Integrand size = 18, antiderivative size = 312 \[ \int \left (d+e x^2\right )^4 (a+b \text {arcsinh}(c x)) \, dx=-\frac {b \left (315 c^8 d^4-420 c^6 d^3 e+378 c^4 d^2 e^2-180 c^2 d e^3+35 e^4\right ) \sqrt {1+c^2 x^2}}{315 c^9}-\frac {4 b e \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \left (1+c^2 x^2\right )^{3/2}}{945 c^9}-\frac {2 b e^2 \left (63 c^4 d^2-90 c^2 d e+35 e^2\right ) \left (1+c^2 x^2\right )^{5/2}}{525 c^9}-\frac {4 b \left (9 c^2 d-7 e\right ) e^3 \left (1+c^2 x^2\right )^{7/2}}{441 c^9}-\frac {b e^4 \left (1+c^2 x^2\right )^{9/2}}{81 c^9}+d^4 x (a+b \text {arcsinh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arcsinh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arcsinh}(c x)) \]
-4/945*b*e*(105*c^6*d^3-189*c^4*d^2*e+135*c^2*d*e^2-35*e^3)*(c^2*x^2+1)^(3 /2)/c^9-2/525*b*e^2*(63*c^4*d^2-90*c^2*d*e+35*e^2)*(c^2*x^2+1)^(5/2)/c^9-4 /441*b*(9*c^2*d-7*e)*e^3*(c^2*x^2+1)^(7/2)/c^9-1/81*b*e^4*(c^2*x^2+1)^(9/2 )/c^9+d^4*x*(a+b*arcsinh(c*x))+4/3*d^3*e*x^3*(a+b*arcsinh(c*x))+6/5*d^2*e^ 2*x^5*(a+b*arcsinh(c*x))+4/7*d*e^3*x^7*(a+b*arcsinh(c*x))+1/9*e^4*x^9*(a+b *arcsinh(c*x))-1/315*b*(315*c^8*d^4-420*c^6*d^3*e+378*c^4*d^2*e^2-180*c^2* d*e^3+35*e^4)*(c^2*x^2+1)^(1/2)/c^9
Time = 0.21 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.83 \[ \int \left (d+e x^2\right )^4 (a+b \text {arcsinh}(c x)) \, dx=\frac {315 a x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )-\frac {b \sqrt {1+c^2 x^2} \left (4480 e^4-320 c^2 e^3 \left (81 d+7 e x^2\right )+48 c^4 e^2 \left (1323 d^2+270 d e x^2+35 e^2 x^4\right )-8 c^6 e \left (11025 d^3+3969 d^2 e x^2+1215 d e^2 x^4+175 e^3 x^6\right )+c^8 \left (99225 d^4+44100 d^3 e x^2+23814 d^2 e^2 x^4+8100 d e^3 x^6+1225 e^4 x^8\right )\right )}{c^9}+315 b x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right ) \text {arcsinh}(c x)}{99225} \]
(315*a*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e ^4*x^8) - (b*Sqrt[1 + c^2*x^2]*(4480*e^4 - 320*c^2*e^3*(81*d + 7*e*x^2) + 48*c^4*e^2*(1323*d^2 + 270*d*e*x^2 + 35*e^2*x^4) - 8*c^6*e*(11025*d^3 + 39 69*d^2*e*x^2 + 1215*d*e^2*x^4 + 175*e^3*x^6) + c^8*(99225*d^4 + 44100*d^3* e*x^2 + 23814*d^2*e^2*x^4 + 8100*d*e^3*x^6 + 1225*e^4*x^8)))/c^9 + 315*b*x *(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e^4*x^8)* ArcSinh[c*x])/99225
Time = 0.70 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6207, 27, 2331, 2389, 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.
