\(\int \frac {(\pi +c^2 \pi x^2)^{5/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx\) [86]

Optimal result

Integrand size = 26, antiderivative size = 166 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=-\frac {b c \pi ^{5/2}}{20 x^4}-\frac {11 b c^3 \pi ^{5/2}}{30 x^2}-\frac {c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x}-\frac {c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 x^5}+\frac {c^5 \pi ^{5/2} (a+b \text {arcsinh}(c x))^2}{2 b}+\frac {23}{15} b c^5 \pi ^{5/2} \log (x) \] Output:

-1/20*b*c*Pi^(5/2)/x^4-11/30*b*c^3*Pi^(5/2)/x^2-c^4*Pi^2*(Pi*c^2*x^2+Pi)^( 
1/2)*(a+b*arcsinh(c*x))/x-1/3*c^2*Pi*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c* 
x))/x^3-1/5*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(c*x))/x^5+1/2*c^5*Pi^(5/2)* 
(a+b*arcsinh(c*x))^2/b+23/15*b*c^5*Pi^(5/2)*ln(x)
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.99 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\frac {\pi ^{5/2} \left (-3 b c x-22 b c^3 x^3-12 a \sqrt {1+c^2 x^2}-44 a c^2 x^2 \sqrt {1+c^2 x^2}-92 a c^4 x^4 \sqrt {1+c^2 x^2}+4 \left (15 a c^5 x^5-b \sqrt {1+c^2 x^2} \left (3+11 c^2 x^2+23 c^4 x^4\right )\right ) \text {arcsinh}(c x)+30 b c^5 x^5 \text {arcsinh}(c x)^2+92 b c^5 x^5 \log (c x)\right )}{60 x^5} \] Input:

Integrate[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^6,x]
 

Output:

(Pi^(5/2)*(-3*b*c*x - 22*b*c^3*x^3 - 12*a*Sqrt[1 + c^2*x^2] - 44*a*c^2*x^2 
*Sqrt[1 + c^2*x^2] - 92*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 4*(15*a*c^5*x^5 - b* 
Sqrt[1 + c^2*x^2]*(3 + 11*c^2*x^2 + 23*c^4*x^4))*ArcSinh[c*x] + 30*b*c^5*x 
^5*ArcSinh[c*x]^2 + 92*b*c^5*x^5*Log[c*x]))/(60*x^5)
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.15, 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, 6222, 244, 2009, 6220, 14, 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.

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

\(\Big \downarrow \) 6222

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

\(\Big \downarrow \) 243

\(\displaystyle \pi c^2 \int \frac {\left (c^2 \pi x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x^4}dx+\frac {1}{10} \pi ^{5/2} b c \int \frac {\left (c^2 x^2+1\right )^2}{x^6}dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 x^5}\)

\(\Big \downarrow \) 49

\(\displaystyle \pi c^2 \int \frac {\left (c^2 \pi x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x^4}dx+\frac {1}{10} \pi ^{5/2} b c \int \left (\frac {c^4}{x^2}+\frac {2 c^2}{x^4}+\frac {1}{x^6}\right )dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 x^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle \pi c^2 \int \frac {\left (c^2 \pi x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x^4}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 x^5}+\frac {1}{10} \pi ^{5/2} b c \left (c^4 \log \left (x^2\right )-\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )\)

\(\Big \downarrow \) 6222

\(\displaystyle \pi c^2 \left (\pi c^2 \int \frac {\sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))}{x^2}dx+\frac {1}{3} \pi ^{3/2} b c \int \frac {c^2 x^2+1}{x^3}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 x^5}+\frac {1}{10} \pi ^{5/2} b c \left (c^4 \log \left (x^2\right )-\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )\)

\(\Big \downarrow \) 244

\(\displaystyle \pi c^2 \left (\pi c^2 \int \frac {\sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))}{x^2}dx+\frac {1}{3} \pi ^{3/2} b c \int \left (\frac {c^2}{x}+\frac {1}{x^3}\right )dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 x^5}+\frac {1}{10} \pi ^{5/2} b c \left (c^4 \log \left (x^2\right )-\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \pi c^2 \left (\pi c^2 \int \frac {\sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))}{x^2}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{3} \pi ^{3/2} b c \left (c^2 \log (x)-\frac {1}{2 x^2}\right )\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 x^5}+\frac {1}{10} \pi ^{5/2} b c \left (c^4 \log \left (x^2\right )-\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )\)

\(\Big \downarrow \) 6220

\(\displaystyle \pi c^2 \left (\pi c^2 \left (\sqrt {\pi } c^2 \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\sqrt {\pi } b c \int \frac {1}{x}dx-\frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{x}\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{3} \pi ^{3/2} b c \left (c^2 \log (x)-\frac {1}{2 x^2}\right )\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 x^5}+\frac {1}{10} \pi ^{5/2} b c \left (c^4 \log \left (x^2\right )-\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )\)

\(\Big \downarrow \) 14

\(\displaystyle \pi c^2 \left (\pi c^2 \left (\sqrt {\pi } c^2 \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{x}+\sqrt {\pi } b c \log (x)\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{3} \pi ^{3/2} b c \left (c^2 \log (x)-\frac {1}{2 x^2}\right )\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 x^5}+\frac {1}{10} \pi ^{5/2} b c \left (c^4 \log \left (x^2\right )-\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )\)

\(\Big \downarrow \) 6198

\(\displaystyle -\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 x^5}+\pi c^2 \left (\pi c^2 \left (-\frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{x}+\frac {\sqrt {\pi } c (a+b \text {arcsinh}(c x))^2}{2 b}+\sqrt {\pi } b c \log (x)\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{3} \pi ^{3/2} b c \left (c^2 \log (x)-\frac {1}{2 x^2}\right )\right )+\frac {1}{10} \pi ^{5/2} b c \left (c^4 \log \left (x^2\right )-\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )\)

Input:

Int[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^6,x]
 

Output:

-1/5*((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^5 + c^2*Pi*(-1/3*((P 
i + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^3 + (b*c*Pi^(3/2)*(-1/2*1/x^ 
2 + c^2*Log[x]))/3 + c^2*Pi*(-((Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) 
)/x) + (c*Sqrt[Pi]*(a + b*ArcSinh[c*x])^2)/(2*b) + b*c*Sqrt[Pi]*Log[x])) + 
 (b*c*Pi^(5/2)*(-1/2*1/x^4 - (2*c^2)/x^2 + c^4*Log[x^2]))/10
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

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 6220

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1355\) vs. \(2(140)=280\).

Time = 1.02 (sec) , antiderivative size = 1356, normalized size of antiderivative = 8.17

method	result	size
default	\(\text {Expression too large to display}\)	\(1356\)
parts	\(\text {Expression too large to display}\)	\(1356\)

Input:

int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(x*c))/x^6,x,method=_RETURNVERBOSE)
 

Output:

1587*b*Pi^(5/2)/(1035*c^8*x^8+765*c^6*x^6+325*c^4*x^4+75*c^2*x^2+9)*x^8*ar 
csinh(x*c)*c^13-5819/30*b*Pi^(5/2)/(1035*c^8*x^8+765*c^6*x^6+325*c^4*x^4+7 
5*c^2*x^2+9)*x^6*(c^2*x^2+1)*c^11+1173*b*Pi^(5/2)/(1035*c^8*x^8+765*c^6*x^ 
6+325*c^4*x^4+75*c^2*x^2+9)*x^6*arcsinh(x*c)*c^11-12719/60*b*Pi^(5/2)/(103 
5*c^8*x^8+765*c^6*x^6+325*c^4*x^4+75*c^2*x^2+9)*x^4*(c^2*x^2+1)*c^9+1495/3 
*b*Pi^(5/2)/(1035*c^8*x^8+765*c^6*x^6+325*c^4*x^4+75*c^2*x^2+9)*x^4*arcsin 
h(x*c)*c^9-1804/15*b*Pi^(5/2)/(1035*c^8*x^8+765*c^6*x^6+325*c^4*x^4+75*c^2 
*x^2+9)*x^2*(c^2*x^2+1)*c^7-1587*b*Pi^(5/2)/(1035*c^8*x^8+765*c^6*x^6+325* 
c^4*x^4+75*c^2*x^2+9)*x^7*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^12-1932*b*Pi^(5 
/2)/(1035*c^8*x^8+765*c^6*x^6+325*c^4*x^4+75*c^2*x^2+9)*x^5*(c^2*x^2+1)^(1 
/2)*arcsinh(x*c)*c^10-3799/3*b*Pi^(5/2)/(1035*c^8*x^8+765*c^6*x^6+325*c^4* 
x^4+75*c^2*x^2+9)*x^3*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^8-1519/3*b*Pi^(5/2) 
/(1035*c^8*x^8+765*c^6*x^6+325*c^4*x^4+75*c^2*x^2+9)*x*(c^2*x^2+1)^(1/2)*a 
rcsinh(x*c)*c^6-669/5*b*Pi^(5/2)/(1035*c^8*x^8+765*c^6*x^6+325*c^4*x^4+75* 
c^2*x^2+9)/x*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^4-108/5*b*Pi^(5/2)/(1035*c^8 
*x^8+765*c^6*x^6+325*c^4*x^4+75*c^2*x^2+9)/x^3*(c^2*x^2+1)^(1/2)*arcsinh(x 
*c)*c^2+8/15*a*c^6*x*(Pi*c^2*x^2+Pi)^(5/2)-1/5*a/Pi/x^5*(Pi*c^2*x^2+Pi)^(7 
/2)+1/2*b*Pi^(5/2)*c^5*arcsinh(x*c)^2-8/15*a*c^4/Pi/x*(Pi*c^2*x^2+Pi)^(7/2 
)+2/3*a*c^6*Pi*x*(Pi*c^2*x^2+Pi)^(3/2)+a*c^6*Pi^2*x*(Pi*c^2*x^2+Pi)^(1/2)+ 
a*c^6*Pi^3*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(...
 

Fricas [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{6}} \,d x } \] Input:

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^6,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^6, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\text {Timed out} \] Input:

integrate((pi*c**2*x**2+pi)**(5/2)*(a+b*asinh(c*x))/x**6,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^6,x, algorithm="maxim 
a")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^6,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. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}}{x^6} \,d x \] Input:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2))/x^6,x)
 

Output:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2))/x^6, x)
 

Reduce [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\frac {\sqrt {\pi }\, \pi ^{2} \left (-23 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-11 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, a +15 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{6}}d x \right ) b \,x^{5}+30 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{4}}d x \right ) b \,c^{2} x^{5}+15 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{2}}d x \right ) b \,c^{4} x^{5}+15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{5} x^{5}+5 a \,c^{5} x^{5}\right )}{15 x^{5}} \] Input:

int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*asinh(c*x))/x^6,x)
 

Output:

(sqrt(pi)*pi**2*( - 23*sqrt(c**2*x**2 + 1)*a*c**4*x**4 - 11*sqrt(c**2*x**2 
 + 1)*a*c**2*x**2 - 3*sqrt(c**2*x**2 + 1)*a + 15*int((sqrt(c**2*x**2 + 1)* 
asinh(c*x))/x**6,x)*b*x**5 + 30*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**4, 
x)*b*c**2*x**5 + 15*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**2,x)*b*c**4*x* 
*5 + 15*log(sqrt(c**2*x**2 + 1) + c*x)*a*c**5*x**5 + 5*a*c**5*x**5))/(15*x 
**5)