\(\int \frac {x^5 (a+b \text {arcsinh}(c x))}{(\pi +c^2 \pi x^2)^{3/2}} \, dx\) [100]

Optimal result

Integrand size = 26, antiderivative size = 137 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {5 b x}{3 c^5 \pi ^{3/2}}-\frac {b x^3}{9 c^3 \pi ^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{c^6 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{c^6 \pi ^2}+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6 \pi ^3}+\frac {b \arctan (c x)}{c^6 \pi ^{3/2}} \] Output:

5/3*b*x/c^5/Pi^(3/2)-1/9*b*x^3/c^3/Pi^(3/2)-(a+b*arcsinh(c*x))/c^6/Pi/(Pi* 
c^2*x^2+Pi)^(1/2)-2*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(c*x))/c^6/Pi^2+1/3* 
(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x))/c^6/Pi^3+b*arctan(c*x)/c^6/Pi^(3/ 
2)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-24 a-12 a c^2 x^2+3 a c^4 x^4+15 b c x \sqrt {1+c^2 x^2}-b c^3 x^3 \sqrt {1+c^2 x^2}+3 b \left (-8-4 c^2 x^2+c^4 x^4\right ) \text {arcsinh}(c x)+9 b \sqrt {1+c^2 x^2} \arctan (c x)}{9 c^6 \pi ^{3/2} \sqrt {1+c^2 x^2}} \] Input:

Integrate[(x^5*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]
 

Output:

(-24*a - 12*a*c^2*x^2 + 3*a*c^4*x^4 + 15*b*c*x*Sqrt[1 + c^2*x^2] - b*c^3*x 
^3*Sqrt[1 + c^2*x^2] + 3*b*(-8 - 4*c^2*x^2 + c^4*x^4)*ArcSinh[c*x] + 9*b*S 
qrt[1 + c^2*x^2]*ArcTan[c*x])/(9*c^6*Pi^(3/2)*Sqrt[1 + c^2*x^2])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6219, 27, 1467, 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.

\(\displaystyle \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi c^2 x^2+\pi \right )^{3/2}} \, dx\)

\(\Big \downarrow \) 6219

\(\displaystyle -\sqrt {\pi } b c \int -\frac {-c^4 x^4+4 c^2 x^2+8}{3 c^6 \pi ^2 \left (c^2 x^2+1\right )}dx+\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi ^3 c^6}-\frac {2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi ^2 c^6}-\frac {a+b \text {arcsinh}(c x)}{\pi c^6 \sqrt {\pi c^2 x^2+\pi }}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \int \frac {-c^4 x^4+4 c^2 x^2+8}{c^2 x^2+1}dx}{3 \pi ^{3/2} c^5}+\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi ^3 c^6}-\frac {2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi ^2 c^6}-\frac {a+b \text {arcsinh}(c x)}{\pi c^6 \sqrt {\pi c^2 x^2+\pi }}\)

\(\Big \downarrow \) 1467

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi ^3 c^6}-\frac {2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi ^2 c^6}-\frac {a+b \text {arcsinh}(c x)}{\pi c^6 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \left (\frac {3 \arctan (c x)}{c}-\frac {1}{3} c^2 x^3+5 x\right )}{3 \pi ^{3/2} c^5}\)

Input:

Int[(x^5*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]
 

Output:

-((a + b*ArcSinh[c*x])/(c^6*Pi*Sqrt[Pi + c^2*Pi*x^2])) - (2*Sqrt[Pi + c^2* 
Pi*x^2]*(a + b*ArcSinh[c*x]))/(c^6*Pi^2) + ((Pi + c^2*Pi*x^2)^(3/2)*(a + b 
*ArcSinh[c*x]))/(3*c^6*Pi^3) + (b*(5*x - (c^2*x^3)/3 + (3*ArcTan[c*x])/c)) 
/(3*c^5*Pi^(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 1467

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
 x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], 
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e 
 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 2009

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

rule 6219

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi 
nh[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[S 
implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) 
/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.34 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.04

method	result	size
default	\(a \left (\frac {x^{4}}{3 \pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {4 \left (\frac {x^{2}}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {2}{\pi \,c^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{3 c^{2}}\right )+\frac {i b \left (-3 i \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}+i x^{5} c^{5}+12 i \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-14 i x^{3} c^{3}+9 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) x^{2} c^{2}-9 \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right ) x^{2} c^{2}+24 i \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-15 i x c +9 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )-9 \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )\right )}{9 c^{6} \pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right )}\)	\(279\)
parts	\(a \left (\frac {x^{4}}{3 \pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {4 \left (\frac {x^{2}}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {2}{\pi \,c^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{3 c^{2}}\right )+\frac {i b \left (-3 i \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}+i x^{5} c^{5}+12 i \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-14 i x^{3} c^{3}+9 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) x^{2} c^{2}-9 \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right ) x^{2} c^{2}+24 i \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-15 i x c +9 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )-9 \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )\right )}{9 c^{6} \pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right )}\)	\(279\)

