\(\int x (d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx\) [282]

Optimal result

Integrand size = 26, antiderivative size = 330 \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {32 b^2 d^2 \sqrt {d+c^2 d x^2}}{245 c^2}+\frac {16 b^2 d \left (d+c^2 d x^2\right )^{3/2}}{735 c^2}+\frac {12 b^2 \left (d+c^2 d x^2\right )^{5/2}}{1225 c^2}+\frac {2 b^2 \left (d+c^2 d x^2\right )^{7/2}}{343 c^2 d}-\frac {2 b d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{35 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{49 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))^2}{7 c^2 d} \] Output:

32/245*b^2*d^2*(c^2*d*x^2+d)^(1/2)/c^2+16/735*b^2*d*(c^2*d*x^2+d)^(3/2)/c^ 
2+12/1225*b^2*(c^2*d*x^2+d)^(5/2)/c^2+2/343*b^2*(c^2*d*x^2+d)^(7/2)/c^2/d- 
2/7*b*d^2*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c/(c^2*x^2+1)^(1/2)-2/7 
*b*c*d^2*x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)-6/35 
*b*c^3*d^2*x^5*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)-2/ 
49*b*c^5*d^2*x^7*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)+ 
1/7*(c^2*d*x^2+d)^(7/2)*(a+b*arcsinh(c*x))^2/c^2/d
 

Mathematica [A] (verified)

Time = 1.37 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.68 \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \sqrt {d+c^2 d x^2} \left (3675 a^2 \left (1+c^2 x^2\right )^4-210 a b c x \sqrt {1+c^2 x^2} \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )+2 b^2 \left (2161+2918 c^2 x^2+1108 c^4 x^4+426 c^6 x^6+75 c^8 x^8\right )+210 b \left (35 a \left (1+c^2 x^2\right )^4-b c x \sqrt {1+c^2 x^2} \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )\right ) \text {arcsinh}(c x)+3675 b^2 \left (1+c^2 x^2\right )^4 \text {arcsinh}(c x)^2\right )}{25725 c^2 \left (1+c^2 x^2\right )} \] Input:

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

Output:

(d^2*Sqrt[d + c^2*d*x^2]*(3675*a^2*(1 + c^2*x^2)^4 - 210*a*b*c*x*Sqrt[1 + 
c^2*x^2]*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) + 2*b^2*(2161 + 2918*c 
^2*x^2 + 1108*c^4*x^4 + 426*c^6*x^6 + 75*c^8*x^8) + 210*b*(35*a*(1 + c^2*x 
^2)^4 - b*c*x*Sqrt[1 + c^2*x^2]*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) 
)*ArcSinh[c*x] + 3675*b^2*(1 + c^2*x^2)^4*ArcSinh[c*x]^2))/(25725*c^2*(1 + 
 c^2*x^2))
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.67, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6213, 6199, 27, 2331, 2389, 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.

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 6199

\(\displaystyle \frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))^2}{7 c^2 d}-\frac {2 b d^2 \sqrt {c^2 d x^2+d} \left (-b c \int \frac {x \left (5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right )}{35 \sqrt {c^2 x^2+1}}dx+\frac {1}{7} c^6 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 x^5 (a+b \text {arcsinh}(c x))+c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{7 c \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))^2}{7 c^2 d}-\frac {2 b d^2 \sqrt {c^2 d x^2+d} \left (-\frac {1}{35} b c \int \frac {x \left (5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{7} c^6 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 x^5 (a+b \text {arcsinh}(c x))+c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{7 c \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2331

\(\displaystyle \frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))^2}{7 c^2 d}-\frac {2 b d^2 \sqrt {c^2 d x^2+d} \left (-\frac {1}{70} b c \int \frac {5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{7} c^6 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 x^5 (a+b \text {arcsinh}(c x))+c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{7 c \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2389

\(\displaystyle \frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))^2}{7 c^2 d}-\frac {2 b d^2 \sqrt {c^2 d x^2+d} \left (-\frac {1}{70} b c \int \left (5 \left (c^2 x^2+1\right )^{5/2}+6 \left (c^2 x^2+1\right )^{3/2}+8 \sqrt {c^2 x^2+1}+\frac {16}{\sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{7} c^6 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 x^5 (a+b \text {arcsinh}(c x))+c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{7 c \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))^2}{7 c^2 d}-\frac {2 b d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^6 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 x^5 (a+b \text {arcsinh}(c x))+c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^2}+\frac {12 \left (c^2 x^2+1\right )^{5/2}}{5 c^2}+\frac {16 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {32 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{7 c \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

((d + c^2*d*x^2)^(7/2)*(a + b*ArcSinh[c*x])^2)/(7*c^2*d) - (2*b*d^2*Sqrt[d 
 + c^2*d*x^2]*(-1/70*(b*c*((32*Sqrt[1 + c^2*x^2])/c^2 + (16*(1 + c^2*x^2)^ 
(3/2))/(3*c^2) + (12*(1 + c^2*x^2)^(5/2))/(5*c^2) + (10*(1 + c^2*x^2)^(7/2 
))/(7*c^2))) + x*(a + b*ArcSinh[c*x]) + c^2*x^3*(a + b*ArcSinh[c*x]) + (3* 
c^4*x^5*(a + b*ArcSinh[c*x]))/5 + (c^6*x^7*(a + b*ArcSinh[c*x]))/7))/(7*c* 
Sqrt[1 + c^2*x^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 2009

