Integrand size = 21, antiderivative size = 241 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) x \sqrt {1-c^2 x^2}}{3072 c^7}+\frac {b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) x^3 \sqrt {1-c^2 x^2}}{4608 c^5}+\frac {b e \left (64 c^2 d+21 e\right ) x^5 \sqrt {1-c^2 x^2}}{1152 c^3}+\frac {b e^2 x^7 \sqrt {1-c^2 x^2}}{64 c}-\frac {b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \arcsin (c x)}{3072 c^8}+\frac {1}{4} d^2 x^4 (a+b \arcsin (c x))+\frac {1}{3} d e x^6 (a+b \arcsin (c x))+\frac {1}{8} e^2 x^8 (a+b \arcsin (c x)) \]
-1/3072*b*(288*c^4*d^2+320*c^2*d*e+105*e^2)*arcsin(c*x)/c^8+1/4*d^2*x^4*(a +b*arcsin(c*x))+1/3*d*e*x^6*(a+b*arcsin(c*x))+1/8*e^2*x^8*(a+b*arcsin(c*x) )+1/3072*b*(288*c^4*d^2+320*c^2*d*e+105*e^2)*x*(-c^2*x^2+1)^(1/2)/c^7+1/46 08*b*(288*c^4*d^2+320*c^2*d*e+105*e^2)*x^3*(-c^2*x^2+1)^(1/2)/c^5+1/1152*b *e*(64*c^2*d+21*e)*x^5*(-c^2*x^2+1)^(1/2)/c^3+1/64*b*e^2*x^7*(-c^2*x^2+1)^ (1/2)/c
Time = 0.12 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.79 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {384 a c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+b c x \sqrt {1-c^2 x^2} \left (315 e^2+30 c^2 e \left (32 d+7 e x^2\right )+8 c^4 \left (108 d^2+80 d e x^2+21 e^2 x^4\right )+16 c^6 \left (36 d^2 x^2+32 d e x^4+9 e^2 x^6\right )\right )+3 b \left (-288 c^4 d^2-320 c^2 d e-105 e^2+128 c^8 \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right )\right ) \arcsin (c x)}{9216 c^8} \]
(384*a*c^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + b*c*x*Sqrt[1 - c^2*x^2]*( 315*e^2 + 30*c^2*e*(32*d + 7*e*x^2) + 8*c^4*(108*d^2 + 80*d*e*x^2 + 21*e^2 *x^4) + 16*c^6*(36*d^2*x^2 + 32*d*e*x^4 + 9*e^2*x^6)) + 3*b*(-288*c^4*d^2 - 320*c^2*d*e - 105*e^2 + 128*c^8*(6*d^2*x^4 + 8*d*e*x^6 + 3*e^2*x^8))*Arc Sin[c*x])/(9216*c^8)
Time = 0.42 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5230, 27, 1590, 25, 363, 262, 262, 223}
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.
| \(\displaystyle \int x^3 \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5230 |
| \(\displaystyle -b c \int \frac {x^4 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{24 \sqrt {1-c^2 x^2}}dx+\frac {1}{4} d^2 x^4 (a+b \arcsin (c x))+\frac {1}{3} d e x^6 (a+b \arcsin (c x))+\frac {1}{8} e^2 x^8 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{24} b c \int \frac {x^4 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{4} d^2 x^4 (a+b \arcsin (c x))+\frac {1}{3} d e x^6 (a+b \arcsin (c x))+\frac {1}{8} e^2 x^8 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 1590 |
| \(\displaystyle -\frac {1}{24} b c \left (-\frac {\int -\frac {x^4 \left (48 c^2 d^2+e \left (64 d c^2+21 e\right ) x^2\right )}{\sqrt {1-c^2 x^2}}dx}{8 c^2}-\frac {3 e^2 x^7 \sqrt {1-c^2 x^2}}{8 c^2}\right )+\frac {1}{4} d^2 x^4 (a+b \arcsin (c x))+\frac {1}{3} d e x^6 (a+b \arcsin (c x))+\frac {1}{8} e^2 x^8 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{24} b c \left (\frac {\int \frac {x^4 \left (48 c^2 d^2+e \left (64 d c^2+21 e\right ) x^2\right )}{\sqrt {1-c^2 x^2}}dx}{8 c^2}-\frac {3 e^2 x^7 \sqrt {1-c^2 x^2}}{8 c^2}\right )+\frac {1}{4} d^2 x^4 (a+b \arcsin (c x))+\frac {1}{3} d e x^6 (a+b \arcsin (c x))+\frac {1}{8} e^2 x^8 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle -\frac {1}{24} b c \left (\frac {\frac {\left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx}{6 c^2}-\frac {e x^5 \sqrt {1-c^2 x^2} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}-\frac {3 e^2 x^7 \sqrt {1-c^2 x^2}}{8 c^2}\right )+\frac {1}{4} d^2 x^4 (a+b \arcsin (c x))+\frac {1}{3} d e x^6 (a+b \arcsin (c x))+\frac {1}{8} e^2 x^8 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {1}{24} b c \left (\frac {\frac {\left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )}{6 c^2}-\frac {e x^5 \sqrt {1-c^2 x^2} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}-\frac {3 e^2 x^7 \sqrt {1-c^2 x^2}}{8 c^2}\right )+\frac {1}{4} d^2 x^4 (a+b \arcsin (c x))+\frac {1}{3} d e x^6 (a+b \arcsin (c x))+\frac {1}{8} e^2 x^8 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {1}{24} b c \left (\frac {\frac {\left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )}{6 c^2}-\frac {e x^5 \sqrt {1-c^2 x^2} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}-\frac {3 e^2 x^7 \sqrt {1-c^2 x^2}}{8 c^2}\right )+\frac {1}{4} d^2 x^4 (a+b \arcsin (c x))+\frac {1}{3} d e x^6 (a+b \arcsin (c x))+\frac {1}{8} e^2 x^8 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{4} d^2 x^4 (a+b \arcsin (c x))+\frac {1}{3} d e x^6 (a+b \arcsin (c x))+\frac {1}{8} e^2 x^8 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {\frac {\left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right ) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{6 c^2}-\frac {e x^5 \sqrt {1-c^2 x^2} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}-\frac {3 e^2 x^7 \sqrt {1-c^2 x^2}}{8 c^2}\right )\) |
(d^2*x^4*(a + b*ArcSin[c*x]))/4 + (d*e*x^6*(a + b*ArcSin[c*x]))/3 + (e^2*x ^8*(a + b*ArcSin[c*x]))/8 - (b*c*((-3*e^2*x^7*Sqrt[1 - c^2*x^2])/(8*c^2) + (-1/6*(e*(64*c^2*d + 21*e)*x^5*Sqrt[1 - c^2*x^2])/c^2 + ((288*c^4*d^2 + 3 20*c^2*d*e + 105*e^2)*(-1/4*(x^3*Sqrt[1 - c^2*x^2])/c^2 + (3*(-1/2*(x*Sqrt [1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/(4*c^2)))/(6*c^2))/(8*c^2)))/24
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ (q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q + 1)) Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
Int[((a_.) + ArcSin[(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*ArcSin[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] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
Time = 0.27 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.22
| method | result | size |
| parts | \(a \left (\frac {1}{8} e^{2} x^{8}+\frac {1}{3} d e \,x^{6}+\frac {1}{4} d^{2} x^{4}\right )+\frac {b \left (\frac {c^{4} \arcsin \left (c x \right ) e^{2} x^{8}}{8}+\frac {c^{4} \arcsin \left (c x \right ) e d \,x^{6}}{3}+\frac {\arcsin \left (c x \right ) d^{2} c^{4} x^{4}}{4}-\frac {3 e^{2} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )+6 d^{2} c^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+8 d \,c^{2} e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{24 c^{4}}\right )}{c^{4}}\) | \(293\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{4} d^{2} c^{8} x^{4}+\frac {1}{3} d \,c^{8} e \,x^{6}+\frac {1}{8} e^{2} c^{8} x^{8}\right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d^{2} c^{8} x^{4}}{4}+\frac {\arcsin \left (c x \right ) d \,c^{8} e \,x^{6}}{3}+\frac {\arcsin \left (c x \right ) e^{2} c^{8} x^{8}}{8}-\frac {e^{2} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8}-\frac {d^{2} c^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-\frac {d \,c^{2} e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{3}\right )}{c^{4}}}{c^{4}}\) | \(303\) |
| default | \(\frac {\frac {a \left (\frac {1}{4} d^{2} c^{8} x^{4}+\frac {1}{3} d \,c^{8} e \,x^{6}+\frac {1}{8} e^{2} c^{8} x^{8}\right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d^{2} c^{8} x^{4}}{4}+\frac {\arcsin \left (c x \right ) d \,c^{8} e \,x^{6}}{3}+\frac {\arcsin \left (c x \right ) e^{2} c^{8} x^{8}}{8}-\frac {e^{2} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8}-\frac {d^{2} c^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-\frac {d \,c^{2} e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{3}\right )}{c^{4}}}{c^{4}}\) | \(303\) |
a*(1/8*e^2*x^8+1/3*d*e*x^6+1/4*d^2*x^4)+b/c^4*(1/8*c^4*arcsin(c*x)*e^2*x^8 +1/3*c^4*arcsin(c*x)*e*d*x^6+1/4*arcsin(c*x)*d^2*c^4*x^4-1/24/c^4*(3*e^2*( -1/8*c^7*x^7*(-c^2*x^2+1)^(1/2)-7/48*c^5*x^5*(-c^2*x^2+1)^(1/2)-35/192*c^3 *x^3*(-c^2*x^2+1)^(1/2)-35/128*c*x*(-c^2*x^2+1)^(1/2)+35/128*arcsin(c*x))+ 6*d^2*c^4*(-1/4*c^3*x^3*(-c^2*x^2+1)^(1/2)-3/8*c*x*(-c^2*x^2+1)^(1/2)+3/8* arcsin(c*x))+8*d*c^2*e*(-1/6*c^5*x^5*(-c^2*x^2+1)^(1/2)-5/24*c^3*x^3*(-c^2 *x^2+1)^(1/2)-5/16*c*x*(-c^2*x^2+1)^(1/2)+5/16*arcsin(c*x))))
Time = 0.26 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.89 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {1152 \, a c^{8} e^{2} x^{8} + 3072 \, a c^{8} d e x^{6} + 2304 \, a c^{8} d^{2} x^{4} + 3 \, {\left (384 \, b c^{8} e^{2} x^{8} + 1024 \, b c^{8} d e x^{6} + 768 \, b c^{8} d^{2} x^{4} - 288 \, b c^{4} d^{2} - 320 \, b c^{2} d e - 105 \, b e^{2}\right )} \arcsin \left (c x\right ) + {\left (144 \, b c^{7} e^{2} x^{7} + 8 \, {\left (64 \, b c^{7} d e + 21 \, b c^{5} e^{2}\right )} x^{5} + 2 \, {\left (288 \, b c^{7} d^{2} + 320 \, b c^{5} d e + 105 \, b c^{3} e^{2}\right )} x^{3} + 3 \, {\left (288 \, b c^{5} d^{2} + 320 \, b c^{3} d e + 105 \, b c e^{2}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{9216 \, c^{8}} \]
1/9216*(1152*a*c^8*e^2*x^8 + 3072*a*c^8*d*e*x^6 + 2304*a*c^8*d^2*x^4 + 3*( 384*b*c^8*e^2*x^8 + 1024*b*c^8*d*e*x^6 + 768*b*c^8*d^2*x^4 - 288*b*c^4*d^2 - 320*b*c^2*d*e - 105*b*e^2)*arcsin(c*x) + (144*b*c^7*e^2*x^7 + 8*(64*b*c ^7*d*e + 21*b*c^5*e^2)*x^5 + 2*(288*b*c^7*d^2 + 320*b*c^5*d*e + 105*b*c^3* e^2)*x^3 + 3*(288*b*c^5*d^2 + 320*b*c^3*d*e + 105*b*c*e^2)*x)*sqrt(-c^2*x^ 2 + 1))/c^8
Time = 1.01 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.59 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\begin {cases} \frac {a d^{2} x^{4}}{4} + \frac {a d e x^{6}}{3} + \frac {a e^{2} x^{8}}{8} + \frac {b d^{2} x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b d e x^{6} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b e^{2} x^{8} \operatorname {asin}{\left (c x \right )}}{8} + \frac {b d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b d e x^{5} \sqrt {- c^{2} x^{2} + 1}}{18 c} + \frac {b e^{2} x^{7} \sqrt {- c^{2} x^{2} + 1}}{64 c} + \frac {3 b d^{2} x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {5 b d e x^{3} \sqrt {- c^{2} x^{2} + 1}}{72 c^{3}} + \frac {7 b e^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{384 c^{3}} - \frac {3 b d^{2} \operatorname {asin}{\left (c x \right )}}{32 c^{4}} + \frac {5 b d e x \sqrt {- c^{2} x^{2} + 1}}{48 c^{5}} + \frac {35 b e^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{1536 c^{5}} - \frac {5 b d e \operatorname {asin}{\left (c x \right )}}{48 c^{6}} + \frac {35 b e^{2} x \sqrt {- c^{2} x^{2} + 1}}{1024 c^{7}} - \frac {35 b e^{2} \operatorname {asin}{\left (c x \right )}}{1024 c^{8}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{2} x^{4}}{4} + \frac {d e x^{6}}{3} + \frac {e^{2} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*d**2*x**4/4 + a*d*e*x**6/3 + a*e**2*x**8/8 + b*d**2*x**4*asin (c*x)/4 + b*d*e*x**6*asin(c*x)/3 + b*e**2*x**8*asin(c*x)/8 + b*d**2*x**3*s qrt(-c**2*x**2 + 1)/(16*c) + b*d*e*x**5*sqrt(-c**2*x**2 + 1)/(18*c) + b*e* *2*x**7*sqrt(-c**2*x**2 + 1)/(64*c) + 3*b*d**2*x*sqrt(-c**2*x**2 + 1)/(32* c**3) + 5*b*d*e*x**3*sqrt(-c**2*x**2 + 1)/(72*c**3) + 7*b*e**2*x**5*sqrt(- c**2*x**2 + 1)/(384*c**3) - 3*b*d**2*asin(c*x)/(32*c**4) + 5*b*d*e*x*sqrt( -c**2*x**2 + 1)/(48*c**5) + 35*b*e**2*x**3*sqrt(-c**2*x**2 + 1)/(1536*c**5 ) - 5*b*d*e*asin(c*x)/(48*c**6) + 35*b*e**2*x*sqrt(-c**2*x**2 + 1)/(1024*c **7) - 35*b*e**2*asin(c*x)/(1024*c**8), Ne(c, 0)), (a*(d**2*x**4/4 + d*e*x **6/3 + e**2*x**8/8), True))
Time = 0.27 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.18 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{2} + \frac {1}{144} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b d e + \frac {1}{3072} \, {\left (384 \, x^{8} \arcsin \left (c x\right ) + {\left (\frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{8}} - \frac {105 \, \arcsin \left (c x\right )}{c^{9}}\right )} c\right )} b e^{2} \]
1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 1/32*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x) /c^5)*c)*b*d^2 + 1/144*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin (c*x)/c^7)*c)*b*d*e + 1/3072*(384*x^8*arcsin(c*x) + (48*sqrt(-c^2*x^2 + 1) *x^7/c^2 + 56*sqrt(-c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(-c^2*x^2 + 1)*x^3/c^6 + 105*sqrt(-c^2*x^2 + 1)*x/c^8 - 105*arcsin(c*x)/c^9)*c)*b*e^2
Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (217) = 434\).
Time = 0.29 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.07 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} x}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} \arcsin \left (c x\right )}{4 \, c^{4}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} x}{32 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d e x}{18 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d e \arcsin \left (c x\right )}{3 \, c^{6}} - \frac {13 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e x}{72 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b e^{2} x}{64 \, c^{7}} + \frac {5 \, b d^{2} \arcsin \left (c x\right )}{32 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d e \arcsin \left (c x\right )}{c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b e^{2} \arcsin \left (c x\right )}{8 \, c^{8}} + \frac {11 \, \sqrt {-c^{2} x^{2} + 1} b d e x}{48 \, c^{5}} + \frac {25 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{2} x}{384 \, c^{7}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d e \arcsin \left (c x\right )}{c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b e^{2} \arcsin \left (c x\right )}{2 \, c^{8}} - \frac {163 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2} x}{1536 \, c^{7}} + \frac {11 \, b d e \arcsin \left (c x\right )}{48 \, c^{6}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b e^{2} \arcsin \left (c x\right )}{4 \, c^{8}} + \frac {93 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} x}{1024 \, c^{7}} + \frac {{\left (c^{2} x^{2} - 1\right )} b e^{2} \arcsin \left (c x\right )}{2 \, c^{8}} + \frac {93 \, b e^{2} \arcsin \left (c x\right )}{1024 \, c^{8}} \]
1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 - 1/16*(-c^2*x^2 + 1)^(3/2)* b*d^2*x/c^3 + 1/4*(c^2*x^2 - 1)^2*b*d^2*arcsin(c*x)/c^4 + 5/32*sqrt(-c^2*x ^2 + 1)*b*d^2*x/c^3 + 1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d*e*x/c^5 + 1/2*(c^2*x^2 - 1)*b*d^2*arcsin(c*x)/c^4 + 1/3*(c^2*x^2 - 1)^3*b*d*e*arcs in(c*x)/c^6 - 13/72*(-c^2*x^2 + 1)^(3/2)*b*d*e*x/c^5 + 1/64*(c^2*x^2 - 1)^ 3*sqrt(-c^2*x^2 + 1)*b*e^2*x/c^7 + 5/32*b*d^2*arcsin(c*x)/c^4 + (c^2*x^2 - 1)^2*b*d*e*arcsin(c*x)/c^6 + 1/8*(c^2*x^2 - 1)^4*b*e^2*arcsin(c*x)/c^8 + 11/48*sqrt(-c^2*x^2 + 1)*b*d*e*x/c^5 + 25/384*(c^2*x^2 - 1)^2*sqrt(-c^2*x^ 2 + 1)*b*e^2*x/c^7 + (c^2*x^2 - 1)*b*d*e*arcsin(c*x)/c^6 + 1/2*(c^2*x^2 - 1)^3*b*e^2*arcsin(c*x)/c^8 - 163/1536*(-c^2*x^2 + 1)^(3/2)*b*e^2*x/c^7 + 1 1/48*b*d*e*arcsin(c*x)/c^6 + 3/4*(c^2*x^2 - 1)^2*b*e^2*arcsin(c*x)/c^8 + 9 3/1024*sqrt(-c^2*x^2 + 1)*b*e^2*x/c^7 + 1/2*(c^2*x^2 - 1)*b*e^2*arcsin(c*x )/c^8 + 93/1024*b*e^2*arcsin(c*x)/c^8
Timed out. \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \]