Integrand size = 26, antiderivative size = 254 \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b d x^2 \sqrt {d+c^2 d x^2}}{32 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c^3 \sqrt {1+c^2 x^2}} \]
1/6*x^3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))+1/16*d*x*(a+b*arcsinh(c*x)) *(c^2*d*x^2+d)^(1/2)/c^2+1/8*d*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)- 1/32*b*d*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-7/96*b*c*d*x^4*(c^2*d *x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/36*b*c^3*d*x^6*(c^2*d*x^2+d)^(1/2)/(c^2* x^2+1)^(1/2)-1/32*d*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c^3/(c^2*x^ 2+1)^(1/2)
Time = 0.64 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.99 \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {48 a c d x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (3+14 c^2 x^2+8 c^4 x^4\right )-144 a d^{3/2} \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-18 b d \sqrt {d+c^2 d x^2} \left (8 \text {arcsinh}(c x)^2+\cosh (4 \text {arcsinh}(c x))-4 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )+b d \sqrt {d+c^2 d x^2} \left (72 \text {arcsinh}(c x)^2+18 \cosh (2 \text {arcsinh}(c x))+9 \cosh (4 \text {arcsinh}(c x))-2 \cosh (6 \text {arcsinh}(c x))+12 \text {arcsinh}(c x) (-3 \sinh (2 \text {arcsinh}(c x))-3 \sinh (4 \text {arcsinh}(c x))+\sinh (6 \text {arcsinh}(c x)))\right )}{2304 c^3 \sqrt {1+c^2 x^2}} \]
(48*a*c*d*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(3 + 14*c^2*x^2 + 8*c^4* x^4) - 144*a*d^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d* x^2]] - 18*b*d*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x] ] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]) + b*d*Sqrt[d + c^2*d*x^2]*(72*Arc Sinh[c*x]^2 + 18*Cosh[2*ArcSinh[c*x]] + 9*Cosh[4*ArcSinh[c*x]] - 2*Cosh[6* ArcSinh[c*x]] + 12*ArcSinh[c*x]*(-3*Sinh[2*ArcSinh[c*x]] - 3*Sinh[4*ArcSin h[c*x]] + Sinh[6*ArcSinh[c*x]])))/(2304*c^3*Sqrt[1 + c^2*x^2])
Time = 0.91 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6223, 244, 2009, 6221, 15, 6227, 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.
| \(\displaystyle \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))dx-\frac {b c d \sqrt {c^2 d x^2+d} \int x^3 \left (c^2 x^2+1\right )dx}{6 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))dx-\frac {b c d \sqrt {c^2 d x^2+d} \int \left (c^2 x^5+x^3\right )dx}{6 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))dx+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right ) \sqrt {c^2 d x^2+d}}{6 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6221 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \int x^3dx}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right ) \sqrt {c^2 d x^2+d}}{6 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right ) \sqrt {c^2 d x^2+d}}{6 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right ) \sqrt {c^2 d x^2+d}}{6 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right ) \sqrt {c^2 d x^2+d}}{6 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{2} d \left (\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {\sqrt {c^2 d x^2+d} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 \sqrt {c^2 x^2+1}}-\frac {b c x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\right )-\frac {b c d \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right ) \sqrt {c^2 d x^2+d}}{6 \sqrt {c^2 x^2+1}}\) |
-1/6*(b*c*d*Sqrt[d + c^2*d*x^2]*(x^4/4 + (c^2*x^6)/6))/Sqrt[1 + c^2*x^2] + (x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/6 + (d*(-1/16*(b*c*x^4*S qrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (x^3*Sqrt[d + c^2*d*x^2]*(a + b*Ar cSinh[c*x]))/4 + (Sqrt[d + c^2*d*x^2]*(-1/4*(b*x^2)/c + (x*Sqrt[1 + c^2*x^ 2]*(a + b*ArcSinh[c*x]))/(2*c^2) - (a + b*ArcSinh[c*x])^2/(4*b*c^3)))/(4*S qrt[1 + c^2*x^2])))/2
3.2.29.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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]
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]
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 + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt [1 + c^2*x^2]] Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] , x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] I nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d , e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
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 + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f* x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(798\) vs. \(2(218)=436\).
