\(\int x (d+e x^2)^2 (a+b \arcsin (c x)) \, dx\) [435]

Optimal result

Integrand size = 19, antiderivative size = 177 \[ \int x \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {b \left (24 c^4 d^2+18 c^2 d e+5 e^2\right ) x \sqrt {1-c^2 x^2}}{96 c^5}+\frac {b e \left (18 c^2 d+5 e\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^2 x^5 \sqrt {1-c^2 x^2}}{36 c}-\frac {b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \arcsin (c x)}{96 c^6 e}+\frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{6 e} \] Output:

1/96*b*(24*c^4*d^2+18*c^2*d*e+5*e^2)*x*(-c^2*x^2+1)^(1/2)/c^5+1/144*b*e*(1 
8*c^2*d+5*e)*x^3*(-c^2*x^2+1)^(1/2)/c^3+1/36*b*e^2*x^5*(-c^2*x^2+1)^(1/2)/ 
c-1/96*b*(2*c^2*d+e)*(8*c^4*d^2+8*c^2*d*e+5*e^2)*arcsin(c*x)/c^6/e+1/6*(e* 
x^2+d)^3*(a+b*arcsin(c*x))/e
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.90 \[ \int x \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {c x \left (48 a c^5 x \left (3 d^2+3 d e x^2+e^2 x^4\right )+b \sqrt {1-c^2 x^2} \left (15 e^2+2 c^2 e \left (27 d+5 e x^2\right )+4 c^4 \left (18 d^2+9 d e x^2+2 e^2 x^4\right )\right )\right )+3 b \left (-24 c^4 d^2-18 c^2 d e-5 e^2+16 c^6 \left (3 d^2 x^2+3 d e x^4+e^2 x^6\right )\right ) \arcsin (c x)}{288 c^6} \] Input:

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

Output:

(c*x*(48*a*c^5*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) + b*Sqrt[1 - c^2*x^2]*(15*e 
^2 + 2*c^2*e*(27*d + 5*e*x^2) + 4*c^4*(18*d^2 + 9*d*e*x^2 + 2*e^2*x^4))) + 
 3*b*(-24*c^4*d^2 - 18*c^2*d*e - 5*e^2 + 16*c^6*(3*d^2*x^2 + 3*d*e*x^4 + e 
^2*x^6))*ArcSin[c*x])/(288*c^6)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5228, 318, 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.

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

\(\Big \downarrow \) 5228

\(\displaystyle \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{6 e}-\frac {b c \int \frac {\left (e x^2+d\right )^3}{\sqrt {1-c^2 x^2}}dx}{6 e}\)

\(\Big \downarrow \) 318

\(\displaystyle \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{6 e}-\frac {b c \left (-\frac {\int -\frac {\left (e x^2+d\right ) \left (5 e \left (2 d c^2+e\right ) x^2+d \left (6 d c^2+e\right )\right )}{\sqrt {1-c^2 x^2}}dx}{6 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{6 c^2}\right )}{6 e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{6 e}-\frac {b c \left (\frac {\int \frac {\left (e x^2+d\right ) \left (5 e \left (2 d c^2+e\right ) x^2+d \left (6 d c^2+e\right )\right )}{\sqrt {1-c^2 x^2}}dx}{6 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{6 c^2}\right )}{6 e}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{6 e}-\frac {b c \left (\frac {-\frac {\int -\frac {e \left (44 d^2 c^4+44 d e c^2+15 e^2\right ) x^2+d \left (24 d^2 c^4+14 d e c^2+5 e^2\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {5 e x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{6 c^2}\right )}{6 e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{6 e}-\frac {b c \left (\frac {\frac {\int \frac {e \left (44 d^2 c^4+44 d e c^2+15 e^2\right ) x^2+d \left (24 d^2 c^4+14 d e c^2+5 e^2\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {5 e x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{6 c^2}\right )}{6 e}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{6 e}-\frac {b c \left (\frac {\frac {\frac {3 \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (44 c^4 d^2+44 c^2 d e+15 e^2\right )}{2 c^2}}{4 c^2}-\frac {5 e x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{6 c^2}\right )}{6 e}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{6 e}-\frac {b c \left (\frac {\frac {\frac {3 \arcsin (c x) \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right )}{2 c^3}-\frac {e x \sqrt {1-c^2 x^2} \left (44 c^4 d^2+44 c^2 d e+15 e^2\right )}{2 c^2}}{4 c^2}-\frac {5 e x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{6 c^2}\right )}{6 e}\)

