Integrand size = 24, antiderivative size = 206 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {52 b^2 d x}{225 c^2}+\frac {26}{675} b^2 d x^3+\frac {2}{125} b^2 c^2 d x^5+\frac {8 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c^3}-\frac {4 b d x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c}+\frac {2 b d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{15 c^3}-\frac {2 b d \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c^3}+\frac {2}{15} d x^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2 \] Output:
-52/225*b^2*d*x/c^2+26/675*b^2*d*x^3+2/125*b^2*c^2*d*x^5+8/45*b*d*(c^2*x^2 +1)^(1/2)*(a+b*arcsinh(c*x))/c^3-4/45*b*d*x^2*(c^2*x^2+1)^(1/2)*(a+b*arcsi nh(c*x))/c+2/15*b*d*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c^3-2/25*b*d*(c^2 *x^2+1)^(5/2)*(a+b*arcsinh(c*x))/c^3+2/15*d*x^3*(a+b*arcsinh(c*x))^2+1/5*d *x^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2
Time = 0.17 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.86 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (225 a^2 c^3 x^3 \left (5+3 c^2 x^2\right )-30 a b \sqrt {1+c^2 x^2} \left (-26+13 c^2 x^2+9 c^4 x^4\right )+2 b^2 c x \left (-390+65 c^2 x^2+27 c^4 x^4\right )-30 b \left (-15 a c^3 x^3 \left (5+3 c^2 x^2\right )+b \sqrt {1+c^2 x^2} \left (-26+13 c^2 x^2+9 c^4 x^4\right )\right ) \text {arcsinh}(c x)+225 b^2 c^3 x^3 \left (5+3 c^2 x^2\right ) \text {arcsinh}(c x)^2\right )}{3375 c^3} \] Input:
Integrate[x^2*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2,x]
Output:
(d*(225*a^2*c^3*x^3*(5 + 3*c^2*x^2) - 30*a*b*Sqrt[1 + c^2*x^2]*(-26 + 13*c ^2*x^2 + 9*c^4*x^4) + 2*b^2*c*x*(-390 + 65*c^2*x^2 + 27*c^4*x^4) - 30*b*(- 15*a*c^3*x^3*(5 + 3*c^2*x^2) + b*Sqrt[1 + c^2*x^2]*(-26 + 13*c^2*x^2 + 9*c ^4*x^4))*ArcSinh[c*x] + 225*b^2*c^3*x^3*(5 + 3*c^2*x^2)*ArcSinh[c*x]^2))/( 3375*c^3)
Time = 1.27 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6223, 6191, 6219, 27, 2009, 6227, 15, 6213, 24}
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.
| \(\displaystyle \int x^2 \left (c^2 d x^2+d\right ) (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle -\frac {2}{5} b c d \int x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {2}{5} d \int x^2 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{5} d x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {2}{5} d \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{5} b c d \int x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{5} d x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle \frac {2}{5} d \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{5} b c d \left (-b c \int -\frac {-3 c^4 x^4-c^2 x^2+2}{15 c^4}dx+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}\right )+\frac {1}{5} d x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} d \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{5} b c d \left (\frac {b \int \left (-3 c^4 x^4-c^2 x^2+2\right )dx}{15 c^3}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}\right )+\frac {1}{5} d x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{5} d \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )+\frac {1}{5} d x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c d \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {2}{5} d \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {b \int x^2dx}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}\right )\right )+\frac {1}{5} d x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c d \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {2}{5} d \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )\right )+\frac {1}{5} d x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c d \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {2}{5} d \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )\right )+\frac {1}{5} d x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c d \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{5} d x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{5} d \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \left (\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )\right )-\frac {2}{5} b c d \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\) |
Input:
Int[x^2*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2,x]
Output:
(d*x^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/5 - (2*b*c*d*((b*(2*x - (c^2* x^3)/3 - (3*c^4*x^5)/5))/(15*c^3) - ((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c* x]))/(3*c^4) + ((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^4)))/5 + (2 *d*((x^3*(a + b*ArcSinh[c*x])^2)/3 - (2*b*c*(-1/9*(b*x^3)/c + (x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2) - (2*(-((b*x)/c) + (Sqrt[1 + c^2* x^2]*(a + b*ArcSinh[c*x]))/c^2))/(3*c^2)))/3))/5
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
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]
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])
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]
Time = 1.46 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.20
| method | result | size |
| parts | \(a^{2} d \left (\frac {1}{5} x^{5} c^{2}+\frac {1}{3} x^{3}\right )+\frac {b^{2} d \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{5}-\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c}{15}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{15}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}-\frac {856 x c}{3375}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {22 x c \left (c^{2} x^{2}+1\right )}{3375}+\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )}{c^{3}}+\frac {2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {13 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {c^{2} x^{2}+1}}{225}-\frac {x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{25}\right )}{c^{3}}\) | \(247\) |
| derivativedivides | \(\frac {a^{2} d \left (\frac {1}{5} x^{5} c^{5}+\frac {1}{3} x^{3} c^{3}\right )+b^{2} d \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{5}-\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c}{15}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{15}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}-\frac {856 x c}{3375}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {22 x c \left (c^{2} x^{2}+1\right )}{3375}+\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {13 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {c^{2} x^{2}+1}}{225}-\frac {x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{25}\right )}{c^{3}}\) | \(248\) |
| default | \(\frac {a^{2} d \left (\frac {1}{5} x^{5} c^{5}+\frac {1}{3} x^{3} c^{3}\right )+b^{2} d \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{5}-\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c}{15}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{15}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}-\frac {856 x c}{3375}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {22 x c \left (c^{2} x^{2}+1\right )}{3375}+\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {13 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {c^{2} x^{2}+1}}{225}-\frac {x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{25}\right )}{c^{3}}\) | \(248\) |
| orering | \(\frac {\left (1647 c^{8} x^{8}+4862 c^{6} x^{6}-4033 c^{4} x^{4}-7800 c^{2} x^{2}-3120\right ) \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{3375 x \,c^{4} \left (c^{2} x^{2}+1\right )^{2}}-\frac {\left (324 c^{6} x^{6}+893 c^{4} x^{4}-2665 c^{2} x^{2}-1950\right ) \left (2 x \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+2 x^{3} c^{2} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {2 x^{2} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{3375 x^{2} c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\left (27 c^{4} x^{4}+65 c^{2} x^{2}-390\right ) \left (2 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+10 x^{2} c^{2} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {8 x \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+\frac {8 x^{3} c^{3} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x^{2} \left (c^{2} d \,x^{2}+d \right ) b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{3} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{3375 x \,c^{4}}\) | \(400\) |
Input:
int(x^2*(c^2*d*x^2+d)*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
a^2*d*(1/5*x^5*c^2+1/3*x^3)+b^2*d/c^3*(1/5*arcsinh(x*c)^2*x*c*(c^2*x^2+1)^ 2-2/15*arcsinh(x*c)^2*x*c-1/15*arcsinh(x*c)^2*x*c*(c^2*x^2+1)-2/25*arcsinh (x*c)*(c^2*x^2+1)^(5/2)-856/3375*x*c+2/125*x*c*(c^2*x^2+1)^2+22/3375*x*c*( c^2*x^2+1)+4/15*arcsinh(x*c)*(c^2*x^2+1)^(1/2)+2/45*arcsinh(x*c)*(c^2*x^2+ 1)^(3/2))+2*a*b*d/c^3*(1/5*arcsinh(x*c)*x^5*c^5+1/3*arcsinh(x*c)*x^3*c^3-1 3/225*x^2*c^2*(c^2*x^2+1)^(1/2)+26/225*(c^2*x^2+1)^(1/2)-1/25*x^4*c^4*(c^2 *x^2+1)^(1/2))
Time = 0.09 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.09 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} d x^{5} + 5 \, {\left (225 \, a^{2} + 26 \, b^{2}\right )} c^{3} d x^{3} - 780 \, b^{2} c d x + 225 \, {\left (3 \, b^{2} c^{5} d x^{5} + 5 \, b^{2} c^{3} d x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (45 \, a b c^{5} d x^{5} + 75 \, a b c^{3} d x^{3} - {\left (9 \, b^{2} c^{4} d x^{4} + 13 \, b^{2} c^{2} d x^{2} - 26 \, b^{2} d\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 30 \, {\left (9 \, a b c^{4} d x^{4} + 13 \, a b c^{2} d x^{2} - 26 \, a b d\right )} \sqrt {c^{2} x^{2} + 1}}{3375 \, c^{3}} \] Input:
integrate(x^2*(c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
1/3375*(27*(25*a^2 + 2*b^2)*c^5*d*x^5 + 5*(225*a^2 + 26*b^2)*c^3*d*x^3 - 7 80*b^2*c*d*x + 225*(3*b^2*c^5*d*x^5 + 5*b^2*c^3*d*x^3)*log(c*x + sqrt(c^2* x^2 + 1))^2 + 30*(45*a*b*c^5*d*x^5 + 75*a*b*c^3*d*x^3 - (9*b^2*c^4*d*x^4 + 13*b^2*c^2*d*x^2 - 26*b^2*d)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 30*(9*a*b*c^4*d*x^4 + 13*a*b*c^2*d*x^2 - 26*a*b*d)*sqrt(c^2*x^2 + 1) )/c^3
Time = 0.59 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.52 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{2} d x^{5}}{5} + \frac {a^{2} d x^{3}}{3} + \frac {2 a b c^{2} d x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {2 a b c d x^{4} \sqrt {c^{2} x^{2} + 1}}{25} + \frac {2 a b d x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {26 a b d x^{2} \sqrt {c^{2} x^{2} + 1}}{225 c} + \frac {52 a b d \sqrt {c^{2} x^{2} + 1}}{225 c^{3}} + \frac {b^{2} c^{2} d x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {2 b^{2} c^{2} d x^{5}}{125} - \frac {2 b^{2} c d x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{25} + \frac {b^{2} d x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {26 b^{2} d x^{3}}{675} - \frac {26 b^{2} d x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{225 c} - \frac {52 b^{2} d x}{225 c^{2}} + \frac {52 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{3}}{3} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(c**2*d*x**2+d)*(a+b*asinh(c*x))**2,x)
Output:
Piecewise((a**2*c**2*d*x**5/5 + a**2*d*x**3/3 + 2*a*b*c**2*d*x**5*asinh(c* x)/5 - 2*a*b*c*d*x**4*sqrt(c**2*x**2 + 1)/25 + 2*a*b*d*x**3*asinh(c*x)/3 - 26*a*b*d*x**2*sqrt(c**2*x**2 + 1)/(225*c) + 52*a*b*d*sqrt(c**2*x**2 + 1)/ (225*c**3) + b**2*c**2*d*x**5*asinh(c*x)**2/5 + 2*b**2*c**2*d*x**5/125 - 2 *b**2*c*d*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/25 + b**2*d*x**3*asinh(c*x)* *2/3 + 26*b**2*d*x**3/675 - 26*b**2*d*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/ (225*c) - 52*b**2*d*x/(225*c**2) + 52*b**2*d*sqrt(c**2*x**2 + 1)*asinh(c*x )/(225*c**3), Ne(c, 0)), (a**2*d*x**3/3, True))
Time = 0.05 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.68 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{5} \, b^{2} c^{2} d x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{5} \, a^{2} c^{2} d x^{5} + \frac {1}{3} \, b^{2} d x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{2} d - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{2} d + \frac {1}{3} \, a^{2} d x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} d \] Input:
integrate(x^2*(c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
1/5*b^2*c^2*d*x^5*arcsinh(c*x)^2 + 1/5*a^2*c^2*d*x^5 + 1/3*b^2*d*x^3*arcsi nh(c*x)^2 + 2/75*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*s qrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*a*b*c^2*d - 2/1125* (15*(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^ 2*x^2 + 1)/c^6)*c*arcsinh(c*x) - (9*c^4*x^5 - 20*c^2*x^3 + 120*x)/c^4)*b^2 *c^2*d + 1/3*a^2*d*x^3 + 2/9*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^ 2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*d - 2/27*(3*c*(sqrt(c^2*x^2 + 1)*x^2 /c^2 - 2*sqrt(c^2*x^2 + 1)/c^4)*arcsinh(c*x) - (c^2*x^3 - 6*x)/c^2)*b^2*d
Exception generated. \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^2*(c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right ) \,d x \] Input:
int(x^2*(a + b*asinh(c*x))^2*(d + c^2*d*x^2),x)
Output:
int(x^2*(a + b*asinh(c*x))^2*(d + c^2*d*x^2), x)
\[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (90 \mathit {asinh} \left (c x \right ) a b \,c^{5} x^{5}+150 \mathit {asinh} \left (c x \right ) a b \,c^{3} x^{3}-18 \sqrt {c^{2} x^{2}+1}\, a b \,c^{4} x^{4}-26 \sqrt {c^{2} x^{2}+1}\, a b \,c^{2} x^{2}+52 \sqrt {c^{2} x^{2}+1}\, a b +225 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+225 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+45 a^{2} c^{5} x^{5}+75 a^{2} c^{3} x^{3}\right )}{225 c^{3}} \] Input:
int(x^2*(c^2*d*x^2+d)*(a+b*asinh(c*x))^2,x)
Output:
(d*(90*asinh(c*x)*a*b*c**5*x**5 + 150*asinh(c*x)*a*b*c**3*x**3 - 18*sqrt(c **2*x**2 + 1)*a*b*c**4*x**4 - 26*sqrt(c**2*x**2 + 1)*a*b*c**2*x**2 + 52*sq rt(c**2*x**2 + 1)*a*b + 225*int(asinh(c*x)**2*x**4,x)*b**2*c**5 + 225*int( asinh(c*x)**2*x**2,x)*b**2*c**3 + 45*a**2*c**5*x**5 + 75*a**2*c**3*x**3))/ (225*c**3)