Time = 0.22 (sec) , antiderivative size = 799, normalized size of antiderivative = 3.15
| method | result | size |
| default | \(\frac {a x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{6 c^{2} d}-\frac {a x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{24 c^{2}}-\frac {a d x \sqrt {c^{2} d \,x^{2}+d}}{16 c^{2}}-\frac {a \,d^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{16 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} d}{32 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (32 c^{7} x^{7}+32 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}+64 c^{5} x^{5}+48 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+38 c^{3} x^{3}+18 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+6 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+6 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{2304 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{512 c^{3} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{256 c^{3} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{512 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (32 c^{7} x^{7}-32 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}+64 c^{5} x^{5}-48 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+38 c^{3} x^{3}-18 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+6 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+6 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{2304 c^{3} \left (c^{2} x^{2}+1\right )}\right )\) | \(799\) |
| parts | \(\frac {a x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{6 c^{2} d}-\frac {a x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{24 c^{2}}-\frac {a d x \sqrt {c^{2} d \,x^{2}+d}}{16 c^{2}}-\frac {a \,d^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{16 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} d}{32 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (32 c^{7} x^{7}+32 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}+64 c^{5} x^{5}+48 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+38 c^{3} x^{3}+18 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+6 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+6 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{2304 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{512 c^{3} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{256 c^{3} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{512 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (32 c^{7} x^{7}-32 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}+64 c^{5} x^{5}-48 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+38 c^{3} x^{3}-18 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+6 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+6 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{2304 c^{3} \left (c^{2} x^{2}+1\right )}\right )\) | \(799\) |
1/6*a*x*(c^2*d*x^2+d)^(5/2)/c^2/d-1/24*a/c^2*x*(c^2*d*x^2+d)^(3/2)-1/16*a/ c^2*d*x*(c^2*d*x^2+d)^(1/2)-1/16*a/c^2*d^2*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d *x^2+d)^(1/2))/(c^2*d)^(1/2)+b*(-1/32*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1 /2)/c^3*arcsinh(c*x)^2*d+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7+32*c^6*x ^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5+48*c^4*x^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3+18 *c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x+(c^2*x^2+1)^(1/2))*(-1+6*arcsinh(c*x))*d/ c^3/(c^2*x^2+1)+1/512*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5+8*c^4*x^4*(c^2*x^2+ 1)^(1/2)+12*c^3*x^3+8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x+(c^2*x^2+1)^(1/2))*( -1+4*arcsinh(c*x))*d/c^3/(c^2*x^2+1)-1/256*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^ 3+2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x+(c^2*x^2+1)^(1/2))*(-1+2*arcsinh(c*x)) *d/c^3/(c^2*x^2+1)-1/256*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2*x^2*(c^2*x ^2+1)^(1/2)+2*c*x-(c^2*x^2+1)^(1/2))*(1+2*arcsinh(c*x))*d/c^3/(c^2*x^2+1)+ 1/512*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5-8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3* x^3-8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x-(c^2*x^2+1)^(1/2))*(1+4*arcsinh(c*x) )*d/c^3/(c^2*x^2+1)+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7-32*c^6*x^6*(c ^2*x^2+1)^(1/2)+64*c^5*x^5-48*c^4*x^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3-18*c^2* x^2*(c^2*x^2+1)^(1/2)+6*c*x-(c^2*x^2+1)^(1/2))*(1+6*arcsinh(c*x))*d/c^3/(c ^2*x^2+1))
\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2} \,d x } \]
integral((a*c^2*d*x^4 + a*d*x^2 + (b*c^2*d*x^4 + b*d*x^2)*arcsinh(c*x))*sq rt(c^2*d*x^2 + d), x)
\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^{2} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )\, dx \]
Exception generated. \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]