Integrand size = 26, antiderivative size = 174 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {4 b^2 \sqrt {d+c^2 d x^2}}{9 c^2}+\frac {2 b^2 \left (d+c^2 d x^2\right )^{3/2}}{27 c^2 d}-\frac {2 b x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c \sqrt {1+c^2 x^2}}-\frac {2 b c x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d} \] Output:
4/9*b^2*(c^2*d*x^2+d)^(1/2)/c^2+2/27*b^2*(c^2*d*x^2+d)^(3/2)/c^2/d-2/3*b*x *(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c/(c^2*x^2+1)^(1/2)-2/9*b*c*x^3*(c ^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)+1/3*(c^2*d*x^2+d)^( 3/2)*(a+b*arcsinh(c*x))^2/c^2/d
Time = 0.42 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.95 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d+c^2 d x^2} \left (-6 a b c x \sqrt {1+c^2 x^2} \left (3+c^2 x^2\right )+9 \left (a+a c^2 x^2\right )^2+2 b^2 \left (7+8 c^2 x^2+c^4 x^4\right )+6 b \left (3 a \left (1+c^2 x^2\right )^2-b c x \sqrt {1+c^2 x^2} \left (3+c^2 x^2\right )\right ) \text {arcsinh}(c x)+9 \left (b+b c^2 x^2\right )^2 \text {arcsinh}(c x)^2\right )}{27 c^2 \left (1+c^2 x^2\right )} \] Input:
Integrate[x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]
Output:
(Sqrt[d + c^2*d*x^2]*(-6*a*b*c*x*Sqrt[1 + c^2*x^2]*(3 + c^2*x^2) + 9*(a + a*c^2*x^2)^2 + 2*b^2*(7 + 8*c^2*x^2 + c^4*x^4) + 6*b*(3*a*(1 + c^2*x^2)^2 - b*c*x*Sqrt[1 + c^2*x^2]*(3 + c^2*x^2))*ArcSinh[c*x] + 9*(b + b*c^2*x^2)^ 2*ArcSinh[c*x]^2))/(27*c^2*(1 + c^2*x^2))
Time = 0.47 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.83, 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, 353, 53, 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.
| \(\displaystyle \int x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c^2 d x^2+d} \int \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{3 c \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6199 |
| \(\displaystyle \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c^2 d x^2+d} \left (-b c \int \frac {x \left (c^2 x^2+3\right )}{3 \sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 c \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c^2 d x^2+d} \left (-\frac {1}{3} b c \int \frac {x \left (c^2 x^2+3\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 c \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c^2 d x^2+d} \left (-\frac {1}{6} b c \int \frac {c^2 x^2+3}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 c \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c^2 d x^2+d} \left (-\frac {1}{6} b c \int \left (\sqrt {c^2 x^2+1}+\frac {2}{\sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 c \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 c \sqrt {c^2 x^2+1}}\) |
Input:
Int[x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]
Output:
((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(3*c^2*d) - (2*b*Sqrt[d + c ^2*d*x^2]*(-1/6*(b*c*((4*Sqrt[1 + c^2*x^2])/c^2 + (2*(1 + c^2*x^2)^(3/2))/ (3*c^2))) + x*(a + b*ArcSinh[c*x]) + (c^2*x^3*(a + b*ArcSinh[c*x]))/3))/(3 *c*Sqrt[1 + c^2*x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(421\) vs. \(2(150)=300\).
Time = 0.89 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.43
| method | result | size |
| orering | \(\frac {\left (19 c^{6} x^{6}+71 c^{4} x^{4}+48 c^{2} x^{2}+14\right ) \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{27 c^{4} \left (c^{2} x^{2}+1\right ) x^{2}}-\frac {2 \left (3 c^{4} x^{4}+16 c^{2} x^{2}+7\right ) \left (\sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d}{\sqrt {c^{2} d \,x^{2}+d}}+\frac {2 x \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{27 c^{4} x^{2}}+\frac {\left (c^{2} x^{2}+7\right ) \left (c^{2} x^{2}+1\right ) \left (\frac {3 \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d x}{\sqrt {c^{2} d \,x^{2}+d}}+\frac {4 \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}-\frac {x^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{4} d^{2}}{\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {4 x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{3} d b}{\sqrt {c^{2} d \,x^{2}+d}\, \sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} x \sqrt {c^{2} d \,x^{2}+d}\, c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{2} \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 c^{4} x}\) | \(422\) |
| default | \(\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (9 \operatorname {arcsinh}\left (x c \right )^{2}-6 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{216 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (9 \operatorname {arcsinh}\left (x c \right )^{2}+6 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{216 c^{2} \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}\right )\) | \(657\) |
