)*sqrt( - c*i 
*x + 1)*asinh(c*x)**2,x)*b**2*c))/(24*c)