Integrand size = 24, antiderivative size = 202 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {16 b d^3 \sqrt {1+c^2 x^2}}{315 c^3}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{3/2}}{945 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{5/2}}{525 c^3}+\frac {b d^3 \left (1+c^2 x^2\right )^{7/2}}{441 c^3}-\frac {b d^3 \left (1+c^2 x^2\right )^{9/2}}{81 c^3}+\frac {1}{3} d^3 x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^2 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{9} c^6 d^3 x^9 (a+b \text {arcsinh}(c x)) \] Output:
16/315*b*d^3*(c^2*x^2+1)^(1/2)/c^3+8/945*b*d^3*(c^2*x^2+1)^(3/2)/c^3+2/525 *b*d^3*(c^2*x^2+1)^(5/2)/c^3+1/441*b*d^3*(c^2*x^2+1)^(7/2)/c^3-1/81*b*d^3* (c^2*x^2+1)^(9/2)/c^3+1/3*d^3*x^3*(a+b*arcsinh(c*x))+3/5*c^2*d^3*x^5*(a+b* arcsinh(c*x))+3/7*c^4*d^3*x^7*(a+b*arcsinh(c*x))+1/9*c^6*d^3*x^9*(a+b*arcs inh(c*x))
Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.67 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^3 \left (315 a c^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right )-b \sqrt {1+c^2 x^2} \left (-5258+2629 c^2 x^2+6297 c^4 x^4+4675 c^6 x^6+1225 c^8 x^8\right )+315 b c^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right ) \text {arcsinh}(c x)\right )}{99225 c^3} \] Input:
Integrate[x^2*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]),x]
Output:
(d^3*(315*a*c^3*x^3*(105 + 189*c^2*x^2 + 135*c^4*x^4 + 35*c^6*x^6) - b*Sqr t[1 + c^2*x^2]*(-5258 + 2629*c^2*x^2 + 6297*c^4*x^4 + 4675*c^6*x^6 + 1225* c^8*x^8) + 315*b*c^3*x^3*(105 + 189*c^2*x^2 + 135*c^4*x^4 + 35*c^6*x^6)*Ar cSinh[c*x]))/(99225*c^3)
Time = 0.57 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6218, 27, 2331, 2123, 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.
| \(\displaystyle \int x^2 \left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6218 |
| \(\displaystyle -b c \int \frac {d^3 x^3 \left (35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right )}{315 \sqrt {c^2 x^2+1}}dx+\frac {1}{9} c^6 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^2 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{315} b c d^3 \int \frac {x^3 \left (35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{9} c^6 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^2 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {1}{630} b c d^3 \int \frac {x^2 \left (35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right )}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{9} c^6 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^2 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle -\frac {1}{630} b c d^3 \int \left (\frac {35 \left (c^2 x^2+1\right )^{7/2}}{c^2}-\frac {5 \left (c^2 x^2+1\right )^{5/2}}{c^2}-\frac {6 \left (c^2 x^2+1\right )^{3/2}}{c^2}-\frac {8 \sqrt {c^2 x^2+1}}{c^2}-\frac {16}{c^2 \sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{9} c^6 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^2 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{9} c^6 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^2 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c d^3 \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^4}-\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^4}-\frac {12 \left (c^2 x^2+1\right )^{5/2}}{5 c^4}-\frac {16 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {32 \sqrt {c^2 x^2+1}}{c^4}\right )\) |
Input:
Int[x^2*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]),x]
Output:
-1/630*(b*c*d^3*((-32*Sqrt[1 + c^2*x^2])/c^4 - (16*(1 + c^2*x^2)^(3/2))/(3 *c^4) - (12*(1 + c^2*x^2)^(5/2))/(5*c^4) - (10*(1 + c^2*x^2)^(7/2))/(7*c^4 ) + (70*(1 + c^2*x^2)^(9/2))/(9*c^4))) + (d^3*x^3*(a + b*ArcSinh[c*x]))/3 + (3*c^2*d^3*x^5*(a + b*ArcSinh[c*x]))/5 + (3*c^4*d^3*x^7*(a + b*ArcSinh[c *x]))/7 + (c^6*d^3*x^9*(a + b*ArcSinh[c*x]))/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(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, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Time = 0.43 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.91
| method | result | size |
| parts | \(d^{3} a \left (\frac {1}{9} c^{6} x^{9}+\frac {3}{7} c^{4} x^{7}+\frac {3}{5} x^{5} c^{2}+\frac {1}{3} x^{3}\right )+\frac {d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{9}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {2629 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{99225}+\frac {5258 \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2099 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{33075}-\frac {187 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {x^{8} c^{8} \sqrt {c^{2} x^{2}+1}}{81}\right )}{c^{3}}\) | \(183\) |
| derivativedivides | \(\frac {d^{3} a \left (\frac {1}{9} c^{9} x^{9}+\frac {3}{7} x^{7} c^{7}+\frac {3}{5} x^{5} c^{5}+\frac {1}{3} x^{3} c^{3}\right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{9}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {2629 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{99225}+\frac {5258 \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2099 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{33075}-\frac {187 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {x^{8} c^{8} \sqrt {c^{2} x^{2}+1}}{81}\right )}{c^{3}}\) | \(187\) |
| default | \(\frac {d^{3} a \left (\frac {1}{9} c^{9} x^{9}+\frac {3}{7} x^{7} c^{7}+\frac {3}{5} x^{5} c^{5}+\frac {1}{3} x^{3} c^{3}\right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{9}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {2629 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{99225}+\frac {5258 \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2099 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{33075}-\frac {187 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {x^{8} c^{8} \sqrt {c^{2} x^{2}+1}}{81}\right )}{c^{3}}\) | \(187\) |
| orering | \(\frac {\left (20825 c^{10} x^{10}+82375 c^{8} x^{8}+119261 c^{6} x^{6}+66701 c^{4} x^{4}-36806 c^{2} x^{2}-10516\right ) \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{99225 c^{4} \left (c^{2} x^{2}+1\right )^{3} x}-\frac {\left (1225 c^{8} x^{8}+4675 c^{6} x^{6}+6297 c^{4} x^{4}+2629 c^{2} x^{2}-5258\right ) \left (2 x \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+6 x^{3} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d +\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{3} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{99225 c^{4} \left (c^{2} x^{2}+1\right )^{2} x^{2}}\) | \(218\) |
Input:
int(x^2*(c^2*d*x^2+d)^3*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
d^3*a*(1/9*c^6*x^9+3/7*c^4*x^7+3/5*x^5*c^2+1/3*x^3)+d^3*b/c^3*(1/9*arcsinh (x*c)*x^9*c^9+3/7*arcsinh(x*c)*x^7*c^7+3/5*arcsinh(x*c)*x^5*c^5+1/3*arcsin h(x*c)*x^3*c^3-2629/99225*x^2*c^2*(c^2*x^2+1)^(1/2)+5258/99225*(c^2*x^2+1) ^(1/2)-2099/33075*x^4*c^4*(c^2*x^2+1)^(1/2)-187/3969*x^6*c^6*(c^2*x^2+1)^( 1/2)-1/81*x^8*c^8*(c^2*x^2+1)^(1/2))
Time = 0.09 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.94 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {11025 \, a c^{9} d^{3} x^{9} + 42525 \, a c^{7} d^{3} x^{7} + 59535 \, a c^{5} d^{3} x^{5} + 33075 \, a c^{3} d^{3} x^{3} + 315 \, {\left (35 \, b c^{9} d^{3} x^{9} + 135 \, b c^{7} d^{3} x^{7} + 189 \, b c^{5} d^{3} x^{5} + 105 \, b c^{3} d^{3} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (1225 \, b c^{8} d^{3} x^{8} + 4675 \, b c^{6} d^{3} x^{6} + 6297 \, b c^{4} d^{3} x^{4} + 2629 \, b c^{2} d^{3} x^{2} - 5258 \, b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{99225 \, c^{3}} \] Input:
integrate(x^2*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="fricas")
Output:
1/99225*(11025*a*c^9*d^3*x^9 + 42525*a*c^7*d^3*x^7 + 59535*a*c^5*d^3*x^5 + 33075*a*c^3*d^3*x^3 + 315*(35*b*c^9*d^3*x^9 + 135*b*c^7*d^3*x^7 + 189*b*c ^5*d^3*x^5 + 105*b*c^3*d^3*x^3)*log(c*x + sqrt(c^2*x^2 + 1)) - (1225*b*c^8 *d^3*x^8 + 4675*b*c^6*d^3*x^6 + 6297*b*c^4*d^3*x^4 + 2629*b*c^2*d^3*x^2 - 5258*b*d^3)*sqrt(c^2*x^2 + 1))/c^3
Time = 1.18 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.31 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{6} d^{3} x^{9}}{9} + \frac {3 a c^{4} d^{3} x^{7}}{7} + \frac {3 a c^{2} d^{3} x^{5}}{5} + \frac {a d^{3} x^{3}}{3} + \frac {b c^{6} d^{3} x^{9} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {b c^{5} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1}}{81} + \frac {3 b c^{4} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {187 b c^{3} d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{3969} + \frac {3 b c^{2} d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {2099 b c d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{33075} + \frac {b d^{3} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {2629 b d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{99225 c} + \frac {5258 b d^{3} \sqrt {c^{2} x^{2} + 1}}{99225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{3}}{3} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(c**2*d*x**2+d)**3*(a+b*asinh(c*x)),x)
Output:
Piecewise((a*c**6*d**3*x**9/9 + 3*a*c**4*d**3*x**7/7 + 3*a*c**2*d**3*x**5/ 5 + a*d**3*x**3/3 + b*c**6*d**3*x**9*asinh(c*x)/9 - b*c**5*d**3*x**8*sqrt( c**2*x**2 + 1)/81 + 3*b*c**4*d**3*x**7*asinh(c*x)/7 - 187*b*c**3*d**3*x**6 *sqrt(c**2*x**2 + 1)/3969 + 3*b*c**2*d**3*x**5*asinh(c*x)/5 - 2099*b*c*d** 3*x**4*sqrt(c**2*x**2 + 1)/33075 + b*d**3*x**3*asinh(c*x)/3 - 2629*b*d**3* x**2*sqrt(c**2*x**2 + 1)/(99225*c) + 5258*b*d**3*sqrt(c**2*x**2 + 1)/(9922 5*c**3), Ne(c, 0)), (a*d**3*x**3/3, True))
Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (174) = 348\).
Time = 0.04 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.92 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{9} \, a c^{6} d^{3} x^{9} + \frac {3}{7} \, a c^{4} d^{3} x^{7} + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{6} d^{3} + \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {3}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{4} d^{3} + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{3} \, a d^{3} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{3} \] Input:
integrate(x^2*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="maxima")
Output:
1/9*a*c^6*d^3*x^9 + 3/7*a*c^4*d^3*x^7 + 1/2835*(315*x^9*arcsinh(c*x) - (35 *sqrt(c^2*x^2 + 1)*x^8/c^2 - 40*sqrt(c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(c^2*x^ 2 + 1)*x^4/c^6 - 64*sqrt(c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(c^2*x^2 + 1)/c^10 )*c)*b*c^6*d^3 + 3/5*a*c^2*d^3*x^5 + 3/245*(35*x^7*arcsinh(c*x) - (5*sqrt( c^2*x^2 + 1)*x^6/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x ^2/c^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c)*b*c^4*d^3 + 1/25*(15*x^5*arcsinh(c*x ) - (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)*b*c^2*d^3 + 1/3*a*d^3*x^3 + 1/9*(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))*b*d^3
Exception generated. \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^2*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),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. \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \] Input:
int(x^2*(a + b*asinh(c*x))*(d + c^2*d*x^2)^3,x)
Output:
int(x^2*(a + b*asinh(c*x))*(d + c^2*d*x^2)^3, x)
Time = 0.19 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.92 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^{3} \left (11025 \mathit {asinh} \left (c x \right ) b \,c^{9} x^{9}+42525 \mathit {asinh} \left (c x \right ) b \,c^{7} x^{7}+59535 \mathit {asinh} \left (c x \right ) b \,c^{5} x^{5}+33075 \mathit {asinh} \left (c x \right ) b \,c^{3} x^{3}-1225 \sqrt {c^{2} x^{2}+1}\, b \,c^{8} x^{8}-4675 \sqrt {c^{2} x^{2}+1}\, b \,c^{6} x^{6}-6297 \sqrt {c^{2} x^{2}+1}\, b \,c^{4} x^{4}-2629 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}+5258 \sqrt {c^{2} x^{2}+1}\, b +11025 a \,c^{9} x^{9}+42525 a \,c^{7} x^{7}+59535 a \,c^{5} x^{5}+33075 a \,c^{3} x^{3}\right )}{99225 c^{3}} \] Input:
int(x^2*(c^2*d*x^2+d)^3*(a+b*asinh(c*x)),x)
Output:
(d**3*(11025*asinh(c*x)*b*c**9*x**9 + 42525*asinh(c*x)*b*c**7*x**7 + 59535 *asinh(c*x)*b*c**5*x**5 + 33075*asinh(c*x)*b*c**3*x**3 - 1225*sqrt(c**2*x* *2 + 1)*b*c**8*x**8 - 4675*sqrt(c**2*x**2 + 1)*b*c**6*x**6 - 6297*sqrt(c** 2*x**2 + 1)*b*c**4*x**4 - 2629*sqrt(c**2*x**2 + 1)*b*c**2*x**2 + 5258*sqrt (c**2*x**2 + 1)*b + 11025*a*c**9*x**9 + 42525*a*c**7*x**7 + 59535*a*c**5*x **5 + 33075*a*c**3*x**3))/(99225*c**3)