\(\int \frac {(\pi +c^2 \pi x^2)^{5/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx\) [84]

Optimal result

Integrand size = 26, antiderivative size = 179 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {1}{16} b c^3 \pi ^{5/2} x^2+\frac {1}{4} b c^5 \pi ^{5/2} x^4-\frac {5}{16} b c \pi ^{5/2} \left (1+c^2 x^2\right )^2+\frac {15}{8} c^2 \pi ^2 x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {5}{4} c^2 \pi x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+\frac {15 c \pi ^{5/2} (a+b \text {arcsinh}(c x))^2}{16 b}+b c \pi ^{5/2} \log (x) \] Output:

1/16*b*c^3*Pi^(5/2)*x^2+1/4*b*c^5*Pi^(5/2)*x^4-5/16*b*c*Pi^(5/2)*(c^2*x^2+ 
1)^2+15/8*c^2*Pi^2*x*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(c*x))+5/4*c^2*Pi*x 
*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x))-(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsi 
nh(c*x))/x+15/16*c*Pi^(5/2)*(a+b*arcsinh(c*x))^2/b+b*c*Pi^(5/2)*ln(x)
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.94 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {\pi ^{5/2} \left (-128 a \sqrt {1+c^2 x^2}+144 a c^2 x^2 \sqrt {1+c^2 x^2}+32 a c^4 x^4 \sqrt {1+c^2 x^2}+120 b c x \text {arcsinh}(c x)^2-32 b c x \cosh (2 \text {arcsinh}(c x))-b c x \cosh (4 \text {arcsinh}(c x))+128 b c x \log (c x)+4 \text {arcsinh}(c x) \left (60 a c x-32 b \sqrt {1+c^2 x^2}+16 b c x \sinh (2 \text {arcsinh}(c x))+b c x \sinh (4 \text {arcsinh}(c x))\right )\right )}{128 x} \] Input:

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

Output:

(Pi^(5/2)*(-128*a*Sqrt[1 + c^2*x^2] + 144*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 32 
*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 120*b*c*x*ArcSinh[c*x]^2 - 32*b*c*x*Cosh[2* 
ArcSinh[c*x]] - b*c*x*Cosh[4*ArcSinh[c*x]] + 128*b*c*x*Log[c*x] + 4*ArcSin 
h[c*x]*(60*a*c*x - 32*b*Sqrt[1 + c^2*x^2] + 16*b*c*x*Sinh[2*ArcSinh[c*x]] 
+ b*c*x*Sinh[4*ArcSinh[c*x]])))/(128*x)
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6222, 243, 49, 2009, 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.

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

\(\Big \downarrow \) 6222

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6201

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6200

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6198

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

Input:

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

Output:

-(((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x) + 5*c^2*Pi*(-1/4*(b*c* 
Pi^(3/2)*(x^2/2 + (c^2*x^4)/4)) + (x*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSin 
h[c*x]))/4 + (3*Pi*(-1/4*(b*c*Sqrt[Pi]*x^2) + (x*Sqrt[Pi + c^2*Pi*x^2]*(a 
+ b*ArcSinh[c*x]))/2 + (Sqrt[Pi]*(a + b*ArcSinh[c*x])^2)/(4*b*c)))/4) + (b 
*c*Pi^(5/2)*(2*c^2*x^2 + (c^4*x^4)/2 + Log[x^2]))/2
 

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 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 6222

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

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.46

method	result	size
default	\(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{\pi x}+a \,c^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}+\frac {5 a \,c^{2} \pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {15 a \,c^{2} \pi ^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8}+\frac {15 a \,c^{2} \pi ^{3} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {5}{2}} \left (32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-8 x^{5} c^{5}+144 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-72 x^{3} c^{3}+120 \operatorname {arcsinh}\left (x c \right )^{2} x c +128 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -128 x c \,\operatorname {arcsinh}\left (x c \right )-128 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-33 x c \right )}{128 x}\)	\(261\)
parts	\(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{\pi x}+a \,c^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}+\frac {5 a \,c^{2} \pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {15 a \,c^{2} \pi ^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8}+\frac {15 a \,c^{2} \pi ^{3} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {5}{2}} \left (32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-8 x^{5} c^{5}+144 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-72 x^{3} c^{3}+120 \operatorname {arcsinh}\left (x c \right )^{2} x c +128 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -128 x c \,\operatorname {arcsinh}\left (x c \right )-128 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-33 x c \right )}{128 x}\)	\(261\)

Input:

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

Output:

-a/Pi/x*(Pi*c^2*x^2+Pi)^(7/2)+a*c^2*x*(Pi*c^2*x^2+Pi)^(5/2)+5/4*a*c^2*Pi*x 
*(Pi*c^2*x^2+Pi)^(3/2)+15/8*a*c^2*Pi^2*x*(Pi*c^2*x^2+Pi)^(1/2)+15/8*a*c^2* 
Pi^3*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+1/12 
8*b*Pi^(5/2)*(32*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^4*c^4-8*x^5*c^5+144*(c^2 
*x^2+1)^(1/2)*arcsinh(x*c)*x^2*c^2-72*x^3*c^3+120*arcsinh(x*c)^2*x*c+128*l 
n((x*c+(c^2*x^2+1)^(1/2))^2-1)*x*c-128*x*c*arcsinh(x*c)-128*arcsinh(x*c)*( 
c^2*x^2+1)^(1/2)-33*x*c)/x
 

Fricas [F]

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

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

Output:

integral(sqrt(pi + pi*c^2*x^2)*(pi^2*a*c^4*x^4 + 2*pi^2*a*c^2*x^2 + pi^2*a 
 + (pi^2*b*c^4*x^4 + 2*pi^2*b*c^2*x^2 + pi^2*b)*arcsinh(c*x))/x^2, x)
 

Sympy [F]

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

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

Output:

pi**(5/2)*(Integral(2*a*c**2*sqrt(c**2*x**2 + 1), x) + Integral(a*sqrt(c** 
2*x**2 + 1)/x**2, x) + Integral(a*c**4*x**2*sqrt(c**2*x**2 + 1), x) + Inte 
gral(2*b*c**2*sqrt(c**2*x**2 + 1)*asinh(c*x), x) + Integral(b*sqrt(c**2*x* 
*2 + 1)*asinh(c*x)/x**2, x) + Integral(b*c**4*x**2*sqrt(c**2*x**2 + 1)*asi 
nh(c*x), x))
 

Maxima [F(-2)]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {\sqrt {\pi }\, \pi ^{2} \left (2 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+9 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-8 \sqrt {c^{2} x^{2}+1}\, a +8 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{2}}d x \right ) b x +8 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{4} x +16 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) b \,c^{2} x +15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a c x -10 a c x \right )}{8 x} \] Input:

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

Output:

(sqrt(pi)*pi**2*(2*sqrt(c**2*x**2 + 1)*a*c**4*x**4 + 9*sqrt(c**2*x**2 + 1) 
*a*c**2*x**2 - 8*sqrt(c**2*x**2 + 1)*a + 8*int((sqrt(c**2*x**2 + 1)*asinh( 
c*x))/x**2,x)*b*x + 8*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*b*c**4*x 
+ 16*int(sqrt(c**2*x**2 + 1)*asinh(c*x),x)*b*c**2*x + 15*log(sqrt(c**2*x** 
2 + 1) + c*x)*a*c*x - 10*a*c*x))/(8*x)