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

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

rule 2389

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand 
[Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p 
, 0] || EqQ[n, 1])
 

rule 6199

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb 
ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSinh[c*x])   u, 
 x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; 
FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
 

rule 6213

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

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.47

method	result	size
orering	\(\frac {\left (9525 c^{10} x^{10}+41691 c^{8} x^{8}+76515 c^{6} x^{6}+124979 c^{4} x^{4}+26152 c^{2} x^{2}+4322\right ) \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{25725 c^{4} \left (c^{2} x^{2}+1\right )^{3} x^{2}}-\frac {2 \left (675 c^{8} x^{8}+3108 c^{6} x^{6}+6352 c^{4} x^{4}+14480 c^{2} x^{2}+2161\right ) \left (\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+5 x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {2 x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{25725 c^{4} x^{2} \left (c^{2} x^{2}+1\right )^{2}}+\frac {\left (75 c^{6} x^{6}+351 c^{4} x^{4}+757 c^{2} x^{2}+2161\right ) \left (15 c^{2} d x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {4 \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+15 x^{3} \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{4} d^{2}+\frac {20 x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{3} d b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{25725 c^{4} \left (c^{2} x^{2}+1\right ) x}\)	\(485\)
default	\(\text {Expression too large to display}\)	\(1773\)
parts	\(\text {Expression too large to display}\)	\(1773\)

Input:

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

Output:

1/25725*(9525*c^10*x^10+41691*c^8*x^8+76515*c^6*x^6+124979*c^4*x^4+26152*c 
^2*x^2+4322)/c^4/(c^2*x^2+1)^3/x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^ 
2-2/25725*(675*c^8*x^8+3108*c^6*x^6+6352*c^4*x^4+14480*c^2*x^2+2161)/c^4/x 
^2/(c^2*x^2+1)^2*((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^2+5*x^2*(c^2*d*x^ 
2+d)^(3/2)*(a+b*arcsinh(x*c))^2*c^2*d+2*x*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh 
(x*c))*b*c/(c^2*x^2+1)^(1/2))+1/25725*(75*c^6*x^6+351*c^4*x^4+757*c^2*x^2+ 
2161)/c^4/(c^2*x^2+1)/x*(15*c^2*d*x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(x*c)) 
^2+4*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))*b*c/(c^2*x^2+1)^(1/2)+15*x^3*( 
c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))^2*c^4*d^2+20*x^2*(c^2*d*x^2+d)^(3/2) 
*(a+b*arcsinh(x*c))*c^3*d*b/(c^2*x^2+1)^(1/2)+2*x*(c^2*d*x^2+d)^(5/2)*b^2* 
c^2/(c^2*x^2+1)-2*x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))*b*c^3/(c^2*x^ 
2+1)^(3/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.35 \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {3675 \, {\left (b^{2} c^{8} d^{2} x^{8} + 4 \, b^{2} c^{6} d^{2} x^{6} + 6 \, b^{2} c^{4} d^{2} x^{4} + 4 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 210 \, {\left (35 \, a b c^{8} d^{2} x^{8} + 140 \, a b c^{6} d^{2} x^{6} + 210 \, a b c^{4} d^{2} x^{4} + 140 \, a b c^{2} d^{2} x^{2} + 35 \, a b d^{2} - {\left (5 \, b^{2} c^{7} d^{2} x^{7} + 21 \, b^{2} c^{5} d^{2} x^{5} + 35 \, b^{2} c^{3} d^{2} x^{3} + 35 \, b^{2} c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (75 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{8} d^{2} x^{8} + 12 \, {\left (1225 \, a^{2} + 71 \, b^{2}\right )} c^{6} d^{2} x^{6} + 2 \, {\left (11025 \, a^{2} + 1108 \, b^{2}\right )} c^{4} d^{2} x^{4} + 4 \, {\left (3675 \, a^{2} + 1459 \, b^{2}\right )} c^{2} d^{2} x^{2} + {\left (3675 \, a^{2} + 4322 \, b^{2}\right )} d^{2} - 210 \, {\left (5 \, a b c^{7} d^{2} x^{7} + 21 \, a b c^{5} d^{2} x^{5} + 35 \, a b c^{3} d^{2} x^{3} + 35 \, a b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{25725 \, {\left (c^{4} x^{2} + c^{2}\right )}} \] Input:

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

Output:

1/25725*(3675*(b^2*c^8*d^2*x^8 + 4*b^2*c^6*d^2*x^6 + 6*b^2*c^4*d^2*x^4 + 4 
*b^2*c^2*d^2*x^2 + b^2*d^2)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1 
))^2 + 210*(35*a*b*c^8*d^2*x^8 + 140*a*b*c^6*d^2*x^6 + 210*a*b*c^4*d^2*x^4 
 + 140*a*b*c^2*d^2*x^2 + 35*a*b*d^2 - (5*b^2*c^7*d^2*x^7 + 21*b^2*c^5*d^2* 
x^5 + 35*b^2*c^3*d^2*x^3 + 35*b^2*c*d^2*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x 
^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (75*(49*a^2 + 2*b^2)*c^8*d^2*x^8 + 
12*(1225*a^2 + 71*b^2)*c^6*d^2*x^6 + 2*(11025*a^2 + 1108*b^2)*c^4*d^2*x^4 
+ 4*(3675*a^2 + 1459*b^2)*c^2*d^2*x^2 + (3675*a^2 + 4322*b^2)*d^2 - 210*(5 
*a*b*c^7*d^2*x^7 + 21*a*b*c^5*d^2*x^5 + 35*a*b*c^3*d^2*x^3 + 35*a*b*c*d^2* 
x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^4*x^2 + c^2)
 

Sympy [F]

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

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

Output:

Integral(x*(d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.83 \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} b^{2} \operatorname {arsinh}\left (c x\right )^{2}}{7 \, c^{2} d} + \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a b \operatorname {arsinh}\left (c x\right )}{7 \, c^{2} d} + \frac {2}{25725} \, b^{2} {\left (\frac {75 \, \sqrt {c^{2} x^{2} + 1} c^{4} d^{\frac {7}{2}} x^{6} + 351 \, \sqrt {c^{2} x^{2} + 1} c^{2} d^{\frac {7}{2}} x^{4} + 757 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {7}{2}} x^{2} + \frac {2161 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {7}{2}}}{c^{2}}}{d} - \frac {105 \, {\left (5 \, c^{6} d^{\frac {7}{2}} x^{7} + 21 \, c^{4} d^{\frac {7}{2}} x^{5} + 35 \, c^{2} d^{\frac {7}{2}} x^{3} + 35 \, d^{\frac {7}{2}} x\right )} \operatorname {arsinh}\left (c x\right )}{c d}\right )} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a^{2}}{7 \, c^{2} d} - \frac {2 \, {\left (5 \, c^{6} d^{\frac {7}{2}} x^{7} + 21 \, c^{4} d^{\frac {7}{2}} x^{5} + 35 \, c^{2} d^{\frac {7}{2}} x^{3} + 35 \, d^{\frac {7}{2}} x\right )} a b}{245 \, c d} \] Input:

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

Output:

1/7*(c^2*d*x^2 + d)^(7/2)*b^2*arcsinh(c*x)^2/(c^2*d) + 2/7*(c^2*d*x^2 + d) 
^(7/2)*a*b*arcsinh(c*x)/(c^2*d) + 2/25725*b^2*((75*sqrt(c^2*x^2 + 1)*c^4*d 
^(7/2)*x^6 + 351*sqrt(c^2*x^2 + 1)*c^2*d^(7/2)*x^4 + 757*sqrt(c^2*x^2 + 1) 
*d^(7/2)*x^2 + 2161*sqrt(c^2*x^2 + 1)*d^(7/2)/c^2)/d - 105*(5*c^6*d^(7/2)* 
x^7 + 21*c^4*d^(7/2)*x^5 + 35*c^2*d^(7/2)*x^3 + 35*d^(7/2)*x)*arcsinh(c*x) 
/(c*d)) + 1/7*(c^2*d*x^2 + d)^(7/2)*a^2/(c^2*d) - 2/245*(5*c^6*d^(7/2)*x^7 
 + 21*c^4*d^(7/2)*x^5 + 35*c^2*d^(7/2)*x^3 + 35*d^(7/2)*x)*a*b/(c*d)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d}\, d^{2} \left (\sqrt {c^{2} x^{2}+1}\, a^{2} c^{6} x^{6}+3 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}+3 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, a^{2}+14 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{5}d x \right ) a b \,c^{6}+28 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) a b \,c^{4}+14 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) a b \,c^{2}+7 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}+14 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}+7 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x d x \right ) b^{2} c^{2}\right )}{7 c^{2}} \] Input:

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

Output:

(sqrt(d)*d**2*(sqrt(c**2*x**2 + 1)*a**2*c**6*x**6 + 3*sqrt(c**2*x**2 + 1)* 
a**2*c**4*x**4 + 3*sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 + sqrt(c**2*x**2 + 1 
)*a**2 + 14*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**5,x)*a*b*c**6 + 28*int(s 
qrt(c**2*x**2 + 1)*asinh(c*x)*x**3,x)*a*b*c**4 + 14*int(sqrt(c**2*x**2 + 1 
)*asinh(c*x)*x,x)*a*b*c**2 + 7*int(sqrt(c**2*x**2 + 1)*asinh(c*x)**2*x**5, 
x)*b**2*c**6 + 14*int(sqrt(c**2*x**2 + 1)*asinh(c*x)**2*x**3,x)*b**2*c**4 
+ 7*int(sqrt(c**2*x**2 + 1)*asinh(c*x)**2*x,x)*b**2*c**2))/(7*c**2)