Integrand size = 30, antiderivative size = 1228 \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {3 b d^2 f^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}+\frac {2 b d^2 g^3 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^3 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {15 b d^2 f g^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d+c^2 d x^2}}{256 \sqrt {1+c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c d^2 g^3 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^3 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15 d^2 f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 c^4}+\frac {5 d^2 f^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {1+c^2 x^2}} \]
15/128*d^2*f*g^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+5/16*d^2*f*g ^2*x^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+3/8*d^2*f*g^2*x^ 3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+3/7*d^2*f^2*g*(c^2* x^2+1)^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+2/63*b*d^2*g^3*x*(c^2* d*x^2+d)^(1/2)/c^3/(c^2*x^2+1)^(1/2)-25/96*b*c*d^2*f^3*x^2*(c^2*d*x^2+d)^( 1/2)/(c^2*x^2+1)^(1/2)-1/189*b*d^2*g^3*x^3*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+ 1)^(1/2)-5/96*b*c^3*d^2*f^3*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/21 *b*c*d^2*g^3*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-19/441*b*c^3*d^2*g^ 3*x^7*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/81*b*c^5*d^2*g^3*x^9*(c^2*d* x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+5/32*d^2*f^3*(a+b*arcsinh(c*x))^2*(c^2*d*x^ 2+d)^(1/2)/b/c/(c^2*x^2+1)^(1/2)+5/16*d^2*f^3*x*(a+b*arcsinh(c*x))*(c^2*d* x^2+d)^(1/2)+5/24*d^2*f^3*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^( 1/2)+1/6*d^2*f^3*x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)-1/ 7*d^2*g^3*(c^2*x^2+1)^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^4+1/9*d^2 *g^3*(c^2*x^2+1)^4*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^4-1/36*b*d^2*f ^3*(c^2*x^2+1)^(5/2)*(c^2*d*x^2+d)^(1/2)/c+15/64*d^2*f*g^2*x^3*(a+b*arcsin h(c*x))*(c^2*d*x^2+d)^(1/2)-3/7*b*d^2*f^2*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x ^2+1)^(1/2)-15/256*b*d^2*f*g^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2) -3/7*b*c*d^2*f^2*g*x^3*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-59/256*b*c*d^ 2*f*g^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-9/35*b*c^3*d^2*f^2*g*...
Time = 2.22 (sec) , antiderivative size = 810, normalized size of antiderivative = 0.66 \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^3 \left (1+c^2 x^2\right ) \left (-20160 a \sqrt {1+c^2 x^2} \left (-256 g^3+c^2 g \left (3456 f^2+945 f g x+128 g^2 x^2\right )+16 c^8 x^5 \left (84 f^3+216 f^2 g x+189 f g^2 x^2+56 g^3 x^3\right )+8 c^6 x^3 \left (546 f^3+1296 f^2 g x+1071 f g^2 x^2+304 g^3 x^3\right )+6 c^4 x \left (924 f^3+1728 f^2 g x+1239 f g^2 x^2+320 g^3 x^3\right )\right )+b \left (-315 c g^2 (7539 f+16384 