| \(\displaystyle \int \left (d+e x^2\right )^4 (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6207 |
| \(\displaystyle -b c \int \frac {x \left (35 e^4 x^8+180 d e^3 x^6+378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4\right )}{315 \sqrt {c^2 x^2+1}}dx+d^4 x (a+b \text {arcsinh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arcsinh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{315} b c \int \frac {x \left (35 e^4 x^8+180 d e^3 x^6+378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4\right )}{\sqrt {c^2 x^2+1}}dx+d^4 x (a+b \text {arcsinh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arcsinh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {1}{630} b c \int \frac {35 e^4 x^8+180 d e^3 x^6+378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4}{\sqrt {c^2 x^2+1}}dx^2+d^4 x (a+b \text {arcsinh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arcsinh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle -\frac {1}{630} b c \int \left (\frac {35 \left (c^2 x^2+1\right )^{7/2} e^4}{c^8}+\frac {20 \left (9 c^2 d-7 e\right ) \left (c^2 x^2+1\right )^{5/2} e^3}{c^8}+\frac {6 \left (63 d^2 c^4-90 d e c^2+35 e^2\right ) \left (c^2 x^2+1\right )^{3/2} e^2}{c^8}+\frac {4 \left (105 d^3 c^6-189 d^2 e c^4+135 d e^2 c^2-35 e^3\right ) \sqrt {c^2 x^2+1} e}{c^8}+\frac {315 d^4 c^8-420 d^3 e c^6+378 d^2 e^2 c^4-180 d e^3 c^2+35 e^4}{c^8 \sqrt {c^2 x^2+1}}\right )dx^2+d^4 x (a+b \text {arcsinh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arcsinh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d^4 x (a+b \text {arcsinh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arcsinh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {40 e^3 \left (c^2 x^2+1\right )^{7/2} \left (9 c^2 d-7 e\right )}{7 c^{10}}+\frac {70 e^4 \left (c^2 x^2+1\right )^{9/2}}{9 c^{10}}+\frac {12 e^2 \left (c^2 x^2+1\right )^{5/2} \left (63 c^4 d^2-90 c^2 d e+35 e^2\right )}{5 c^{10}}+\frac {8 e \left (c^2 x^2+1\right )^{3/2} \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right )}{3 c^{10}}+\frac {2 \sqrt {c^2 x^2+1} \left (315 c^8 d^4-420 c^6 d^3 e+378 c^4 d^2 e^2-180 c^2 d e^3+35 e^4\right )}{c^{10}}\right )\) |
-1/630*(b*c*((2*(315*c^8*d^4 - 420*c^6*d^3*e + 378*c^4*d^2*e^2 - 180*c^2*d *e^3 + 35*e^4)*Sqrt[1 + c^2*x^2])/c^10 + (8*e*(105*c^6*d^3 - 189*c^4*d^2*e + 135*c^2*d*e^2 - 35*e^3)*(1 + c^2*x^2)^(3/2))/(3*c^10) + (12*e^2*(63*c^4 *d^2 - 90*c^2*d*e + 35*e^2)*(1 + c^2*x^2)^(5/2))/(5*c^10) + (40*(9*c^2*d - 7*e)*e^3*(1 + c^2*x^2)^(7/2))/(7*c^10) + (70*e^4*(1 + c^2*x^2)^(9/2))/(9* c^10))) + d^4*x*(a + b*ArcSinh[c*x]) + (4*d^3*e*x^3*(a + b*ArcSinh[c*x]))/ 3 + (6*d^2*e^2*x^5*(a + b*ArcSinh[c*x]))/5 + (4*d*e^3*x^7*(a + b*ArcSinh[c *x]))/7 + (e^4*x^9*(a + b*ArcSinh[c*x]))/9
3.7.6.3.1 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[(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[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(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}, x] && NeQ[e, c^2*d] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
Time = 0.30 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.36
| method | result | size |
