3.1.11 \(\int (d+e x) (a+b \arcsin (c x))^2 \, dx\) [11]

3.1.11.1 Optimal result

Integrand size = 16, antiderivative size = 142 \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=-2 b^2 d x-\frac {1}{4} b^2 e x^2+\frac {2 b d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {b e x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}-\frac {d^2 (a+b \arcsin (c x))^2}{2 e}-\frac {e (a+b \arcsin (c x))^2}{4 c^2}+\frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e} \]

output

-2*b^2*d*x-1/4*b^2*e*x^2-1/2*d^2*(a+b*arcsin(c*x))^2/e-1/4*e*(a+b*arcsin(c 
*x))^2/c^2+1/2*(e*x+d)^2*(a+b*arcsin(c*x))^2/e+2*b*d*(a+b*arcsin(c*x))*(-c 
^2*x^2+1)^(1/2)/c+1/2*b*e*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c
 

3.1.11.2 Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.08 \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=\frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e}-\frac {b \left (2 b d e x+\frac {1}{4} b e^2 x^2-\frac {2 d e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}-\frac {e^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}+\frac {d^2 (a+b \arcsin (c x))^2}{2 b}+\frac {e^2 (a+b \arcsin (c x))^2}{4 b c^2}\right )}{e} \]

input

Integrate[(d + e*x)*(a + b*ArcSin[c*x])^2,x]
 

output

((d + e*x)^2*(a + b*ArcSin[c*x])^2)/(2*e) - (b*(2*b*d*e*x + (b*e^2*x^2)/4 
- (2*d*e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c - (e^2*x*Sqrt[1 - c^2*x^ 
2]*(a + b*ArcSin[c*x]))/(2*c) + (d^2*(a + b*ArcSin[c*x])^2)/(2*b) + (e^2*( 
a + b*ArcSin[c*x])^2)/(4*b*c^2)))/e
 

3.1.11.3 Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, 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.

\(\displaystyle \int (d+e x) (a+b \arcsin (c x))^2 \, dx\)

\(\Big \downarrow \) 5242

\(\displaystyle \frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e}-\frac {b c \int \frac {(d+e x)^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{e}\)

\(\Big \downarrow \) 5262

\(\displaystyle \frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e}-\frac {b c \int \left (\frac {(a+b \arcsin (c x)) d^2}{\sqrt {1-c^2 x^2}}+\frac {2 e x (a+b \arcsin (c x)) d}{\sqrt {1-c^2 x^2}}+\frac {e^2 x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}\right )dx}{e}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e}-\frac {b c \left (\frac {e^2 (a+b \arcsin (c x))^2}{4 b c^3}-\frac {2 d e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}-\frac {e^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {d^2 (a+b \arcsin (c x))^2}{2 b c}+\frac {2 b d e x}{c}+\frac {b e^2 x^2}{4 c}\right )}{e}\)

input

Int[(d + e*x)*(a + b*ArcSin[c*x])^2,x]
 

output

((d + e*x)^2*(a + b*ArcSin[c*x])^2)/(2*e) - (b*c*((2*b*d*e*x)/c + (b*e^2*x 
^2)/(4*c) - (2*d*e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^2 - (e^2*x*Sqr 
t[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c^2) + (d^2*(a + b*ArcSin[c*x])^2)/ 
(2*b*c) + (e^2*(a + b*ArcSin[c*x])^2)/(4*b*c^3)))/e
 

3.1.11.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5242

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]
 

rule 5262

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))
 

3.1.11.4 Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.33

method	result	size
parts	\(a^{2} \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b^{2} \left (\frac {e \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4 c}+d \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c}+\frac {2 a b \left (\frac {c \arcsin \left (c x \right ) x^{2} e}{2}+\arcsin \left (c x \right ) d c x -\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )-2 d c \sqrt {-c^{2} x^{2}+1}}{2 c}\right )}{c}\)	\(189\)
derivativedivides	\(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {e \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}\right )}{c}+\frac {2 a b \left (\arcsin \left (c x \right ) c^{2} x d +\frac {\arcsin \left (c x \right ) c^{2} e \,x^{2}}{2}-\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}+d c \sqrt {-c^{2} x^{2}+1}\right )}{c}}{c}\)	\(198\)
default	\(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {e \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}\right )}{c}+\frac {2 a b \left (\arcsin \left (c x \right ) c^{2} x d +\frac {\arcsin \left (c x \right ) c^{2} e \,x^{2}}{2}-\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}+d c \sqrt {-c^{2} x^{2}+1}\right )}{c}}{c}\)	\(198\)

input

int((e*x+d)*(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)
 

output

a^2*(1/2*e*x^2+d*x)+b^2/c*(1/4*e*(2*arcsin(c*x)^2*x^2*c^2+2*(-c^2*x^2+1)^( 
1/2)*arcsin(c*x)*x*c-arcsin(c*x)^2-c^2*x^2)/c+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*(1/2*c*arcsin(c*x)*x^2*e+arcsin( 
c*x)*d*c*x-1/2/c*(e*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))-2*d*c*(- 
c^2*x^2+1)^(1/2)))
 

3.1.11.5 Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.10 \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=\frac {{\left (2 \, a^{2} - b^{2}\right )} c^{2} e x^{2} + 4 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{2} d x + {\left (2 \, b^{2} c^{2} e x^{2} + 4 \, b^{2} c^{2} d x - b^{2} e\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (2 \, a b c^{2} e x^{2} + 4 \, a b c^{2} d x - a b e\right )} \arcsin \left (c x\right ) + 2 \, {\left (a b c e x + 4 \, a b c d + {\left (b^{2} c e x + 4 \, b^{2} c d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{4 \, c^{2}} \]

input

integrate((e*x+d)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")
 

output

1/4*((2*a^2 - b^2)*c^2*e*x^2 + 4*(a^2 - 2*b^2)*c^2*d*x + (2*b^2*c^2*e*x^2 
+ 4*b^2*c^2*d*x - b^2*e)*arcsin(c*x)^2 + 2*(2*a*b*c^2*e*x^2 + 4*a*b*c^2*d* 
x - a*b*e)*arcsin(c*x) + 2*(a*b*c*e*x + 4*a*b*c*d + (b^2*c*e*x + 4*b^2*c*d 
)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^2
 

3.1.11.6 Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.64 \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=\begin {cases} a^{2} d x + \frac {a^{2} e x^{2}}{2} + 2 a b d x \operatorname {asin}{\left (c x \right )} + a b e x^{2} \operatorname {asin}{\left (c x \right )} + \frac {2 a b d \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {a b e x \sqrt {- c^{2} x^{2} + 1}}{2 c} - \frac {a b e \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + b^{2} d x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d x + \frac {b^{2} e x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {b^{2} e x^{2}}{4} + \frac {2 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {b^{2} e x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} - \frac {b^{2} e \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a^{2} \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]

input

integrate((e*x+d)*(a+b*asin(c*x))**2,x)
 

output

Piecewise((a**2*d*x + a**2*e*x**2/2 + 2*a*b*d*x*asin(c*x) + a*b*e*x**2*asi 
n(c*x) + 2*a*b*d*sqrt(-c**2*x**2 + 1)/c + a*b*e*x*sqrt(-c**2*x**2 + 1)/(2* 
c) - a*b*e*asin(c*x)/(2*c**2) + b**2*d*x*asin(c*x)**2 - 2*b**2*d*x + b**2* 
e*x**2*asin(c*x)**2/2 - b**2*e*x**2/4 + 2*b**2*d*sqrt(-c**2*x**2 + 1)*asin 
(c*x)/c + b**2*e*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(2*c) - b**2*e*asin(c*x) 
**2/(4*c**2), Ne(c, 0)), (a**2*(d*x + e*x**2/2), True))
 

3.1.11.7 Maxima [F]

\[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=\int { {\left (e x + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]

input

integrate((e*x+d)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")
 

output

b^2*d*x*arcsin(c*x)^2 + 1/2*a^2*e*x^2 + 1/2*(2*x^2*arcsin(c*x) + c*(sqrt(- 
c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a*b*e + 1/2*(x^2*arctan2(c*x, sqrt( 
c*x + 1)*sqrt(-c*x + 1))^2 + 2*c*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*x^ 
2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*x^2 - 1), x))*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
 

3.1.11.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.72 \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=b^{2} d x \arcsin \left (c x\right )^{2} + 2 \, a b d x \arcsin \left (c x\right ) + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} e x \arcsin \left (c x\right )}{2 \, c} + a^{2} d x - 2 \, b^{2} d x + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} a b e x}{2 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a b e \arcsin \left (c x\right )}{c^{2}} + \frac {b^{2} e \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} e}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e}{4 \, c^{2}} + \frac {a b e \arcsin \left (c x\right )}{2 \, c^{2}} - \frac {b^{2} e}{8 \, c^{2}} \]

input

integrate((e*x+d)*(a+b*arcsin(c*x))^2,x, algorithm="giac")
 

output

b^2*d*x*arcsin(c*x)^2 + 2*a*b*d*x*arcsin(c*x) + 1/2*sqrt(-c^2*x^2 + 1)*b^2 
*e*x*arcsin(c*x)/c + a^2*d*x - 2*b^2*d*x + 1/2*(c^2*x^2 - 1)*b^2*e*arcsin( 
c*x)^2/c^2 + 1/2*sqrt(-c^2*x^2 + 1)*a*b*e*x/c + 2*sqrt(-c^2*x^2 + 1)*b^2*d 
*arcsin(c*x)/c + (c^2*x^2 - 1)*a*b*e*arcsin(c*x)/c^2 + 1/4*b^2*e*arcsin(c* 
x)^2/c^2 + 2*sqrt(-c^2*x^2 + 1)*a*b*d/c + 1/2*(c^2*x^2 - 1)*a^2*e/c^2 - 1/ 
4*(c^2*x^2 - 1)*b^2*e/c^2 + 1/2*a*b*e*arcsin(c*x)/c^2 - 1/8*b^2*e/c^2
 

3.1.11.9 Mupad [F(-1)]

Timed out. \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right ) \,d x \]

input

int((a + b*asin(c*x))^2*(d + e*x),x)
 

output

int((a + b*asin(c*x))^2*(d + e*x), x)