g x)+30240 c^5 x^2 \left (1848 f^3+2304 f^2 g x+1239 f g^2 x^2+256 g^3 x^3\right )+3360 c^3 \left (6279 f^3+20736 f^2 g x+2835 f g^2 x^2+256 g^3 x^3\right )+2304 c^7 x^4 \left (9555 f^3+18144 f^2 g x+12495 f g^2 x^2+3040 g^3 x^3\right )+640 c^9 x^6 \left (7056 f^3+15552 f^2 g x+11907 f g^2 x^2+3136 g^3 x^3\right )\right )\right )+3175200 b c d^3 f \left (8 c^2 f^2-3 g^2\right ) \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+6350400 a c d^{5/2} f \left (8 c^2 f^2-3 g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+2520 b d^3 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (27648 c^2 f^2 g \sqrt {1+c^2 x^2}-2048 g^3 \sqrt {1+c^2 x^2}+82944 c^4 f^2 g x^2 \sqrt {1+c^2 x^2}+1024 c^2 g^3 x^2 \sqrt {1+c^2 x^2}+82944 c^6 f^2 g x^4 \sqrt {1+c^2 x^2}+15360 c^4 g^3 x^4 \sqrt {1+c^2 x^2}+27648 c^8 f^2 g x^6 \sqrt {1+c^2 x^2}+19456 c^6 g^3 x^6 \sqrt {1+c^2 x^2}+7168 c^8 g^3 x^8 \sqrt {1+c^2 x^2}+3024 c f \left (5 c^2 f^2-g^2\right ) \sinh (2 \text {arcsinh}(c x))+1512 c f \left (2 c^2 f^2+g^2\right ) \sinh (4 \text {arcsinh}(c x))+336 c^3 f^3 \sinh (6 \text {arcsinh}(c x))+1008 c f g^2 \sinh (6 \text {arcsinh}(c x))+189 c f g^2 \sinh (8 \text {arcsinh}(c x))\right )}{162570240 c^4 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \]
(-(d^3*(1 + c^2*x^2)*(-20160*a*Sqrt[1 + c^2*x^2]*(-256*g^3 + c^2*g*(3456*f ^2 + 945*f*g*x + 128*g^2*x^2) + 16*c^8*x^5*(84*f^3 + 216*f^2*g*x + 189*f*g ^2*x^2 + 56*g^3*x^3) + 8*c^6*x^3*(546*f^3 + 1296*f^2*g*x + 1071*f*g^2*x^2 + 304*g^3*x^3) + 6*c^4*x*(924*f^3 + 1728*f^2*g*x + 1239*f*g^2*x^2 + 320*g^ 3*x^3)) + b*(-315*c*g^2*(7539*f + 16384*g*x) + 30240*c^5*x^2*(1848*f^3 + 2 304*f^2*g*x + 1239*f*g^2*x^2 + 256*g^3*x^3) + 3360*c^3*(6279*f^3 + 20736*f ^2*g*x + 2835*f*g^2*x^2 + 256*g^3*x^3) + 2304*c^7*x^4*(9555*f^3 + 18144*f^ 2*g*x + 12495*f*g^2*x^2 + 3040*g^3*x^3) + 640*c^9*x^6*(7056*f^3 + 15552*f^ 2*g*x + 11907*f*g^2*x^2 + 3136*g^3*x^3)))) + 3175200*b*c*d^3*f*(8*c^2*f^2 - 3*g^2)*(1 + c^2*x^2)*ArcSinh[c*x]^2 + 6350400*a*c*d^(5/2)*f*(8*c^2*f^2 - 3*g^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 2520*b*d^3*(1 + c^2*x^2)*ArcSinh[c*x]*(27648*c^2*f^2*g*Sqrt [1 + c^2*x^2] - 2048*g^3*Sqrt[1 + c^2*x^2] + 82944*c^4*f^2*g*x^2*Sqrt[1 + c^2*x^2] + 1024*c^2*g^3*x^2*Sqrt[1 + c^2*x^2] + 82944*c^6*f^2*g*x^4*Sqrt[1 + c^2*x^2] + 15360*c^4*g^3*x^4*Sqrt[1 + c^2*x^2] + 27648*c^8*f^2*g*x^6*Sq rt[1 + c^2*x^2] + 19456*c^6*g^3*x^6*Sqrt[1 + c^2*x^2] + 7168*c^8*g^3*x^8*S qrt[1 + c^2*x^2] + 3024*c*f*(5*c^2*f^2 - g^2)*Sinh[2*ArcSinh[c*x]] + 1512* c*f*(2*c^2*f^2 + g^2)*Sinh[4*ArcSinh[c*x]] + 336*c^3*f^3*Sinh[6*ArcSinh[c* x]] + 1008*c*f*g^2*Sinh[6*ArcSinh[c*x]] + 189*c*f*g^2*Sinh[8*ArcSinh[c*x]] ))/(162570240*c^4*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2])
Time = 1.42 (sec) , antiderivative size = 617, normalized size of antiderivative = 0.50, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6260, 6253, 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.
