3.6.46 \(\int (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) [546]

3.6.46.1 Optimal result

Integrand size = 35, antiderivative size = 344 \[ \int (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {25 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{96 \left (1+c^2 x^2\right )^{5/2}}-\frac {5 b c^3 x^4 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{96 \left (1+c^2 x^2\right )^{5/2}}-\frac {b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt {1+c^2 x^2}}{36 c}+\frac {1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))}{16 \left (1+c^2 x^2\right )^2}+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))}{24 \left (1+c^2 x^2\right )}+\frac {5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{32 b c \left (1+c^2 x^2\right )^{5/2}} \]

output

-25/96*b*c*x^2*(d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(5/2)/(c^2*x^2+1)^(5/2)-5/96* 
b*c^3*x^4*(d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(5/2)/(c^2*x^2+1)^(5/2)+1/6*x*(d+I 
*c*d*x)^(5/2)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))+5/16*x*(d+I*c*d*x)^(5/2 
)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^2+5/24*x*(d+I*c*d*x)^(5 
/2)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)+5/32*(d+I*c*d*x)^(5/2 
)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))^2/b/c/(c^2*x^2+1)^(5/2)-1/36*b*(d+I 
*c*d*x)^(5/2)*(f-I*c*f*x)^(5/2)*(c^2*x^2+1)^(1/2)/c
 

3.6.46.2 Mathematica [A] (verified)

Time = 2.76 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.40 \[ \int (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1584 a c d^2 f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+1248 a c^3 d^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+384 a c^5 d^2 f^2 x^5 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+360 b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2-270 b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))-27 b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (4 \text {arcsinh}(c x))-2 b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (6 \text {arcsinh}(c x))+720 a d^{5/2} f^{5/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+12 b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) (45 \sinh (2 \text {arcsinh}(c x))+9 \sinh (4 \text {arcsinh}(c x))+\sinh (6 \text {arcsinh}(c x)))}{2304 c \sqrt {1+c^2 x^2}} \]

input

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

output

(1584*a*c*d^2*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] 
+ 1248*a*c^3*d^2*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2* 
x^2] + 384*a*c^5*d^2*f^2*x^5*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + 
c^2*x^2] + 360*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^ 
2 - 270*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] 
 - 27*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] - 
 2*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[6*ArcSinh[c*x]] + 72 
0*a*d^(5/2)*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]] + 12*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I 
*c*f*x]*ArcSinh[c*x]*(45*Sinh[2*ArcSinh[c*x]] + 9*Sinh[4*ArcSinh[c*x]] + S 
inh[6*ArcSinh[c*x]]))/(2304*c*Sqrt[1 + c^2*x^2])
 

3.6.46.3 Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.58, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {6211, 6201, 241, 6201, 244, 2009, 6200, 15, 6198}

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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 6201

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

\(\Big \downarrow \) 241

\(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{\left (c^2 x^2+1\right )^{5/2}}\)

\(\Big \downarrow \) 6201

\(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int x \left (c^2 x^2+1\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{\left (c^2 x^2+1\right )^{5/2}}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int \left (c^2 x^3+x\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{\left (c^2 x^2+1\right )^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{\left (c^2 x^2+1\right )^{5/2}}\)

\(\Big \downarrow \) 6200

\(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{\left (c^2 x^2+1\right )^{5/2}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{\left (c^2 x^2+1\right )^{5/2}}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{\left (c^2 x^2+1\right )^{5/2}}\)

input

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

output

((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(-1/36*(b*(1 + c^2*x^2)^3)/c + (x 
*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/6 + (5*(-1/4*(b*c*(x^2/2 + (c^2 
*x^4)/4)) + (x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/4 + (3*(-1/4*(b*c 
*x^2) + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/2 + (a + b*ArcSinh[c*x] 
)^2/(4*b*c)))/4))/6))/(1 + c^2*x^2)^(5/2)
 

3.6.46.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6200

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 
/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[(a + b*ArcSinh[c*x])^n/Sq 
rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* 
x^2]]   Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
 

rule 6201

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + 
(Simp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x 
], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[x* 
(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, 
b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 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]
 

3.6.46.4 Maple [F]

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

input

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

output

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

3.6.46.5 Fricas [F]

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

input

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

output

integral((b*c^4*d^2*f^2*x^4 + 2*b*c^2*d^2*f^2*x^2 + b*d^2*f^2)*sqrt(I*c*d* 
x + d)*sqrt(-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1)) + (a*c^4*d^2*f^2*x^ 
4 + 2*a*c^2*d^2*f^2*x^2 + a*d^2*f^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f), 
 x)
 

3.6.46.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.6.46.7 Maxima [F(-2)]

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

input

integrate((d+I*c*d*x)^(5/2)*(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.
 

3.6.46.8 Giac [F(-2)]

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

input

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

3.6.46.9 Mupad [F(-1)]

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

input

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

output

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