Integrand size = 21, antiderivative size = 185 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^3} \, dx=-\frac {b c d^2 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^2 x \sqrt {1-c^2 x^2}}{4 c}-\frac {b e^2 \arcsin (c x)}{4 c^2}-i b d e \arcsin (c x)^2-\frac {d^2 (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \arcsin (c x))+2 b d e \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-2 b d e \arcsin (c x) \log (x)+2 d e (a+b \arcsin (c x)) \log (x)-i b d e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \]
-1/4*b*e^2*arcsin(c*x)/c^2-I*b*d*e*arcsin(c*x)^2-1/2*d^2*(a+b*arcsin(c*x)) /x^2+1/2*e^2*x^2*(a+b*arcsin(c*x))+2*b*d*e*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x ^2+1)^(1/2))^2)-2*b*d*e*arcsin(c*x)*ln(x)+2*d*e*(a+b*arcsin(c*x))*ln(x)-I* b*d*e*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*b*c*d^2*(-c^2*x^2+1)^(1/ 2)/x+1/4*b*e^2*x*(-c^2*x^2+1)^(1/2)/c
Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.99 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^3} \, dx=\frac {1}{2} \left (-\frac {a d^2}{x^2}+a e^2 x^2-\frac {b c d^2 \sqrt {1-c^2 x^2}}{x}+\frac {b e^2 x \sqrt {1-c^2 x^2}}{2 c}-2 i b d e \arcsin (c x)^2+\frac {b e^2 \arctan \left (\frac {c x}{1-\sqrt {1-c^2 x^2}}\right )}{c^2}+b \arcsin (c x) \left (-\frac {d^2}{x^2}+e^2 x^2+4 d e \log \left (1-e^{2 i \arcsin (c x)}\right )\right )+4 a d e \log (x)-2 i b d e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right ) \]
(-((a*d^2)/x^2) + a*e^2*x^2 - (b*c*d^2*Sqrt[1 - c^2*x^2])/x + (b*e^2*x*Sqr t[1 - c^2*x^2])/(2*c) - (2*I)*b*d*e*ArcSin[c*x]^2 + (b*e^2*ArcTan[(c*x)/(1 - Sqrt[1 - c^2*x^2])])/c^2 + b*ArcSin[c*x]*(-(d^2/x^2) + e^2*x^2 + 4*d*e* Log[1 - E^((2*I)*ArcSin[c*x])]) + 4*a*d*e*Log[x] - (2*I)*b*d*e*PolyLog[2, E^((2*I)*ArcSin[c*x])])/2
Time = 0.63 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.05, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^3} \, dx\) |
| \(\Big \downarrow \) 5230 |
| \(\displaystyle -b c \int -\frac {\frac {d^2}{x^2}-4 e \log (x) d-e^2 x^2}{2 \sqrt {1-c^2 x^2}}dx-\frac {d^2 (a+b \arcsin (c x))}{2 x^2}+2 d e \log (x) (a+b \arcsin (c x))+\frac {1}{2} e^2 x^2 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} b c \int \frac {\frac {d^2}{x^2}-4 e \log (x) d-e^2 x^2}{\sqrt {1-c^2 x^2}}dx-\frac {d^2 (a+b \arcsin (c x))}{2 x^2}+2 d e \log (x) (a+b \arcsin (c x))+\frac {1}{2} e^2 x^2 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} b c \int \left (\frac {d^2}{x^2 \sqrt {1-c^2 x^2}}-\frac {4 e \log (x) d}{\sqrt {1-c^2 x^2}}-\frac {e^2 x^2}{\sqrt {1-c^2 x^2}}\right )dx-\frac {d^2 (a+b \arcsin (c x))}{2 x^2}+2 d e \log (x) (a+b \arcsin (c x))+\frac {1}{2} e^2 x^2 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^2 (a+b \arcsin (c x))}{2 x^2}+2 d e \log (x) (a+b \arcsin (c x))+\frac {1}{2} e^2 x^2 (a+b \arcsin (c x))+\frac {1}{2} b c \left (-\frac {e^2 \arcsin (c x)}{2 c^3}-\frac {2 i d e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{c}-\frac {2 i d e \arcsin (c x)^2}{c}+\frac {4 d e \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{c}-\frac {4 d e \log (x) \arcsin (c x)}{c}-\frac {d^2 \sqrt {1-c^2 x^2}}{x}+\frac {e^2 x \sqrt {1-c^2 x^2}}{2 c^2}\right )\) |
-1/2*(d^2*(a + b*ArcSin[c*x]))/x^2 + (e^2*x^2*(a + b*ArcSin[c*x]))/2 + 2*d *e*(a + b*ArcSin[c*x])*Log[x] + (b*c*(-((d^2*Sqrt[1 - c^2*x^2])/x) + (e^2* x*Sqrt[1 - c^2*x^2])/(2*c^2) - (e^2*ArcSin[c*x])/(2*c^3) - ((2*I)*d*e*ArcS in[c*x]^2)/c + (4*d*e*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])])/c - (4*d *e*ArcSin[c*x]*Log[x])/c - ((2*I)*d*e*PolyLog[2, E^((2*I)*ArcSin[c*x])])/c ))/2
