Integrand size = 18, antiderivative size = 156 \[ \int \left (d+e x^2\right ) (a+b \arcsin (c x))^2 \, dx=-2 b^2 d x-\frac {4 b^2 e x}{9 c^2}-\frac {2}{27} b^2 e x^3+\frac {2 b d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {4 b e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3}+\frac {2 b e x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c}+d x (a+b \arcsin (c x))^2+\frac {1}{3} e x^3 (a+b \arcsin (c x))^2 \]
-2*b^2*d*x-4/9*b^2*e*x/c^2-2/27*b^2*e*x^3+d*x*(a+b*arcsin(c*x))^2+1/3*e*x^ 3*(a+b*arcsin(c*x))^2+2*b*d*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c+4/9*b*e *(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+2/9*b*e*x^2*(a+b*arcsin(c*x))*(- c^2*x^2+1)^(1/2)/c
Time = 0.17 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95 \[ \int \left (d+e x^2\right ) (a+b \arcsin (c x))^2 \, dx=d x (a+b \arcsin (c x))^2+\frac {1}{3} e x^3 (a+b \arcsin (c x))^2-2 b d \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}\right )-\frac {2}{27} b e \left (b x^3-\frac {3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {6 \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{c}\right ) \]
d*x*(a + b*ArcSin[c*x])^2 + (e*x^3*(a + b*ArcSin[c*x])^2)/3 - 2*b*d*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c) - (2*b*e*(b*x^3 - (3*x^2*Sqrt[ 1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (6*((b*x)/c - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^2))/c))/27
Time = 0.45 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5172, 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.
| \(\displaystyle \int \left (d+e x^2\right ) (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5172 |
| \(\displaystyle \int \left (d (a+b \arcsin (c x))^2+e x^2 (a+b \arcsin (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {2 b e x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c}+\frac {4 b e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3}+d x (a+b \arcsin (c x))^2+\frac {1}{3} e x^3 (a+b \arcsin (c x))^2-\frac {4 b^2 e x}{9 c^2}-2 b^2 d x-\frac {2}{27} b^2 e x^3\) |
-2*b^2*d*x - (4*b^2*e*x)/(9*c^2) - (2*b^2*e*x^3)/27 + (2*b*d*Sqrt[1 - c^2* x^2]*(a + b*ArcSin[c*x]))/c + (4*b*e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) )/(9*c^3) + (2*b*e*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + d*x* (a + b*ArcSin[c*x])^2 + (e*x^3*(a + b*ArcSin[c*x])^2)/3
3.7.61.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G tQ[p, 0] || IGtQ[n, 0])
Time = 0.24 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.42
| method | result | size |
| parts | \(a^{2} \left (\frac {1}{3} x^{3} e +d x \right )+\frac {b^{2} \left (\frac {e \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27 c^{2}}+d \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c}+\frac {2 a b \left (\frac {c \arcsin \left (c x \right ) x^{3} e}{3}+\arcsin \left (c x \right ) d c x -\frac {e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )-3 d \,c^{2} \sqrt {-c^{2} x^{2}+1}}{3 c^{2}}\right )}{c}\) | \(221\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (d \,c^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {e \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\arcsin \left (c x \right ) d \,c^{3} x +\frac {\arcsin \left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+d \,c^{2} \sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}}{c}\) | \(232\) |
| default | \(\frac {\frac {a^{2} \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (d \,c^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {e \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\arcsin \left (c x \right ) d \,c^{3} x +\frac {\arcsin \left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+d \,c^{2} \sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}}{c}\) | \(232\) |
a^2*(1/3*x^3*e+d*x)+b^2/c*(1/27*e*(9*c^3*x^3*arcsin(c*x)^2+6*(-c^2*x^2+1)^ (1/2)*arcsin(c*x)*x^2*c^2-2*c^3*x^3+12*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-12*c *x)/c^2+d*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)))+2*a* b/c*(1/3*c*arcsin(c*x)*x^3*e+arcsin(c*x)*d*c*x-1/3/c^2*(e*(-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^2*(-c^2*x^2+1)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.13 \[ \int \left (d+e x^2\right ) (a+b \arcsin (c x))^2 \, dx=\frac {{\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{3} e x^{3} + 9 \, {\left (b^{2} c^{3} e x^{3} + 3 \, b^{2} c^{3} d x\right )} \arcsin \left (c x\right )^{2} + 3 \, {\left (9 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{3} d - 4 \, b^{2} c e\right )} x + 18 \, {\left (a b c^{3} e x^{3} + 3 \, a b c^{3} d x\right )} \arcsin \left (c x\right ) + 6 \, {\left (a b c^{2} e x^{2} + 9 \, a b c^{2} d + 2 \, a b e + {\left (b^{2} c^{2} e x^{2} + 9 \, b^{2} c^{2} d + 2 \, b^{2} e\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c^{3}} \]
1/27*((9*a^2 - 2*b^2)*c^3*e*x^3 + 9*(b^2*c^3*e*x^3 + 3*b^2*c^3*d*x)*arcsin (c*x)^2 + 3*(9*(a^2 - 2*b^2)*c^3*d - 4*b^2*c*e)*x + 18*(a*b*c^3*e*x^3 + 3* a*b*c^3*d*x)*arcsin(c*x) + 6*(a*b*c^2*e*x^2 + 9*a*b*c^2*d + 2*a*b*e + (b^2 *c^2*e*x^2 + 9*b^2*c^2*d + 2*b^2*e)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^3
Time = 0.31 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.79 \[ \int \left (d+e x^2\right ) (a+b \arcsin (c x))^2 \, dx=\begin {cases} a^{2} d x + \frac {a^{2} e x^{3}}{3} + 2 a b d x \operatorname {asin}{\left (c x \right )} + \frac {2 a b e x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {2 a b d \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {2 a b e x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {4 a b e \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d x + \frac {b^{2} e x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {2 b^{2} e x^{3}}{27} + \frac {2 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {2 b^{2} e x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c} - \frac {4 b^{2} e x}{9 c^{2}} + \frac {4 b^{2} e \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\a^{2} \left (d x + \frac {e x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a**2*d*x + a**2*e*x**3/3 + 2*a*b*d*x*asin(c*x) + 2*a*b*e*x**3*a sin(c*x)/3 + 2*a*b*d*sqrt(-c**2*x**2 + 1)/c + 2*a*b*e*x**2*sqrt(-c**2*x**2 + 1)/(9*c) + 4*a*b*e*sqrt(-c**2*x**2 + 1)/(9*c**3) + b**2*d*x*asin(c*x)** 2 - 2*b**2*d*x + b**2*e*x**3*asin(c*x)**2/3 - 2*b**2*e*x**3/27 + 2*b**2*d* sqrt(-c**2*x**2 + 1)*asin(c*x)/c + 2*b**2*e*x**2*sqrt(-c**2*x**2 + 1)*asin (c*x)/(9*c) - 4*b**2*e*x/(9*c**2) + 4*b**2*e*sqrt(-c**2*x**2 + 1)*asin(c*x )/(9*c**3), Ne(c, 0)), (a**2*(d*x + e*x**3/3), True))
Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.42 \[ \int \left (d+e x^2\right ) (a+b \arcsin (c x))^2 \, dx=\frac {1}{3} \, b^{2} e x^{3} \arcsin \left (c x\right )^{2} + \frac {1}{3} \, a^{2} e x^{3} + b^{2} d x \arcsin \left (c x\right )^{2} + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b e + \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} e - 2 \, b^{2} d {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d}{c} \]
1/3*b^2*e*x^3*arcsin(c*x)^2 + 1/3*a^2*e*x^3 + b^2*d*x*arcsin(c*x)^2 + 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 + 2/27*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1) /c^4)*arcsin(c*x) - (c^2*x^3 + 6*x)/c^2)*b^2*e - 2*b^2*d*(x - sqrt(-c^2*x^ 2 + 1)*arcsin(c*x)/c) + a^2*d*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1)) *a*b*d/c
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (140) = 280\).
Time = 0.32 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.83 \[ \int \left (d+e x^2\right ) (a+b \arcsin (c x))^2 \, dx=\frac {1}{3} \, a^{2} e x^{3} + b^{2} d x \arcsin \left (c x\right )^{2} + 2 \, a b d x \arcsin \left (c x\right ) + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + a^{2} d x - 2 \, b^{2} d x + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b e x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b^{2} e x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{c} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} e x}{27 \, c^{2}} + \frac {2 \, a b e x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d}{c} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} e \arcsin \left (c x\right )}{9 \, c^{3}} - \frac {14 \, b^{2} e x}{27 \, c^{2}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b e}{9 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} e \arcsin \left (c x\right )}{3 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b e}{3 \, c^{3}} \]
1/3*a^2*e*x^3 + b^2*d*x*arcsin(c*x)^2 + 2*a*b*d*x*arcsin(c*x) + 1/3*(c^2*x ^2 - 1)*b^2*e*x*arcsin(c*x)^2/c^2 + a^2*d*x - 2*b^2*d*x + 2/3*(c^2*x^2 - 1 )*a*b*e*x*arcsin(c*x)/c^2 + 1/3*b^2*e*x*arcsin(c*x)^2/c^2 + 2*sqrt(-c^2*x^ 2 + 1)*b^2*d*arcsin(c*x)/c - 2/27*(c^2*x^2 - 1)*b^2*e*x/c^2 + 2/3*a*b*e*x* arcsin(c*x)/c^2 + 2*sqrt(-c^2*x^2 + 1)*a*b*d/c - 2/9*(-c^2*x^2 + 1)^(3/2)* b^2*e*arcsin(c*x)/c^3 - 14/27*b^2*e*x/c^2 - 2/9*(-c^2*x^2 + 1)^(3/2)*a*b*e /c^3 + 2/3*sqrt(-c^2*x^2 + 1)*b^2*e*arcsin(c*x)/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*a*b*e/c^3
Timed out. \[ \int \left (d+e x^2\right ) (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right ) \,d x \]