| parts | \(\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (9 \operatorname {arcsinh}\left (x c \right )^{2}-6 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{216 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (9 \operatorname {arcsinh}\left (x c \right )^{2}+6 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{216 c^{2} \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}\right )\) | \(657\) |
Input:
int(x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
1/27*(19*c^6*x^6+71*c^4*x^4+48*c^2*x^2+14)/c^4/(c^2*x^2+1)/x^2*(c^2*d*x^2+ d)^(1/2)*(a+b*arcsinh(x*c))^2-2/27*(3*c^4*x^4+16*c^2*x^2+7)/c^4/x^2*((c^2* d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))^2+x^2/(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x *c))^2*c^2*d+2*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))*b*c/(c^2*x^2+1)^(1 /2))+1/27*(c^2*x^2+7)/c^4*(c^2*x^2+1)/x*(3/(c^2*d*x^2+d)^(1/2)*(a+b*arcsin h(x*c))^2*c^2*d*x+4*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))*b*c/(c^2*x^2+1) ^(1/2)-x^3/(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(x*c))^2*c^4*d^2+4*x^2/(c^2*d*x ^2+d)^(1/2)*(a+b*arcsinh(x*c))*c^3*d*b/(c^2*x^2+1)^(1/2)+2*b^2*x*(c^2*d*x^ 2+d)^(1/2)*c^2/(c^2*x^2+1)-2*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))*b* c^3/(c^2*x^2+1)^(3/2))
Time = 0.11 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.43 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {9 \, {\left (b^{2} c^{4} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (3 \, a b c^{4} x^{4} + 6 \, a b c^{2} x^{2} + 3 \, a b - {\left (b^{2} c^{3} x^{3} + 3 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} x^{4} + 2 \, {\left (9 \, a^{2} + 8 \, b^{2}\right )} c^{2} x^{2} + 9 \, a^{2} + 14 \, b^{2} - 6 \, {\left (a b c^{3} x^{3} + 3 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{27 \, {\left (c^{4} x^{2} + c^{2}\right )}} \] Input:
integrate(x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas" )
Output:
1/27*(9*(b^2*c^4*x^4 + 2*b^2*c^2*x^2 + b^2)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6*(3*a*b*c^4*x^4 + 6*a*b*c^2*x^2 + 3*a*b - (b^2*c^3 *x^3 + 3*b^2*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^ 2*x^2 + 1)) + ((9*a^2 + 2*b^2)*c^4*x^4 + 2*(9*a^2 + 8*b^2)*c^2*x^2 + 9*a^2 + 14*b^2 - 6*(a*b*c^3*x^3 + 3*a*b*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^4*x^2 + c^2)
\[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \] Input:
integrate(x*(c**2*d*x**2+d)**(1/2)*(a+b*asinh(c*x))**2,x)
Output:
Integral(x*sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))**2, x)
Time = 0.05 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.05 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {2}{27} \, b^{2} {\left (\frac {\sqrt {c^{2} x^{2} + 1} d^{\frac {3}{2}} x^{2} + \frac {7 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {3}{2}}}{c^{2}}}{d} - \frac {3 \, {\left (c^{2} d^{\frac {3}{2}} x^{3} + 3 \, d^{\frac {3}{2}} x\right )} \operatorname {arsinh}\left (c x\right )}{c d}\right )} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b^{2} \operatorname {arsinh}\left (c x\right )^{2}}{3 \, c^{2} d} + \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a b \operatorname {arsinh}\left (c x\right )}{3 \, c^{2} d} - \frac {2 \, {\left (c^{2} d^{\frac {3}{2}} x^{3} + 3 \, d^{\frac {3}{2}} x\right )} a b}{9 \, c d} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a^{2}}{3 \, c^{2} d} \] Input:
integrate(x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxima" )
Output:
2/27*b^2*((sqrt(c^2*x^2 + 1)*d^(3/2)*x^2 + 7*sqrt(c^2*x^2 + 1)*d^(3/2)/c^2 )/d - 3*(c^2*d^(3/2)*x^3 + 3*d^(3/2)*x)*arcsinh(c*x)/(c*d)) + 1/3*(c^2*d*x ^2 + d)^(3/2)*b^2*arcsinh(c*x)^2/(c^2*d) + 2/3*(c^2*d*x^2 + d)^(3/2)*a*b*a rcsinh(c*x)/(c^2*d) - 2/9*(c^2*d^(3/2)*x^3 + 3*d^(3/2)*x)*a*b/(c*d) + 1/3* (c^2*d*x^2 + d)^(3/2)*a^2/(c^2*d)
Exception generated. \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(c^2*d*x^2+d)^(1/2)*(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
Timed out. \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d\,c^2\,x^2+d} \,d x \] Input:
int(x*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(1/2),x)
Output:
int(x*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(1/2), x)
\[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d}\, \left (\sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, a^{2}+6 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) a b \,c^{2}+3 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x d x \right ) b^{2} c^{2}\right )}{3 c^{2}} \] Input:
int(x*(c^2*d*x^2+d)^(1/2)*(a+b*asinh(c*x))^2,x)
Output:
(sqrt(d)*(sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 + sqrt(c**2*x**2 + 1)*a**2 + 6*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)*a*b*c**2 + 3*int(sqrt(c**2*x**2 + 1)*asinh(c*x)**2*x,x)*b**2*c**2))/(3*c**2)