Integrand size = 21, antiderivative size = 105 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^2 \, dx=-\frac {b^2 e (c+d x)^2}{4 d}+\frac {b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{2 d}-\frac {e (a+b \arcsin (c+d x))^2}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^2}{2 d} \]
-1/4*b^2*e*(d*x+c)^2/d-1/4*e*(a+b*arcsin(d*x+c))^2/d+1/2*e*(d*x+c)^2*(a+b* arcsin(d*x+c))^2/d+1/2*b*e*(d*x+c)*(a+b*arcsin(d*x+c))*(1-(d*x+c)^2)^(1/2) /d
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.82 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^2 \, dx=-\frac {e \left (b^2 (c+d x)^2-2 b (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))+(a+b \arcsin (c+d x))^2-2 (c+d x)^2 (a+b \arcsin (c+d x))^2\right )}{4 d} \]
-1/4*(e*(b^2*(c + d*x)^2 - 2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcS in[c + d*x]) + (a + b*ArcSin[c + d*x])^2 - 2*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2))/d
Time = 0.39 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5304, 27, 5138, 5210, 15, 5152}
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.
| \(\displaystyle \int (c e+d e x) (a+b \arcsin (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int e (c+d x) (a+b \arcsin (c+d x))^2d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int (c+d x) (a+b \arcsin (c+d x))^2d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^2-b \int \frac {(c+d x)^2 (a+b \arcsin (c+d x))}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^2-b \left (\frac {1}{2} \int \frac {a+b \arcsin (c+d x)}{\sqrt {1-(c+d x)^2}}d(c+d x)+\frac {1}{2} b \int (c+d x)d(c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^2-b \left (\frac {1}{2} \int \frac {a+b \arcsin (c+d x)}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) (a+b \arcsin (c+d x))+\frac {1}{4} b (c+d x)^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^2-b \left (-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) (a+b \arcsin (c+d x))+\frac {(a+b \arcsin (c+d x))^2}{4 b}+\frac {1}{4} b (c+d x)^2\right )\right )}{d}\) |
(e*(((c + d*x)^2*(a + b*ArcSin[c + d*x])^2)/2 - b*((b*(c + d*x)^2)/4 - ((c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/2 + (a + b*ArcSin[c + d*x])^2/(4*b))))/d
3.2.91.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.46 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.39
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} e \left (d x +c \right )^{2}}{2}+b^{2} e \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}+\frac {\arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}-\frac {\arcsin \left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{2}}{4}\right )+2 e a b \left (\frac {\left (d x +c \right )^{2} \arcsin \left (d x +c \right )}{2}+\frac {\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{4}-\frac {\arcsin \left (d x +c \right )}{4}\right )}{d}\) | \(146\) |
| default | \(\frac {\frac {a^{2} e \left (d x +c \right )^{2}}{2}+b^{2} e \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}+\frac {\arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}-\frac {\arcsin \left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{2}}{4}\right )+2 e a b \left (\frac {\left (d x +c \right )^{2} \arcsin \left (d x +c \right )}{2}+\frac {\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{4}-\frac {\arcsin \left (d x +c \right )}{4}\right )}{d}\) | \(146\) |
| parts | \(a^{2} e \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {b^{2} e \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}+\frac {\arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}-\frac {\arcsin \left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{2}}{4}\right )}{d}+\frac {2 e a b \left (\frac {\left (d x +c \right )^{2} \arcsin \left (d x +c \right )}{2}+\frac {\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{4}-\frac {\arcsin \left (d x +c \right )}{4}\right )}{d}\) | \(150\) |
1/d*(1/2*a^2*e*(d*x+c)^2+b^2*e*(1/2*((d*x+c)^2-1)*arcsin(d*x+c)^2+1/2*arcs in(d*x+c)*((d*x+c)*(1-(d*x+c)^2)^(1/2)+arcsin(d*x+c))-1/4*arcsin(d*x+c)^2- 1/4*(d*x+c)^2)+2*e*a*b*(1/2*(d*x+c)^2*arcsin(d*x+c)+1/4*(d*x+c)*(1-(d*x+c) ^2)^(1/2)-1/4*arcsin(d*x+c)))
Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.79 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^2 \, dx=\frac {{\left (2 \, a^{2} - b^{2}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{2} - b^{2}\right )} c d e x + {\left (2 \, b^{2} d^{2} e x^{2} + 4 \, b^{2} c d e x + {\left (2 \, b^{2} c^{2} - b^{2}\right )} e\right )} \arcsin \left (d x + c\right )^{2} + 2 \, {\left (2 \, a b d^{2} e x^{2} + 4 \, a b c d e x + {\left (2 \, a b c^{2} - a b\right )} e\right )} \arcsin \left (d x + c\right ) + 2 \, {\left (a b d e x + a b c e + {\left (b^{2} d e x + b^{2} c e\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{4 \, d} \]
1/4*((2*a^2 - b^2)*d^2*e*x^2 + 2*(2*a^2 - b^2)*c*d*e*x + (2*b^2*d^2*e*x^2 + 4*b^2*c*d*e*x + (2*b^2*c^2 - b^2)*e)*arcsin(d*x + c)^2 + 2*(2*a*b*d^2*e* x^2 + 4*a*b*c*d*e*x + (2*a*b*c^2 - a*b)*e)*arcsin(d*x + c) + 2*(a*b*d*e*x + a*b*c*e + (b^2*d*e*x + b^2*c*e)*arcsin(d*x + c))*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1))/d
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (88) = 176\).
Time = 0.21 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.19 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^2 \, dx=\begin {cases} a^{2} c e x + \frac {a^{2} d e x^{2}}{2} + \frac {a b c^{2} e \operatorname {asin}{\left (c + d x \right )}}{d} + 2 a b c e x \operatorname {asin}{\left (c + d x \right )} + \frac {a b c e \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{2 d} + a b d e x^{2} \operatorname {asin}{\left (c + d x \right )} + \frac {a b e x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{2} - \frac {a b e \operatorname {asin}{\left (c + d x \right )}}{2 d} + \frac {b^{2} c^{2} e \operatorname {asin}^{2}{\left (c + d x \right )}}{2 d} + b^{2} c e x \operatorname {asin}^{2}{\left (c + d x \right )} - \frac {b^{2} c e x}{2} + \frac {b^{2} c e \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{2 d} + \frac {b^{2} d e x^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{2} - \frac {b^{2} d e x^{2}}{4} + \frac {b^{2} e x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{2} - \frac {b^{2} e \operatorname {asin}^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {asin}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \]
Piecewise((a**2*c*e*x + a**2*d*e*x**2/2 + a*b*c**2*e*asin(c + d*x)/d + 2*a *b*c*e*x*asin(c + d*x) + a*b*c*e*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(2* d) + a*b*d*e*x**2*asin(c + d*x) + a*b*e*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/2 - a*b*e*asin(c + d*x)/(2*d) + b**2*c**2*e*asin(c + d*x)**2/(2*d) + b**2*c*e*x*asin(c + d*x)**2 - b**2*c*e*x/2 + b**2*c*e*sqrt(-c**2 - 2*c*d* x - d**2*x**2 + 1)*asin(c + d*x)/(2*d) + b**2*d*e*x**2*asin(c + d*x)**2/2 - b**2*d*e*x**2/4 + b**2*e*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/2 - b**2*e*asin(c + d*x)**2/(4*d), Ne(d, 0)), (c*e*x*(a + b*asin(c) )**2, True))
\[ \int (c e+d e x) (a+b \arcsin (c+d x))^2 \, dx=\int { {\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
1/2*a^2*d*e*x^2 + 1/2*(2*x^2*arcsin(d*x + c) + d*(3*c^2*arcsin(-(d^2*x + c *d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^3 + sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1 )*x/d^2 - (c^2 - 1)*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d ^3 - 3*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c/d^3))*a*b*d*e + a^2*c*e*x + 2* ((d*x + c)*arcsin(d*x + c) + sqrt(-(d*x + c)^2 + 1))*a*b*c*e/d + 1/2*(b^2* d*e*x^2 + 2*b^2*c*e*x)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + integrate((b^2*d^2*e*x^2 + 2*b^2*c*d*e*x)*sqrt(d*x + c + 1)*sqrt(- d*x - c + 1)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))/(d^2*x ^2 + 2*c*d*x + c^2 - 1), x)
Time = 0.32 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.75 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^2 \, dx=\frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{2} e \arcsin \left (d x + c\right )^{2}}{2 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{2} e \arcsin \left (d x + c\right )}{2 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} a b e \arcsin \left (d x + c\right )}{d} + \frac {b^{2} e \arcsin \left (d x + c\right )^{2}}{4 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b e}{2 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} e}{2 \, d} - \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{2} e}{4 \, d} + \frac {a b e \arcsin \left (d x + c\right )}{2 \, d} - \frac {b^{2} e}{8 \, d} \]
1/2*((d*x + c)^2 - 1)*b^2*e*arcsin(d*x + c)^2/d + 1/2*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^2*e*arcsin(d*x + c)/d + ((d*x + c)^2 - 1)*a*b*e*arcsin(d*x + c)/d + 1/4*b^2*e*arcsin(d*x + c)^2/d + 1/2*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a*b*e/d + 1/2*((d*x + c)^2 - 1)*a^2*e/d - 1/4*((d*x + c)^2 - 1)*b^2*e/ d + 1/2*a*b*e*arcsin(d*x + c)/d - 1/8*b^2*e/d
Timed out. \[ \int (c e+d e x) (a+b \arcsin (c+d x))^2 \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2 \,d x \]