Input:

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

Output:

((d + e*x^2)^3*(a + b*ArcSin[c*x]))/(6*e) - (b*c*(-1/6*(e*x*Sqrt[1 - c^2*x 
^2]*(d + e*x^2)^2)/c^2 + ((-5*e*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2]*(d + e*x 
^2))/(4*c^2) + (-1/2*(e*(44*c^4*d^2 + 44*c^2*d*e + 15*e^2)*x*Sqrt[1 - c^2* 
x^2])/c^2 + (3*(2*c^2*d + e)*(8*c^4*d^2 + 8*c^2*d*e + 5*e^2)*ArcSin[c*x])/ 
(2*c^3))/(4*c^2))/(6*c^2)))/(6*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 = 253, normalized size of antiderivative = 1.43

method	result	size
parts	\(\frac {a \left (e \,x^{2}+d \right )^{3}}{6 e}+\frac {b \left (\frac {c^{2} e^{2} \arcsin \left (c x \right ) x^{6}}{6}+\frac {c^{2} e \arcsin \left (c x \right ) x^{4} d}{2}+\frac {\arcsin \left (c x \right ) c^{2} x^{2} d^{2}}{2}+\frac {c^{2} \arcsin \left (c x \right ) d^{3}}{6 e}-\frac {c^{6} d^{3} \arcsin \left (c x \right )+e^{3} \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )+3 d \,c^{2} e^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+3 d^{2} c^{4} e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{6 c^{4} e}\right )}{c^{2}}\)	\(253\)
derivativedivides	\(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b \left (\frac {\arcsin \left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\arcsin \left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {e \arcsin \left (c x \right ) c^{6} d \,x^{4}}{2}+\frac {e^{2} \arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{6} d^{3} \arcsin \left (c x \right )+e^{3} \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )+3 d \,c^{2} e^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+3 d^{2} c^{4} e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{6 e}\right )}{c^{4}}}{c^{2}}\)	\(264\)
default	\(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b \left (\frac {\arcsin \left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\arcsin \left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {e \arcsin \left (c x \right ) c^{6} d \,x^{4}}{2}+\frac {e^{2} \arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{6} d^{3} \arcsin \left (c x \right )+e^{3} \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )+3 d \,c^{2} e^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+3 d^{2} c^{4} e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{6 e}\right )}{c^{4}}}{c^{2}}\)	\(264\)
orering	\(\frac {\left (88 x^{8} e^{3} c^{6}+380 x^{6} e^{2} c^{6} d +684 x^{4} e \,c^{6} d^{2}+10 x^{6} e^{3} c^{4}+216 c^{6} d^{3} x^{2}+92 x^{4} e^{2} c^{4} d -414 x^{2} e \,c^{4} d^{2}+25 x^{4} e^{3} c^{2}-144 d^{3} c^{4}-319 x^{2} e^{2} c^{2} d -108 c^{2} d^{2} e -90 x^{2} e^{3}-30 d \,e^{2}\right ) \left (a +b \arcsin \left (c x \right )\right )}{288 \left (e \,x^{2}+d \right ) c^{6}}-\frac {\left (8 e^{2} x^{4} c^{4}+36 c^{4} d e \,x^{2}+72 c^{4} d^{2}+10 c^{2} e^{2} x^{2}+54 c^{2} d e +15 e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (\left (e \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )+4 x^{2} \left (e \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right ) e +\frac {x \left (e \,x^{2}+d \right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{288 c^{6} \left (e \,x^{2}+d \right )^{2}}\)	\(302\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.03 \[ \int x \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {48 \, a c^{6} e^{2} x^{6} + 144 \, a c^{6} d e x^{4} + 144 \, a c^{6} d^{2} x^{2} + 3 \, {\left (16 \, b c^{6} e^{2} x^{6} + 48 \, b c^{6} d e x^{4} + 48 \, b c^{6} d^{2} x^{2} - 24 \, b c^{4} d^{2} - 18 \, b c^{2} d e - 5 \, b e^{2}\right )} \arcsin \left (c x\right ) + {\left (8 \, b c^{5} e^{2} x^{5} + 2 \, {\left (18 \, b c^{5} d e + 5 \, b c^{3} e^{2}\right )} x^{3} + 3 \, {\left (24 \, b c^{5} d^{2} + 18 \, b c^{3} d e + 5 \, b c e^{2}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{6}} \] Input:

