\(\int \frac {(\pi +c^2 \pi x^2)^{3/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx\) [75]

Optimal result

Integrand size = 26, antiderivative size = 129 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=-\frac {b c \pi ^{3/2}}{42 x^6}-\frac {2 b c^3 \pi ^{3/2}}{35 x^4}-\frac {b c^5 \pi ^{3/2}}{70 x^2}-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{7 \pi x^7}+\frac {2 c^2 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{35 \pi x^5}-\frac {2}{35} b c^7 \pi ^{3/2} \log (x) \] Output:

-1/42*b*c*Pi^(3/2)/x^6-2/35*b*c^3*Pi^(3/2)/x^4-1/70*b*c^5*Pi^(3/2)/x^2-1/7 
*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(c*x))/Pi/x^7+2/35*c^2*(Pi*c^2*x^2+Pi)^ 
(5/2)*(a+b*arcsinh(c*x))/Pi/x^5-2/35*b*c^7*Pi^(3/2)*ln(x)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.30 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=-\frac {\pi ^{3/2} \left (25 b c x+60 b c^3 x^3+15 b c^5 x^5-147 b c^7 x^7+150 a \sqrt {1+c^2 x^2}+240 a c^2 x^2 \sqrt {1+c^2 x^2}+30 a c^4 x^4 \sqrt {1+c^2 x^2}-60 a c^6 x^6 \sqrt {1+c^2 x^2}-30 b \left (1+c^2 x^2\right )^{5/2} \left (-5+2 c^2 x^2\right ) \text {arcsinh}(c x)+60 b c^7 x^7 \log (x)\right )}{1050 x^7} \] Input:

Integrate[((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^8,x]
 

Output:

-1/1050*(Pi^(3/2)*(25*b*c*x + 60*b*c^3*x^3 + 15*b*c^5*x^5 - 147*b*c^7*x^7 
+ 150*a*Sqrt[1 + c^2*x^2] + 240*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 30*a*c^4*x^4 
*Sqrt[1 + c^2*x^2] - 60*a*c^6*x^6*Sqrt[1 + c^2*x^2] - 30*b*(1 + c^2*x^2)^( 
5/2)*(-5 + 2*c^2*x^2)*ArcSinh[c*x] + 60*b*c^7*x^7*Log[x]))/x^7
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.87, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6219, 27, 354, 85, 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.

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

\(\Big \downarrow \) 6219

\(\displaystyle -\sqrt {\pi } b c \int -\frac {\pi \left (5-2 c^2 x^2\right ) \left (c^2 x^2+1\right )^2}{35 x^7}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{7 \pi x^7}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{35 \pi x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{35} \pi ^{3/2} b c \int \frac {\left (5-2 c^2 x^2\right ) \left (c^2 x^2+1\right )^2}{x^7}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{7 \pi x^7}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{35 \pi x^5}\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{70} \pi ^{3/2} b c \int \frac {\left (5-2 c^2 x^2\right ) \left (c^2 x^2+1\right )^2}{x^8}dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{7 \pi x^7}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{35 \pi x^5}\)

\(\Big \downarrow \) 85

\(\displaystyle \frac {1}{70} \pi ^{3/2} b c \int \left (-\frac {2 c^6}{x^2}+\frac {c^4}{x^4}+\frac {8 c^2}{x^6}+\frac {5}{x^8}\right )dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{7 \pi x^7}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{35 \pi x^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{7 \pi x^7}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{35 \pi x^5}+\frac {1}{70} \pi ^{3/2} b c \left (-2 c^6 \log \left (x^2\right )-\frac {c^4}{x^2}-\frac {4 c^2}{x^4}-\frac {5}{3 x^6}\right )\)

Input:

Int[((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^8,x]
 

Output:

-1/7*((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(Pi*x^7) + (2*c^2*(Pi 
+ c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(35*Pi*x^5) + (b*c*Pi^(3/2)*(-5/ 
(3*x^6) - (4*c^2)/x^4 - c^4/x^2 - 2*c^6*Log[x^2]))/70
 

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 85

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : 
> Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, 
 d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* 
f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n 
 + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 
1])
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1827\) vs. \(2(105)=210\).

Time = 1.18 (sec) , antiderivative size = 1828, normalized size of antiderivative = 14.17

method	result	size
default	\(\text {Expression too large to display}\)	\(1828\)
parts	\(\text {Expression too large to display}\)	\(1828\)

Input:

int((Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(x*c))/x^8,x,method=_RETURNVERBOSE)
 

Output:

-2*b*Pi^(3/2)/(35*c^10*x^10+35*c^8*x^8-70*c^6*x^6-154*c^4*x^4-105*c^2*x^2- 
25)*x^10*arcsinh(x*c)*c^17+6/35*b*Pi^(3/2)/(35*c^10*x^10+35*c^8*x^8-70*c^6 
*x^6-154*c^4*x^4-105*c^2*x^2-25)*x^8*(c^2*x^2+1)*c^15-2*b*Pi^(3/2)/(35*c^1 
0*x^10+35*c^8*x^8-70*c^6*x^6-154*c^4*x^4-105*c^2*x^2-25)*x^8*arcsinh(x*c)* 
c^15-121/210*b*Pi^(3/2)/(35*c^10*x^10+35*c^8*x^8-70*c^6*x^6-154*c^4*x^4-10 
5*c^2*x^2-25)*x^6*(c^2*x^2+1)*c^13+4*b*Pi^(3/2)/(35*c^10*x^10+35*c^8*x^8-7 
0*c^6*x^6-154*c^4*x^4-105*c^2*x^2-25)*x^6*arcsinh(x*c)*c^13-202/105*b*Pi^( 
3/2)/(35*c^10*x^10+35*c^8*x^8-70*c^6*x^6-154*c^4*x^4-105*c^2*x^2-25)*x^4*( 
c^2*x^2+1)*c^11+44/5*b*Pi^(3/2)/(35*c^10*x^10+35*c^8*x^8-70*c^6*x^6-154*c^ 
4*x^4-105*c^2*x^2-25)*x^4*arcsinh(x*c)*c^11+19/210*b*Pi^(3/2)/(35*c^10*x^1 
0+35*c^8*x^8-70*c^6*x^6-154*c^4*x^4-105*c^2*x^2-25)*x^2*(c^2*x^2+1)*c^9+6* 
b*Pi^(3/2)/(35*c^10*x^10+35*c^8*x^8-70*c^6*x^6-154*c^4*x^4-105*c^2*x^2-25) 
*x^2*arcsinh(x*c)*c^9+281/42*b*Pi^(3/2)/(35*c^10*x^10+35*c^8*x^8-70*c^6*x^ 
6-154*c^4*x^4-105*c^2*x^2-25)/x^2*(c^2*x^2+1)*c^5+10/3*b*Pi^(3/2)/(35*c^10 
*x^10+35*c^8*x^8-70*c^6*x^6-154*c^4*x^4-105*c^2*x^2-25)/x^4*(c^2*x^2+1)*c^ 
3+1637/35*b*Pi^(3/2)/(35*c^10*x^10+35*c^8*x^8-70*c^6*x^6-154*c^4*x^4-105*c 
^2*x^2-25)/x*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^6+327/7*b*Pi^(3/2)/(35*c^10* 
x^10+35*c^8*x^8-70*c^6*x^6-154*c^4*x^4-105*c^2*x^2-25)/x^3*(c^2*x^2+1)^(1/ 
2)*arcsinh(x*c)*c^4+145/7*b*Pi^(3/2)/(35*c^10*x^10+35*c^8*x^8-70*c^6*x^6-1 
54*c^4*x^4-105*c^2*x^2-25)/x^5*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^2+2*b*P...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (105) = 210\).

Time = 0.17 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.37 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=\frac {6 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (2 \, \pi b c^{8} x^{8} + \pi b c^{6} x^{6} - 9 \, \pi b c^{4} x^{4} - 13 \, \pi b c^{2} x^{2} - 5 \, \pi b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 6 \, \sqrt {\pi } {\left (\pi b c^{9} x^{9} + \pi b c^{7} x^{7}\right )} \log \left (\frac {\pi + \pi c^{2} x^{6} + \pi c^{2} x^{2} + \pi x^{4} - \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )}}{c^{2} x^{4} + x^{2}}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (12 \, \pi a c^{8} x^{8} + 6 \, \pi a c^{6} x^{6} - 54 \, \pi a c^{4} x^{4} - 78 \, \pi a c^{2} x^{2} - 30 \, \pi a - {\left (3 \, \pi b c^{5} x^{5} - \pi {\left (3 \, b c^{5} + 12 \, b c^{3} + 5 \, b c\right )} x^{7} + 12 \, \pi b c^{3} x^{3} + 5 \, \pi b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{210 \, {\left (c^{2} x^{9} + x^{7}\right )}} \] Input:

integrate((pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x))/x^8,x, algorithm="frica 
s")
 

Output:

1/210*(6*sqrt(pi + pi*c^2*x^2)*(2*pi*b*c^8*x^8 + pi*b*c^6*x^6 - 9*pi*b*c^4 
*x^4 - 13*pi*b*c^2*x^2 - 5*pi*b)*log(c*x + sqrt(c^2*x^2 + 1)) + 6*sqrt(pi) 
*(pi*b*c^9*x^9 + pi*b*c^7*x^7)*log((pi + pi*c^2*x^6 + pi*c^2*x^2 + pi*x^4 
- sqrt(pi)*sqrt(pi + pi*c^2*x^2)*sqrt(c^2*x^2 + 1)*(x^4 - 1))/(c^2*x^4 + x 
^2)) + sqrt(pi + pi*c^2*x^2)*(12*pi*a*c^8*x^8 + 6*pi*a*c^6*x^6 - 54*pi*a*c 
^4*x^4 - 78*pi*a*c^2*x^2 - 30*pi*a - (3*pi*b*c^5*x^5 - pi*(3*b*c^5 + 12*b* 
c^3 + 5*b*c)*x^7 + 12*pi*b*c^3*x^3 + 5*pi*b*c*x)*sqrt(c^2*x^2 + 1)))/(c^2* 
x^9 + x^7)
 

Sympy [F]

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

integrate((pi*c**2*x**2+pi)**(3/2)*(a+b*asinh(c*x))/x**8,x)
 

Output:

pi**(3/2)*(Integral(a*sqrt(c**2*x**2 + 1)/x**8, x) + Integral(a*c**2*sqrt( 
c**2*x**2 + 1)/x**6, x) + Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**8, 
x) + Integral(b*c**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**6, x))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.13 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=-\frac {1}{210} \, {\left (12 \, \pi ^{\frac {3}{2}} c^{6} \log \left (x\right ) + \frac {3 \, \pi ^{\frac {3}{2}} c^{4} x^{4} + 12 \, \pi ^{\frac {3}{2}} c^{2} x^{2} + 5 \, \pi ^{\frac {3}{2}}}{x^{6}}\right )} b c + \frac {1}{35} \, b {\left (\frac {2 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} c^{2}}{\pi x^{5}} - \frac {5 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}}{\pi x^{7}}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{35} \, a {\left (\frac {2 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} c^{2}}{\pi x^{5}} - \frac {5 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}}{\pi x^{7}}\right )} \] Input:

integrate((pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x))/x^8,x, algorithm="maxim 
a")
 

Output:

-1/210*(12*pi^(3/2)*c^6*log(x) + (3*pi^(3/2)*c^4*x^4 + 12*pi^(3/2)*c^2*x^2 
 + 5*pi^(3/2))/x^6)*b*c + 1/35*b*(2*(pi + pi*c^2*x^2)^(5/2)*c^2/(pi*x^5) - 
 5*(pi + pi*c^2*x^2)^(5/2)/(pi*x^7))*arcsinh(c*x) + 1/35*a*(2*(pi + pi*c^2 
*x^2)^(5/2)*c^2/(pi*x^5) - 5*(pi + pi*c^2*x^2)^(5/2)/(pi*x^7))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x))/x^8,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. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}}{x^8} \,d x \] Input:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2))/x^8,x)
 

Output:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2))/x^8, x)
 

Reduce [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=\frac {\sqrt {\pi }\, \pi \left (2 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} x^{6}-\sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-8 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-5 \sqrt {c^{2} x^{2}+1}\, a +35 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{8}}d x \right ) b \,x^{7}+35 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{6}}d x \right ) b \,c^{2} x^{7}-2 a \,c^{7} x^{7}\right )}{35 x^{7}} \] Input:

int((Pi*c^2*x^2+Pi)^(3/2)*(a+b*asinh(c*x))/x^8,x)
 

Output:

(sqrt(pi)*pi*(2*sqrt(c**2*x**2 + 1)*a*c**6*x**6 - sqrt(c**2*x**2 + 1)*a*c* 
*4*x**4 - 8*sqrt(c**2*x**2 + 1)*a*c**2*x**2 - 5*sqrt(c**2*x**2 + 1)*a + 35 
*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**8,x)*b*x**7 + 35*int((sqrt(c**2*x 
**2 + 1)*asinh(c*x))/x**6,x)*b*c**2*x**7 - 2*a*c**7*x**7))/(35*x**7)