| \(\displaystyle \int \left (c^2 d x^2+d\right )^{5/2} (f+g x)^3 (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6260 |
| \(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \int (f+g x)^3 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \int \left (\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x)) f^3+3 g x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x)) f^2+3 g^2 x^2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x)) f+g^3 x^3 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))\right )dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \left (-\frac {15 f g^2 (a+b \text {arcsinh}(c x))^2}{256 b c^3}+\frac {1}{6} f^3 x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{24} f^3 x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {5}{16} f^3 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {3 f^2 g \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {15 f g^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {3}{8} f g^2 x^3 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{16} f g^2 x^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {15}{64} f g^2 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {g^3 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {g^3 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {5 f^3 (a+b \text {arcsinh}(c x))^2}{32 b c}-\frac {3}{49} b c^5 f^2 g x^7-\frac {3}{64} b c^5 f g^2 x^8-\frac {1}{81} b c^5 g^3 x^9-\frac {5}{96} b c^3 f^3 x^4-\frac {9}{35} b c^3 f^2 g x^5-\frac {17}{96} b c^3 f g^2 x^6-\frac {19}{441} b c^3 g^3 x^7+\frac {2 b g^3 x}{63 c^3}-\frac {b f^3 \left (c^2 x^2+1\right )^3}{36 c}-\frac {25}{96} b c f^3 x^2-\frac {3}{7} b c f^2 g x^3-\frac {3 b f^2 g x}{7 c}-\frac {59}{256} b c f g^2 x^4-\frac {15 b f g^2 x^2}{256 c}-\frac {1}{21} b c g^3 x^5-\frac {b g^3 x^3}{189 c}\right )}{\sqrt {c^2 x^2+1}}\) |
(d^2*Sqrt[d + c^2*d*x^2]*((-3*b*f^2*g*x)/(7*c) + (2*b*g^3*x)/(63*c^3) - (2 5*b*c*f^3*x^2)/96 - (15*b*f*g^2*x^2)/(256*c) - (3*b*c*f^2*g*x^3)/7 - (b*g^ 3*x^3)/(189*c) - (5*b*c^3*f^3*x^4)/96 - (59*b*c*f*g^2*x^4)/256 - (9*b*c^3* f^2*g*x^5)/35 - (b*c*g^3*x^5)/21 - (17*b*c^3*f*g^2*x^6)/96 - (3*b*c^5*f^2* g*x^7)/49 - (19*b*c^3*g^3*x^7)/441 - (3*b*c^5*f*g^2*x^8)/64 - (b*c^5*g^3*x ^9)/81 - (b*f^3*(1 + c^2*x^2)^3)/(36*c) + (5*f^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/16 + (15*f*g^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/ (128*c^2) + (15*f*g^2*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/64 + (5* f^3*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/24 + (5*f*g^2*x^3*(1 + c^2 *x^2)^(3/2)*(a + b*ArcSinh[c*x]))/16 + (f^3*x*(1 + c^2*x^2)^(5/2)*(a + b*A rcSinh[c*x]))/6 + (3*f*g^2*x^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/8 + (3*f^2*g*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^2) - (g^3*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^4) + (g^3*(1 + c^2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(9*c^4) + (5*f^3*(a + b*ArcSinh[c*x])^2)/(32*b*c) - (1 5*f*g^2*(a + b*ArcSinh[c*x])^2)/(256*b*c^3)))/Sqrt[1 + c^2*x^2]
3.1.43.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n , 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 + c^2*x^2) ^p] Int[(f + g*x)^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IntegerQ [p - 1/2] && !GtQ[d, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2883\) vs. \(2(1080)=2160\).
