Integrand size = 21, antiderivative size = 653 \[ \int \frac {x^4 (a+b \arcsin (c x))}{d+e x^2} \, dx=-\frac {a d x}{e^2}-\frac {b d \sqrt {1-c^2 x^2}}{c e^2}+\frac {b \sqrt {1-c^2 x^2}}{3 c^3 e}-\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3 e}-\frac {b d x \arcsin (c x)}{e^2}+\frac {x^3 (a+b \arcsin (c x))}{3 e}+\frac {(-d)^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}+\frac {i b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {i b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}+\frac {i b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {i b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}} \]
-a*d*x/e^2-1/9*b*(-c^2*x^2+1)^(3/2)/c^3/e-b*d*x*arcsin(c*x)/e^2+1/3*x^3*(a +b*arcsin(c*x))/e+1/2*(-d)^(3/2)*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1 )^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/e^(5/2)-1/2*(-d)^(3/2)* (a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)- (c^2*d+e)^(1/2)))/e^(5/2)+1/2*(-d)^(3/2)*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c ^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/e^(5/2)-1/2*(-d )^(3/2)*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d )^(1/2)+(c^2*d+e)^(1/2)))/e^(5/2)+1/2*I*b*(-d)^(3/2)*polylog(2,-(I*c*x+(-c ^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/e^(5/2)-1/2*I*b *(-d)^(3/2)*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-( c^2*d+e)^(1/2)))/e^(5/2)+1/2*I*b*(-d)^(3/2)*polylog(2,-(I*c*x+(-c^2*x^2+1) ^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/e^(5/2)-1/2*I*b*(-d)^(3/ 2)*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^ (1/2)))/e^(5/2)-b*d*(-c^2*x^2+1)^(1/2)/c/e^2+1/3*b*(-c^2*x^2+1)^(1/2)/c^3/ e
Time = 0.67 (sec) , antiderivative size = 515, normalized size of antiderivative = 0.79 \[ \int \frac {x^4 (a+b \arcsin (c x))}{d+e x^2} \, dx=-\frac {a d x}{e^2}+\frac {a x^3}{3 e}+\frac {a d^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2}}+\frac {b \left (-\frac {4 d \sqrt {e} \left (\sqrt {1-c^2 x^2}+c x \arcsin (c x)\right )}{c}+\frac {4 e^{3/2} \left (\sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+3 c^3 x^3 \arcsin (c x)\right )}{9 c^3}+d^{3/2} \left (-\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )-2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )+d^{3/2} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )}{4 e^{5/2}} \]
-((a*d*x)/e^2) + (a*x^3)/(3*e) + (a*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e ^(5/2) + (b*((-4*d*Sqrt[e]*(Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]))/c + (4*e ^(3/2)*(Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 3*c^3*x^3*ArcSin[c*x]))/(9*c^3) + d^(3/2)*(-(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSi n[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x ]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))) - 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[ c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] - 2*PolyLog[2, -((Sqrt[e]*E^(I*Ar cSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]) + d^(3/2)*(ArcSin[c*x]*(ArcSi n[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c ^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2* d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2* d + e])])))/(4*e^(5/2))
Time = 1.40 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5232, 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.
| \(\displaystyle \int \frac {x^4 (a+b \arcsin (c x))}{d+e x^2} \, dx\) |
| \(\Big \downarrow \) 5232 |
| \(\displaystyle \int \left (\frac {d^2 (a+b \arcsin (c x))}{e^2 \left (d+e x^2\right )}-\frac {d (a+b \arcsin (c x))}{e^2}+\frac {x^2 (a+b \arcsin (c x))}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(-d)^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}+\frac {x^3 (a+b \arcsin (c x))}{3 e}-\frac {a d x}{e^2}+\frac {i b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e^{5/2}}-\frac {i b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e^{5/2}}+\frac {i b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e^{5/2}}-\frac {i b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e^{5/2}}-\frac {b d x \arcsin (c x)}{e^2}-\frac {b d \sqrt {1-c^2 x^2}}{c e^2}-\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3 e}+\frac {b \sqrt {1-c^2 x^2}}{3 c^3 e}\) |
-((a*d*x)/e^2) - (b*d*Sqrt[1 - c^2*x^2])/(c*e^2) + (b*Sqrt[1 - c^2*x^2])/( 3*c^3*e) - (b*(1 - c^2*x^2)^(3/2))/(9*c^3*e) - (b*d*x*ArcSin[c*x])/e^2 + ( x^3*(a + b*ArcSin[c*x]))/(3*e) + ((-d)^(3/2)*(a + b*ArcSin[c*x])*Log[1 - ( Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^(5/2)) - ((-d)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c *Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^(5/2)) + ((-d)^(3/2)*(a + b*ArcSin[c*x ])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/ (2*e^(5/2)) - ((-d)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin [c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^(5/2)) + ((I/2)*b*(-d)^(3/ 2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e ]))])/e^(5/2) - ((I/2)*b*(-d)^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x])) /(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/e^(5/2) + ((I/2)*b*(-d)^(3/2)*PolyLog[ 2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/e^(5/ 2) - ((I/2)*b*(-d)^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[ -d] + Sqrt[c^2*d + e])])/e^(5/2)
3.7.24.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, ( f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 13.20 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.58
| method | result | size |
| parts | \(\frac {a \,x^{3}}{3 e}-\frac {a d x}{e^{2}}+\frac {a \,d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}-\frac {b d \sqrt {-c^{2} x^{2}+1}}{c \,e^{2}}-\frac {b d x \arcsin \left (c x \right )}{e^{2}}+\frac {b \sqrt {-c^{2} x^{2}+1}}{4 c^{3} e}+\frac {b \arcsin \left (c x \right ) x}{4 c^{2} e}+\frac {b c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e -2 c^{2} d -e \right )}\right )}{2 e^{2}}+\frac {b c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e -2 c^{2} d -e}\right )}{2 e^{2}}-\frac {b \cos \left (3 \arcsin \left (c x \right )\right )}{36 c^{3} e}-\frac {b \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{12 c^{3} e}\) | \(376\) |
| derivativedivides | \(\frac {-\frac {a \,c^{5} d x}{e^{2}}+\frac {a \,c^{5} x^{3}}{3 e}+\frac {a \,c^{5} d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}-\frac {b \,c^{4} \sqrt {-c^{2} x^{2}+1}\, d}{e^{2}}-\frac {b \,c^{5} \arcsin \left (c x \right ) d x}{e^{2}}+\frac {b \,c^{2} \sqrt {-c^{2} x^{2}+1}}{4 e}+\frac {b \,c^{3} \arcsin \left (c x \right ) x}{4 e}-\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e^{2}}-\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e^{2}}-\frac {b \,c^{2} \cos \left (3 \arcsin \left (c x \right )\right )}{36 e}-\frac {b \,c^{2} \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{12 e}}{c^{5}}\) | \(394\) |
| default | \(\frac {-\frac {a \,c^{5} d x}{e^{2}}+\frac {a \,c^{5} x^{3}}{3 e}+\frac {a \,c^{5} d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}-\frac {b \,c^{4} \sqrt {-c^{2} x^{2}+1}\, d}{e^{2}}-\frac {b \,c^{5} \arcsin \left (c x \right ) d x}{e^{2}}+\frac {b \,c^{2} \sqrt {-c^{2} x^{2}+1}}{4 e}+\frac {b \,c^{3} \arcsin \left (c x \right ) x}{4 e}-\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e^{2}}-\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e^{2}}-\frac {b \,c^{2} \cos \left (3 \arcsin \left (c x \right )\right )}{36 e}-\frac {b \,c^{2} \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{12 e}}{c^{5}}\) | \(394\) |
1/3*a/e*x^3-a*d*x/e^2+a*d^2/e^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-b*d*(- c^2*x^2+1)^(1/2)/c/e^2-b*d*x*arcsin(c*x)/e^2+1/4*b*(-c^2*x^2+1)^(1/2)/c^3/ e+1/4*b/c^2/e*arcsin(c*x)*x+1/2*b*c/e^2*d^2*sum(1/_R1/(_R1^2*e-2*c^2*d-e)* (I*arcsin(c*x)*ln((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-I*c*x-(-c ^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(-4*c^2*d-2*e)*_Z^2+e))+1/2*b*c/e ^2*d^2*sum(_R1/(_R1^2*e-2*c^2*d-e)*(I*arcsin(c*x)*ln((_R1-I*c*x-(-c^2*x^2+ 1)^(1/2))/_R1)+dilog((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^ 4+(-4*c^2*d-2*e)*_Z^2+e))-1/36*b/c^3/e*cos(3*arcsin(c*x))-1/12*b/c^3*arcsi n(c*x)/e*sin(3*arcsin(c*x))
\[ \int \frac {x^4 (a+b \arcsin (c x))}{d+e x^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{4}}{e x^{2} + d} \,d x } \]
\[ \int \frac {x^4 (a+b \arcsin (c x))}{d+e x^2} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \]
Exception generated. \[ \int \frac {x^4 (a+b \arcsin (c x))}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Exception generated. \[ \int \frac {x^4 (a+b \arcsin (c x))}{d+e x^2} \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int \frac {x^4 (a+b \arcsin (c x))}{d+e x^2} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{e\,x^2+d} \,d x \]