3.7.12.3.1 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[((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 = 0.62 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.33
| method | result | size |
| parts | \(a \left (\frac {e^{2} x^{2}}{2}-\frac {d^{2}}{2 x^{2}}+2 d e \ln \left (x \right )\right )-i b d e \arcsin \left (c x \right )^{2}+\frac {b \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{4 c}+\frac {b \,e^{2} \arcsin \left (c x \right ) x^{2}}{2}-\frac {b \,e^{2} \arcsin \left (c x \right )}{4 c^{2}}+\frac {i b \,c^{2} d^{2}}{2}-\frac {b c \,d^{2} \sqrt {-c^{2} x^{2}+1}}{2 x}-\frac {b \,d^{2} \arcsin \left (c x \right )}{2 x^{2}}+2 b e d \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i b e d \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 b e d \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i b e d \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\) | \(246\) |
| derivativedivides | \(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}-\frac {i b d e \arcsin \left (c x \right )^{2}}{c^{2}}+\frac {b \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{4 c^{3}}+\frac {b \arcsin \left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \,e^{2} \arcsin \left (c x \right )}{4 c^{4}}+\frac {i d^{2} b}{2}-\frac {b \,d^{2} \sqrt {-c^{2} x^{2}+1}}{2 c x}-\frac {b \arcsin \left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {2 b \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) d e \arcsin \left (c x \right )}{c^{2}}+\frac {2 b \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right ) d e \arcsin \left (c x \right )}{c^{2}}-\frac {2 i b \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right ) d e}{c^{2}}-\frac {2 i b \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right ) d e}{c^{2}}\right )\) | \(281\) |
| default | \(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}-\frac {i b d e \arcsin \left (c x \right )^{2}}{c^{2}}+\frac {b \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{4 c^{3}}+\frac {b \arcsin \left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \,e^{2} \arcsin \left (c x \right )}{4 c^{4}}+\frac {i d^{2} b}{2}-\frac {b \,d^{2} \sqrt {-c^{2} x^{2}+1}}{2 c x}-\frac {b \arcsin \left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {2 b \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) d e \arcsin \left (c x \right )}{c^{2}}+\frac {2 b \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right ) d e \arcsin \left (c x \right )}{c^{2}}-\frac {2 i b \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right ) d e}{c^{2}}-\frac {2 i b \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right ) d e}{c^{2}}\right )\) | \(281\) |
a*(1/2*e^2*x^2-1/2*d^2/x^2+2*d*e*ln(x))-I*b*d*e*arcsin(c*x)^2+1/4*b*e^2*x* (-c^2*x^2+1)^(1/2)/c+1/2*b*e^2*arcsin(c*x)*x^2-1/4*b*e^2*arcsin(c*x)/c^2+1 /2*I*b*c^2*d^2-1/2*b*c*d^2*(-c^2*x^2+1)^(1/2)/x-1/2*b*d^2/x^2*arcsin(c*x)+ 2*b*e*d*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*I*b*e*d*polylog(2,-I* c*x-(-c^2*x^2+1)^(1/2))+2*b*e*d*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2)) -2*I*b*e*d*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))
\[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
integral((a*e^2*x^4 + 2*a*d*e*x^2 + a*d^2 + (b*e^2*x^4 + 2*b*d*e*x^2 + b*d ^2)*arcsin(c*x))/x^3, x)
\[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{3}}\, dx \]
\[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
1/2*a*e^2*x^2 - 1/2*b*d^2*(sqrt(-c^2*x^2 + 1)*c/x + arcsin(c*x)/x^2) + 2*a *d*e*log(x) - 1/2*a*d^2/x^2 + integrate((b*e^2*x^2 + 2*b*d*e)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x, x)
\[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2}{x^3} \,d x \]