Time = 1.05 (sec) , antiderivative size = 2884, normalized size of antiderivative = 2.35
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2884\) |
| parts | \(\text {Expression too large to display}\) | \(2884\) |
a*(f^3*(1/6*x*(c^2*d*x^2+d)^(5/2)+5/6*d*(1/4*x*(c^2*d*x^2+d)^(3/2)+3/4*d*( 1/2*x*(c^2*d*x^2+d)^(1/2)+1/2*d*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/ 2))/(c^2*d)^(1/2))))+g^3*(1/9*x^2*(c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4*(c^ 2*d*x^2+d)^(7/2))+3*f*g^2*(1/8*x*(c^2*d*x^2+d)^(7/2)/c^2/d-1/8/c^2*(1/6*x* (c^2*d*x^2+d)^(5/2)+5/6*d*(1/4*x*(c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*(c^2*d*x ^2+d)^(1/2)+1/2*d*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1 /2)))))+3/7*f^2*g*(c^2*d*x^2+d)^(7/2)/c^2/d)+b*(5/256*(d*(c^2*x^2+1))^(1/2 )*f*arcsinh(c*x)^2*(8*c^2*f^2-3*g^2)*d^2/(c^2*x^2+1)^(1/2)/c^3+1/41472*(d* (c^2*x^2+1))^(1/2)*(256*c^10*x^10+256*c^9*x^9*(c^2*x^2+1)^(1/2)+704*c^8*x^ 8+576*c^7*x^7*(c^2*x^2+1)^(1/2)+688*c^6*x^6+432*c^5*x^5*(c^2*x^2+1)^(1/2)+ 280*c^4*x^4+120*c^3*x^3*(c^2*x^2+1)^(1/2)+41*c^2*x^2+9*c*x*(c^2*x^2+1)^(1/ 2)+1)*g^3*(-1+9*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)+3/16384*(d*(c^2*x^2+1))^ (1/2)*(128*c^9*x^9+128*c^8*x^8*(c^2*x^2+1)^(1/2)+320*c^7*x^7+256*c^6*x^6*( c^2*x^2+1)^(1/2)+272*c^5*x^5+160*c^4*x^4*(c^2*x^2+1)^(1/2)+88*c^3*x^3+32*c ^2*x^2*(c^2*x^2+1)^(1/2)+8*c*x+(c^2*x^2+1)^(1/2))*f*g^2*(-1+8*arcsinh(c*x) )*d^2/c^3/(c^2*x^2+1)+3/25088*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8+64*c^7*x^7 *(c^2*x^2+1)^(1/2)+144*c^6*x^6+112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4+5 6*c^3*x^3*(c^2*x^2+1)^(1/2)+25*c^2*x^2+7*c*x*(c^2*x^2+1)^(1/2)+1)*g*(28*ar csinh(c*x)*c^2*f^2-4*c^2*f^2+7*arcsinh(c*x)*g^2-g^2)*d^2/c^4/(c^2*x^2+1)+1 /2304*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7+32*c^6*x^6*(c^2*x^2+1)^(1/2)+64...
\[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]
integral((a*c^4*d^2*g^3*x^7 + 3*a*c^4*d^2*f*g^2*x^6 + 3*a*d^2*f^2*g*x + a* d^2*f^3 + (3*a*c^4*d^2*f^2*g + 2*a*c^2*d^2*g^3)*x^5 + (a*c^4*d^2*f^3 + 6*a *c^2*d^2*f*g^2)*x^4 + (6*a*c^2*d^2*f^2*g + a*d^2*g^3)*x^3 + (2*a*c^2*d^2*f ^3 + 3*a*d^2*f*g^2)*x^2 + (b*c^4*d^2*g^3*x^7 + 3*b*c^4*d^2*f*g^2*x^6 + 3*b *d^2*f^2*g*x + b*d^2*f^3 + (3*b*c^4*d^2*f^2*g + 2*b*c^2*d^2*g^3)*x^5 + (b* c^4*d^2*f^3 + 6*b*c^2*d^2*f*g^2)*x^4 + (6*b*c^2*d^2*f^2*g + b*d^2*g^3)*x^3 + (2*b*c^2*d^2*f^3 + 3*b*d^2*f*g^2)*x^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d ), x)
Timed out. \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Timed out} \]
Exception generated. \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \]