Integrand size = 21, antiderivative size = 190 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^2} \, dx=\frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \sqrt {1-c^2 x^2}}{5 c^5}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^{3/2}}{15 c^5}+\frac {b e^3 \left (1-c^2 x^2\right )^{5/2}}{25 c^5}-\frac {d^3 (a+b \arcsin (c x))}{x}+3 d^2 e x (a+b \arcsin (c x))+d e^2 x^3 (a+b \arcsin (c x))+\frac {1}{5} e^3 x^5 (a+b \arcsin (c x))-b c d^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \]
-1/15*b*e^2*(5*c^2*d+2*e)*(-c^2*x^2+1)^(3/2)/c^5+1/25*b*e^3*(-c^2*x^2+1)^( 5/2)/c^5-d^3*(a+b*arcsin(c*x))/x+3*d^2*e*x*(a+b*arcsin(c*x))+d*e^2*x^3*(a+ b*arcsin(c*x))+1/5*e^3*x^5*(a+b*arcsin(c*x))-b*c*d^3*arctanh((-c^2*x^2+1)^ (1/2))+1/5*b*e*(15*c^4*d^2+5*c^2*d*e+e^2)*(-c^2*x^2+1)^(1/2)/c^5
Time = 0.14 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^2} \, dx=-\frac {a d^3}{x}+3 a d^2 e x+a d e^2 x^3+\frac {1}{5} a e^3 x^5+\frac {b e \sqrt {1-c^2 x^2} \left (8 e^2+2 c^2 e \left (25 d+2 e x^2\right )+c^4 \left (225 d^2+25 d e x^2+3 e^2 x^4\right )\right )}{75 c^5}+\frac {b \left (-5 d^3+15 d^2 e x^2+5 d e^2 x^4+e^3 x^6\right ) \arcsin (c x)}{5 x}+b c d^3 \log (x)-b c d^3 \log \left (1+\sqrt {1-c^2 x^2}\right ) \]
-((a*d^3)/x) + 3*a*d^2*e*x + a*d*e^2*x^3 + (a*e^3*x^5)/5 + (b*e*Sqrt[1 - c ^2*x^2]*(8*e^2 + 2*c^2*e*(25*d + 2*e*x^2) + c^4*(225*d^2 + 25*d*e*x^2 + 3* e^2*x^4)))/(75*c^5) + (b*(-5*d^3 + 15*d^2*e*x^2 + 5*d*e^2*x^4 + e^3*x^6)*A rcSin[c*x])/(5*x) + b*c*d^3*Log[x] - b*c*d^3*Log[1 + Sqrt[1 - c^2*x^2]]
Time = 0.57 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5230, 27, 2331, 2123, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^2} \, dx\) |
| \(\Big \downarrow \) 5230 |
| \(\displaystyle -b c \int -\frac {-e^3 x^6-5 d e^2 x^4-15 d^2 e x^2+5 d^3}{5 x \sqrt {1-c^2 x^2}}dx-\frac {d^3 (a+b \arcsin (c x))}{x}+3 d^2 e x (a+b \arcsin (c x))+d e^2 x^3 (a+b \arcsin (c x))+\frac {1}{5} e^3 x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} b c \int \frac {-e^3 x^6-5 d e^2 x^4-15 d^2 e x^2+5 d^3}{x \sqrt {1-c^2 x^2}}dx-\frac {d^3 (a+b \arcsin (c x))}{x}+3 d^2 e x (a+b \arcsin (c x))+d e^2 x^3 (a+b \arcsin (c x))+\frac {1}{5} e^3 x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {1}{10} b c \int \frac {-e^3 x^6-5 d e^2 x^4-15 d^2 e x^2+5 d^3}{x^2 \sqrt {1-c^2 x^2}}dx^2-\frac {d^3 (a+b \arcsin (c x))}{x}+3 d^2 e x (a+b \arcsin (c x))+d e^2 x^3 (a+b \arcsin (c x))+\frac {1}{5} e^3 x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \frac {1}{10} b c \int \left (\frac {5 d^3}{x^2 \sqrt {1-c^2 x^2}}-\frac {e^3 \left (1-c^2 x^2\right )^{3/2}}{c^4}+\frac {e^2 \left (5 d c^2+2 e\right ) \sqrt {1-c^2 x^2}}{c^4}-\frac {e \left (15 d^2 c^4+5 d e c^2+e^2\right )}{c^4 \sqrt {1-c^2 x^2}}\right )dx^2-\frac {d^3 (a+b \arcsin (c x))}{x}+3 d^2 e x (a+b \arcsin (c x))+d e^2 x^3 (a+b \arcsin (c x))+\frac {1}{5} e^3 x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 (a+b \arcsin (c x))}{x}+3 d^2 e x (a+b \arcsin (c x))+d e^2 x^3 (a+b \arcsin (c x))+\frac {1}{5} e^3 x^5 (a+b \arcsin (c x))+\frac {1}{10} b c \left (-10 d^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {2 e^2 \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+2 e\right )}{3 c^6}+\frac {2 e^3 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {2 e \sqrt {1-c^2 x^2} \left (15 c^4 d^2+5 c^2 d e+e^2\right )}{c^6}\right )\) |
-((d^3*(a + b*ArcSin[c*x]))/x) + 3*d^2*e*x*(a + b*ArcSin[c*x]) + d*e^2*x^3 *(a + b*ArcSin[c*x]) + (e^3*x^5*(a + b*ArcSin[c*x]))/5 + (b*c*((2*e*(15*c^ 4*d^2 + 5*c^2*d*e + e^2)*Sqrt[1 - c^2*x^2])/c^6 - (2*e^2*(5*c^2*d + 2*e)*( 1 - c^2*x^2)^(3/2))/(3*c^6) + (2*e^3*(1 - c^2*x^2)^(5/2))/(5*c^6) - 10*d^3 *ArcTanh[Sqrt[1 - c^2*x^2]]))/10
3.7.20.3.1 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[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
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 = 0.20 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.32
| method | result | size |
| parts | \(a \left (\frac {e^{3} x^{5}}{5}+d \,e^{2} x^{3}+3 d^{2} e x -\frac {d^{3}}{x}\right )+b c \left (\frac {\arcsin \left (c x \right ) e^{3} x^{5}}{5 c}+\frac {\arcsin \left (c x \right ) d \,e^{2} x^{3}}{c}+\frac {3 \arcsin \left (c x \right ) d^{2} e x}{c}-\frac {\arcsin \left (c x \right ) d^{3}}{c x}-\frac {e^{3} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )+5 c^{6} d^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+5 c^{2} d \,e^{2} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )-15 c^{4} d^{2} e \sqrt {-c^{2} x^{2}+1}}{5 c^{6}}\right )\) | \(250\) |
| derivativedivides | \(c \left (\frac {a \left (3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {b \left (3 \arcsin \left (c x \right ) c^{5} d^{2} e x +\arcsin \left (c x \right ) c^{5} d \,e^{2} x^{3}+\frac {\arcsin \left (c x \right ) e^{3} c^{5} x^{5}}{5}-\frac {\arcsin \left (c x \right ) c^{5} d^{3}}{x}-\frac {e^{3} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-c^{6} d^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )-c^{2} d \,e^{2} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+3 c^{4} d^{2} e \sqrt {-c^{2} x^{2}+1}\right )}{c^{6}}\right )\) | \(264\) |
| default | \(c \left (\frac {a \left (3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {b \left (3 \arcsin \left (c x \right ) c^{5} d^{2} e x +\arcsin \left (c x \right ) c^{5} d \,e^{2} x^{3}+\frac {\arcsin \left (c x \right ) e^{3} c^{5} x^{5}}{5}-\frac {\arcsin \left (c x \right ) c^{5} d^{3}}{x}-\frac {e^{3} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-c^{6} d^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )-c^{2} d \,e^{2} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+3 c^{4} d^{2} e \sqrt {-c^{2} x^{2}+1}\right )}{c^{6}}\right )\) | \(264\) |
a*(1/5*e^3*x^5+d*e^2*x^3+3*d^2*e*x-d^3/x)+b*c*(1/5/c*arcsin(c*x)*e^3*x^5+1 /c*arcsin(c*x)*d*e^2*x^3+3/c*arcsin(c*x)*d^2*e*x-arcsin(c*x)*d^3/c/x-1/5/c ^6*(e^3*(-1/5*c^4*x^4*(-c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(-c^2*x^2+1)^(1/2)-8 /15*(-c^2*x^2+1)^(1/2))+5*c^6*d^3*arctanh(1/(-c^2*x^2+1)^(1/2))+5*c^2*d*e^ 2*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))-15*c^4*d^2*e*(- c^2*x^2+1)^(1/2)))
Time = 0.32 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.26 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^2} \, dx=\frac {30 \, a c^{5} e^{3} x^{6} + 150 \, a c^{5} d e^{2} x^{4} - 75 \, b c^{6} d^{3} x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) + 75 \, b c^{6} d^{3} x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 450 \, a c^{5} d^{2} e x^{2} - 150 \, a c^{5} d^{3} + 30 \, {\left (b c^{5} e^{3} x^{6} + 5 \, b c^{5} d e^{2} x^{4} + 15 \, b c^{5} d^{2} e x^{2} - 5 \, b c^{5} d^{3}\right )} \arcsin \left (c x\right ) + 2 \, {\left (3 \, b c^{4} e^{3} x^{5} + {\left (25 \, b c^{4} d e^{2} + 4 \, b c^{2} e^{3}\right )} x^{3} + {\left (225 \, b c^{4} d^{2} e + 50 \, b c^{2} d e^{2} + 8 \, b e^{3}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{150 \, c^{5} x} \]
1/150*(30*a*c^5*e^3*x^6 + 150*a*c^5*d*e^2*x^4 - 75*b*c^6*d^3*x*log(sqrt(-c ^2*x^2 + 1) + 1) + 75*b*c^6*d^3*x*log(sqrt(-c^2*x^2 + 1) - 1) + 450*a*c^5* d^2*e*x^2 - 150*a*c^5*d^3 + 30*(b*c^5*e^3*x^6 + 5*b*c^5*d*e^2*x^4 + 15*b*c ^5*d^2*e*x^2 - 5*b*c^5*d^3)*arcsin(c*x) + 2*(3*b*c^4*e^3*x^5 + (25*b*c^4*d *e^2 + 4*b*c^2*e^3)*x^3 + (225*b*c^4*d^2*e + 50*b*c^2*d*e^2 + 8*b*e^3)*x)* sqrt(-c^2*x^2 + 1))/(c^5*x)
Time = 3.10 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.45 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^2} \, dx=- \frac {a d^{3}}{x} + 3 a d^{2} e x + a d e^{2} x^{3} + \frac {a e^{3} x^{5}}{5} + b c d^{3} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - b c d e^{2} \left (\begin {cases} - \frac {x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c^{2}} - \frac {2 \sqrt {- c^{2} x^{2} + 1}}{3 c^{4}} & \text {for}\: c^{2} \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) - \frac {b c e^{3} \left (\begin {cases} - \frac {x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c^{2}} - \frac {4 x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{4}} - \frac {8 \sqrt {- c^{2} x^{2} + 1}}{15 c^{6}} & \text {for}\: c^{2} \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right )}{5} - \frac {b d^{3} \operatorname {asin}{\left (c x \right )}}{x} + 3 b d^{2} e \left (\begin {cases} 0 & \text {for}\: c = 0 \\x \operatorname {asin}{\left (c x \right )} + \frac {\sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right ) + b d e^{2} x^{3} \operatorname {asin}{\left (c x \right )} + \frac {b e^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} \]
-a*d**3/x + 3*a*d**2*e*x + a*d*e**2*x**3 + a*e**3*x**5/5 + b*c*d**3*Piecew ise((-acosh(1/(c*x)), 1/Abs(c**2*x**2) > 1), (I*asin(1/(c*x)), True)) - b* c*d*e**2*Piecewise((-x**2*sqrt(-c**2*x**2 + 1)/(3*c**2) - 2*sqrt(-c**2*x** 2 + 1)/(3*c**4), Ne(c**2, 0)), (x**4/4, True)) - b*c*e**3*Piecewise((-x**4 *sqrt(-c**2*x**2 + 1)/(5*c**2) - 4*x**2*sqrt(-c**2*x**2 + 1)/(15*c**4) - 8 *sqrt(-c**2*x**2 + 1)/(15*c**6), Ne(c**2, 0)), (x**6/6, True))/5 - b*d**3* asin(c*x)/x + 3*b*d**2*e*Piecewise((0, Eq(c, 0)), (x*asin(c*x) + sqrt(-c** 2*x**2 + 1)/c, True)) + b*d*e**2*x**3*asin(c*x) + b*e**3*x**5*asin(c*x)/5
Time = 0.27 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.27 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^2} \, dx=\frac {1}{5} \, a e^{3} x^{5} + a d e^{2} x^{3} - {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} b d^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b e^{3} + 3 \, a d^{2} e x + \frac {3 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{2} e}{c} - \frac {a d^{3}}{x} \]
1/5*a*e^3*x^5 + a*d*e^2*x^3 - (c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x )) + arcsin(c*x)/x)*b*d^3 + 1/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1) *x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*d*e^2 + 1/75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^ 2*x^2 + 1)/c^6)*c)*b*e^3 + 3*a*d^2*e*x + 3*(c*x*arcsin(c*x) + sqrt(-c^2*x^ 2 + 1))*b*d^2*e/c - a*d^3/x
Leaf count of result is larger than twice the leaf count of optimal. 10769 vs. \(2 (174) = 348\).
Time = 11.90 (sec) , antiderivative size = 10769, normalized size of antiderivative = 56.68 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^2} \, dx=\text {Too large to display} \]
-1/2*b*c^18*d^3*x^12*arcsin(c*x)/((c^16*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^14*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^12*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^10*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*c^8*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^6*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^12) - 1/2*a*c^18*d^3*x^12/((c^16*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^14*x^ 9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^12*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10 *c^10*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*c^8*x^3/(sqrt(-c^2*x^2 + 1) + 1)^ 3 + c^6*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^12) + b*c^17* d^3*x^11*log(abs(c)*abs(x))/((c^16*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^ 14*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^12*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^10*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*c^8*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^6*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^11) - b* c^17*d^3*x^11*log(sqrt(-c^2*x^2 + 1) + 1)/((c^16*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^14*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^12*x^7/(sqrt(-c^2*x ^2 + 1) + 1)^7 + 10*c^10*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*c^8*x^3/(sqrt( -c^2*x^2 + 1) + 1)^3 + c^6*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^11) - 3*b*c^16*d^3*x^10*arcsin(c*x)/((c^16*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^14*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^12*x^7/(sqrt(-c^2*x ^2 + 1) + 1)^7 + 10*c^10*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*c^8*x^3/(sqrt( -c^2*x^2 + 1) + 1)^3 + c^6*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 +...
Timed out. \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arcsin (c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x^2} \,d x \]