Input:

int(x^5*(a+b*arcsinh(x*c))/(Pi*c^2*x^2+Pi)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

a*(1/3*x^4/Pi/c^2/(Pi*c^2*x^2+Pi)^(1/2)-4/3/c^2*(x^2/Pi/c^2/(Pi*c^2*x^2+Pi 
)^(1/2)+2/Pi/c^4/(Pi*c^2*x^2+Pi)^(1/2)))+1/9*I*b*(-3*I*arcsinh(x*c)*(c^2*x 
^2+1)^(1/2)*x^4*c^4+I*x^5*c^5+12*I*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^2*c^2- 
14*I*x^3*c^3+9*ln(x*c+(c^2*x^2+1)^(1/2)+I)*x^2*c^2-9*ln(x*c+(c^2*x^2+1)^(1 
/2)-I)*x^2*c^2+24*I*arcsinh(x*c)*(c^2*x^2+1)^(1/2)-15*I*x*c+9*ln(x*c+(c^2* 
x^2+1)^(1/2)+I)-9*ln(x*c+(c^2*x^2+1)^(1/2)-I))/c^6/Pi^(3/2)/(c^2*x^2+1)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.43 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {9 \, \sqrt {\pi } {\left (b c^{2} x^{2} + b\right )} \arctan \left (-\frac {2 \, \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} c x}{\pi - \pi c^{4} x^{4}}\right ) - 6 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} - 4 \, b c^{2} x^{2} - 8 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} - 15 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 24 \, a\right )}}{18 \, {\left (\pi ^{2} c^{8} x^{2} + \pi ^{2} c^{6}\right )}} \] Input:

integrate(x^5*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="frica 
s")
 

Output:

-1/18*(9*sqrt(pi)*(b*c^2*x^2 + b)*arctan(-2*sqrt(pi)*sqrt(pi + pi*c^2*x^2) 
*sqrt(c^2*x^2 + 1)*c*x/(pi - pi*c^4*x^4)) - 6*sqrt(pi + pi*c^2*x^2)*(b*c^4 
*x^4 - 4*b*c^2*x^2 - 8*b)*log(c*x + sqrt(c^2*x^2 + 1)) - 2*sqrt(pi + pi*c^ 
2*x^2)*(3*a*c^4*x^4 - 12*a*c^2*x^2 - (b*c^3*x^3 - 15*b*c*x)*sqrt(c^2*x^2 + 
 1) - 24*a))/(pi^2*c^8*x^2 + pi^2*c^6)
 

Sympy [F]

\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a x^{5}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{5} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \] Input:

integrate(x**5*(a+b*asinh(c*x))/(pi*c**2*x**2+pi)**(3/2),x)
 

Output:

(Integral(a*x**5/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x) 
 + Integral(b*x**5*asinh(c*x)/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x 
**2 + 1)), x))/pi**(3/2)
 

Maxima [F]

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

integrate(x^5*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="maxim 
a")
 

Output:

1/3*a*(x^4/(pi*sqrt(pi + pi*c^2*x^2)*c^2) - 4*x^2/(pi*sqrt(pi + pi*c^2*x^2 
)*c^4) - 8/(pi*sqrt(pi + pi*c^2*x^2)*c^6)) + 1/3*b*((sqrt(pi)*c^4*x^4 - 4* 
sqrt(pi)*c^2*x^2 - 8*sqrt(pi))*log(c*x + sqrt(c^2*x^2 + 1))/(pi^2*sqrt(c^2 
*x^2 + 1)*c^6) - integrate((sqrt(pi)*c^4*x^4 - 4*sqrt(pi)*c^2*x^2 - 8*sqrt 
(pi))/(sqrt(c^2*x^2 + 1)*x), x)/(pi^2*c^6) + 3*integrate(1/3*(sqrt(pi)*c^4 
*x^4 - 4*sqrt(pi)*c^2*x^2 - 8*sqrt(pi))/(pi^2*c^9*x^4 + pi^2*c^7*x^2 + (pi 
^2*c^8*x^3 + pi^2*c^6*x)*sqrt(c^2*x^2 + 1)), x))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^5*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="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. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \] Input:

int((x^5*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(3/2),x)
 

Output:

int((x^5*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-8 \sqrt {c^{2} x^{2}+1}\, a +3 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{5}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{8} x^{2}+3 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{5}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{6}}{3 \sqrt {\pi }\, c^{6} \pi \left (c^{2} x^{2}+1\right )} \] Input:

int(x^5*(a+b*asinh(c*x))/(Pi*c^2*x^2+Pi)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

(sqrt(c**2*x**2 + 1)*a*c**4*x**4 - 4*sqrt(c**2*x**2 + 1)*a*c**2*x**2 - 8*s 
qrt(c**2*x**2 + 1)*a + 3*int((asinh(c*x)*x**5)/(sqrt(c**2*x**2 + 1)*c**2*x 
**2 + sqrt(c**2*x**2 + 1)),x)*b*c**8*x**2 + 3*int((asinh(c*x)*x**5)/(sqrt( 
c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**6)/(3*sqrt(pi)*c** 
6*pi*(c**2*x**2 + 1))