\(\int x^5 (\pi +c^2 \pi x^2)^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) [77]

Optimal result

Integrand size = 26, antiderivative size = 189 \[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {8 b \pi ^{5/2} x}{693 c^5}+\frac {4 b \pi ^{5/2} x^3}{2079 c^3}-\frac {b \pi ^{5/2} x^5}{1155 c}-\frac {113 b c \pi ^{5/2} x^7}{4851}-\frac {23}{891} b c^3 \pi ^{5/2} x^9-\frac {1}{121} b c^5 \pi ^{5/2} x^{11}+\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6 \pi }-\frac {2 \left (\pi +c^2 \pi x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6 \pi ^2}+\frac {\left (\pi +c^2 \pi x^2\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6 \pi ^3} \] Output:

-8/693*b*Pi^(5/2)*x/c^5+4/2079*b*Pi^(5/2)*x^3/c^3-1/1155*b*Pi^(5/2)*x^5/c- 
113/4851*b*c*Pi^(5/2)*x^7-23/891*b*c^3*Pi^(5/2)*x^9-1/121*b*c^5*Pi^(5/2)*x 
^11+1/7*(Pi*c^2*x^2+Pi)^(7/2)*(a+b*arcsinh(c*x))/c^6/Pi-2/9*(Pi*c^2*x^2+Pi 
)^(9/2)*(a+b*arcsinh(c*x))/c^6/Pi^2+1/11*(Pi*c^2*x^2+Pi)^(11/2)*(a+b*arcsi 
nh(c*x))/c^6/Pi^3
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.70 \[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\pi ^{5/2} \left (3465 a \left (1+c^2 x^2\right )^{7/2} \left (8-28 c^2 x^2+63 c^4 x^4\right )-b c x \left (27720-4620 c^2 x^2+2079 c^4 x^4+55935 c^6 x^6+61985 c^8 x^8+19845 c^{10} x^{10}\right )+3465 b \left (1+c^2 x^2\right )^{7/2} \left (8-28 c^2 x^2+63 c^4 x^4\right ) \text {arcsinh}(c x)\right )}{2401245 c^6} \] Input:

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

Output:

(Pi^(5/2)*(3465*a*(1 + c^2*x^2)^(7/2)*(8 - 28*c^2*x^2 + 63*c^4*x^4) - b*c* 
x*(27720 - 4620*c^2*x^2 + 2079*c^4*x^4 + 55935*c^6*x^6 + 61985*c^8*x^8 + 1 
9845*c^10*x^10) + 3465*b*(1 + c^2*x^2)^(7/2)*(8 - 28*c^2*x^2 + 63*c^4*x^4) 
*ArcSinh[c*x]))/(2401245*c^6)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.87, 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.

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

\(\Big \downarrow \) 6219

\(\displaystyle -\sqrt {\pi } b c \int \frac {\pi ^2 \left (c^2 x^2+1\right )^3 \left (63 c^4 x^4-28 c^2 x^2+8\right )}{693 c^6}dx+\frac {\left (\pi c^2 x^2+\pi \right )^{11/2} (a+b \text {arcsinh}(c x))}{11 \pi ^3 c^6}-\frac {2 \left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^2 c^6}+\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi c^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\pi ^{5/2} b \int \left (c^2 x^2+1\right )^3 \left (63 c^4 x^4-28 c^2 x^2+8\right )dx}{693 c^5}+\frac {\left (\pi c^2 x^2+\pi \right )^{11/2} (a+b \text {arcsinh}(c x))}{11 \pi ^3 c^6}-\frac {2 \left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^2 c^6}+\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi c^6}\)

\(\Big \downarrow \) 1467

\(\displaystyle -\frac {\pi ^{5/2} b \int \left (63 c^{10} x^{10}+161 c^8 x^8+113 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8\right )dx}{693 c^5}+\frac {\left (\pi c^2 x^2+\pi \right )^{11/2} (a+b \text {arcsinh}(c x))}{11 \pi ^3 c^6}-\frac {2 \left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^2 c^6}+\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi c^6}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (\pi c^2 x^2+\pi \right )^{11/2} (a+b \text {arcsinh}(c x))}{11 \pi ^3 c^6}-\frac {2 \left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^2 c^6}+\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi c^6}-\frac {\pi ^{5/2} b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\)

Input:

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

Output:

-1/693*(b*Pi^(5/2)*(8*x - (4*c^2*x^3)/3 + (3*c^4*x^5)/5 + (113*c^6*x^7)/7 
+ (161*c^8*x^9)/9 + (63*c^10*x^11)/11))/c^5 + ((Pi + c^2*Pi*x^2)^(7/2)*(a 
+ b*ArcSinh[c*x]))/(7*c^6*Pi) - (2*(Pi + c^2*Pi*x^2)^(9/2)*(a + b*ArcSinh[ 
c*x]))/(9*c^6*Pi^2) + ((Pi + c^2*Pi*x^2)^(11/2)*(a + b*ArcSinh[c*x]))/(11* 
c^6*Pi^3)
 

