\(\int (d+i c d x)^{3/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) [214]

Optimal result

Integrand size = 35, antiderivative size = 483 \[ \int (d+i c d x)^{3/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {i b d f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}}{5 \sqrt {1+c^2 x^2}}-\frac {3 b c d f^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{16 \sqrt {1+c^2 x^2}}+\frac {2 i b c^2 d f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{15 \sqrt {1+c^2 x^2}}+\frac {i b c^4 d f^2 x^5 \sqrt {d+i c d x} \sqrt {f-i c f x}}{25 \sqrt {1+c^2 x^2}}-\frac {b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )^{3/2}}{16 c}+\frac {3}{8} d f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))+\frac {1}{4} d f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))-\frac {i d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{5 c}+\frac {3 d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}} \] Output:

1/5*I*b*d*f^2*x*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)-3/16 
*b*c*d*f^2*x^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)+2/15* 
I*b*c^2*d*f^2*x^3*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)+1/ 
25*I*b*c^4*d*f^2*x^5*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2) 
-1/16*b*d*f^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(c^2*x^2+1)^(3/2)/c+3/8* 
d*f^2*x*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))+1/4*d*f^2*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))-1/5*I* 
d*f^2*(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)) 
/c+3/16*d*f^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))^2/b/c 
/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 2.84 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.41 \[ \int (d+i c d x)^{3/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1200 i b c d f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-1920 i a d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+6000 a c d f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-3840 i a c^2 d f^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+2400 a c^3 d f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-1920 i a c^4 d f^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+1800 b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2-1200 b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))-75 b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (4 \text {arcsinh}(c x))+3600 a d^{3/2} f^{5/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+200 i b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (3 \text {arcsinh}(c x))+60 b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (-10 i \cosh (3 \text {arcsinh}(c x))-2 i \cosh (5 \text {arcsinh}(c x))+5 \left (-4 i \sqrt {1+c^2 x^2}+8 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))\right )\right )+24 i b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (5 \text {arcsinh}(c x))}{9600 c \sqrt {1+c^2 x^2}} \] Input:

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

Output:

((1200*I)*b*c*d*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (1920*I)*a*d*f 
^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 6000*a*c*d*f^2* 
x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (3840*I)*a*c^2*d 
*f^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 2400*a*c^ 
3*d*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (1920* 
I)*a*c^4*d*f^2*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 
 1800*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 - 1200*b* 
d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - 75*b*d*f^ 
2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 3600*a*d^(3/2 
)*f^(5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x 
]*Sqrt[f - I*c*f*x]] + (200*I)*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] 
*Sinh[3*ArcSinh[c*x]] + 60*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Arc 
Sinh[c*x]*((-10*I)*Cosh[3*ArcSinh[c*x]] - (2*I)*Cosh[5*ArcSinh[c*x]] + 5*( 
(-4*I)*Sqrt[1 + c^2*x^2] + 8*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])) 
 + (24*I)*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[5*ArcSinh[c*x]] 
)/(9600*c*Sqrt[1 + c^2*x^2])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.41, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6253

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*((I/5)*b*x - (5*b*c*x^2)/16 + ( 
(2*I)/15)*b*c^2*x^3 - (b*c^3*x^4)/16 + (I/25)*b*c^4*x^5 + (3*x*Sqrt[1 + c^ 
2*x^2]*(a + b*ArcSinh[c*x]))/8 + (x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x 
]))/4 - ((I/5)*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/c + (3*(a + b*Arc 
Sinh[c*x])^2)/(16*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. 1352 vs. \(2 (395 ) = 790\).

Time = 5.11 (sec) , antiderivative size = 1353, normalized size of antiderivative = 2.80

method	result	size
default	\(\text {Expression too large to display}\)	\(1353\)
parts	\(\text {Expression too large to display}\)	\(1353\)

Input:

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

Output:

1/5*I*a/c/f*(d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(7/2)+3/20*I*a*d/c/f*(d+I*c*d*x) 
^(1/2)*(f-I*c*f*x)^(7/2)-1/20*I*a*d/c*(f-I*c*f*x)^(5/2)*(d+I*c*d*x)^(1/2)- 
1/8*I*a*d*f/c*(f-I*c*f*x)^(3/2)*(d+I*c*d*x)^(1/2)-3/8*I*a*d*f^2/c*(f-I*c*f 
*x)^(1/2)*(d+I*c*d*x)^(1/2)+3/8*a*d^2*f^3*((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*(3/16*(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)^2*f^2*d-1/800*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)*(-1+5* 
arcsinh(x*c))*f^2*d/(c^2*x^2+1)/c+1/256*(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))*(-1+4*arcsinh(x*c))*f^2*d/(c^2*x^2+1)/ 
c-1/96*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)*(-1+3*arcsinh(x*c))*f^ 
2*d/(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))*(-1+2*arcsinh(x*c))* 
f^2*d/(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)-1)*f^2*d/(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)+1)*f^2*d/(c^2*x^2+1)/c+1/16*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c...
 

Fricas [F]

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

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

Output:

integral((-I*b*c^3*d*f^2*x^3 + b*c^2*d*f^2*x^2 - I*b*c*d*f^2*x + b*d*f^2)* 
sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1)) + (-I*a* 
c^3*d*f^2*x^3 + a*c^2*d*f^2*x^2 - I*a*c*d*f^2*x + a*d*f^2)*sqrt(I*c*d*x + 
d)*sqrt(-I*c*f*x + f), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

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

Giac [F(-2)]

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

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

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

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

Output:

(sqrt(f)*sqrt(d)*d*f**2*(30*asin(sqrt( - c*i*x + 1)/sqrt(2))*a*i - 8*sqrt( 
c*i*x + 1)*sqrt( - c*i*x + 1)*a*c**4*i*x**4 + 10*sqrt(c*i*x + 1)*sqrt( - c 
*i*x + 1)*a*c**3*x**3 - 16*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*a*c**2*i*x** 
2 + 25*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*a*c*x - 8*sqrt(c*i*x + 1)*sqrt( 
- c*i*x + 1)*a*i - 40*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh(c*x)*x* 
*3,x)*b*c**4*i + 40*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh(c*x)*x**2 
,x)*b*c**3 - 40*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh(c*x)*x,x)*b*c 
**2*i + 40*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh(c*x),x)*b*c))/(40* 
c)