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

Output:

1/288*(48*a*c^6*e^2*x^6 + 144*a*c^6*d*e*x^4 + 144*a*c^6*d^2*x^2 + 3*(16*b* 
c^6*e^2*x^6 + 48*b*c^6*d*e*x^4 + 48*b*c^6*d^2*x^2 - 24*b*c^4*d^2 - 18*b*c^ 
2*d*e - 5*b*e^2)*arcsin(c*x) + (8*b*c^5*e^2*x^5 + 2*(18*b*c^5*d*e + 5*b*c^ 
3*e^2)*x^3 + 3*(24*b*c^5*d^2 + 18*b*c^3*d*e + 5*b*c*e^2)*x)*sqrt(-c^2*x^2 
+ 1))/c^6
 

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.69 \[ \int x \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\begin {cases} \frac {a d^{2} x^{2}}{2} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{6}}{6} + \frac {b d^{2} x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b d e x^{4} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e^{2} x^{6} \operatorname {asin}{\left (c x \right )}}{6} + \frac {b d^{2} x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b d e x^{3} \sqrt {- c^{2} x^{2} + 1}}{8 c} + \frac {b e^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{36 c} - \frac {b d^{2} \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {3 b d e x \sqrt {- c^{2} x^{2} + 1}}{16 c^{3}} + \frac {5 b e^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{144 c^{3}} - \frac {3 b d e \operatorname {asin}{\left (c x \right )}}{16 c^{4}} + \frac {5 b e^{2} x \sqrt {- c^{2} x^{2} + 1}}{96 c^{5}} - \frac {5 b e^{2} \operatorname {asin}{\left (c x \right )}}{96 c^{6}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{2} x^{2}}{2} + \frac {d e x^{4}}{2} + \frac {e^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*d**2*x**2/2 + a*d*e*x**4/2 + a*e**2*x**6/6 + b*d**2*x**2*asin 
(c*x)/2 + b*d*e*x**4*asin(c*x)/2 + b*e**2*x**6*asin(c*x)/6 + b*d**2*x*sqrt 
(-c**2*x**2 + 1)/(4*c) + b*d*e*x**3*sqrt(-c**2*x**2 + 1)/(8*c) + b*e**2*x* 
*5*sqrt(-c**2*x**2 + 1)/(36*c) - b*d**2*asin(c*x)/(4*c**2) + 3*b*d*e*x*sqr 
t(-c**2*x**2 + 1)/(16*c**3) + 5*b*e**2*x**3*sqrt(-c**2*x**2 + 1)/(144*c**3 
) - 3*b*d*e*asin(c*x)/(16*c**4) + 5*b*e**2*x*sqrt(-c**2*x**2 + 1)/(96*c**5 
) - 5*b*e**2*asin(c*x)/(96*c**6), Ne(c, 0)), (a*(d**2*x**2/2 + d*e*x**4/2 
+ e**2*x**6/6), True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.26 \[ \int x \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {1}{6} \, a e^{2} x^{6} + \frac {1}{2} \, a d e x^{4} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d e + \frac {1}{288} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b e^{2} \] Input:

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

Output:

1/6*a*e^2*x^6 + 1/2*a*d*e*x^4 + 1/2*a*d^2*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^2 + 1/16*(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*arcs 
in(c*x)/c^5)*c)*b*d*e + 1/288*(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*e^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (161) = 322\).

