3.6.42 \(\int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx\) [542]

3.6.42.1 Optimal result

Integrand size = 32, antiderivative size = 222 \[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=-\frac {1}{4} b^2 x \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {b c x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}} \]

output

-1/4*b^2*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)+1/2*x*(c*d*x+d)^(1/2)*(-c*e*x+ 
e)^(1/2)*(a+b*arcsin(c*x))^2+1/4*b^2*arcsin(c*x)*(c*d*x+d)^(1/2)*(-c*e*x+e 
)^(1/2)/c/(-c^2*x^2+1)^(1/2)-1/2*b*c*x^2*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2) 
*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)+1/6*(a+b*arcsin(c*x))^3*(c*d*x+d)^(1/ 
2)*(-c*e*x+e)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)
 

3.6.42.2 Mathematica [A] (verified)

Time = 1.83 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.30 \[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\frac {4 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3-12 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+6 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) (b \cos (2 \arcsin (c x))+2 a \sin (2 \arcsin (c x)))+6 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 (2 a+b \sin (2 \arcsin (c x)))+3 \sqrt {d+c d x} \sqrt {e-c e x} \left (4 a^2 c x \sqrt {1-c^2 x^2}+2 a b \cos (2 \arcsin (c x))-b^2 \sin (2 \arcsin (c x))\right )}{24 c \sqrt {1-c^2 x^2}} \]

input

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

output

(4*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 12*a^2*Sqrt[d]*Sqrt 
[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d 
]*Sqrt[e]*(-1 + c^2*x^2))] + 6*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c* 
x]*(b*Cos[2*ArcSin[c*x]] + 2*a*Sin[2*ArcSin[c*x]]) + 6*b*Sqrt[d + c*d*x]*S 
qrt[e - c*e*x]*ArcSin[c*x]^2*(2*a + b*Sin[2*ArcSin[c*x]]) + 3*Sqrt[d + c*d 
*x]*Sqrt[e - c*e*x]*(4*a^2*c*x*Sqrt[1 - c^2*x^2] + 2*a*b*Cos[2*ArcSin[c*x] 
] - b^2*Sin[2*ArcSin[c*x]]))/(24*c*Sqrt[1 - c^2*x^2])
 

3.6.42.3 Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.66, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5178, 5156, 5138, 262, 223, 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.

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

\(\Big \downarrow \) 5178

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

\(\Big \downarrow \) 5156

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5152

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )}{\sqrt {1-c^2 x^2}}\)

input

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

output

(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) 
^2)/2 + (a + b*ArcSin[c*x])^3/(6*b*c) - b*c*((x^2*(a + b*ArcSin[c*x]))/2 - 
 (b*c*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2)))/Sqrt[1 
- c^2*x^2]
 

3.6.42.3.1 Defintions of rubi rules used

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 5138

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]
 

rule 5152

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]
 

rule 5156

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 
)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[(a + b*ArcSin[c*x])^n/Sqrt[ 
1 - c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 
]]   Int[x*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
 

rule 5178

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) 
 + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 - c^2*x^ 
2)^q)   Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 
- e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
 

3.6.42.4 Maple [F]

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

input

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

output

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

3.6.42.5 Fricas [F]

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

input

integrate((c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x, algorith 
m="fricas")
 

output

integral((b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)*sqrt(c*d*x + d)*sqr 
t(-c*e*x + e), x)
 

3.6.42.6 Sympy [F]

\[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\int \sqrt {d \left (c x + 1\right )} \sqrt {- e \left (c x - 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]

input

integrate((c*d*x+d)**(1/2)*(-c*e*x+e)**(1/2)*(a+b*asin(c*x))**2,x)
 

output

Integral(sqrt(d*(c*x + 1))*sqrt(-e*(c*x - 1))*(a + b*asin(c*x))**2, x)
 

3.6.42.7 Maxima [F(-2)]

Exception generated. \[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: ValueError} \]

input

integrate((c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x, algorith 
m="maxima")
 

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.6.42.8 Giac [F]

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

input

integrate((c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x, algorith 
m="giac")
 

output

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

3.6.42.9 Mupad [F(-1)]

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

input

int((a + b*asin(c*x))^2*(d + c*d*x)^(1/2)*(e - c*e*x)^(1/2),x)
 

output

int((a + b*asin(c*x))^2*(d + c*d*x)^(1/2)*(e - c*e*x)^(1/2), x)