Integrand size = 18, antiderivative size = 242 \[ \int (d+e x)^2 (a+b \arcsin (c x))^2 \, dx=-2 b^2 d^2 x-\frac {4 b^2 e^2 x}{9 c^2}-\frac {1}{2} b^2 d e x^2-\frac {2}{27} b^2 e^2 x^3+\frac {2 b d^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {4 b e^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3}+\frac {b d e x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {2 b e^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c}-\frac {d^3 (a+b \arcsin (c x))^2}{3 e}-\frac {d e (a+b \arcsin (c x))^2}{2 c^2}+\frac {(d+e x)^3 (a+b \arcsin (c x))^2}{3 e} \] Output:
-2*b^2*d^2*x-4/9*b^2*e^2*x/c^2-1/2*b^2*d*e*x^2-2/27*b^2*e^2*x^3+2*b*d^2*(- c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c+4/9*b*e^2*(-c^2*x^2+1)^(1/2)*(a+b*arc sin(c*x))/c^3+b*d*e*x*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c+2/9*b*e^2*x^2 *(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c-1/3*d^3*(a+b*arcsin(c*x))^2/e-1/2* d*e*(a+b*arcsin(c*x))^2/c^2+1/3*(e*x+d)^3*(a+b*arcsin(c*x))^2/e
Time = 0.20 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.03 \[ \int (d+e x)^2 (a+b \arcsin (c x))^2 \, dx=\frac {18 a^2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+6 a b \sqrt {1-c^2 x^2} \left (4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )-b^2 c x \left (24 e^2+c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )\right )+6 b \left (-9 a c d e+6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+b \sqrt {1-c^2 x^2} \left (4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )\right ) \arcsin (c x)+9 b^2 c \left (6 c^2 d^2 x+2 c^2 e^2 x^3+3 d e \left (-1+2 c^2 x^2\right )\right ) \arcsin (c x)^2}{54 c^3} \] Input:
Integrate[(d + e*x)^2*(a + b*ArcSin[c*x])^2,x]
Output:
(18*a^2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) + 6*a*b*Sqrt[1 - c^2*x^2]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) - b^2*c*x*(24*e^2 + c^2*(108*d^2 + 27*d*e*x + 4*e^2*x^2)) + 6*b*(-9*a*c*d*e + 6*a*c^3*x*(3*d^2 + 3*d*e*x + e^ 2*x^2) + b*Sqrt[1 - c^2*x^2]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2))) *ArcSin[c*x] + 9*b^2*c*(6*c^2*d^2*x + 2*c^2*e^2*x^3 + 3*d*e*(-1 + 2*c^2*x^ 2))*ArcSin[c*x]^2)/(54*c^3)
Time = 0.81 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5242, 5262, 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.
| \(\displaystyle \int (d+e x)^2 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5242 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \arcsin (c x))^2}{3 e}-\frac {2 b c \int \frac {(d+e x)^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 e}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \arcsin (c x))^2}{3 e}-\frac {2 b c \int \left (\frac {(a+b \arcsin (c x)) d^3}{\sqrt {1-c^2 x^2}}+\frac {3 e x (a+b \arcsin (c x)) d^2}{\sqrt {1-c^2 x^2}}+\frac {3 e^2 x^2 (a+b \arcsin (c x)) d}{\sqrt {1-c^2 x^2}}+\frac {e^3 x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}\right )dx}{3 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \arcsin (c x))^2}{3 e}-\frac {2 b c \left (\frac {3 d e^2 (a+b \arcsin (c x))^2}{4 b c^3}-\frac {3 d^2 e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}-\frac {3 d e^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}-\frac {e^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}-\frac {2 e^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^4}+\frac {d^3 (a+b \arcsin (c x))^2}{2 b c}+\frac {2 b e^3 x}{3 c^3}+\frac {3 b d^2 e x}{c}+\frac {3 b d e^2 x^2}{4 c}+\frac {b e^3 x^3}{9 c}\right )}{3 e}\) |
Input:
Int[(d + e*x)^2*(a + b*ArcSin[c*x])^2,x]
Output:
((d + e*x)^3*(a + b*ArcSin[c*x])^2)/(3*e) - (2*b*c*((3*b*d^2*e*x)/c + (2*b *e^3*x)/(3*c^3) + (3*b*d*e^2*x^2)/(4*c) + (b*e^3*x^3)/(9*c) - (3*d^2*e*Sqr t[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^2 - (2*e^3*Sqrt[1 - c^2*x^2]*(a + b* ArcSin[c*x]))/(3*c^4) - (3*d*e^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/ (2*c^2) - (e^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c^2) + (d^3*( a + b*ArcSin[c*x])^2)/(2*b*c) + (3*d*e^2*(a + b*ArcSin[c*x])^2)/(4*b*c^3)) )/(3*e)
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] & & EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ [n, 0] && (m == 1 || p > 0 || (n == 1 && p > -1) || (m == 2 && p < -2))
Time = 0.45 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.55
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \left (c x e +c d \right )^{3}}{3 c^{2} e}+\frac {b^{2} \left (c^{2} d^{2} \left (\arcsin \left (c x \right )^{2} c x -2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {d c e \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{2}+\frac {e^{2} \left (9 \arcsin \left (c x \right )^{2} c^{3} x^{3}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) c^{3} d^{3}}{3 e}+\arcsin \left (c x \right ) c^{3} d^{2} x +e \arcsin \left (c x \right ) c^{3} d \,x^{2}+\frac {\arcsin \left (c x \right ) c^{3} x^{3} e^{2}}{3}-\frac {c^{3} d^{3} \arcsin \left (c x \right )+e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+3 d c \,e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )-3 d^{2} c^{2} e \sqrt {-c^{2} x^{2}+1}}{3 e}\right )}{c^{2}}}{c}\) | \(374\) |
| default | \(\frac {\frac {a^{2} \left (c x e +c d \right )^{3}}{3 c^{2} e}+\frac {b^{2} \left (c^{2} d^{2} \left (\arcsin \left (c x \right )^{2} c x -2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {d c e \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{2}+\frac {e^{2} \left (9 \arcsin \left (c x \right )^{2} c^{3} x^{3}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) c^{3} d^{3}}{3 e}+\arcsin \left (c x \right ) c^{3} d^{2} x +e \arcsin \left (c x \right ) c^{3} d \,x^{2}+\frac {\arcsin \left (c x \right ) c^{3} x^{3} e^{2}}{3}-\frac {c^{3} d^{3} \arcsin \left (c x \right )+e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+3 d c \,e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )-3 d^{2} c^{2} e \sqrt {-c^{2} x^{2}+1}}{3 e}\right )}{c^{2}}}{c}\) | \(374\) |
| parts | \(\frac {a^{2} \left (e x +d \right )^{3}}{3 e}+\frac {b^{2} \left (18 \arcsin \left (c x \right )^{2} c^{3} x^{3} e^{2}+54 \arcsin \left (c x \right )^{2} c^{3} x^{2} d e +54 \arcsin \left (c x \right )^{2} c^{3} x \,d^{2}+12 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2} e^{2}+54 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x d e +108 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} d^{2}-27 \arcsin \left (c x \right )^{2} c d e -4 c^{3} x^{3} e^{2}-27 c^{3} d e \,x^{2}-108 c^{3} d^{2} x +24 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) e^{2}-24 e^{2} x c \right )}{54 c^{3}}+\frac {2 a b \left (\frac {c \,e^{2} \arcsin \left (c x \right ) x^{3}}{3}+c e \arcsin \left (c x \right ) x^{2} d +\arcsin \left (c x \right ) d^{2} c x +\frac {c \arcsin \left (c x \right ) d^{3}}{3 e}-\frac {c^{3} d^{3} \arcsin \left (c x \right )+e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+3 d c \,e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )-3 d^{2} c^{2} e \sqrt {-c^{2} x^{2}+1}}{3 c^{2} e}\right )}{c}\) | \(383\) |
| orering | \(\frac {\left (38 c^{4} e^{4} x^{5}+206 c^{4} d \,e^{3} x^{4}+531 c^{4} d^{2} e^{2} x^{3}+162 x^{2} e \,d^{3} c^{4}+54 c^{4} d^{4} x +48 c^{2} e^{4} x^{3}-174 c^{2} d \,e^{3} x^{2}-540 c^{2} d^{2} e^{2} x -135 e \,d^{3} c^{2}-96 e^{4} x -24 d \,e^{3}\right ) \left (a +b \arcsin \left (c x \right )\right )^{2}}{54 \left (e x +d \right )^{2} c^{4}}-\frac {\left (24 e^{3} c^{4} x^{5}+143 e^{2} c^{4} x^{4} d +459 e \,c^{4} x^{3} d^{2}+68 e^{3} c^{2} x^{3}-174 x^{2} e^{2} d \,c^{2}-594 c^{2} d^{2} e x -108 d^{3} c^{2}-120 x \,e^{3}-24 d \,e^{2}\right ) \left (2 \left (e x +d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2} e +\frac {2 \left (e x +d \right )^{2} \left (a +b \arcsin \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{108 c^{4} \left (e x +d \right )^{3}}+\frac {x \left (4 c^{2} e^{2} x^{2}+27 c^{2} d e x +108 c^{2} d^{2}+24 e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (2 e^{2} \left (a +b \arcsin \left (c x \right )\right )^{2}+\frac {8 \left (e x +d \right ) \left (a +b \arcsin \left (c x \right )\right ) e b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 \left (e x +d \right )^{2} b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {2 \left (e x +d \right )^{2} \left (a +b \arcsin \left (c x \right )\right ) b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{108 c^{4} \left (e x +d \right )^{2}}\) | \(453\) |
Input:
int((e*x+d)^2*(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/c*(1/3*a^2/c^2*(c*e*x+c*d)^3/e+b^2/c^2*(c^2*d^2*(arcsin(c*x)^2*c*x-2*c*x +2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+1/2*d*c*e*(2*arcsin(c*x)^2*x^2*c^2+2*ar csin(c*x)*(-c^2*x^2+1)^(1/2)*c*x-arcsin(c*x)^2-c^2*x^2)+1/27*e^2*(9*arcsin (c*x)^2*c^3*x^3+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-2*c^3*x^3+12*arcs in(c*x)*(-c^2*x^2+1)^(1/2)-12*c*x))+2*a*b/c^2*(1/3/e*arcsin(c*x)*c^3*d^3+a rcsin(c*x)*c^3*d^2*x+e*arcsin(c*x)*c^3*d*x^2+1/3*arcsin(c*x)*c^3*x^3*e^2-1 /3/e*(c^3*d^3*arcsin(c*x)+e^3*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x ^2+1)^(1/2))+3*d*c*e^2*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))-3*d^2 *c^2*e*(-c^2*x^2+1)^(1/2))))
Time = 0.10 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.20 \[ \int (d+e x)^2 (a+b \arcsin (c x))^2 \, dx=\frac {2 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{3} e^{2} x^{3} + 27 \, {\left (2 \, a^{2} - b^{2}\right )} c^{3} d e x^{2} + 9 \, {\left (2 \, b^{2} c^{3} e^{2} x^{3} + 6 \, b^{2} c^{3} d e x^{2} + 6 \, b^{2} c^{3} d^{2} x - 3 \, b^{2} c d e\right )} \arcsin \left (c x\right )^{2} + 6 \, {\left (9 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{3} d^{2} - 4 \, b^{2} c e^{2}\right )} x + 18 \, {\left (2 \, a b c^{3} e^{2} x^{3} + 6 \, a b c^{3} d e x^{2} + 6 \, a b c^{3} d^{2} x - 3 \, a b c d e\right )} \arcsin \left (c x\right ) + 6 \, {\left (2 \, a b c^{2} e^{2} x^{2} + 9 \, a b c^{2} d e x + 18 \, a b c^{2} d^{2} + 4 \, a b e^{2} + {\left (2 \, b^{2} c^{2} e^{2} x^{2} + 9 \, b^{2} c^{2} d e x + 18 \, b^{2} c^{2} d^{2} + 4 \, b^{2} e^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{54 \, c^{3}} \] Input:
integrate((e*x+d)^2*(a+b*arcsin(c*x))^2,x, algorithm="fricas")
Output:
1/54*(2*(9*a^2 - 2*b^2)*c^3*e^2*x^3 + 27*(2*a^2 - b^2)*c^3*d*e*x^2 + 9*(2* b^2*c^3*e^2*x^3 + 6*b^2*c^3*d*e*x^2 + 6*b^2*c^3*d^2*x - 3*b^2*c*d*e)*arcsi n(c*x)^2 + 6*(9*(a^2 - 2*b^2)*c^3*d^2 - 4*b^2*c*e^2)*x + 18*(2*a*b*c^3*e^2 *x^3 + 6*a*b*c^3*d*e*x^2 + 6*a*b*c^3*d^2*x - 3*a*b*c*d*e)*arcsin(c*x) + 6* (2*a*b*c^2*e^2*x^2 + 9*a*b*c^2*d*e*x + 18*a*b*c^2*d^2 + 4*a*b*e^2 + (2*b^2 *c^2*e^2*x^2 + 9*b^2*c^2*d*e*x + 18*b^2*c^2*d^2 + 4*b^2*e^2)*arcsin(c*x))* sqrt(-c^2*x^2 + 1))/c^3
Time = 0.33 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.88 \[ \int (d+e x)^2 (a+b \arcsin (c x))^2 \, dx=\begin {cases} a^{2} d^{2} x + a^{2} d e x^{2} + \frac {a^{2} e^{2} x^{3}}{3} + 2 a b d^{2} x \operatorname {asin}{\left (c x \right )} + 2 a b d e x^{2} \operatorname {asin}{\left (c x \right )} + \frac {2 a b e^{2} x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {2 a b d^{2} \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {a b d e x \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {2 a b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {a b d e \operatorname {asin}{\left (c x \right )}}{c^{2}} + \frac {4 a b e^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d^{2} x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d^{2} x + b^{2} d e x^{2} \operatorname {asin}^{2}{\left (c x \right )} - \frac {b^{2} d e x^{2}}{2} + \frac {b^{2} e^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {2 b^{2} e^{2} x^{3}}{27} + \frac {2 b^{2} d^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {b^{2} d e x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {2 b^{2} e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c} - \frac {b^{2} d e \operatorname {asin}^{2}{\left (c x \right )}}{2 c^{2}} - \frac {4 b^{2} e^{2} x}{9 c^{2}} + \frac {4 b^{2} e^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**2*(a+b*asin(c*x))**2,x)
Output:
Piecewise((a**2*d**2*x + a**2*d*e*x**2 + a**2*e**2*x**3/3 + 2*a*b*d**2*x*a sin(c*x) + 2*a*b*d*e*x**2*asin(c*x) + 2*a*b*e**2*x**3*asin(c*x)/3 + 2*a*b* d**2*sqrt(-c**2*x**2 + 1)/c + a*b*d*e*x*sqrt(-c**2*x**2 + 1)/c + 2*a*b*e** 2*x**2*sqrt(-c**2*x**2 + 1)/(9*c) - a*b*d*e*asin(c*x)/c**2 + 4*a*b*e**2*sq rt(-c**2*x**2 + 1)/(9*c**3) + b**2*d**2*x*asin(c*x)**2 - 2*b**2*d**2*x + b **2*d*e*x**2*asin(c*x)**2 - b**2*d*e*x**2/2 + b**2*e**2*x**3*asin(c*x)**2/ 3 - 2*b**2*e**2*x**3/27 + 2*b**2*d**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/c + b **2*d*e*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/c + 2*b**2*e**2*x**2*sqrt(-c**2*x **2 + 1)*asin(c*x)/(9*c) - b**2*d*e*asin(c*x)**2/(2*c**2) - 4*b**2*e**2*x/ (9*c**2) + 4*b**2*e**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c**3), Ne(c, 0)), (a**2*(d**2*x + d*e*x**2 + e**2*x**3/3), True))
\[ \int (d+e x)^2 (a+b \arcsin (c x))^2 \, dx=\int { {\left (e x + d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((e*x+d)^2*(a+b*arcsin(c*x))^2,x, algorithm="maxima")
Output:
1/3*a^2*e^2*x^3 + b^2*d^2*x*arcsin(c*x)^2 + a^2*d*e*x^2 + (2*x^2*arcsin(c* x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a*b*d*e + 2/9*(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))*a *b*e^2 - 2*b^2*d^2*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^2*d^2*x + 2* (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d^2/c + 1/3*(b^2*e^2*x^3 + 3*b^ 2*d*e*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + integrate(2/3*(b ^2*c*e^2*x^3 + 3*b^2*c*d*e*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*x^2 - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (218) = 436\).
Time = 0.15 (sec) , antiderivative size = 487, normalized size of antiderivative = 2.01 \[ \int (d+e x)^2 (a+b \arcsin (c x))^2 \, dx=\frac {1}{3} \, a^{2} e^{2} x^{3} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} + 2 \, a b d^{2} x \arcsin \left (c x\right ) + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e^{2} x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} d e x \arcsin \left (c x\right )}{c} + a^{2} d^{2} x - 2 \, b^{2} d^{2} x + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b e^{2} x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d e \arcsin \left (c x\right )^{2}}{c^{2}} + \frac {b^{2} e^{2} x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} a b d e x}{c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{c} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} e^{2} x}{27 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b d e \arcsin \left (c x\right )}{c^{2}} + \frac {2 \, a b e^{2} x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b^{2} d e \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{c} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} e^{2} \arcsin \left (c x\right )}{9 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} d e}{c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d e}{2 \, c^{2}} - \frac {14 \, b^{2} e^{2} x}{27 \, c^{2}} + \frac {a b d e \arcsin \left (c x\right )}{c^{2}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b e^{2}}{9 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} e^{2} \arcsin \left (c x\right )}{3 \, c^{3}} - \frac {b^{2} d e}{4 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b e^{2}}{3 \, c^{3}} \] Input:
integrate((e*x+d)^2*(a+b*arcsin(c*x))^2,x, algorithm="giac")
Output:
1/3*a^2*e^2*x^3 + b^2*d^2*x*arcsin(c*x)^2 + 2*a*b*d^2*x*arcsin(c*x) + 1/3* (c^2*x^2 - 1)*b^2*e^2*x*arcsin(c*x)^2/c^2 + sqrt(-c^2*x^2 + 1)*b^2*d*e*x*a rcsin(c*x)/c + a^2*d^2*x - 2*b^2*d^2*x + 2/3*(c^2*x^2 - 1)*a*b*e^2*x*arcsi n(c*x)/c^2 + (c^2*x^2 - 1)*b^2*d*e*arcsin(c*x)^2/c^2 + 1/3*b^2*e^2*x*arcsi n(c*x)^2/c^2 + sqrt(-c^2*x^2 + 1)*a*b*d*e*x/c + 2*sqrt(-c^2*x^2 + 1)*b^2*d ^2*arcsin(c*x)/c - 2/27*(c^2*x^2 - 1)*b^2*e^2*x/c^2 + 2*(c^2*x^2 - 1)*a*b* d*e*arcsin(c*x)/c^2 + 2/3*a*b*e^2*x*arcsin(c*x)/c^2 + 1/2*b^2*d*e*arcsin(c *x)^2/c^2 + 2*sqrt(-c^2*x^2 + 1)*a*b*d^2/c - 2/9*(-c^2*x^2 + 1)^(3/2)*b^2* e^2*arcsin(c*x)/c^3 + (c^2*x^2 - 1)*a^2*d*e/c^2 - 1/2*(c^2*x^2 - 1)*b^2*d* e/c^2 - 14/27*b^2*e^2*x/c^2 + a*b*d*e*arcsin(c*x)/c^2 - 2/9*(-c^2*x^2 + 1) ^(3/2)*a*b*e^2/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*b^2*e^2*arcsin(c*x)/c^3 - 1/4* b^2*d*e/c^2 + 2/3*sqrt(-c^2*x^2 + 1)*a*b*e^2/c^3
Timed out. \[ \int (d+e x)^2 (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \] Input:
int((a + b*asin(c*x))^2*(d + e*x)^2,x)
Output:
int((a + b*asin(c*x))^2*(d + e*x)^2, x)
\[ \int (d+e x)^2 (a+b \arcsin (c x))^2 \, dx=\frac {18 \mathit {asin} \left (c x \right )^{2} b^{2} c^{3} d^{2} x +18 \mathit {asin} \left (c x \right )^{2} b^{2} c^{3} d e \,x^{2}-9 \mathit {asin} \left (c x \right )^{2} b^{2} c d e +36 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) b^{2} c^{2} d^{2}+18 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) b^{2} c^{2} d e x +36 \mathit {asin} \left (c x \right ) a b \,c^{3} d^{2} x +36 \mathit {asin} \left (c x \right ) a b \,c^{3} d e \,x^{2}+12 \mathit {asin} \left (c x \right ) a b \,c^{3} e^{2} x^{3}-18 \mathit {asin} \left (c x \right ) a b c d e +36 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} d^{2}+18 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} d e x +4 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} e^{2} x^{2}+8 \sqrt {-c^{2} x^{2}+1}\, a b \,e^{2}+18 \left (\int \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3} e^{2}+18 a^{2} c^{3} d^{2} x +18 a^{2} c^{3} d e \,x^{2}+6 a^{2} c^{3} e^{2} x^{3}-36 b^{2} c^{3} d^{2} x -9 b^{2} c^{3} d e \,x^{2}}{18 c^{3}} \] Input:
int((e*x+d)^2*(a+b*asin(c*x))^2,x)
Output:
(18*asin(c*x)**2*b**2*c**3*d**2*x + 18*asin(c*x)**2*b**2*c**3*d*e*x**2 - 9 *asin(c*x)**2*b**2*c*d*e + 36*sqrt( - c**2*x**2 + 1)*asin(c*x)*b**2*c**2*d **2 + 18*sqrt( - c**2*x**2 + 1)*asin(c*x)*b**2*c**2*d*e*x + 36*asin(c*x)*a *b*c**3*d**2*x + 36*asin(c*x)*a*b*c**3*d*e*x**2 + 12*asin(c*x)*a*b*c**3*e* *2*x**3 - 18*asin(c*x)*a*b*c*d*e + 36*sqrt( - c**2*x**2 + 1)*a*b*c**2*d**2 + 18*sqrt( - c**2*x**2 + 1)*a*b*c**2*d*e*x + 4*sqrt( - c**2*x**2 + 1)*a*b *c**2*e**2*x**2 + 8*sqrt( - c**2*x**2 + 1)*a*b*e**2 + 18*int(asin(c*x)**2* x**2,x)*b**2*c**3*e**2 + 18*a**2*c**3*d**2*x + 18*a**2*c**3*d*e*x**2 + 6*a **2*c**3*e**2*x**3 - 36*b**2*c**3*d**2*x - 9*b**2*c**3*d*e*x**2)/(18*c**3)