Time = 0.13 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.98 \[ \int x \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {1}{6} \, a e^{2} x^{6} + \frac {1}{2} \, a d e x^{4} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} x}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} \arcsin \left (c x\right )}{2 \, c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e x}{8 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d^{2}}{2 \, c^{2}} + \frac {b d^{2} \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d e \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d e x}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{2} x}{36 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d e \arcsin \left (c x\right )}{c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b e^{2} \arcsin \left (c x\right )}{6 \, c^{6}} - \frac {13 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2} x}{144 \, c^{5}} + \frac {5 \, b d e \arcsin \left (c x\right )}{16 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b e^{2} \arcsin \left (c x\right )}{2 \, c^{6}} + \frac {11 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} x}{96 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )} b e^{2} \arcsin \left (c x\right )}{2 \, c^{6}} + \frac {11 \, b e^{2} \arcsin \left (c x\right )}{96 \, c^{6}} \] Input:

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

Output:

1/6*a*e^2*x^6 + 1/2*a*d*e*x^4 + 1/4*sqrt(-c^2*x^2 + 1)*b*d^2*x/c + 1/2*(c^ 
2*x^2 - 1)*b*d^2*arcsin(c*x)/c^2 - 1/8*(-c^2*x^2 + 1)^(3/2)*b*d*e*x/c^3 + 
1/2*(c^2*x^2 - 1)*a*d^2/c^2 + 1/4*b*d^2*arcsin(c*x)/c^2 + 1/2*(c^2*x^2 - 1 
)^2*b*d*e*arcsin(c*x)/c^4 + 5/16*sqrt(-c^2*x^2 + 1)*b*d*e*x/c^3 + 1/36*(c^ 
2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e^2*x/c^5 + (c^2*x^2 - 1)*b*d*e*arcsin(c 
*x)/c^4 + 1/6*(c^2*x^2 - 1)^3*b*e^2*arcsin(c*x)/c^6 - 13/144*(-c^2*x^2 + 1 
)^(3/2)*b*e^2*x/c^5 + 5/16*b*d*e*arcsin(c*x)/c^4 + 1/2*(c^2*x^2 - 1)^2*b*e 
^2*arcsin(c*x)/c^6 + 11/96*sqrt(-c^2*x^2 + 1)*b*e^2*x/c^5 + 1/2*(c^2*x^2 - 
 1)*b*e^2*arcsin(c*x)/c^6 + 11/96*b*e^2*arcsin(c*x)/c^6
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.42 \[ \int x \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {144 \mathit {asin} \left (c x \right ) b \,c^{6} d^{2} x^{2}+144 \mathit {asin} \left (c x \right ) b \,c^{6} d e \,x^{4}+48 \mathit {asin} \left (c x \right ) b \,c^{6} e^{2} x^{6}-72 \mathit {asin} \left (c x \right ) b \,c^{4} d^{2}-54 \mathit {asin} \left (c x \right ) b \,c^{2} d e -15 \mathit {asin} \left (c x \right ) b \,e^{2}+72 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} d^{2} x +36 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} d e \,x^{3}+8 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} e^{2} x^{5}+54 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d e x +10 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} e^{2} x^{3}+15 \sqrt {-c^{2} x^{2}+1}\, b c \,e^{2} x +144 a \,c^{6} d^{2} x^{2}+144 a \,c^{6} d e \,x^{4}+48 a \,c^{6} e^{2} x^{6}}{288 c^{6}} \] Input:

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

Output:

(144*asin(c*x)*b*c**6*d**2*x**2 + 144*asin(c*x)*b*c**6*d*e*x**4 + 48*asin( 
c*x)*b*c**6*e**2*x**6 - 72*asin(c*x)*b*c**4*d**2 - 54*asin(c*x)*b*c**2*d*e 
 - 15*asin(c*x)*b*e**2 + 72*sqrt( - c**2*x**2 + 1)*b*c**5*d**2*x + 36*sqrt 
( - c**2*x**2 + 1)*b*c**5*d*e*x**3 + 8*sqrt( - c**2*x**2 + 1)*b*c**5*e**2* 
x**5 + 54*sqrt( - c**2*x**2 + 1)*b*c**3*d*e*x + 10*sqrt( - c**2*x**2 + 1)* 
b*c**3*e**2*x**3 + 15*sqrt( - c**2*x**2 + 1)*b*c*e**2*x + 144*a*c**6*d**2* 
x**2 + 144*a*c**6*d*e*x**4 + 48*a*c**6*e**2*x**6)/(288*c**6)