3.6.40 \(\int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) [540]

3.6.40.1 Optimal result

Integrand size = 35, antiderivative size = 459 \[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {i b d x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b c d x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (1+c^2 x^2\right )^{3/2}}-\frac {2 i b c^2 d x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{15 \left (1+c^2 x^2\right )^{3/2}}-\frac {b c^3 d x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (1+c^2 x^2\right )^{3/2}}-\frac {i b c^4 d x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{25 \left (1+c^2 x^2\right )^{3/2}}+\frac {1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))+\frac {3 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{8 \left (1+c^2 x^2\right )}+\frac {i d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{5 c}+\frac {3 d (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{16 b c \left (1+c^2 x^2\right )^{3/2}} \]

output

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

3.6.40.2 Mathematica [A] (verified)

Time = 3.17 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.49 \[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-1200 i b c d^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x}+1920 i a d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+6000 a c d^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+3840 i a c^2 d^2 f x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+2400 a c^3 d^2 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+1920 i a c^4 d^2 f x^4 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+1800 b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2-1200 b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))-75 b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (4 \text {arcsinh}(c x))+3600 a d^{5/2} f^{3/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )-200 i b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (3 \text {arcsinh}(c x))+60 b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (10 i \cosh (3 \text {arcsinh}(c x))+2 i \cosh (5 \text {arcsinh}(c x))+5 \left (4 i \sqrt {1+c^2 x^2}+8 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))\right )\right )-24 i b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (5 \text {arcsinh}(c x))}{9600 c \sqrt {1+c^2 x^2}} \]

input

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

output

((-1200*I)*b*c*d^2*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (1920*I)*a*d^ 
2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 6000*a*c*d^2*f 
*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^2*f*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^2*f*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^2*f*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] 
+ 1800*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 - 1200*b 
*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - 75*b*d^2 
*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 3600*a*d^(5/ 
2)*f^(3/2)*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^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x 
]*Sinh[3*ArcSinh[c*x]] + 60*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Ar 
cSinh[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^2*f*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])
 

3.6.40.3 Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.43, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6253

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

\(\Big \downarrow \) 2009

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

input

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

output

(d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*((-1/5*I)*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* 
ArcSinh[c*x])^2)/(16*b*c)))/(1 + c^2*x^2)^(3/2)
 

3.6.40.3.1 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]))
 

3.6.40.4 Maple [F]

\[\int \left (i c d x +d \right )^{\frac {5}{2}} \left (-i c f x +f \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )d x\]

input

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

output

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

3.6.40.5 Fricas [F]

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

input

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

output

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

3.6.40.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.6.40.7 Maxima [F(-2)]

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

input

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

output

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

3.6.40.8 Giac [F(-2)]

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

input

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

3.6.40.9 Mupad [F(-1)]

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

input

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

output

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