Integrand size = 24, antiderivative size = 212 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {5}{96} b^2 d^2 x^2+\frac {5 b^2 d^2 \left (1+c^2 x^2\right )^2}{288 c^2}+\frac {b^2 d^2 \left (1+c^2 x^2\right )^3}{108 c^2}-\frac {5 b d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{48 c}-\frac {5 b d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{72 c}-\frac {b d^2 x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{18 c}-\frac {5 d^2 (a+b \text {arcsinh}(c x))^2}{96 c^2}+\frac {d^2 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{6 c^2} \] Output:
5/96*b^2*d^2*x^2+5/288*b^2*d^2*(c^2*x^2+1)^2/c^2+1/108*b^2*d^2*(c^2*x^2+1) ^3/c^2-5/48*b*d^2*x*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c-5/72*b*d^2*x*(c ^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c-1/18*b*d^2*x*(c^2*x^2+1)^(5/2)*(a+b*a rcsinh(c*x))/c-5/96*d^2*(a+b*arcsinh(c*x))^2/c^2+1/6*d^2*(c^2*x^2+1)^3*(a+ b*arcsinh(c*x))^2/c^2
Time = 1.08 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.98 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \left (c x \left (144 a^2 c x \left (3+3 c^2 x^2+c^4 x^4\right )-6 a b \sqrt {1+c^2 x^2} \left (33+26 c^2 x^2+8 c^4 x^4\right )+b^2 c x \left (99+39 c^2 x^2+8 c^4 x^4\right )\right )+6 b \left (-b c x \sqrt {1+c^2 x^2} \left (33+26 c^2 x^2+8 c^4 x^4\right )+3 a \left (11+48 c^2 x^2+48 c^4 x^4+16 c^6 x^6\right )\right ) \text {arcsinh}(c x)+9 b^2 \left (11+48 c^2 x^2+48 c^4 x^4+16 c^6 x^6\right ) \text {arcsinh}(c x)^2\right )}{864 c^2} \] Input:
Integrate[x*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^2*(c*x*(144*a^2*c*x*(3 + 3*c^2*x^2 + c^4*x^4) - 6*a*b*Sqrt[1 + c^2*x^2] *(33 + 26*c^2*x^2 + 8*c^4*x^4) + b^2*c*x*(99 + 39*c^2*x^2 + 8*c^4*x^4)) + 6*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(33 + 26*c^2*x^2 + 8*c^4*x^4)) + 3*a*(11 + 48*c^2*x^2 + 48*c^4*x^4 + 16*c^6*x^6))*ArcSinh[c*x] + 9*b^2*(11 + 48*c^2*x ^2 + 48*c^4*x^4 + 16*c^6*x^6)*ArcSinh[c*x]^2))/(864*c^2)
Time = 0.89 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6213, 6201, 241, 6201, 244, 2009, 6200, 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.
| \(\displaystyle \int x \left (c^2 d x^2+d\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {d^2 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{6 c^2}-\frac {b d^2 \int \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx}{3 c}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle \frac {d^2 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{6 c^2}-\frac {b d^2 \left (\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {1}{6} b c \int x \left (c^2 x^2+1\right )^2dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))\right )}{3 c}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {d^2 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{6 c^2}-\frac {b d^2 \left (\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle \frac {d^2 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{6 c^2}-\frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int x \left (c^2 x^2+1\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {d^2 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{6 c^2}-\frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int \left (c^2 x^3+x\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{6 c^2}-\frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle \frac {d^2 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{6 c^2}-\frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {d^2 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{6 c^2}-\frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {d^2 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{6 c^2}-\frac {b d^2 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )}{3 c}\) |
Input:
Int[x*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^2*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(6*c^2) - (b*d^2*(-1/36*(b*(1 + c^2*x^2)^3)/c + (x*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/6 + (5*(-1 /4*(b*c*(x^2/2 + (c^2*x^4)/4)) + (x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x ]))/4 + (3*(-1/4*(b*c*x^2) + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/2 + (a + b*ArcSinh[c*x])^2/(4*b*c)))/4))/6))/(3*c)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -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_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x* (1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[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]
Time = 1.60 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.17
| method | result | size |
| derivativedivides | \(\frac {\frac {d^{2} a^{2} \left (c^{2} x^{2}+1\right )^{3}}{6}+b^{2} d^{2} \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{3}}{6}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{18}-\frac {5 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{72}-\frac {5 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{48}-\frac {5 \operatorname {arcsinh}\left (x c \right )^{2}}{96}+\frac {\left (c^{2} x^{2}+1\right )^{3}}{108}+\frac {5 \left (c^{2} x^{2}+1\right )^{2}}{288}+\frac {5 c^{2} x^{2}}{96}+\frac {5}{96}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{2}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {11 \,\operatorname {arcsinh}\left (x c \right )}{96}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}-\frac {5 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{144}-\frac {5 \sqrt {c^{2} x^{2}+1}\, x c}{96}\right )}{c^{2}}\) | \(247\) |
| default | \(\frac {\frac {d^{2} a^{2} \left (c^{2} x^{2}+1\right )^{3}}{6}+b^{2} d^{2} \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{3}}{6}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{18}-\frac {5 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{72}-\frac {5 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{48}-\frac {5 \operatorname {arcsinh}\left (x c \right )^{2}}{96}+\frac {\left (c^{2} x^{2}+1\right )^{3}}{108}+\frac {5 \left (c^{2} x^{2}+1\right )^{2}}{288}+\frac {5 c^{2} x^{2}}{96}+\frac {5}{96}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{2}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {11 \,\operatorname {arcsinh}\left (x c \right )}{96}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}-\frac {5 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{144}-\frac {5 \sqrt {c^{2} x^{2}+1}\, x c}{96}\right )}{c^{2}}\) | \(247\) |
| parts | \(\frac {d^{2} a^{2} \left (c^{2} x^{2}+1\right )^{3}}{6 c^{2}}+\frac {b^{2} d^{2} \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{3}}{6}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{18}-\frac {5 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{72}-\frac {5 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{48}-\frac {5 \operatorname {arcsinh}\left (x c \right )^{2}}{96}+\frac {\left (c^{2} x^{2}+1\right )^{3}}{108}+\frac {5 \left (c^{2} x^{2}+1\right )^{2}}{288}+\frac {5 c^{2} x^{2}}{96}+\frac {5}{96}\right )}{c^{2}}+\frac {2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{2}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {11 \,\operatorname {arcsinh}\left (x c \right )}{96}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}-\frac {5 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{144}-\frac {5 \sqrt {c^{2} x^{2}+1}\, x c}{96}\right )}{c^{2}}\) | \(252\) |
| orering | \(\frac {\left (728 c^{8} x^{8}+3251 c^{6} x^{6}+6466 c^{4} x^{4}+3177 c^{2} x^{2}+594\right ) \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{1728 c^{2} \left (c^{2} x^{2}+1\right )^{3}}-\frac {\left (120 c^{6} x^{6}+571 c^{4} x^{4}+1323 c^{2} x^{2}+396\right ) \left (\left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+4 x^{2} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {2 x \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{1728 c^{2} \left (c^{2} x^{2}+1\right )^{2}}+\frac {x \left (8 c^{4} x^{4}+39 c^{2} x^{2}+99\right ) \left (12 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d x +\frac {4 \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+8 x^{3} c^{4} d^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {16 x^{2} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{3} d b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x \left (c^{2} d \,x^{2}+d \right )^{2} b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{2} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{1728 c^{2} \left (c^{2} x^{2}+1\right )}\) | \(435\) |
Input:
int(x*(c^2*d*x^2+d)^2*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
1/c^2*(1/6*d^2*a^2*(c^2*x^2+1)^3+b^2*d^2*(1/6*arcsinh(x*c)^2*(c^2*x^2+1)^3 -1/18*arcsinh(x*c)*x*c*(c^2*x^2+1)^(5/2)-5/72*arcsinh(x*c)*x*c*(c^2*x^2+1) ^(3/2)-5/48*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x*c-5/96*arcsinh(x*c)^2+1/108*( c^2*x^2+1)^3+5/288*(c^2*x^2+1)^2+5/96*c^2*x^2+5/96)+2*d^2*a*b*(1/6*arcsinh (x*c)*x^6*c^6+1/2*arcsinh(x*c)*c^4*x^4+1/2*arcsinh(x*c)*c^2*x^2+11/96*arcs inh(x*c)-1/36*x*c*(c^2*x^2+1)^(5/2)-5/144*x*c*(c^2*x^2+1)^(3/2)-5/96*(c^2* x^2+1)^(1/2)*x*c))
Time = 0.11 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.45 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {8 \, {\left (18 \, a^{2} + b^{2}\right )} c^{6} d^{2} x^{6} + 3 \, {\left (144 \, a^{2} + 13 \, b^{2}\right )} c^{4} d^{2} x^{4} + 9 \, {\left (48 \, a^{2} + 11 \, b^{2}\right )} c^{2} d^{2} x^{2} + 9 \, {\left (16 \, b^{2} c^{6} d^{2} x^{6} + 48 \, b^{2} c^{4} d^{2} x^{4} + 48 \, b^{2} c^{2} d^{2} x^{2} + 11 \, b^{2} d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (48 \, a b c^{6} d^{2} x^{6} + 144 \, a b c^{4} d^{2} x^{4} + 144 \, a b c^{2} d^{2} x^{2} + 33 \, a b d^{2} - {\left (8 \, b^{2} c^{5} d^{2} x^{5} + 26 \, b^{2} c^{3} d^{2} x^{3} + 33 \, b^{2} c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (8 \, a b c^{5} d^{2} x^{5} + 26 \, a b c^{3} d^{2} x^{3} + 33 \, a b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}{864 \, c^{2}} \] Input:
integrate(x*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
1/864*(8*(18*a^2 + b^2)*c^6*d^2*x^6 + 3*(144*a^2 + 13*b^2)*c^4*d^2*x^4 + 9 *(48*a^2 + 11*b^2)*c^2*d^2*x^2 + 9*(16*b^2*c^6*d^2*x^6 + 48*b^2*c^4*d^2*x^ 4 + 48*b^2*c^2*d^2*x^2 + 11*b^2*d^2)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6*(4 8*a*b*c^6*d^2*x^6 + 144*a*b*c^4*d^2*x^4 + 144*a*b*c^2*d^2*x^2 + 33*a*b*d^2 - (8*b^2*c^5*d^2*x^5 + 26*b^2*c^3*d^2*x^3 + 33*b^2*c*d^2*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 6*(8*a*b*c^5*d^2*x^5 + 26*a*b*c^3*d^2 *x^3 + 33*a*b*c*d^2*x)*sqrt(c^2*x^2 + 1))/c^2
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (202) = 404\).
Time = 0.73 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.03 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{6}}{6} + \frac {a^{2} c^{2} d^{2} x^{4}}{2} + \frac {a^{2} d^{2} x^{2}}{2} + \frac {a b c^{4} d^{2} x^{6} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {a b c^{3} d^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{18} + a b c^{2} d^{2} x^{4} \operatorname {asinh}{\left (c x \right )} - \frac {13 a b c d^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{72} + a b d^{2} x^{2} \operatorname {asinh}{\left (c x \right )} - \frac {11 a b d^{2} x \sqrt {c^{2} x^{2} + 1}}{48 c} + \frac {11 a b d^{2} \operatorname {asinh}{\left (c x \right )}}{48 c^{2}} + \frac {b^{2} c^{4} d^{2} x^{6} \operatorname {asinh}^{2}{\left (c x \right )}}{6} + \frac {b^{2} c^{4} d^{2} x^{6}}{108} - \frac {b^{2} c^{3} d^{2} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{18} + \frac {b^{2} c^{2} d^{2} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {13 b^{2} c^{2} d^{2} x^{4}}{288} - \frac {13 b^{2} c d^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{72} + \frac {b^{2} d^{2} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {11 b^{2} d^{2} x^{2}}{96} - \frac {11 b^{2} d^{2} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{48 c} + \frac {11 b^{2} d^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{96 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{2} x^{2}}{2} & \text {otherwise} \end {cases} \] Input:
integrate(x*(c**2*d*x**2+d)**2*(a+b*asinh(c*x))**2,x)
Output:
Piecewise((a**2*c**4*d**2*x**6/6 + a**2*c**2*d**2*x**4/2 + a**2*d**2*x**2/ 2 + a*b*c**4*d**2*x**6*asinh(c*x)/3 - a*b*c**3*d**2*x**5*sqrt(c**2*x**2 + 1)/18 + a*b*c**2*d**2*x**4*asinh(c*x) - 13*a*b*c*d**2*x**3*sqrt(c**2*x**2 + 1)/72 + a*b*d**2*x**2*asinh(c*x) - 11*a*b*d**2*x*sqrt(c**2*x**2 + 1)/(48 *c) + 11*a*b*d**2*asinh(c*x)/(48*c**2) + b**2*c**4*d**2*x**6*asinh(c*x)**2 /6 + b**2*c**4*d**2*x**6/108 - b**2*c**3*d**2*x**5*sqrt(c**2*x**2 + 1)*asi nh(c*x)/18 + b**2*c**2*d**2*x**4*asinh(c*x)**2/2 + 13*b**2*c**2*d**2*x**4/ 288 - 13*b**2*c*d**2*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/72 + b**2*d**2*x* *2*asinh(c*x)**2/2 + 11*b**2*d**2*x**2/96 - 11*b**2*d**2*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(48*c) + 11*b**2*d**2*asinh(c*x)**2/(96*c**2), Ne(c, 0)), (a**2*d**2*x**2/2, True))
Leaf count of result is larger than twice the leaf count of optimal. 621 vs. \(2 (190) = 380\).
