3.3.10 \(\int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx\) [210]

3.3.10.1 Optimal result

Integrand size = 29, antiderivative size = 374 \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {52 b^2 \sqrt {d-c^2 d x^2}}{225 c^4}+\frac {4 a b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {26 b^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 c^4}-\frac {2 b^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^4}+\frac {4 b^2 x \sqrt {d-c^2 d x^2} \arcsin (c x)}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{45 c \sqrt {1-c^2 x^2}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{25 \sqrt {1-c^2 x^2}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{15 c^4}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \]

output

52/225*b^2*(-c^2*d*x^2+d)^(1/2)/c^4+26/675*b^2*(-c^2*x^2+1)*(-c^2*d*x^2+d) 
^(1/2)/c^4-2/125*b^2*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)/c^4-2/15*(a+b*arc 
sin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^4-1/15*x^2*(a+b*arcsin(c*x))^2*(-c^2*d* 
x^2+d)^(1/2)/c^2+1/5*x^4*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)+4/15*a*b 
*x*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+4/15*b^2*x*arcsin(c*x)*(-c^ 
2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+2/45*b*x^3*(a+b*arcsin(c*x))*(-c^2 
*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-2/25*b*c*x^5*(a+b*arcsin(c*x))*(-c^2* 
d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)
 

3.3.10.2 Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.65 \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d-c^2 d x^2} \left (225 a^2 \sqrt {1-c^2 x^2} \left (-2-c^2 x^2+3 c^4 x^4\right )-30 a b c x \left (-30-5 c^2 x^2+9 c^4 x^4\right )-2 b^2 \sqrt {1-c^2 x^2} \left (-428+11 c^2 x^2+27 c^4 x^4\right )-30 b \left (15 a \sqrt {1-c^2 x^2} \left (2+c^2 x^2-3 c^4 x^4\right )+b c x \left (-30-5 c^2 x^2+9 c^4 x^4\right )\right ) \arcsin (c x)+225 b^2 \sqrt {1-c^2 x^2} \left (-2-c^2 x^2+3 c^4 x^4\right ) \arcsin (c x)^2\right )}{3375 c^4 \sqrt {1-c^2 x^2}} \]

input

Integrate[x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2,x]
 

output

(Sqrt[d - c^2*d*x^2]*(225*a^2*Sqrt[1 - c^2*x^2]*(-2 - c^2*x^2 + 3*c^4*x^4) 
 - 30*a*b*c*x*(-30 - 5*c^2*x^2 + 9*c^4*x^4) - 2*b^2*Sqrt[1 - c^2*x^2]*(-42 
8 + 11*c^2*x^2 + 27*c^4*x^4) - 30*b*(15*a*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2 - 
 3*c^4*x^4) + b*c*x*(-30 - 5*c^2*x^2 + 9*c^4*x^4))*ArcSin[c*x] + 225*b^2*S 
qrt[1 - c^2*x^2]*(-2 - c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]^2))/(3375*c^4*Sqrt 
[1 - c^2*x^2])
 

3.3.10.3 Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {5198, 5138, 243, 53, 2009, 5210, 5138, 243, 53, 2009, 5182, 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.

\(\displaystyle \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx\)

\(\Big \downarrow \) 5198

\(\displaystyle -\frac {2 b c \sqrt {d-c^2 d x^2} \int x^4 (a+b \arcsin (c x))dx}{5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 5138

\(\displaystyle -\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{5} b c \int \frac {x^5}{\sqrt {1-c^2 x^2}}dx\right )}{5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx^2\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \int \left (\frac {\left (1-c^2 x^2\right )^{3/2}}{c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {1}{c^4 \sqrt {1-c^2 x^2}}\right )dx^2\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {2 b \int x^2 (a+b \arcsin (c x))dx}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {1-c^2 x^2}}dx\right )}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx^2\right )}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \int \left (\frac {1}{c^2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2}}{c^2}\right )dx^2\right )}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5182

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \left (\frac {2 b \int (a+b \arcsin (c x))dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {2 \left (\frac {2 b \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {1-c^2 x^2}}\)

input

Int[x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2,x]
 

output

(x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/5 - (2*b*c*Sqrt[d - c^2*d* 
x^2]*(-1/10*(b*c*((-2*Sqrt[1 - c^2*x^2])/c^6 + (4*(1 - c^2*x^2)^(3/2))/(3* 
c^6) - (2*(1 - c^2*x^2)^(5/2))/(5*c^6))) + (x^5*(a + b*ArcSin[c*x]))/5))/( 
5*Sqrt[1 - c^2*x^2]) + (Sqrt[d - c^2*d*x^2]*(-1/3*(x^2*Sqrt[1 - c^2*x^2]*( 
a + b*ArcSin[c*x])^2)/c^2 + (2*b*(-1/6*(b*c*((-2*Sqrt[1 - c^2*x^2])/c^4 + 
(2*(1 - c^2*x^2)^(3/2))/(3*c^4))) + (x^3*(a + b*ArcSin[c*x]))/3))/(3*c) + 
(2*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/c^2) + (2*b*(a*x + (b*Sqrt 
[1 - c^2*x^2])/c + b*x*ArcSin[c*x]))/c))/(3*c^2)))/(5*Sqrt[1 - c^2*x^2])
 

3.3.10.3.1 Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

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

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 5182

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

rule 5198

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
(e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcS 
in[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 
 - c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x 
] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[ 
(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, 
 f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 5210

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]
 

3.3.10.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 1165, normalized size of antiderivative = 3.11

method	result	size
default	\(\text {Expression too large to display}\)	\(1165\)
parts	\(\text {Expression too large to display}\)	\(1165\)

input

int(x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)
 

output

a^2*(-1/5*x^2*(-c^2*d*x^2+d)^(3/2)/c^2/d-2/15/d/c^4*(-c^2*d*x^2+d)^(3/2))+ 
b^2*(1/4000*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4-16*I*(-c^2*x^2+1 
)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-5*I*(-c^2*x^2+1 
)^(1/2)*x*c-1)*(10*I*arcsin(c*x)+25*arcsin(c*x)^2-2)/c^4/(c^2*x^2-1)+1/864 
*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*c^3*x^3*(-c^2*x^2+1)^(1/2 
)+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)/c^4/(c 
^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1) 
*(arcsin(c*x)^2-2+2*I*arcsin(c*x))/c^4/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^( 
1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)^2-2-2*I*arcsin(c*x) 
)/c^4/(c^2*x^2-1)+1/864*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^( 
1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*(-6*I*arcsin(c*x)+9 
*arcsin(c*x)^2-2)/c^4/(c^2*x^2-1)+1/4000*(-d*(c^2*x^2-1))^(1/2)*(16*I*c^5* 
x^5*(-c^2*x^2+1)^(1/2)+16*c^6*x^6-20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-28*c^4*x 
^4+5*I*(-c^2*x^2+1)^(1/2)*x*c+13*c^2*x^2-1)*(-10*I*arcsin(c*x)+25*arcsin(c 
*x)^2-2)/c^4/(c^2*x^2-1))+2*a*b*(1/800*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6- 
28*c^4*x^4-16*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1 
/2)*x^3*c^3-5*I*(-c^2*x^2+1)^(1/2)*x*c-1)*(I+5*arcsin(c*x))/c^4/(c^2*x^2-1 
)-1/16*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin 
(c*x)+I)/c^4/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2) 
*x*c+c^2*x^2-1)*(arcsin(c*x)-I)/c^4/(c^2*x^2-1)+1/288*(-d*(c^2*x^2-1))^...
 

3.3.10.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.74 \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {30 \, {\left (9 \, a b c^{5} x^{5} - 5 \, a b c^{3} x^{3} - 30 \, a b c x + {\left (9 \, b^{2} c^{5} x^{5} - 5 \, b^{2} c^{3} x^{3} - 30 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left (27 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{6} x^{6} - 4 \, {\left (225 \, a^{2} - 8 \, b^{2}\right )} c^{4} x^{4} - {\left (225 \, a^{2} - 878 \, b^{2}\right )} c^{2} x^{2} + 225 \, {\left (3 \, b^{2} c^{6} x^{6} - 4 \, b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \arcsin \left (c x\right )^{2} + 450 \, a^{2} - 856 \, b^{2} + 450 \, {\left (3 \, a b c^{6} x^{6} - 4 \, a b c^{4} x^{4} - a b c^{2} x^{2} + 2 \, a b\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{3375 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]

input

integrate(x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2,x, algorithm="frica 
s")
 

output

1/3375*(30*(9*a*b*c^5*x^5 - 5*a*b*c^3*x^3 - 30*a*b*c*x + (9*b^2*c^5*x^5 - 
5*b^2*c^3*x^3 - 30*b^2*c*x)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^ 
2 + 1) + (27*(25*a^2 - 2*b^2)*c^6*x^6 - 4*(225*a^2 - 8*b^2)*c^4*x^4 - (225 
*a^2 - 878*b^2)*c^2*x^2 + 225*(3*b^2*c^6*x^6 - 4*b^2*c^4*x^4 - b^2*c^2*x^2 
 + 2*b^2)*arcsin(c*x)^2 + 450*a^2 - 856*b^2 + 450*(3*a*b*c^6*x^6 - 4*a*b*c 
^4*x^4 - a*b*c^2*x^2 + 2*a*b)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^6*x^2 
- c^4)
 

3.3.10.6 Sympy [F]

\[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int x^{3} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]

input

integrate(x**3*(-c**2*d*x**2+d)**(1/2)*(a+b*asin(c*x))**2,x)
 

output

Integral(x**3*sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))**2, x)
 

3.3.10.7 Maxima [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.83 \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=-\frac {1}{15} \, b^{2} {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \arcsin \left (c x\right )^{2} - \frac {2}{15} \, a b {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \arcsin \left (c x\right ) - \frac {1}{15} \, a^{2} {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} - \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {-c^{2} x^{2} + 1} c^{2} \sqrt {d} x^{4} + 11 \, \sqrt {-c^{2} x^{2} + 1} \sqrt {d} x^{2} - \frac {428 \, \sqrt {-c^{2} x^{2} + 1} \sqrt {d}}{c^{2}}}{c^{2}} + \frac {15 \, {\left (9 \, c^{4} \sqrt {d} x^{5} - 5 \, c^{2} \sqrt {d} x^{3} - 30 \, \sqrt {d} x\right )} \arcsin \left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (9 \, c^{4} \sqrt {d} x^{5} - 5 \, c^{2} \sqrt {d} x^{3} - 30 \, \sqrt {d} x\right )} a b}{225 \, c^{3}} \]

input

integrate(x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxim 
a")
 

output

-1/15*b^2*(3*(-c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(3/2) 
/(c^4*d))*arcsin(c*x)^2 - 2/15*a*b*(3*(-c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) + 
 2*(-c^2*d*x^2 + d)^(3/2)/(c^4*d))*arcsin(c*x) - 1/15*a^2*(3*(-c^2*d*x^2 + 
 d)^(3/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(3/2)/(c^4*d)) - 2/3375*b^2*((2 
7*sqrt(-c^2*x^2 + 1)*c^2*sqrt(d)*x^4 + 11*sqrt(-c^2*x^2 + 1)*sqrt(d)*x^2 - 
 428*sqrt(-c^2*x^2 + 1)*sqrt(d)/c^2)/c^2 + 15*(9*c^4*sqrt(d)*x^5 - 5*c^2*s 
qrt(d)*x^3 - 30*sqrt(d)*x)*arcsin(c*x)/c^3) - 2/225*(9*c^4*sqrt(d)*x^5 - 5 
*c^2*sqrt(d)*x^3 - 30*sqrt(d)*x)*a*b/c^3
 

3.3.10.8 Giac [F(-2)]

Exception generated. \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: TypeError} \]

input

integrate(x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac" 
)
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

3.3.10.9 Mupad [F(-1)]

Timed out. \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]

input

int(x^3*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(1/2),x)
 

output

int(x^3*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(1/2), x)