Integrand size = 21, antiderivative size = 262 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^3} \, dx=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3 b e^2 \left (8 c^2 d+e\right ) \arcsin (c x)}{32 c^4}-\frac {3}{2} i b d^2 e \arcsin (c x)^2-\frac {d^3 (a+b \arcsin (c x))}{2 x^2}+\frac {3}{2} d e^2 x^2 (a+b \arcsin (c x))+\frac {1}{4} e^3 x^4 (a+b \arcsin (c x))+3 b d^2 e \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-3 b d^2 e \arcsin (c x) \log (x)+3 d^2 e (a+b \arcsin (c x)) \log (x)-\frac {3}{2} i b d^2 e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \]
-3/32*b*e^2*(8*c^2*d+e)*arcsin(c*x)/c^4-3/2*I*b*d^2*e*arcsin(c*x)^2-1/2*d^ 3*(a+b*arcsin(c*x))/x^2+3/2*d*e^2*x^2*(a+b*arcsin(c*x))+1/4*e^3*x^4*(a+b*a rcsin(c*x))+3*b*d^2*e*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-3*b*d ^2*e*arcsin(c*x)*ln(x)+3*d^2*e*(a+b*arcsin(c*x))*ln(x)-3/2*I*b*d^2*e*polyl og(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*b*c*d^3*(-c^2*x^2+1)^(1/2)/x+3/32*b *e^2*(8*c^2*d+e)*x*(-c^2*x^2+1)^(1/2)/c^3+1/16*b*e^3*x^3*(-c^2*x^2+1)^(1/2 )/c
Time = 0.32 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^3} \, dx=\frac {1}{32} \left (-\frac {16 a d^3}{x^2}+48 a d e^2 x^2+8 a e^3 x^4-\frac {16 b d^3 \left (c x \sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{x^2}+\frac {b e^3 \left (c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )+8 c^4 x^4 \arcsin (c x)-6 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )}{c^4}+\frac {24 b d e^2 \left (c x \sqrt {1-c^2 x^2}+2 c^2 x^2 \arcsin (c x)-2 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )}{c^2}+96 a d^2 e \log (x)+96 b d^2 e \left (\arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-\frac {1}{2} i \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )\right )\right ) \]
((-16*a*d^3)/x^2 + 48*a*d*e^2*x^2 + 8*a*e^3*x^4 - (16*b*d^3*(c*x*Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/x^2 + (b*e^3*(c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^ 2) + 8*c^4*x^4*ArcSin[c*x] - 6*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])]))/c^ 4 + (24*b*d*e^2*(c*x*Sqrt[1 - c^2*x^2] + 2*c^2*x^2*ArcSin[c*x] - 2*ArcTan[ (c*x)/(-1 + Sqrt[1 - c^2*x^2])]))/c^2 + 96*a*d^2*e*Log[x] + 96*b*d^2*e*(Ar cSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - (I/2)*(ArcSin[c*x]^2 + PolyLog[ 2, E^((2*I)*ArcSin[c*x])])))/32
Time = 1.03 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.02, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^3} \, dx\) |
| \(\Big \downarrow \) 5230 |
| \(\displaystyle -b c \int -\frac {-e^3 x^6-6 d e^2 x^4-12 d^2 e \log (x) x^2+2 d^3}{4 x^2 \sqrt {1-c^2 x^2}}dx-\frac {d^3 (a+b \arcsin (c x))}{2 x^2}+3 d^2 e \log (x) (a+b \arcsin (c x))+\frac {3}{2} d e^2 x^2 (a+b \arcsin (c x))+\frac {1}{4} e^3 x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} b c \int \frac {-e^3 x^6-6 d e^2 x^4-12 d^2 e \log (x) x^2+2 d^3}{x^2 \sqrt {1-c^2 x^2}}dx-\frac {d^3 (a+b \arcsin (c x))}{2 x^2}+3 d^2 e \log (x) (a+b \arcsin (c x))+\frac {3}{2} d e^2 x^2 (a+b \arcsin (c x))+\frac {1}{4} e^3 x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{4} b c \int \left (\frac {-e^3 x^6-6 d e^2 x^4+2 d^3}{x^2 \sqrt {1-c^2 x^2}}-\frac {12 d^2 e \log (x)}{\sqrt {1-c^2 x^2}}\right )dx-\frac {d^3 (a+b \arcsin (c x))}{2 x^2}+3 d^2 e \log (x) (a+b \arcsin (c x))+\frac {3}{2} d e^2 x^2 (a+b \arcsin (c x))+\frac {1}{4} e^3 x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 (a+b \arcsin (c x))}{2 x^2}+3 d^2 e \log (x) (a+b \arcsin (c x))+\frac {3}{2} d e^2 x^2 (a+b \arcsin (c x))+\frac {1}{4} e^3 x^4 (a+b \arcsin (c x))+\frac {1}{4} b c \left (-\frac {3 e^2 \arcsin (c x) \left (8 c^2 d+e\right )}{8 c^5}-\frac {6 i d^2 e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{c}-\frac {6 i d^2 e \arcsin (c x)^2}{c}+\frac {12 d^2 e \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{c}-\frac {12 d^2 e \log (x) \arcsin (c x)}{c}-\frac {2 d^3 \sqrt {1-c^2 x^2}}{x}+\frac {e^3 x^3 \sqrt {1-c^2 x^2}}{4 c^2}+\frac {3 e^2 x \sqrt {1-c^2 x^2} \left (8 c^2 d+e\right )}{8 c^4}\right )\) |
-1/2*(d^3*(a + b*ArcSin[c*x]))/x^2 + (3*d*e^2*x^2*(a + b*ArcSin[c*x]))/2 + (e^3*x^4*(a + b*ArcSin[c*x]))/4 + 3*d^2*e*(a + b*ArcSin[c*x])*Log[x] + (b *c*((-2*d^3*Sqrt[1 - c^2*x^2])/x + (3*e^2*(8*c^2*d + e)*x*Sqrt[1 - c^2*x^2 ])/(8*c^4) + (e^3*x^3*Sqrt[1 - c^2*x^2])/(4*c^2) - (3*e^2*(8*c^2*d + e)*Ar cSin[c*x])/(8*c^5) - ((6*I)*d^2*e*ArcSin[c*x]^2)/c + (12*d^2*e*ArcSin[c*x] *Log[1 - E^((2*I)*ArcSin[c*x])])/c - (12*d^2*e*ArcSin[c*x]*Log[x])/c - ((6 *I)*d^2*e*PolyLog[2, E^((2*I)*ArcSin[c*x])])/c))/4
3.7.21.3.1 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 = 357, normalized size of antiderivative = 1.36
| method | result | size |
| parts | \(a \left (\frac {e^{3} x^{4}}{4}+\frac {3 d \,e^{2} x^{2}}{2}-\frac {d^{3}}{2 x^{2}}+3 d^{2} e \ln \left (x \right )\right )-3 i b \,d^{2} e \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {b c \,d^{3} \sqrt {-c^{2} x^{2}+1}}{2 x}-\frac {b \,d^{3} \arcsin \left (c x \right )}{2 x^{2}}+\frac {b \arcsin \left (c x \right ) e^{3} \cos \left (4 \arcsin \left (c x \right )\right )}{32 c^{4}}+\frac {b \,e^{3} \arcsin \left (c x \right ) x^{2}}{4 c^{2}}-\frac {3 b \,e^{2} d \arcsin \left (c x \right )}{4 c^{2}}+3 b \,d^{2} e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 i b \,d^{2} e \arcsin \left (c x \right )^{2}}{2}-3 i b \,d^{2} e \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\frac {i b \,c^{2} d^{3}}{2}+3 b \,d^{2} e \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\frac {3 b \,e^{2} \arcsin \left (c x \right ) x^{2} d}{2}+\frac {b \,e^{3} \sqrt {-c^{2} x^{2}+1}\, x}{8 c^{3}}-\frac {b \,e^{3} \sin \left (4 \arcsin \left (c x \right )\right )}{128 c^{4}}-\frac {b \,e^{3} \arcsin \left (c x \right )}{8 c^{4}}+\frac {3 b \,e^{2} \sqrt {-c^{2} x^{2}+1}\, x d}{4 c}\) | \(357\) |
| derivativedivides | \(c^{2} \left (\frac {a \left (\frac {3 c^{4} d \,e^{2} x^{2}}{2}+\frac {e^{3} c^{4} x^{4}}{4}-\frac {c^{4} d^{3}}{2 x^{2}}+3 c^{4} d^{2} e \ln \left (c x \right )\right )}{c^{6}}-\frac {3 i b \,d^{2} e \arcsin \left (c x \right )^{2}}{2 c^{2}}-\frac {3 i b \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right ) d^{2} e}{c^{2}}-\frac {b \,d^{3} \sqrt {-c^{2} x^{2}+1}}{2 c x}+\frac {b \,e^{3} x \sqrt {-c^{2} x^{2}+1}}{8 c^{5}}-\frac {3 b d \,e^{2} \arcsin \left (c x \right )}{4 c^{4}}+\frac {b \arcsin \left (c x \right ) e^{3} x^{2}}{4 c^{4}}+\frac {b \arcsin \left (c x \right ) e^{3} \cos \left (4 \arcsin \left (c x \right )\right )}{32 c^{6}}+\frac {3 b \arcsin \left (c x \right ) d \,e^{2} x^{2}}{2 c^{2}}-\frac {b \arcsin \left (c x \right ) d^{3}}{2 c^{2} x^{2}}+\frac {3 b d \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{4 c^{3}}-\frac {b \,e^{3} \arcsin \left (c x \right )}{8 c^{6}}-\frac {b \,e^{3} \sin \left (4 \arcsin \left (c x \right )\right )}{128 c^{6}}+\frac {3 b \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) d^{2} e \arcsin \left (c x \right )}{c^{2}}+\frac {3 b \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right ) d^{2} e \arcsin \left (c x \right )}{c^{2}}-\frac {3 i b \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right ) d^{2} e}{c^{2}}+\frac {i d^{3} b}{2}\right )\) | \(398\) |
| default | \(c^{2} \left (\frac {a \left (\frac {3 c^{4} d \,e^{2} x^{2}}{2}+\frac {e^{3} c^{4} x^{4}}{4}-\frac {c^{4} d^{3}}{2 x^{2}}+3 c^{4} d^{2} e \ln \left (c x \right )\right )}{c^{6}}-\frac {3 i b \,d^{2} e \arcsin \left (c x \right )^{2}}{2 c^{2}}-\frac {3 i b \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right ) d^{2} e}{c^{2}}-\frac {b \,d^{3} \sqrt {-c^{2} x^{2}+1}}{2 c x}+\frac {b \,e^{3} x \sqrt {-c^{2} x^{2}+1}}{8 c^{5}}-\frac {3 b d \,e^{2} \arcsin \left (c x \right )}{4 c^{4}}+\frac {b \arcsin \left (c x \right ) e^{3} x^{2}}{4 c^{4}}+\frac {b \arcsin \left (c x \right ) e^{3} \cos \left (4 \arcsin \left (c x \right )\right )}{32 c^{6}}+\frac {3 b \arcsin \left (c x \right ) d \,e^{2} x^{2}}{2 c^{2}}-\frac {b \arcsin \left (c x \right ) d^{3}}{2 c^{2} x^{2}}+\frac {3 b d \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{4 c^{3}}-\frac {b \,e^{3} \arcsin \left (c x \right )}{8 c^{6}}-\frac {b \,e^{3} \sin \left (4 \arcsin \left (c x \right )\right )}{128 c^{6}}+\frac {3 b \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) d^{2} e \arcsin \left (c x \right )}{c^{2}}+\frac {3 b \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right ) d^{2} e \arcsin \left (c x \right )}{c^{2}}-\frac {3 i b \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right ) d^{2} e}{c^{2}}+\frac {i d^{3} b}{2}\right )\) | \(398\) |
a*(1/4*e^3*x^4+3/2*d*e^2*x^2-1/2*d^3/x^2+3*d^2*e*ln(x))-3*I*b*d^2*e*polylo g(2,I*c*x+(-c^2*x^2+1)^(1/2))-1/2*b*c*d^3*(-c^2*x^2+1)^(1/2)/x-1/2*b*d^3/x ^2*arcsin(c*x)+1/32*b/c^4*arcsin(c*x)*e^3*cos(4*arcsin(c*x))+1/4*b/c^2*e^3 *arcsin(c*x)*x^2-3/4*b/c^2*e^2*d*arcsin(c*x)+3*b*d^2*e*arcsin(c*x)*ln(1+I* c*x+(-c^2*x^2+1)^(1/2))-3/2*I*b*d^2*e*arcsin(c*x)^2-3*I*b*d^2*e*polylog(2, -I*c*x-(-c^2*x^2+1)^(1/2))+1/2*I*b*c^2*d^3+3*b*d^2*e*arcsin(c*x)*ln(1-I*c* x-(-c^2*x^2+1)^(1/2))+3/2*b*e^2*arcsin(c*x)*x^2*d+1/8*b/c^3*e^3*(-c^2*x^2+ 1)^(1/2)*x-1/128*b/c^4*e^3*sin(4*arcsin(c*x))-1/8*b/c^4*e^3*arcsin(c*x)+3/ 4*b/c*e^2*(-c^2*x^2+1)^(1/2)*x*d
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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^3, x)
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x^{3}}\, dx \]
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
1/4*a*e^3*x^4 + 3/2*a*d*e^2*x^2 - 1/2*b*d^3*(sqrt(-c^2*x^2 + 1)*c/x + arcs in(c*x)/x^2) + 3*a*d^2*e*log(x) - 1/2*a*d^3/x^2 + integrate((b*e^3*x^4 + 3 *b*d*e^2*x^2 + 3*b*d^2*e)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x, x)
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x^3} \,d x \]