Time = 0.06 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.93 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx =\text {Too large to display} \] Input:
integrate(x*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
1/6*b^2*c^4*d^2*x^6*arcsinh(c*x)^2 + 1/6*a^2*c^4*d^2*x^6 + 1/2*b^2*c^2*d^2 *x^4*arcsinh(c*x)^2 + 1/2*a^2*c^2*d^2*x^4 + 1/144*(48*x^6*arcsinh(c*x) - ( 8*sqrt(c^2*x^2 + 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(c^2*x ^2 + 1)*x/c^6 - 15*arcsinh(c*x)/c^7)*c)*a*b*c^4*d^2 + 1/864*((8*x^6/c^2 - 15*x^4/c^4 + 45*x^2/c^6 - 45*log(c*x + sqrt(c^2*x^2 + 1))^2/c^8)*c^2 - 6*( 8*sqrt(c^2*x^2 + 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(c^2*x ^2 + 1)*x/c^6 - 15*arcsinh(c*x)/c^7)*c*arcsinh(c*x))*b^2*c^4*d^2 + 1/2*b^2 *d^2*x^2*arcsinh(c*x)^2 + 1/8*(8*x^4*arcsinh(c*x) - (2*sqrt(c^2*x^2 + 1)*x ^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c)*a*b*c^2*d^2 + 1/16*((x^4/c^2 - 3*x^2/c^4 + 3*log(c*x + sqrt(c^2*x^2 + 1))^2/c^6)*c^2 - 2 *(2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x) /c^5)*c*arcsinh(c*x))*b^2*c^2*d^2 + 1/2*a^2*d^2*x^2 + 1/2*(2*x^2*arcsinh(c *x) - c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^3))*a*b*d^2 + 1/4*(c^2*( x^2/c^2 - log(c*x + sqrt(c^2*x^2 + 1))^2/c^4) - 2*c*(sqrt(c^2*x^2 + 1)*x/c ^2 - arcsinh(c*x)/c^3)*arcsinh(c*x))*b^2*d^2
Exception generated. \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(c^2*d*x^2+d)^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. \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \] Input:
int(x*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^2,x)
Output:
int(x*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^2, x)
\[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^{2} \left (72 \mathit {asinh} \left (c x \right )^{2} b^{2} c^{2} x^{2}+36 \mathit {asinh} \left (c x \right )^{2} b^{2}-72 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) b^{2} c x +48 \mathit {asinh} \left (c x \right ) a b \,c^{6} x^{6}+144 \mathit {asinh} \left (c x \right ) a b \,c^{4} x^{4}+144 \mathit {asinh} \left (c x \right ) a b \,c^{2} x^{2}-8 \sqrt {c^{2} x^{2}+1}\, a b \,c^{5} x^{5}-26 \sqrt {c^{2} x^{2}+1}\, a b \,c^{3} x^{3}-33 \sqrt {c^{2} x^{2}+1}\, a b c x +144 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}+288 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}+33 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a b +24 a^{2} c^{6} x^{6}+72 a^{2} c^{4} x^{4}+72 a^{2} c^{2} x^{2}+36 b^{2} c^{2} x^{2}\right )}{144 c^{2}} \] Input:
int(x*(c^2*d*x^2+d)^2*(a+b*asinh(c*x))^2,x)
Output:
(d**2*(72*asinh(c*x)**2*b**2*c**2*x**2 + 36*asinh(c*x)**2*b**2 - 72*sqrt(c **2*x**2 + 1)*asinh(c*x)*b**2*c*x + 48*asinh(c*x)*a*b*c**6*x**6 + 144*asin h(c*x)*a*b*c**4*x**4 + 144*asinh(c*x)*a*b*c**2*x**2 - 8*sqrt(c**2*x**2 + 1 )*a*b*c**5*x**5 - 26*sqrt(c**2*x**2 + 1)*a*b*c**3*x**3 - 33*sqrt(c**2*x**2 + 1)*a*b*c*x + 144*int(asinh(c*x)**2*x**5,x)*b**2*c**6 + 288*int(asinh(c* x)**2*x**3,x)*b**2*c**4 + 33*log(sqrt(c**2*x**2 + 1) + c*x)*a*b + 24*a**2* c**6*x**6 + 72*a**2*c**4*x**4 + 72*a**2*c**2*x**2 + 36*b**2*c**2*x**2))/(1 44*c**2)