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 [A] (verified)

Time = 1.14 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.23

method	result	size
orering	\(\frac {\left (83349 c^{12} x^{12}+299047 c^{10} x^{10}+363737 c^{8} x^{8}+140481 c^{6} x^{6}-7854 c^{4} x^{4}+53592 c^{2} x^{2}+33264\right ) \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{480249 c^{6} \left (c^{2} x^{2}+1\right )^{3}}-\frac {\left (19845 c^{10} x^{10}+61985 c^{8} x^{8}+55935 c^{6} x^{6}+2079 c^{4} x^{4}-4620 c^{2} x^{2}+27720\right ) \left (5 x^{4} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+5 x^{6} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) \pi \,c^{2}+\frac {x^{5} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{2401245 x^{4} c^{6} \left (c^{2} x^{2}+1\right )^{2}}\)	\(233\)
default	\(a \left (\frac {x^{4} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{11 \pi \,c^{2}}-\frac {4 \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{9 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{63 \pi \,c^{4}}\right )}{11 c^{2}}\right )+\frac {b \,\pi ^{\frac {5}{2}} \left (218295 \,\operatorname {arcsinh}\left (x c \right ) x^{12} c^{12}+776160 \,\operatorname {arcsinh}\left (x c \right ) x^{10} c^{10}-19845 \sqrt {c^{2} x^{2}+1}\, x^{11} c^{11}+949410 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}-61985 x^{9} c^{9} \sqrt {c^{2} x^{2}+1}+401940 \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}-55935 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}-3465 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-2079 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+13860 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+4620 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+27720 \,\operatorname {arcsinh}\left (x c \right )-27720 \sqrt {c^{2} x^{2}+1}\, x c \right )}{2401245 c^{6} \sqrt {c^{2} x^{2}+1}}\)	\(286\)
parts	\(a \left (\frac {x^{4} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{11 \pi \,c^{2}}-\frac {4 \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{9 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{63 \pi \,c^{4}}\right )}{11 c^{2}}\right )+\frac {b \,\pi ^{\frac {5}{2}} \left (218295 \,\operatorname {arcsinh}\left (x c \right ) x^{12} c^{12}+776160 \,\operatorname {arcsinh}\left (x c \right ) x^{10} c^{10}-19845 \sqrt {c^{2} x^{2}+1}\, x^{11} c^{11}+949410 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}-61985 x^{9} c^{9} \sqrt {c^{2} x^{2}+1}+401940 \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}-55935 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}-3465 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-2079 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+13860 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+4620 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+27720 \,\operatorname {arcsinh}\left (x c \right )-27720 \sqrt {c^{2} x^{2}+1}\, x c \right )}{2401245 c^{6} \sqrt {c^{2} x^{2}+1}}\)	\(286\)

Input:

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

Output:

1/480249*(83349*c^12*x^12+299047*c^10*x^10+363737*c^8*x^8+140481*c^6*x^6-7 
854*c^4*x^4+53592*c^2*x^2+33264)/c^6/(c^2*x^2+1)^3*(Pi*c^2*x^2+Pi)^(5/2)*( 
a+b*arcsinh(x*c))-1/2401245/x^4*(19845*c^10*x^10+61985*c^8*x^8+55935*c^6*x 
^6+2079*c^4*x^4-4620*c^2*x^2+27720)/c^6/(c^2*x^2+1)^2*(5*x^4*(Pi*c^2*x^2+P 
i)^(5/2)*(a+b*arcsinh(x*c))+5*x^6*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(x*c)) 
*Pi*c^2+x^5*(Pi*c^2*x^2+Pi)^(5/2)*b*c/(c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.58 \[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {3465 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (63 \, \pi ^{2} b c^{12} x^{12} + 224 \, \pi ^{2} b c^{10} x^{10} + 274 \, \pi ^{2} b c^{8} x^{8} + 116 \, \pi ^{2} b c^{6} x^{6} - \pi ^{2} b c^{4} x^{4} + 4 \, \pi ^{2} b c^{2} x^{2} + 8 \, \pi ^{2} b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (218295 \, \pi ^{2} a c^{12} x^{12} + 776160 \, \pi ^{2} a c^{10} x^{10} + 949410 \, \pi ^{2} a c^{8} x^{8} + 401940 \, \pi ^{2} a c^{6} x^{6} - 3465 \, \pi ^{2} a c^{4} x^{4} + 13860 \, \pi ^{2} a c^{2} x^{2} + 27720 \, \pi ^{2} a - {\left (19845 \, \pi ^{2} b c^{11} x^{11} + 61985 \, \pi ^{2} b c^{9} x^{9} + 55935 \, \pi ^{2} b c^{7} x^{7} + 2079 \, \pi ^{2} b c^{5} x^{5} - 4620 \, \pi ^{2} b c^{3} x^{3} + 27720 \, \pi ^{2} b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{2401245 \, {\left (c^{8} x^{2} + c^{6}\right )}} \] Input:

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

Output:

1/2401245*(3465*sqrt(pi + pi*c^2*x^2)*(63*pi^2*b*c^12*x^12 + 224*pi^2*b*c^ 
10*x^10 + 274*pi^2*b*c^8*x^8 + 116*pi^2*b*c^6*x^6 - pi^2*b*c^4*x^4 + 4*pi^ 
2*b*c^2*x^2 + 8*pi^2*b)*log(c*x + sqrt(c^2*x^2 + 1)) + sqrt(pi + pi*c^2*x^ 
2)*(218295*pi^2*a*c^12*x^12 + 776160*pi^2*a*c^10*x^10 + 949410*pi^2*a*c^8* 
x^8 + 401940*pi^2*a*c^6*x^6 - 3465*pi^2*a*c^4*x^4 + 13860*pi^2*a*c^2*x^2 + 
 27720*pi^2*a - (19845*pi^2*b*c^11*x^11 + 61985*pi^2*b*c^9*x^9 + 55935*pi^ 
2*b*c^7*x^7 + 2079*pi^2*b*c^5*x^5 - 4620*pi^2*b*c^3*x^3 + 27720*pi^2*b*c*x 
)*sqrt(c^2*x^2 + 1)))/(c^8*x^2 + c^6)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.13 \[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{693} \, {\left (\frac {63 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} x^{4}}{\pi c^{2}} - \frac {28 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} x^{2}}{\pi c^{4}} + \frac {8 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}}}{\pi c^{6}}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{693} \, {\left (\frac {63 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} x^{4}}{\pi c^{2}} - \frac {28 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} x^{2}}{\pi c^{4}} + \frac {8 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}}}{\pi c^{6}}\right )} a - \frac {{\left (19845 \, \pi ^{\frac {5}{2}} c^{10} x^{11} + 61985 \, \pi ^{\frac {5}{2}} c^{8} x^{9} + 55935 \, \pi ^{\frac {5}{2}} c^{6} x^{7} + 2079 \, \pi ^{\frac {5}{2}} c^{4} x^{5} - 4620 \, \pi ^{\frac {5}{2}} c^{2} x^{3} + 27720 \, \pi ^{\frac {5}{2}} x\right )} b}{2401245 \, c^{5}} \] Input:

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

Output:

1/693*(63*(pi + pi*c^2*x^2)^(7/2)*x^4/(pi*c^2) - 28*(pi + pi*c^2*x^2)^(7/2 
)*x^2/(pi*c^4) + 8*(pi + pi*c^2*x^2)^(7/2)/(pi*c^6))*b*arcsinh(c*x) + 1/69 
3*(63*(pi + pi*c^2*x^2)^(7/2)*x^4/(pi*c^2) - 28*(pi + pi*c^2*x^2)^(7/2)*x^ 
2/(pi*c^4) + 8*(pi + pi*c^2*x^2)^(7/2)/(pi*c^6))*a - 1/2401245*(19845*pi^( 
5/2)*c^10*x^11 + 61985*pi^(5/2)*c^8*x^9 + 55935*pi^(5/2)*c^6*x^7 + 2079*pi 
^(5/2)*c^4*x^5 - 4620*pi^(5/2)*c^2*x^3 + 27720*pi^(5/2)*x)*b/c^5
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {\pi }\, \pi ^{2} \left (63 \sqrt {c^{2} x^{2}+1}\, a \,c^{10} x^{10}+161 \sqrt {c^{2} x^{2}+1}\, a \,c^{8} x^{8}+113 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} x^{6}+3 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+8 \sqrt {c^{2} x^{2}+1}\, a +693 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{9}d x \right ) b \,c^{10}+1386 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{7}d x \right ) b \,c^{8}+693 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{5}d x \right ) b \,c^{6}\right )}{693 c^{6}} \] Input:

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

Output:

(sqrt(pi)*pi**2*(63*sqrt(c**2*x**2 + 1)*a*c**10*x**10 + 161*sqrt(c**2*x**2 
 + 1)*a*c**8*x**8 + 113*sqrt(c**2*x**2 + 1)*a*c**6*x**6 + 3*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 + 693*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**9,x)*b*c**10 + 1386*int( 
sqrt(c**2*x**2 + 1)*asinh(c*x)*x**7,x)*b*c**8 + 693*int(sqrt(c**2*x**2 + 1 
)*asinh(c*x)*x**5,x)*b*c**6))/(693*c**6)