| parts | \(a \left (\frac {1}{9} e^{4} x^{9}+\frac {4}{7} d \,e^{3} x^{7}+\frac {6}{5} d^{2} e^{2} x^{5}+\frac {4}{3} d^{3} e \,x^{3}+d^{4} x \right )+\frac {b \left (\frac {c \,\operatorname {arcsinh}\left (c x \right ) e^{4} x^{9}}{9}+\frac {4 c \,\operatorname {arcsinh}\left (c x \right ) d \,e^{3} x^{7}}{7}+\frac {6 c \,\operatorname {arcsinh}\left (c x \right ) d^{2} e^{2} x^{5}}{5}+\frac {4 c \,\operatorname {arcsinh}\left (c x \right ) d^{3} e \,x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) d^{4} c x -\frac {35 e^{4} \left (\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{63}+\frac {16 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{315}+\frac {128 \sqrt {c^{2} x^{2}+1}}{315}\right )+315 d^{4} c^{8} \sqrt {c^{2} x^{2}+1}+180 d \,c^{2} e^{3} \left (\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{35}+\frac {8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {c^{2} x^{2}+1}}{35}\right )+378 d^{2} c^{4} e^{2} \left (\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )+420 d^{3} c^{6} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{315 c^{8}}\right )}{c}\) | \(425\) |
| derivativedivides | \(\frac {\frac {a \left (d^{4} c^{9} x +\frac {4}{3} d^{3} c^{9} e \,x^{3}+\frac {6}{5} d^{2} c^{9} e^{2} x^{5}+\frac {4}{7} d \,c^{9} e^{3} x^{7}+\frac {1}{9} e^{4} c^{9} x^{9}\right )}{c^{8}}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) d^{4} c^{9} x +\frac {4 \,\operatorname {arcsinh}\left (c x \right ) d^{3} c^{9} e \,x^{3}}{3}+\frac {6 \,\operatorname {arcsinh}\left (c x \right ) d^{2} c^{9} e^{2} x^{5}}{5}+\frac {4 \,\operatorname {arcsinh}\left (c x \right ) d \,c^{9} e^{3} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (c x \right ) e^{4} c^{9} x^{9}}{9}-\frac {e^{4} \left (\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{63}+\frac {16 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{315}+\frac {128 \sqrt {c^{2} x^{2}+1}}{315}\right )}{9}-d^{4} c^{8} \sqrt {c^{2} x^{2}+1}-\frac {4 d^{3} c^{6} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-\frac {6 d^{2} c^{4} e^{2} \left (\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}-\frac {4 d \,c^{2} e^{3} \left (\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{35}+\frac {8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {c^{2} x^{2}+1}}{35}\right )}{7}\right )}{c^{8}}}{c}\) | \(451\) |
| default | \(\frac {\frac {a \left (d^{4} c^{9} x +\frac {4}{3} d^{3} c^{9} e \,x^{3}+\frac {6}{5} d^{2} c^{9} e^{2} x^{5}+\frac {4}{7} d \,c^{9} e^{3} x^{7}+\frac {1}{9} e^{4} c^{9} x^{9}\right )}{c^{8}}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) d^{4} c^{9} x +\frac {4 \,\operatorname {arcsinh}\left (c x \right ) d^{3} c^{9} e \,x^{3}}{3}+\frac {6 \,\operatorname {arcsinh}\left (c x \right ) d^{2} c^{9} e^{2} x^{5}}{5}+\frac {4 \,\operatorname {arcsinh}\left (c x \right ) d \,c^{9} e^{3} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (c x \right ) e^{4} c^{9} x^{9}}{9}-\frac {e^{4} \left (\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{63}+\frac {16 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{315}+\frac {128 \sqrt {c^{2} x^{2}+1}}{315}\right )}{9}-d^{4} c^{8} \sqrt {c^{2} x^{2}+1}-\frac {4 d^{3} c^{6} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-\frac {6 d^{2} c^{4} e^{2} \left (\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}-\frac {4 d \,c^{2} e^{3} \left (\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{35}+\frac {8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {c^{2} x^{2}+1}}{35}\right )}{7}\right )}{c^{8}}}{c}\) | \(451\) |
a*(1/9*e^4*x^9+4/7*d*e^3*x^7+6/5*d^2*e^2*x^5+4/3*d^3*e*x^3+d^4*x)+b/c*(1/9 *c*arcsinh(c*x)*e^4*x^9+4/7*c*arcsinh(c*x)*d*e^3*x^7+6/5*c*arcsinh(c*x)*d^ 2*e^2*x^5+4/3*c*arcsinh(c*x)*d^3*e*x^3+arcsinh(c*x)*d^4*c*x-1/315/c^8*(35* e^4*(1/9*c^8*x^8*(c^2*x^2+1)^(1/2)-8/63*c^6*x^6*(c^2*x^2+1)^(1/2)+16/105*c ^4*x^4*(c^2*x^2+1)^(1/2)-64/315*c^2*x^2*(c^2*x^2+1)^(1/2)+128/315*(c^2*x^2 +1)^(1/2))+315*d^4*c^8*(c^2*x^2+1)^(1/2)+180*d*c^2*e^3*(1/7*c^6*x^6*(c^2*x ^2+1)^(1/2)-6/35*c^4*x^4*(c^2*x^2+1)^(1/2)+8/35*c^2*x^2*(c^2*x^2+1)^(1/2)- 16/35*(c^2*x^2+1)^(1/2))+378*d^2*c^4*e^2*(1/5*c^4*x^4*(c^2*x^2+1)^(1/2)-4/ 15*c^2*x^2*(c^2*x^2+1)^(1/2)+8/15*(c^2*x^2+1)^(1/2))+420*d^3*c^6*e*(1/3*c^ 2*x^2*(c^2*x^2+1)^(1/2)-2/3*(c^2*x^2+1)^(1/2))))
Time = 0.27 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.07 \[ \int \left (d+e x^2\right )^4 (a+b \text {arcsinh}(c x)) \, dx=\frac {11025 \, a c^{9} e^{4} x^{9} + 56700 \, a c^{9} d e^{3} x^{7} + 119070 \, a c^{9} d^{2} e^{2} x^{5} + 132300 \, a c^{9} d^{3} e x^{3} + 99225 \, a c^{9} d^{4} x + 315 \, {\left (35 \, b c^{9} e^{4} x^{9} + 180 \, b c^{9} d e^{3} x^{7} + 378 \, b c^{9} d^{2} e^{2} x^{5} + 420 \, b c^{9} d^{3} e x^{3} + 315 \, b c^{9} d^{4} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (1225 \, b c^{8} e^{4} x^{8} + 99225 \, b c^{8} d^{4} - 88200 \, b c^{6} d^{3} e + 63504 \, b c^{4} d^{2} e^{2} - 25920 \, b c^{2} d e^{3} + 100 \, {\left (81 \, b c^{8} d e^{3} - 14 \, b c^{6} e^{4}\right )} x^{6} + 4480 \, b e^{4} + 6 \, {\left (3969 \, b c^{8} d^{2} e^{2} - 1620 \, b c^{6} d e^{3} + 280 \, b c^{4} e^{4}\right )} x^{4} + 4 \, {\left (11025 \, b c^{8} d^{3} e - 7938 \, b c^{6} d^{2} e^{2} + 3240 \, b c^{4} d e^{3} - 560 \, b c^{2} e^{4}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{99225 \, c^{9}} \]
1/99225*(11025*a*c^9*e^4*x^9 + 56700*a*c^9*d*e^3*x^7 + 119070*a*c^9*d^2*e^ 2*x^5 + 132300*a*c^9*d^3*e*x^3 + 99225*a*c^9*d^4*x + 315*(35*b*c^9*e^4*x^9 + 180*b*c^9*d*e^3*x^7 + 378*b*c^9*d^2*e^2*x^5 + 420*b*c^9*d^3*e*x^3 + 315 *b*c^9*d^4*x)*log(c*x + sqrt(c^2*x^2 + 1)) - (1225*b*c^8*e^4*x^8 + 99225*b *c^8*d^4 - 88200*b*c^6*d^3*e + 63504*b*c^4*d^2*e^2 - 25920*b*c^2*d*e^3 + 1 00*(81*b*c^8*d*e^3 - 14*b*c^6*e^4)*x^6 + 4480*b*e^4 + 6*(3969*b*c^8*d^2*e^ 2 - 1620*b*c^6*d*e^3 + 280*b*c^4*e^4)*x^4 + 4*(11025*b*c^8*d^3*e - 7938*b* c^6*d^2*e^2 + 3240*b*c^4*d*e^3 - 560*b*c^2*e^4)*x^2)*sqrt(c^2*x^2 + 1))/c^ 9
Time = 1.20 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.90 \[ \int \left (d+e x^2\right )^4 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} a d^{4} x + \frac {4 a d^{3} e x^{3}}{3} + \frac {6 a d^{2} e^{2} x^{5}}{5} + \frac {4 a d e^{3} x^{7}}{7} + \frac {a e^{4} x^{9}}{9} + b d^{4} x \operatorname {asinh}{\left (c x \right )} + \frac {4 b d^{3} e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} + \frac {6 b d^{2} e^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} + \frac {4 b d e^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} + \frac {b e^{4} x^{9} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {b d^{4} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {4 b d^{3} e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} - \frac {6 b d^{2} e^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25 c} - \frac {4 b d e^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{49 c} - \frac {b e^{4} x^{8} \sqrt {c^{2} x^{2} + 1}}{81 c} + \frac {8 b d^{3} e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + \frac {8 b d^{2} e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{25 c^{3}} + \frac {24 b d e^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{245 c^{3}} + \frac {8 b e^{4} x^{6} \sqrt {c^{2} x^{2} + 1}}{567 c^{3}} - \frac {16 b d^{2} e^{2} \sqrt {c^{2} x^{2} + 1}}{25 c^{5}} - \frac {32 b d e^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{245 c^{5}} - \frac {16 b e^{4} x^{4} \sqrt {c^{2} x^{2} + 1}}{945 c^{5}} + \frac {64 b d e^{3} \sqrt {c^{2} x^{2} + 1}}{245 c^{7}} + \frac {64 b e^{4} x^{2} \sqrt {c^{2} x^{2} + 1}}{2835 c^{7}} - \frac {128 b e^{4} \sqrt {c^{2} x^{2} + 1}}{2835 c^{9}} & \text {for}\: c \neq 0 \\a \left (d^{4} x + \frac {4 d^{3} e x^{3}}{3} + \frac {6 d^{2} e^{2} x^{5}}{5} + \frac {4 d e^{3} x^{7}}{7} + \frac {e^{4} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*d**4*x + 4*a*d**3*e*x**3/3 + 6*a*d**2*e**2*x**5/5 + 4*a*d*e** 3*x**7/7 + a*e**4*x**9/9 + b*d**4*x*asinh(c*x) + 4*b*d**3*e*x**3*asinh(c*x )/3 + 6*b*d**2*e**2*x**5*asinh(c*x)/5 + 4*b*d*e**3*x**7*asinh(c*x)/7 + b*e **4*x**9*asinh(c*x)/9 - b*d**4*sqrt(c**2*x**2 + 1)/c - 4*b*d**3*e*x**2*sqr t(c**2*x**2 + 1)/(9*c) - 6*b*d**2*e**2*x**4*sqrt(c**2*x**2 + 1)/(25*c) - 4 *b*d*e**3*x**6*sqrt(c**2*x**2 + 1)/(49*c) - b*e**4*x**8*sqrt(c**2*x**2 + 1 )/(81*c) + 8*b*d**3*e*sqrt(c**2*x**2 + 1)/(9*c**3) + 8*b*d**2*e**2*x**2*sq rt(c**2*x**2 + 1)/(25*c**3) + 24*b*d*e**3*x**4*sqrt(c**2*x**2 + 1)/(245*c* *3) + 8*b*e**4*x**6*sqrt(c**2*x**2 + 1)/(567*c**3) - 16*b*d**2*e**2*sqrt(c **2*x**2 + 1)/(25*c**5) - 32*b*d*e**3*x**2*sqrt(c**2*x**2 + 1)/(245*c**5) - 16*b*e**4*x**4*sqrt(c**2*x**2 + 1)/(945*c**5) + 64*b*d*e**3*sqrt(c**2*x* *2 + 1)/(245*c**7) + 64*b*e**4*x**2*sqrt(c**2*x**2 + 1)/(2835*c**7) - 128* b*e**4*sqrt(c**2*x**2 + 1)/(2835*c**9), Ne(c, 0)), (a*(d**4*x + 4*d**3*e*x **3/3 + 6*d**2*e**2*x**5/5 + 4*d*e**3*x**7/7 + e**4*x**9/9), True))
Time = 0.20 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.33 \[ \int \left (d+e x^2\right )^4 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{9} \, a e^{4} x^{9} + \frac {4}{7} \, a d e^{3} x^{7} + \frac {6}{5} \, a d^{2} e^{2} x^{5} + \frac {4}{3} \, a d^{3} e x^{3} + \frac {4}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{3} e + \frac {2}{25} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{2} e^{2} + \frac {4}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b d e^{3} + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b e^{4} + a d^{4} x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{4}}{c} \]
1/9*a*e^4*x^9 + 4/7*a*d*e^3*x^7 + 6/5*a*d^2*e^2*x^5 + 4/3*a*d^3*e*x^3 + 4/ 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*e + 2/25*(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*d^2*e^2 + 4/ 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 *d*e^3 + 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*e^4 + a*d^4*x + (c*x*arc sinh(c*x) - sqrt(c^2*x^2 + 1))*b*d^4/c
Exception generated. \[ \int \left (d+e x^2\right )^4 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int \left (d+e x^2\right )^4 (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^4 \,d x \]