\(\int \frac {(d+e x^2)^3 (a+b \arcsin (c x))}{x} \, dx\) [446]

Optimal result

Integrand size = 21, antiderivative size = 357 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x} \, dx=\frac {3 b d^2 e x \sqrt {1-c^2 x^2}}{4 c}+\frac {9 b d e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {5 b e^3 x \sqrt {1-c^2 x^2}}{96 c^5}+\frac {3 b d e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {5 b e^3 x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^3 x^5 \sqrt {1-c^2 x^2}}{36 c}-\frac {3 b d^2 e \arcsin (c x)}{4 c^2}-\frac {9 b d e^2 \arcsin (c x)}{32 c^4}-\frac {5 b e^3 \arcsin (c x)}{96 c^6}-\frac {1}{2} i b d^3 \arcsin (c x)^2+\frac {3}{2} d^2 e x^2 (a+b \arcsin (c x))+\frac {3}{4} d e^2 x^4 (a+b \arcsin (c x))+\frac {1}{6} e^3 x^6 (a+b \arcsin (c x))+b d^3 \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-b d^3 \arcsin (c x) \log (x)+d^3 (a+b \arcsin (c x)) \log (x)-\frac {1}{2} i b d^3 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \] Output:

3/4*b*d^2*e*x*(-c^2*x^2+1)^(1/2)/c+9/32*b*d*e^2*x*(-c^2*x^2+1)^(1/2)/c^3+5 
/96*b*e^3*x*(-c^2*x^2+1)^(1/2)/c^5+3/16*b*d*e^2*x^3*(-c^2*x^2+1)^(1/2)/c+5 
/144*b*e^3*x^3*(-c^2*x^2+1)^(1/2)/c^3+1/36*b*e^3*x^5*(-c^2*x^2+1)^(1/2)/c- 
3/4*b*d^2*e*arcsin(c*x)/c^2-9/32*b*d*e^2*arcsin(c*x)/c^4-5/96*b*e^3*arcsin 
(c*x)/c^6-1/2*I*b*d^3*arcsin(c*x)^2+3/2*d^2*e*x^2*(a+b*arcsin(c*x))+3/4*d* 
e^2*x^4*(a+b*arcsin(c*x))+1/6*e^3*x^6*(a+b*arcsin(c*x))+b*d^3*arcsin(c*x)* 
ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-b*d^3*arcsin(c*x)*ln(x)+d^3*(a+b*arcsin 
(c*x))*ln(x)-1/2*I*b*d^3*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x} \, dx=\frac {1}{12} \left (18 a d^2 e x^2+9 a d e^2 x^4+2 a e^3 x^6+18 b d^2 e x^2 \arcsin (c x)+9 b d e^2 x^4 \arcsin (c x)+2 b e^3 x^6 \arcsin (c x)+\frac {b e^3 \left (c x \sqrt {1-c^2 x^2} \left (15+10 c^2 x^2+8 c^4 x^4\right )-30 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )}{24 c^6}+\frac {9 b d e^2 \left (c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )-6 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )}{8 c^4}+\frac {9 b d^2 e \left (c x \sqrt {1-c^2 x^2}-2 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )}{c^2}+12 b d^3 \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+12 a d^3 \log (x)-6 i b d^3 \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )\right ) \] Input:

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

Output:

(18*a*d^2*e*x^2 + 9*a*d*e^2*x^4 + 2*a*e^3*x^6 + 18*b*d^2*e*x^2*ArcSin[c*x] 
 + 9*b*d*e^2*x^4*ArcSin[c*x] + 2*b*e^3*x^6*ArcSin[c*x] + (b*e^3*(c*x*Sqrt[ 
1 - c^2*x^2]*(15 + 10*c^2*x^2 + 8*c^4*x^4) - 30*ArcTan[(c*x)/(-1 + Sqrt[1 
- c^2*x^2])]))/(24*c^6) + (9*b*d*e^2*(c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2 
) - 6*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])]))/(8*c^4) + (9*b*d^2*e*(c*x*S 
qrt[1 - c^2*x^2] - 2*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])]))/c^2 + 12*b*d 
^3*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 12*a*d^3*Log[x] - (6*I)*b* 
d^3*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c*x])]))/12
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5230, 27, 7293, 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.

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

\(\Big \downarrow \) 5230

\(\displaystyle -b c \int \frac {2 e^3 x^6+9 d e^2 x^4+18 d^2 e x^2+12 d^3 \log (x)}{12 \sqrt {1-c^2 x^2}}dx+d^3 \log (x) (a+b \arcsin (c x))+\frac {3}{2} d^2 e x^2 (a+b \arcsin (c x))+\frac {3}{4} d e^2 x^4 (a+b \arcsin (c x))+\frac {1}{6} e^3 x^6 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{12} b c \int \frac {2 e^3 x^6+9 d e^2 x^4+18 d^2 e x^2+12 d^3 \log (x)}{\sqrt {1-c^2 x^2}}dx+d^3 \log (x) (a+b \arcsin (c x))+\frac {3}{2} d^2 e x^2 (a+b \arcsin (c x))+\frac {3}{4} d e^2 x^4 (a+b \arcsin (c x))+\frac {1}{6} e^3 x^6 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{12} b c \int \left (\frac {2 e^3 x^6}{\sqrt {1-c^2 x^2}}+\frac {9 d e^2 x^4}{\sqrt {1-c^2 x^2}}+\frac {18 d^2 e x^2}{\sqrt {1-c^2 x^2}}+\frac {12 d^3 \log (x)}{\sqrt {1-c^2 x^2}}\right )dx+d^3 \log (x) (a+b \arcsin (c x))+\frac {3}{2} d^2 e x^2 (a+b \arcsin (c x))+\frac {3}{4} d e^2 x^4 (a+b \arcsin (c x))+\frac {1}{6} e^3 x^6 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle d^3 \log (x) (a+b \arcsin (c x))+\frac {3}{2} d^2 e x^2 (a+b \arcsin (c x))+\frac {3}{4} d e^2 x^4 (a+b \arcsin (c x))+\frac {1}{6} e^3 x^6 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {5 e^3 \arcsin (c x)}{8 c^7}+\frac {27 d e^2 \arcsin (c x)}{8 c^5}+\frac {9 d^2 e \arcsin (c x)}{c^3}+\frac {6 i d^3 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{c}+\frac {6 i d^3 \arcsin (c x)^2}{c}-\frac {12 d^3 \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{c}+\frac {12 d^3 \log (x) \arcsin (c x)}{c}-\frac {9 d^2 e x \sqrt {1-c^2 x^2}}{c^2}-\frac {9 d e^2 x^3 \sqrt {1-c^2 x^2}}{4 c^2}-\frac {e^3 x^5 \sqrt {1-c^2 x^2}}{3 c^2}-\frac {5 e^3 x \sqrt {1-c^2 x^2}}{8 c^6}-\frac {27 d e^2 x \sqrt {1-c^2 x^2}}{8 c^4}-\frac {5 e^3 x^3 \sqrt {1-c^2 x^2}}{12 c^4}\right )\)

Input:

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

Output:

(3*d^2*e*x^2*(a + b*ArcSin[c*x]))/2 + (3*d*e^2*x^4*(a + b*ArcSin[c*x]))/4 
+ (e^3*x^6*(a + b*ArcSin[c*x]))/6 + d^3*(a + b*ArcSin[c*x])*Log[x] - (b*c* 
((-9*d^2*e*x*Sqrt[1 - c^2*x^2])/c^2 - (27*d*e^2*x*Sqrt[1 - c^2*x^2])/(8*c^ 
4) - (5*e^3*x*Sqrt[1 - c^2*x^2])/(8*c^6) - (9*d*e^2*x^3*Sqrt[1 - c^2*x^2]) 
/(4*c^2) - (5*e^3*x^3*Sqrt[1 - c^2*x^2])/(12*c^4) - (e^3*x^5*Sqrt[1 - c^2* 
x^2])/(3*c^2) + (9*d^2*e*ArcSin[c*x])/c^3 + (27*d*e^2*ArcSin[c*x])/(8*c^5) 
 + (5*e^3*ArcSin[c*x])/(8*c^7) + ((6*I)*d^3*ArcSin[c*x]^2)/c - (12*d^3*Arc 
Sin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])])/c + (12*d^3*ArcSin[c*x]*Log[x])/c 
 + ((6*I)*d^3*PolyLog[2, E^((2*I)*ArcSin[c*x])])/c))/12
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5230

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]))
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 2.13 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.96

method	result	size
parts	\(a \left (\frac {e^{3} x^{6}}{6}+\frac {3 d \,e^{2} x^{4}}{4}+\frac {3 d^{2} e \,x^{2}}{2}+d^{3} \ln \left (x \right )\right )+b \left (-\frac {i \arcsin \left (c x \right )^{2} d^{3}}{2}+d^{3} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+d^{3} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d^{3} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d^{3} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right ) e^{3} \cos \left (6 \arcsin \left (c x \right )\right )}{192 c^{6}}+\frac {e^{3} \sin \left (6 \arcsin \left (c x \right )\right )}{1152 c^{6}}+\frac {\arcsin \left (c x \right ) e^{2} \left (3 c^{2} d +e \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{32 c^{6}}-\frac {3 e^{2} \sin \left (4 \arcsin \left (c x \right )\right ) d}{128 c^{4}}-\frac {e^{3} \sin \left (4 \arcsin \left (c x \right )\right )}{128 c^{6}}-\frac {e \arcsin \left (c x \right ) \left (48 c^{4} d^{2}+24 c^{2} d e +5 e^{2}\right ) \cos \left (2 \arcsin \left (c x \right )\right )}{64 c^{6}}+\frac {3 e \sin \left (2 \arcsin \left (c x \right )\right ) d^{2}}{8 c^{2}}+\frac {3 e^{2} \sin \left (2 \arcsin \left (c x \right )\right ) d}{16 c^{4}}+\frac {5 e^{3} \sin \left (2 \arcsin \left (c x \right )\right )}{128 c^{6}}\right )\)	\(341\)
derivativedivides	\(\frac {a \left (\frac {3 c^{6} d^{2} e \,x^{2}}{2}+\frac {3 c^{6} d \,e^{2} x^{4}}{4}+\frac {e^{3} x^{6} c^{6}}{6}+c^{6} d^{3} \ln \left (c x \right )\right )}{c^{6}}+\frac {b \left (-\frac {i c^{6} d^{3} \arcsin \left (c x \right )^{2}}{2}+c^{6} d^{3} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+c^{6} d^{3} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i c^{6} d^{3} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-i c^{6} d^{3} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {e^{3} \arcsin \left (c x \right ) \cos \left (6 \arcsin \left (c x \right )\right )}{192}+\frac {e^{3} \sin \left (6 \arcsin \left (c x \right )\right )}{1152}+\frac {e^{2} \arcsin \left (c x \right ) \left (3 c^{2} d +e \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{32}-\frac {3 c^{2} d \,e^{2} \sin \left (4 \arcsin \left (c x \right )\right )}{128}-\frac {e^{3} \sin \left (4 \arcsin \left (c x \right )\right )}{128}-\frac {e \arcsin \left (c x \right ) \left (48 c^{4} d^{2}+24 c^{2} d e +5 e^{2}\right ) \cos \left (2 \arcsin \left (c x \right )\right )}{64}+\frac {3 c^{4} d^{2} e \sin \left (2 \arcsin \left (c x \right )\right )}{8}+\frac {3 c^{2} d \,e^{2} \sin \left (2 \arcsin \left (c x \right )\right )}{16}+\frac {5 e^{3} \sin \left (2 \arcsin \left (c x \right )\right )}{128}\right )}{c^{6}}\)	\(358\)
default	\(\frac {a \left (\frac {3 c^{6} d^{2} e \,x^{2}}{2}+\frac {3 c^{6} d \,e^{2} x^{4}}{4}+\frac {e^{3} x^{6} c^{6}}{6}+c^{6} d^{3} \ln \left (c x \right )\right )}{c^{6}}+\frac {b \left (-\frac {i c^{6} d^{3} \arcsin \left (c x \right )^{2}}{2}+c^{6} d^{3} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+c^{6} d^{3} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i c^{6} d^{3} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-i c^{6} d^{3} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {e^{3} \arcsin \left (c x \right ) \cos \left (6 \arcsin \left (c x \right )\right )}{192}+\frac {e^{3} \sin \left (6 \arcsin \left (c x \right )\right )}{1152}+\frac {e^{2} \arcsin \left (c x \right ) \left (3 c^{2} d +e \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{32}-\frac {3 c^{2} d \,e^{2} \sin \left (4 \arcsin \left (c x \right )\right )}{128}-\frac {e^{3} \sin \left (4 \arcsin \left (c x \right )\right )}{128}-\frac {e \arcsin \left (c x \right ) \left (48 c^{4} d^{2}+24 c^{2} d e +5 e^{2}\right ) \cos \left (2 \arcsin \left (c x \right )\right )}{64}+\frac {3 c^{4} d^{2} e \sin \left (2 \arcsin \left (c x \right )\right )}{8}+\frac {3 c^{2} d \,e^{2} \sin \left (2 \arcsin \left (c x \right )\right )}{16}+\frac {5 e^{3} \sin \left (2 \arcsin \left (c x \right )\right )}{128}\right )}{c^{6}}\)	\(358\)

Input:

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

Output:

a*(1/6*e^3*x^6+3/4*d*e^2*x^4+3/2*d^2*e*x^2+d^3*ln(x))+b*(-1/2*I*arcsin(c*x 
)^2*d^3+d^3*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+d^3*arcsin(c*x)*ln( 
1-I*c*x-(-c^2*x^2+1)^(1/2))-I*d^3*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-I*d 
^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-1/192*arcsin(c*x)*e^3/c^6*cos(6*arc 
sin(c*x))+1/1152*e^3/c^6*sin(6*arcsin(c*x))+1/32*arcsin(c*x)*e^2*(3*c^2*d+ 
e)/c^6*cos(4*arcsin(c*x))-3/128*e^2/c^4*sin(4*arcsin(c*x))*d-1/128*e^3/c^6 
*sin(4*arcsin(c*x))-1/64*e*arcsin(c*x)*(48*c^4*d^2+24*c^2*d*e+5*e^2)/c^6*c 
os(2*arcsin(c*x))+3/8*e/c^2*sin(2*arcsin(c*x))*d^2+3/16*e^2/c^4*sin(2*arcs 
in(c*x))*d+5/128*e^3/c^6*sin(2*arcsin(c*x)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/6*a*e^3*x^6 + 3/4*a*d*e^2*x^4 + 3/2*a*d^2*e*x^2 + a*d^3*log(x) + integra 
te((b*e^3*x^6 + 3*b*d*e^2*x^4 + 3*b*d^2*e*x^2 + b*d^3)*arctan2(c*x, sqrt(c 
*x + 1)*sqrt(-c*x + 1))/x, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

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
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x} \, dx=\frac {432 \mathit {asin} \left (c x \right ) b \,c^{6} d^{2} e \,x^{2}+216 \mathit {asin} \left (c x \right ) b \,c^{6} d \,e^{2} x^{4}+48 \mathit {asin} \left (c x \right ) b \,c^{6} e^{3} x^{6}-216 \mathit {asin} \left (c x \right ) b \,c^{4} d^{2} e -81 \mathit {asin} \left (c x \right ) b \,c^{2} d \,e^{2}-15 \mathit {asin} \left (c x \right ) b \,e^{3}+216 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} d^{2} e x +54 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} d \,e^{2} x^{3}+8 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} e^{3} x^{5}+81 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d \,e^{2} x +10 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} e^{3} x^{3}+15 \sqrt {-c^{2} x^{2}+1}\, b c \,e^{3} x +288 \left (\int \frac {\mathit {asin} \left (c x \right )}{x}d x \right ) b \,c^{6} d^{3}+288 \,\mathrm {log}\left (x \right ) a \,c^{6} d^{3}+432 a \,c^{6} d^{2} e \,x^{2}+216 a \,c^{6} d \,e^{2} x^{4}+48 a \,c^{6} e^{3} x^{6}}{288 c^{6}} \] Input:

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

Output:

(432*asin(c*x)*b*c**6*d**2*e*x**2 + 216*asin(c*x)*b*c**6*d*e**2*x**4 + 48* 
asin(c*x)*b*c**6*e**3*x**6 - 216*asin(c*x)*b*c**4*d**2*e - 81*asin(c*x)*b* 
c**2*d*e**2 - 15*asin(c*x)*b*e**3 + 216*sqrt( - c**2*x**2 + 1)*b*c**5*d**2 
*e*x + 54*sqrt( - c**2*x**2 + 1)*b*c**5*d*e**2*x**3 + 8*sqrt( - c**2*x**2 
+ 1)*b*c**5*e**3*x**5 + 81*sqrt( - c**2*x**2 + 1)*b*c**3*d*e**2*x + 10*sqr 
t( - c**2*x**2 + 1)*b*c**3*e**3*x**3 + 15*sqrt( - c**2*x**2 + 1)*b*c*e**3* 
x + 288*int(asin(c*x)/x,x)*b*c**6*d**3 + 288*log(x)*a*c**6*d**3 + 432*a*c* 
*6*d**2*e*x**2 + 216*a*c**6*d*e**2*x**4 + 48*a*c**6*e**3*x**6)/(288*c**6)