Integrand size = 24, antiderivative size = 226 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=-\frac {16 b d^3 \sqrt {1+c^2 x^2}}{1155 c^5}-\frac {8 b d^3 \left (1+c^2 x^2\right )^{3/2}}{3465 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{5/2}}{1925 c^5}-\frac {b d^3 \left (1+c^2 x^2\right )^{7/2}}{1617 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{9/2}}{297 c^5}-\frac {b d^3 \left (1+c^2 x^2\right )^{11/2}}{121 c^5}+\frac {1}{5} d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^2 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {1}{11} c^6 d^3 x^{11} (a+b \text {arcsinh}(c x)) \]
-8/3465*b*d^3*(c^2*x^2+1)^(3/2)/c^5-2/1925*b*d^3*(c^2*x^2+1)^(5/2)/c^5-1/1 617*b*d^3*(c^2*x^2+1)^(7/2)/c^5+4/297*b*d^3*(c^2*x^2+1)^(9/2)/c^5-1/121*b* d^3*(c^2*x^2+1)^(11/2)/c^5+1/5*d^3*x^5*(a+b*arcsinh(c*x))+3/7*c^2*d^3*x^7* (a+b*arcsinh(c*x))+1/3*c^4*d^3*x^9*(a+b*arcsinh(c*x))+1/11*c^6*d^3*x^11*(a +b*arcsinh(c*x))-16/1155*b*d^3*(c^2*x^2+1)^(1/2)/c^5
Time = 0.10 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.63 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^3 \left (3465 a c^5 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right )-b \sqrt {1+c^2 x^2} \left (50488-25244 c^2 x^2+18933 c^4 x^4+117625 c^6 x^6+111475 c^8 x^8+33075 c^{10} x^{10}\right )+3465 b c^5 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right ) \text {arcsinh}(c x)\right )}{4002075 c^5} \]
(d^3*(3465*a*c^5*x^5*(231 + 495*c^2*x^2 + 385*c^4*x^4 + 105*c^6*x^6) - b*S qrt[1 + c^2*x^2]*(50488 - 25244*c^2*x^2 + 18933*c^4*x^4 + 117625*c^6*x^6 + 111475*c^8*x^8 + 33075*c^10*x^10) + 3465*b*c^5*x^5*(231 + 495*c^2*x^2 + 3 85*c^4*x^4 + 105*c^6*x^6)*ArcSinh[c*x]))/(4002075*c^5)
Time = 0.61 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.93, 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.
| \(\displaystyle \int x^4 \left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6218 |
| \(\displaystyle -b c \int \frac {d^3 x^5 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right )}{1155 \sqrt {c^2 x^2+1}}dx+\frac {1}{11} c^6 d^3 x^{11} (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^2 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^3 x^5 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c d^3 \int \frac {x^5 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right )}{\sqrt {c^2 x^2+1}}dx}{1155}+\frac {1}{11} c^6 d^3 x^{11} (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^2 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^3 x^5 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {b c d^3 \int \frac {x^4 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right )}{\sqrt {c^2 x^2+1}}dx^2}{2310}+\frac {1}{11} c^6 d^3 x^{11} (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^2 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^3 x^5 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle -\frac {b c d^3 \int \left (\frac {105 \left (c^2 x^2+1\right )^{9/2}}{c^4}-\frac {140 \left (c^2 x^2+1\right )^{7/2}}{c^4}+\frac {5 \left (c^2 x^2+1\right )^{5/2}}{c^4}+\frac {6 \left (c^2 x^2+1\right )^{3/2}}{c^4}+\frac {8 \sqrt {c^2 x^2+1}}{c^4}+\frac {16}{c^4 \sqrt {c^2 x^2+1}}\right )dx^2}{2310}+\frac {1}{11} c^6 d^3 x^{11} (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^2 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^3 x^5 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{11} c^6 d^3 x^{11} (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^2 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^3 x^5 (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (\frac {210 \left (c^2 x^2+1\right )^{11/2}}{11 c^6}-\frac {280 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}+\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {12 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {16 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {32 \sqrt {c^2 x^2+1}}{c^6}\right )}{2310}\) |
-1/2310*(b*c*d^3*((32*Sqrt[1 + c^2*x^2])/c^6 + (16*(1 + c^2*x^2)^(3/2))/(3 *c^6) + (12*(1 + c^2*x^2)^(5/2))/(5*c^6) + (10*(1 + c^2*x^2)^(7/2))/(7*c^6 ) - (280*(1 + c^2*x^2)^(9/2))/(9*c^6) + (210*(1 + c^2*x^2)^(11/2))/(11*c^6 ))) + (d^3*x^5*(a + b*ArcSinh[c*x]))/5 + (3*c^2*d^3*x^7*(a + b*ArcSinh[c*x ]))/7 + (c^4*d^3*x^9*(a + b*ArcSinh[c*x]))/3 + (c^6*d^3*x^11*(a + b*ArcSin h[c*x]))/11
3.1.19.3.1 Defintions of rubi rules used
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 = 0.21 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.89
| method | result | size |
| parts | \(d^{3} a \left (\frac {1}{11} c^{6} x^{11}+\frac {1}{3} c^{4} x^{9}+\frac {3}{7} c^{2} x^{7}+\frac {1}{5} x^{5}\right )+\frac {d^{3} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{11} x^{11}}{11}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {91 c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {4705 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {c^{10} x^{10} \sqrt {c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(202\) |
| derivativedivides | \(\frac {d^{3} a \left (\frac {1}{11} c^{11} x^{11}+\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{11} x^{11}}{11}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {91 c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {4705 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {c^{10} x^{10} \sqrt {c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(206\) |
| default | \(\frac {d^{3} a \left (\frac {1}{11} c^{11} x^{11}+\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{11} x^{11}}{11}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {91 c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {4705 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {c^{10} x^{10} \sqrt {c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(206\) |
d^3*a*(1/11*c^6*x^11+1/3*c^4*x^9+3/7*c^2*x^7+1/5*x^5)+d^3*b/c^5*(1/11*arcs inh(c*x)*c^11*x^11+1/3*arcsinh(c*x)*c^9*x^9+3/7*arcsinh(c*x)*c^7*x^7+1/5*a rcsinh(c*x)*c^5*x^5-91/3267*c^8*x^8*(c^2*x^2+1)^(1/2)-4705/160083*c^6*x^6* (c^2*x^2+1)^(1/2)-6311/1334025*c^4*x^4*(c^2*x^2+1)^(1/2)+25244/4002075*c^2 *x^2*(c^2*x^2+1)^(1/2)-50488/4002075*(c^2*x^2+1)^(1/2)-1/121*c^10*x^10*(c^ 2*x^2+1)^(1/2))
Time = 0.29 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.89 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {363825 \, a c^{11} d^{3} x^{11} + 1334025 \, a c^{9} d^{3} x^{9} + 1715175 \, a c^{7} d^{3} x^{7} + 800415 \, a c^{5} d^{3} x^{5} + 3465 \, {\left (105 \, b c^{11} d^{3} x^{11} + 385 \, b c^{9} d^{3} x^{9} + 495 \, b c^{7} d^{3} x^{7} + 231 \, b c^{5} d^{3} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (33075 \, b c^{10} d^{3} x^{10} + 111475 \, b c^{8} d^{3} x^{8} + 117625 \, b c^{6} d^{3} x^{6} + 18933 \, b c^{4} d^{3} x^{4} - 25244 \, b c^{2} d^{3} x^{2} + 50488 \, b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{4002075 \, c^{5}} \]
1/4002075*(363825*a*c^11*d^3*x^11 + 1334025*a*c^9*d^3*x^9 + 1715175*a*c^7* d^3*x^7 + 800415*a*c^5*d^3*x^5 + 3465*(105*b*c^11*d^3*x^11 + 385*b*c^9*d^3 *x^9 + 495*b*c^7*d^3*x^7 + 231*b*c^5*d^3*x^5)*log(c*x + sqrt(c^2*x^2 + 1)) - (33075*b*c^10*d^3*x^10 + 111475*b*c^8*d^3*x^8 + 117625*b*c^6*d^3*x^6 + 18933*b*c^4*d^3*x^4 - 25244*b*c^2*d^3*x^2 + 50488*b*d^3)*sqrt(c^2*x^2 + 1) )/c^5
Time = 2.23 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.28 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{6} d^{3} x^{11}}{11} + \frac {a c^{4} d^{3} x^{9}}{3} + \frac {3 a c^{2} d^{3} x^{7}}{7} + \frac {a d^{3} x^{5}}{5} + \frac {b c^{6} d^{3} x^{11} \operatorname {asinh}{\left (c x \right )}}{11} - \frac {b c^{5} d^{3} x^{10} \sqrt {c^{2} x^{2} + 1}}{121} + \frac {b c^{4} d^{3} x^{9} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {91 b c^{3} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1}}{3267} + \frac {3 b c^{2} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {4705 b c d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{160083} + \frac {b d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {6311 b d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{1334025 c} + \frac {25244 b d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{4002075 c^{3}} - \frac {50488 b d^{3} \sqrt {c^{2} x^{2} + 1}}{4002075 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{5}}{5} & \text {otherwise} \end {cases} \]
Piecewise((a*c**6*d**3*x**11/11 + a*c**4*d**3*x**9/3 + 3*a*c**2*d**3*x**7/ 7 + a*d**3*x**5/5 + b*c**6*d**3*x**11*asinh(c*x)/11 - b*c**5*d**3*x**10*sq rt(c**2*x**2 + 1)/121 + b*c**4*d**3*x**9*asinh(c*x)/3 - 91*b*c**3*d**3*x** 8*sqrt(c**2*x**2 + 1)/3267 + 3*b*c**2*d**3*x**7*asinh(c*x)/7 - 4705*b*c*d* *3*x**6*sqrt(c**2*x**2 + 1)/160083 + b*d**3*x**5*asinh(c*x)/5 - 6311*b*d** 3*x**4*sqrt(c**2*x**2 + 1)/(1334025*c) + 25244*b*d**3*x**2*sqrt(c**2*x**2 + 1)/(4002075*c**3) - 50488*b*d**3*sqrt(c**2*x**2 + 1)/(4002075*c**5), Ne( c, 0)), (a*d**3*x**5/5, True))
Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (194) = 388\).
Time = 0.20 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.06 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{11} \, a c^{6} d^{3} x^{11} + \frac {1}{3} \, a c^{4} d^{3} x^{9} + \frac {3}{7} \, a c^{2} d^{3} x^{7} + \frac {1}{7623} \, {\left (693 \, x^{11} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {63 \, \sqrt {c^{2} x^{2} + 1} x^{10}}{c^{2}} - \frac {70 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{4}} + \frac {80 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{6}} - \frac {96 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{10}} - \frac {256 \, \sqrt {c^{2} x^{2} + 1}}{c^{12}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{945} \, {\left (315 \, x^{9} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{4} d^{3} + \frac {1}{5} \, a d^{3} x^{5} + \frac {3}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{3} \]
1/11*a*c^6*d^3*x^11 + 1/3*a*c^4*d^3*x^9 + 3/7*a*c^2*d^3*x^7 + 1/7623*(693* x^11*arcsinh(c*x) - (63*sqrt(c^2*x^2 + 1)*x^10/c^2 - 70*sqrt(c^2*x^2 + 1)* x^8/c^4 + 80*sqrt(c^2*x^2 + 1)*x^6/c^6 - 96*sqrt(c^2*x^2 + 1)*x^4/c^8 + 12 8*sqrt(c^2*x^2 + 1)*x^2/c^10 - 256*sqrt(c^2*x^2 + 1)/c^12)*c)*b*c^6*d^3 + 1/945*(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^4*d^3 + 1/5*a*d^3*x^5 + 3/245*(3 5*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^2*d ^3 + 1/75*(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*d^3
Exception generated. \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \]