\(\int x (d+e x^2)^3 (a+b \arcsin (c x)) \, dx\) [444]

Optimal result

Integrand size = 19, antiderivative size = 251 \[ \int x \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {b \left (256 c^6 d^3+288 c^4 d^2 e+160 c^2 d e^2+35 e^3\right ) x \sqrt {1-c^2 x^2}}{1024 c^7}+\frac {b e \left (288 c^4 d^2+160 c^2 d e+35 e^2\right ) x^3 \sqrt {1-c^2 x^2}}{1536 c^5}+\frac {b e^2 \left (32 c^2 d+7 e\right ) x^5 \sqrt {1-c^2 x^2}}{384 c^3}+\frac {b e^3 x^7 \sqrt {1-c^2 x^2}}{64 c}-\frac {b \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right ) \arcsin (c x)}{1024 c^8 e}+\frac {\left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e} \] Output:

1/1024*b*(256*c^6*d^3+288*c^4*d^2*e+160*c^2*d*e^2+35*e^3)*x*(-c^2*x^2+1)^( 
1/2)/c^7+1/1536*b*e*(288*c^4*d^2+160*c^2*d*e+35*e^2)*x^3*(-c^2*x^2+1)^(1/2 
)/c^5+1/384*b*e^2*(32*c^2*d+7*e)*x^5*(-c^2*x^2+1)^(1/2)/c^3+1/64*b*e^3*x^7 
*(-c^2*x^2+1)^(1/2)/c-1/1024*b*(128*c^8*d^4+256*c^6*d^3*e+288*c^4*d^2*e^2+ 
160*c^2*d*e^3+35*e^4)*arcsin(c*x)/c^8/e+1/8*(e*x^2+d)^4*(a+b*arcsin(c*x))/ 
e
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.92 \[ \int x \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {c x \left (384 a c^7 x \left (4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right )+b \sqrt {1-c^2 x^2} \left (105 e^3+10 c^2 e^2 \left (48 d+7 e x^2\right )+8 c^4 e \left (108 d^2+40 d e x^2+7 e^2 x^4\right )+16 c^6 \left (48 d^3+36 d^2 e x^2+16 d e^2 x^4+3 e^3 x^6\right )\right )\right )+3 b \left (-256 c^6 d^3-288 c^4 d^2 e-160 c^2 d e^2-35 e^3+128 c^8 \left (4 d^3 x^2+6 d^2 e x^4+4 d e^2 x^6+e^3 x^8\right )\right ) \arcsin (c x)}{3072 c^8} \] Input:

Integrate[x*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]
 

Output:

(c*x*(384*a*c^7*x*(4*d^3 + 6*d^2*e*x^2 + 4*d*e^2*x^4 + e^3*x^6) + b*Sqrt[1 
 - c^2*x^2]*(105*e^3 + 10*c^2*e^2*(48*d + 7*e*x^2) + 8*c^4*e*(108*d^2 + 40 
*d*e*x^2 + 7*e^2*x^4) + 16*c^6*(48*d^3 + 36*d^2*e*x^2 + 16*d*e^2*x^4 + 3*e 
^3*x^6))) + 3*b*(-256*c^6*d^3 - 288*c^4*d^2*e - 160*c^2*d*e^2 - 35*e^3 + 1 
28*c^8*(4*d^3*x^2 + 6*d^2*e*x^4 + 4*d*e^2*x^6 + e^3*x^8))*ArcSin[c*x])/(30 
72*c^8)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {5228, 318, 25, 403, 25, 403, 25, 299, 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.

\(\displaystyle \int x \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx\)

\(\Big \downarrow \) 5228

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e}-\frac {b c \int \frac {\left (e x^2+d\right )^4}{\sqrt {1-c^2 x^2}}dx}{8 e}\)

\(\Big \downarrow \) 318

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e}-\frac {b c \left (-\frac {\int -\frac {\left (e x^2+d\right )^2 \left (7 e \left (2 d c^2+e\right ) x^2+d \left (8 d c^2+e\right )\right )}{\sqrt {1-c^2 x^2}}dx}{8 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e}-\frac {b c \left (\frac {\int \frac {\left (e x^2+d\right )^2 \left (7 e \left (2 d c^2+e\right ) x^2+d \left (8 d c^2+e\right )\right )}{\sqrt {1-c^2 x^2}}dx}{8 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e}-\frac {b c \left (\frac {-\frac {\int -\frac {\left (e x^2+d\right ) \left (e \left (104 d^2 c^4+104 d e c^2+35 e^2\right ) x^2+d \left (48 d^2 c^4+20 d e c^2+7 e^2\right )\right )}{\sqrt {1-c^2 x^2}}dx}{6 c^2}-\frac {7 e x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e}-\frac {b c \left (\frac {\frac {\int \frac {\left (e x^2+d\right ) \left (e \left (104 d^2 c^4+104 d e c^2+35 e^2\right ) x^2+d \left (48 d^2 c^4+20 d e c^2+7 e^2\right )\right )}{\sqrt {1-c^2 x^2}}dx}{6 c^2}-\frac {7 e x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e}-\frac {b c \left (\frac {\frac {-\frac {\int -\frac {5 e \left (2 d c^2+e\right ) \left (40 d^2 c^4+40 d e c^2+21 e^2\right ) x^2+d \left (192 d^3 c^6+184 d^2 e c^4+132 d e^2 c^2+35 e^3\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}-\frac {7 e x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e}-\frac {b c \left (\frac {\frac {\frac {\int \frac {5 e \left (2 d c^2+e\right ) \left (40 d^2 c^4+40 d e c^2+21 e^2\right ) x^2+d \left (192 d^3 c^6+184 d^2 e c^4+132 d e^2 c^2+35 e^3\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}-\frac {7 e x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e}-\frac {b c \left (\frac {\frac {\frac {\frac {3 \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right ) \int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {5 e x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{2 c^2}}{4 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}-\frac {7 e x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e}-\frac {b c \left (\frac {\frac {\frac {\frac {3 \arcsin (c x) \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right )}{2 c^3}-\frac {5 e x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{2 c^2}}{4 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}-\frac {7 e x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e}\)

Input:

Int[x*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]
 

Output:

((d + e*x^2)^4*(a + b*ArcSin[c*x]))/(8*e) - (b*c*(-1/8*(e*x*Sqrt[1 - c^2*x 
^2]*(d + e*x^2)^3)/c^2 + ((-7*e*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2]*(d + e*x 
^2)^2)/(6*c^2) + (-1/4*(e*(104*c^4*d^2 + 104*c^2*d*e + 35*e^2)*x*Sqrt[1 - 
c^2*x^2]*(d + e*x^2))/c^2 + ((-5*e*(2*c^2*d + e)*(40*c^4*d^2 + 40*c^2*d*e 
+ 21*e^2)*x*Sqrt[1 - c^2*x^2])/(2*c^2) + (3*(128*c^8*d^4 + 256*c^6*d^3*e + 
 288*c^4*d^2*e^2 + 160*c^2*d*e^3 + 35*e^4)*ArcSin[c*x])/(2*c^3))/(4*c^2))/ 
(6*c^2))/(8*c^2)))/(8*e)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 223

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]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

rule 5228

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])/(2*e*(p + 1))), x] 
 - Simp[b*(c/(2*e*(p + 1)))   Int[(d + e*x^2)^(p + 1)/Sqrt[1 - c^2*x^2], x] 
, x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.45

method	result	size
parts	\(\frac {a \left (e \,x^{2}+d \right )^{4}}{8 e}+\frac {b \left (\frac {c^{2} e^{3} \arcsin \left (c x \right ) x^{8}}{8}+\frac {c^{2} e^{2} \arcsin \left (c x \right ) x^{6} d}{2}+\frac {3 c^{2} e \arcsin \left (c x \right ) x^{4} d^{2}}{4}+\frac {\arcsin \left (c x \right ) c^{2} x^{2} d^{3}}{2}+\frac {c^{2} \arcsin \left (c x \right ) d^{4}}{8 e}-\frac {c^{8} d^{4} \arcsin \left (c x \right )+e^{4} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )+4 d \,c^{2} e^{3} \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )+6 d^{2} c^{4} e^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+4 d^{3} c^{6} e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{8 c^{6} e}\right )}{c^{2}}\)	\(365\)
derivativedivides	\(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{8 c^{6} e}+\frac {b \left (\frac {\arcsin \left (c x \right ) c^{8} d^{4}}{8 e}+\frac {\arcsin \left (c x \right ) c^{8} d^{3} x^{2}}{2}+\frac {3 e \arcsin \left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e^{2} \arcsin \left (c x \right ) c^{8} d \,x^{6}}{2}+\frac {e^{3} \arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {c^{8} d^{4} \arcsin \left (c x \right )+e^{4} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )+4 d \,c^{2} e^{3} \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )+6 d^{2} c^{4} e^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+4 d^{3} c^{6} e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{8 e}\right )}{c^{6}}}{c^{2}}\)	\(376\)
default	\(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{8 c^{6} e}+\frac {b \left (\frac {\arcsin \left (c x \right ) c^{8} d^{4}}{8 e}+\frac {\arcsin \left (c x \right ) c^{8} d^{3} x^{2}}{2}+\frac {3 e \arcsin \left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e^{2} \arcsin \left (c x \right ) c^{8} d \,x^{6}}{2}+\frac {e^{3} \arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {c^{8} d^{4} \arcsin \left (c x \right )+e^{4} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )+4 d \,c^{2} e^{3} \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )+6 d^{2} c^{4} e^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+4 d^{3} c^{6} e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{8 e}\right )}{c^{6}}}{c^{2}}\)	\(376\)
orering	\(\frac {\left (720 c^{8} e^{4} x^{10}+3760 c^{8} d \,e^{3} x^{8}+8128 c^{8} d^{2} e^{2} x^{6}+56 c^{6} e^{4} x^{8}+9792 c^{8} d^{3} e \,x^{4}+456 c^{6} d \,e^{3} x^{6}+2304 c^{8} d^{4} x^{2}+2080 c^{6} d^{2} e^{2} x^{4}+98 c^{4} e^{4} x^{6}-5856 c^{6} d^{3} e \,x^{2}+1134 c^{4} d \,e^{3} x^{4}-1536 c^{6} d^{4}-6752 c^{4} d^{2} e^{2} x^{2}+245 c^{2} e^{4} x^{4}-1728 c^{4} d^{3} e -3805 c^{2} d \,e^{3} x^{2}-960 c^{2} d^{2} e^{2}-840 e^{4} x^{2}-210 d \,e^{3}\right ) \left (a +b \arcsin \left (c x \right )\right )}{3072 \left (e \,x^{2}+d \right ) c^{8}}-\frac {\left (48 e^{3} x^{6} c^{6}+256 c^{6} d \,e^{2} x^{4}+576 c^{6} d^{2} e \,x^{2}+56 c^{4} e^{3} x^{4}+768 c^{6} d^{3}+320 c^{4} d \,e^{2} x^{2}+864 c^{4} d^{2} e +70 c^{2} e^{3} x^{2}+480 c^{2} d \,e^{2}+105 e^{3}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (\left (e \,x^{2}+d \right )^{3} \left (a +b \arcsin \left (c x \right )\right )+6 x^{2} \left (e \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right ) e +\frac {x \left (e \,x^{2}+d \right )^{3} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{3072 c^{8} \left (e \,x^{2}+d \right )^{3}}\)	\(428\)

Input:

int(x*(e*x^2+d)^3*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
 

Output:

1/8*a*(e*x^2+d)^4/e+b/c^2*(1/8*c^2*e^3*arcsin(c*x)*x^8+1/2*c^2*e^2*arcsin( 
c*x)*x^6*d+3/4*c^2*e*arcsin(c*x)*x^4*d^2+1/2*arcsin(c*x)*c^2*x^2*d^3+1/8*c 
^2/e*arcsin(c*x)*d^4-1/8/c^6/e*(c^8*d^4*arcsin(c*x)+e^4*(-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))+4*d*c^2*e^3*(-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))+6*d^2*c^4*e^2*(-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))+4*d^3*c^6*e*(-1/2*c*x*(-c 
^2*x^2+1)^(1/2)+1/2*arcsin(c*x))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.09 \[ \int x \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {384 \, a c^{8} e^{3} x^{8} + 1536 \, a c^{8} d e^{2} x^{6} + 2304 \, a c^{8} d^{2} e x^{4} + 1536 \, a c^{8} d^{3} x^{2} + 3 \, {\left (128 \, b c^{8} e^{3} x^{8} + 512 \, b c^{8} d e^{2} x^{6} + 768 \, b c^{8} d^{2} e x^{4} + 512 \, b c^{8} d^{3} x^{2} - 256 \, b c^{6} d^{3} - 288 \, b c^{4} d^{2} e - 160 \, b c^{2} d e^{2} - 35 \, b e^{3}\right )} \arcsin \left (c x\right ) + {\left (48 \, b c^{7} e^{3} x^{7} + 8 \, {\left (32 \, b c^{7} d e^{2} + 7 \, b c^{5} e^{3}\right )} x^{5} + 2 \, {\left (288 \, b c^{7} d^{2} e + 160 \, b c^{5} d e^{2} + 35 \, b c^{3} e^{3}\right )} x^{3} + 3 \, {\left (256 \, b c^{7} d^{3} + 288 \, b c^{5} d^{2} e + 160 \, b c^{3} d e^{2} + 35 \, b c e^{3}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{3072 \, c^{8}} \] Input:

integrate(x*(e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="fricas")
 

Output:

1/3072*(384*a*c^8*e^3*x^8 + 1536*a*c^8*d*e^2*x^6 + 2304*a*c^8*d^2*e*x^4 + 
1536*a*c^8*d^3*x^2 + 3*(128*b*c^8*e^3*x^8 + 512*b*c^8*d*e^2*x^6 + 768*b*c^ 
8*d^2*e*x^4 + 512*b*c^8*d^3*x^2 - 256*b*c^6*d^3 - 288*b*c^4*d^2*e - 160*b* 
c^2*d*e^2 - 35*b*e^3)*arcsin(c*x) + (48*b*c^7*e^3*x^7 + 8*(32*b*c^7*d*e^2 
+ 7*b*c^5*e^3)*x^5 + 2*(288*b*c^7*d^2*e + 160*b*c^5*d*e^2 + 35*b*c^3*e^3)* 
x^3 + 3*(256*b*c^7*d^3 + 288*b*c^5*d^2*e + 160*b*c^3*d*e^2 + 35*b*c*e^3)*x 
)*sqrt(-c^2*x^2 + 1))/c^8
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (238) = 476\).

Time = 0.90 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.92 \[ \int x \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\begin {cases} \frac {a d^{3} x^{2}}{2} + \frac {3 a d^{2} e x^{4}}{4} + \frac {a d e^{2} x^{6}}{2} + \frac {a e^{3} x^{8}}{8} + \frac {b d^{3} x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {3 b d^{2} e x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b d e^{2} x^{6} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e^{3} x^{8} \operatorname {asin}{\left (c x \right )}}{8} + \frac {b d^{3} x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {3 b d^{2} e x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b d e^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{12 c} + \frac {b e^{3} x^{7} \sqrt {- c^{2} x^{2} + 1}}{64 c} - \frac {b d^{3} \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {9 b d^{2} e x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {5 b d e^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{48 c^{3}} + \frac {7 b e^{3} x^{5} \sqrt {- c^{2} x^{2} + 1}}{384 c^{3}} - \frac {9 b d^{2} e \operatorname {asin}{\left (c x \right )}}{32 c^{4}} + \frac {5 b d e^{2} x \sqrt {- c^{2} x^{2} + 1}}{32 c^{5}} + \frac {35 b e^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{1536 c^{5}} - \frac {5 b d e^{2} \operatorname {asin}{\left (c x \right )}}{32 c^{6}} + \frac {35 b e^{3} x \sqrt {- c^{2} x^{2} + 1}}{1024 c^{7}} - \frac {35 b e^{3} \operatorname {asin}{\left (c x \right )}}{1024 c^{8}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{3} x^{2}}{2} + \frac {3 d^{2} e x^{4}}{4} + \frac {d e^{2} x^{6}}{2} + \frac {e^{3} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \] Input:

integrate(x*(e*x**2+d)**3*(a+b*asin(c*x)),x)
 

Output:

Piecewise((a*d**3*x**2/2 + 3*a*d**2*e*x**4/4 + a*d*e**2*x**6/2 + a*e**3*x* 
*8/8 + b*d**3*x**2*asin(c*x)/2 + 3*b*d**2*e*x**4*asin(c*x)/4 + b*d*e**2*x* 
*6*asin(c*x)/2 + b*e**3*x**8*asin(c*x)/8 + b*d**3*x*sqrt(-c**2*x**2 + 1)/( 
4*c) + 3*b*d**2*e*x**3*sqrt(-c**2*x**2 + 1)/(16*c) + b*d*e**2*x**5*sqrt(-c 
**2*x**2 + 1)/(12*c) + b*e**3*x**7*sqrt(-c**2*x**2 + 1)/(64*c) - b*d**3*as 
in(c*x)/(4*c**2) + 9*b*d**2*e*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 5*b*d*e** 
2*x**3*sqrt(-c**2*x**2 + 1)/(48*c**3) + 7*b*e**3*x**5*sqrt(-c**2*x**2 + 1) 
/(384*c**3) - 9*b*d**2*e*asin(c*x)/(32*c**4) + 5*b*d*e**2*x*sqrt(-c**2*x** 
2 + 1)/(32*c**5) + 35*b*e**3*x**3*sqrt(-c**2*x**2 + 1)/(1536*c**5) - 5*b*d 
*e**2*asin(c*x)/(32*c**6) + 35*b*e**3*x*sqrt(-c**2*x**2 + 1)/(1024*c**7) - 
 35*b*e**3*asin(c*x)/(1024*c**8), Ne(c, 0)), (a*(d**3*x**2/2 + 3*d**2*e*x* 
*4/4 + d*e**2*x**6/2 + e**3*x**8/8), True))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.37 \[ \int x \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {1}{8} \, a e^{3} x^{8} + \frac {1}{2} \, a d e^{2} x^{6} + \frac {3}{4} \, a d^{2} e x^{4} + \frac {1}{2} \, a d^{3} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{3} + \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{2} e + \frac {1}{96} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b d e^{2} + \frac {1}{3072} \, {\left (384 \, x^{8} \arcsin \left (c x\right ) + {\left (\frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{8}} - \frac {105 \, \arcsin \left (c x\right )}{c^{9}}\right )} c\right )} b e^{3} \] Input:

integrate(x*(e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="maxima")
 

Output:

1/8*a*e^3*x^8 + 1/2*a*d*e^2*x^6 + 3/4*a*d^2*e*x^4 + 1/2*a*d^3*x^2 + 1/4*(2 
*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*b*d^3 + 
 3/32*(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*e + 1/96*(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^2 + 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 + 7 
0*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^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (231) = 462\).

Time = 0.14 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.38 \[ \int x \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx =\text {Too large to display} \] Input:

integrate(x*(e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="giac")
 

Output:

1/8*a*e^3*x^8 + 1/2*a*d*e^2*x^6 + 3/4*a*d^2*e*x^4 + 1/4*sqrt(-c^2*x^2 + 1) 
*b*d^3*x/c + 1/2*(c^2*x^2 - 1)*b*d^3*arcsin(c*x)/c^2 - 3/16*(-c^2*x^2 + 1) 
^(3/2)*b*d^2*e*x/c^3 + 1/2*(c^2*x^2 - 1)*a*d^3/c^2 + 1/4*b*d^3*arcsin(c*x) 
/c^2 + 3/4*(c^2*x^2 - 1)^2*b*d^2*e*arcsin(c*x)/c^4 + 15/32*sqrt(-c^2*x^2 + 
 1)*b*d^2*e*x/c^3 + 1/12*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d*e^2*x/c^5 
+ 3/2*(c^2*x^2 - 1)*b*d^2*e*arcsin(c*x)/c^4 + 1/2*(c^2*x^2 - 1)^3*b*d*e^2* 
arcsin(c*x)/c^6 - 13/48*(-c^2*x^2 + 1)^(3/2)*b*d*e^2*x/c^5 + 1/64*(c^2*x^2 
 - 1)^3*sqrt(-c^2*x^2 + 1)*b*e^3*x/c^7 + 15/32*b*d^2*e*arcsin(c*x)/c^4 + 3 
/2*(c^2*x^2 - 1)^2*b*d*e^2*arcsin(c*x)/c^6 + 1/8*(c^2*x^2 - 1)^4*b*e^3*arc 
sin(c*x)/c^8 + 11/32*sqrt(-c^2*x^2 + 1)*b*d*e^2*x/c^5 + 25/384*(c^2*x^2 - 
1)^2*sqrt(-c^2*x^2 + 1)*b*e^3*x/c^7 + 3/2*(c^2*x^2 - 1)*b*d*e^2*arcsin(c*x 
)/c^6 + 1/2*(c^2*x^2 - 1)^3*b*e^3*arcsin(c*x)/c^8 - 163/1536*(-c^2*x^2 + 1 
)^(3/2)*b*e^3*x/c^7 + 11/32*b*d*e^2*arcsin(c*x)/c^6 + 3/4*(c^2*x^2 - 1)^2* 
b*e^3*arcsin(c*x)/c^8 + 93/1024*sqrt(-c^2*x^2 + 1)*b*e^3*x/c^7 + 1/2*(c^2* 
x^2 - 1)*b*e^3*arcsin(c*x)/c^8 + 93/1024*b*e^3*arcsin(c*x)/c^8
 

Mupad [F(-1)]

Timed out. \[ \int x \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\int x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \] Input:

int(x*(a + b*asin(c*x))*(d + e*x^2)^3,x)
 

Output:

int(x*(a + b*asin(c*x))*(d + e*x^2)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.59 \[ \int x \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {1536 \mathit {asin} \left (c x \right ) b \,c^{8} d^{3} x^{2}+2304 \mathit {asin} \left (c x \right ) b \,c^{8} d^{2} e \,x^{4}+1536 \mathit {asin} \left (c x \right ) b \,c^{8} d \,e^{2} x^{6}+384 \mathit {asin} \left (c x \right ) b \,c^{8} e^{3} x^{8}-768 \mathit {asin} \left (c x \right ) b \,c^{6} d^{3}-864 \mathit {asin} \left (c x \right ) b \,c^{4} d^{2} e -480 \mathit {asin} \left (c x \right ) b \,c^{2} d \,e^{2}-105 \mathit {asin} \left (c x \right ) b \,e^{3}+768 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} d^{3} x +576 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} d^{2} e \,x^{3}+256 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} d \,e^{2} x^{5}+48 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} e^{3} x^{7}+864 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} d^{2} e x +320 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} d \,e^{2} x^{3}+56 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} e^{3} x^{5}+480 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d \,e^{2} x +70 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} e^{3} x^{3}+105 \sqrt {-c^{2} x^{2}+1}\, b c \,e^{3} x +1536 a \,c^{8} d^{3} x^{2}+2304 a \,c^{8} d^{2} e \,x^{4}+1536 a \,c^{8} d \,e^{2} x^{6}+384 a \,c^{8} e^{3} x^{8}}{3072 c^{8}} \] Input:

int(x*(e*x^2+d)^3*(a+b*asin(c*x)),x)
 

Output:

(1536*asin(c*x)*b*c**8*d**3*x**2 + 2304*asin(c*x)*b*c**8*d**2*e*x**4 + 153 
6*asin(c*x)*b*c**8*d*e**2*x**6 + 384*asin(c*x)*b*c**8*e**3*x**8 - 768*asin 
(c*x)*b*c**6*d**3 - 864*asin(c*x)*b*c**4*d**2*e - 480*asin(c*x)*b*c**2*d*e 
**2 - 105*asin(c*x)*b*e**3 + 768*sqrt( - c**2*x**2 + 1)*b*c**7*d**3*x + 57 
6*sqrt( - c**2*x**2 + 1)*b*c**7*d**2*e*x**3 + 256*sqrt( - c**2*x**2 + 1)*b 
*c**7*d*e**2*x**5 + 48*sqrt( - c**2*x**2 + 1)*b*c**7*e**3*x**7 + 864*sqrt( 
 - c**2*x**2 + 1)*b*c**5*d**2*e*x + 320*sqrt( - c**2*x**2 + 1)*b*c**5*d*e* 
*2*x**3 + 56*sqrt( - c**2*x**2 + 1)*b*c**5*e**3*x**5 + 480*sqrt( - c**2*x* 
*2 + 1)*b*c**3*d*e**2*x + 70*sqrt( - c**2*x**2 + 1)*b*c**3*e**3*x**3 + 105 
*sqrt( - c**2*x**2 + 1)*b*c*e**3*x + 1536*a*c**8*d**3*x**2 + 2304*a*c**8*d 
**2*e*x**4 + 1536*a*c**8*d*e**2*x**6 + 384*a*c**8*e**3*x**8)/(3072*c**8)