3.1.85 \(\int x^2 (d-c^2 d x^2)^{5/2} (a+b \arcsin (c x)) \, dx\) [85]

3.1.85.1 Optimal result

Integrand size = 27, antiderivative size = 351 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}} \]

output

5/48*d*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))+1/8*x^3*(-c^2*d*x^2+d)^( 
5/2)*(a+b*arcsin(c*x))-5/128*d^2*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/ 
c^2+5/64*d^2*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+5/256*b*d^2*x^2*(- 
c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-59/768*b*c*d^2*x^4*(-c^2*d*x^2+d)^ 
(1/2)/(-c^2*x^2+1)^(1/2)+17/288*b*c^3*d^2*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x 
^2+1)^(1/2)-1/64*b*c^5*d^2*x^8*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+5/2 
56*d^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)
 

3.1.85.2 Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.56 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (45 a^2+b^2 c^2 x^2 \left (45-177 c^2 x^2+136 c^4 x^4-36 c^6 x^6\right )+6 a b c x \sqrt {1-c^2 x^2} \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )+6 b \left (15 a+b c x \sqrt {1-c^2 x^2} \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )\right ) \arcsin (c x)+45 b^2 \arcsin (c x)^2\right )}{2304 b c^3 \sqrt {1-c^2 x^2}} \]

input

Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]
 

output

(d^2*Sqrt[d - c^2*d*x^2]*(45*a^2 + b^2*c^2*x^2*(45 - 177*c^2*x^2 + 136*c^4 
*x^4 - 36*c^6*x^6) + 6*a*b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2*x^2 - 136* 
c^4*x^4 + 48*c^6*x^6) + 6*b*(15*a + b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2 
*x^2 - 136*c^4*x^4 + 48*c^6*x^6))*ArcSin[c*x] + 45*b^2*ArcSin[c*x]^2))/(23 
04*b*c^3*Sqrt[1 - c^2*x^2])
 

3.1.85.3 Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5202, 243, 49, 2009, 5202, 244, 2009, 5198, 15, 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.

\(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx\)

\(\Big \downarrow \) 5202

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right )^2dx}{8 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^2 \left (1-c^2 x^2\right )^2dx^2}{16 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (c^4 x^6-2 c^2 x^4+x^2\right )dx^2}{16 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5202

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right )dx}{6 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int \left (x^3-c^2 x^5\right )dx}{6 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))dx+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5198

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^3dx}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {5}{8} d \left (\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{2} d \left (\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {\sqrt {d-c^2 d x^2} \left (\frac {(a+b \arcsin (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\right )-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

input

Int[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]
 

output

-1/16*(b*c*d^2*Sqrt[d - c^2*d*x^2]*(x^4/2 - (2*c^2*x^6)/3 + (c^4*x^8)/4))/ 
Sqrt[1 - c^2*x^2] + (x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/8 + (5 
*d*(-1/6*(b*c*d*Sqrt[d - c^2*d*x^2]*(x^4/4 - (c^2*x^6)/6))/Sqrt[1 - c^2*x^ 
2] + (x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/6 + (d*(-1/16*(b*c*x^ 
4*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] + (x^3*Sqrt[d - c^2*d*x^2]*(a + b 
*ArcSin[c*x]))/4 + (Sqrt[d - c^2*d*x^2]*((b*x^2)/(4*c) - (x*Sqrt[1 - c^2*x 
^2]*(a + b*ArcSin[c*x]))/(2*c^2) + (a + b*ArcSin[c*x])^2/(4*b*c^3)))/(4*Sq 
rt[1 - c^2*x^2])))/2))/8
 

3.1.85.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 49

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}, x] 
&& IGtQ[m, 0] && IGtQ[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 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

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 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 5202

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

3.1.85.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 907, normalized size of antiderivative = 2.58

method	result	size
default	\(-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{8 c^{2} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{48 c^{2}}+\frac {5 a d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{192 c^{2}}+\frac {5 a \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{2}}+\frac {5 a \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d^{2}}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-128 i \sqrt {-c^{2} x^{2}+1}\, x^{8} c^{8}+128 c^{9} x^{9}+256 i \sqrt {-c^{2} x^{2}+1}\, c^{6} x^{6}-320 c^{7} x^{7}-160 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+272 c^{5} x^{5}+32 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-88 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+8 c x \right ) \left (8 \arcsin \left (c x \right )+i\right ) d^{2}}{16384 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arcsin \left (c x \right )\right ) d^{2}}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (73 i+312 \arcsin \left (c x \right )\right ) \cos \left (7 \arcsin \left (c x \right )\right ) d^{2}}{147456 c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (55 i+456 \arcsin \left (c x \right )\right ) \sin \left (7 \arcsin \left (c x \right )\right ) d^{2}}{147456 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (13 i+12 \arcsin \left (c x \right )\right ) \cos \left (5 \arcsin \left (c x \right )\right ) d^{2}}{9216 c^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (i+12 \arcsin \left (c x \right )\right ) \sin \left (5 \arcsin \left (c x \right )\right ) d^{2}}{9216 c^{3} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (4 \arcsin \left (c x \right )+i\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d^{2}}{1024 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (5 i+4 \arcsin \left (c x \right )\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d^{2}}{1024 c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(907\)
parts	\(-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{8 c^{2} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{48 c^{2}}+\frac {5 a d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{192 c^{2}}+\frac {5 a \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{2}}+\frac {5 a \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d^{2}}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-128 i \sqrt {-c^{2} x^{2}+1}\, x^{8} c^{8}+128 c^{9} x^{9}+256 i \sqrt {-c^{2} x^{2}+1}\, c^{6} x^{6}-320 c^{7} x^{7}-160 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+272 c^{5} x^{5}+32 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-88 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+8 c x \right ) \left (8 \arcsin \left (c x \right )+i\right ) d^{2}}{16384 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arcsin \left (c x \right )\right ) d^{2}}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (73 i+312 \arcsin \left (c x \right )\right ) \cos \left (7 \arcsin \left (c x \right )\right ) d^{2}}{147456 c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (55 i+456 \arcsin \left (c x \right )\right ) \sin \left (7 \arcsin \left (c x \right )\right ) d^{2}}{147456 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (13 i+12 \arcsin \left (c x \right )\right ) \cos \left (5 \arcsin \left (c x \right )\right ) d^{2}}{9216 c^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (i+12 \arcsin \left (c x \right )\right ) \sin \left (5 \arcsin \left (c x \right )\right ) d^{2}}{9216 c^{3} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (4 \arcsin \left (c x \right )+i\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d^{2}}{1024 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (5 i+4 \arcsin \left (c x \right )\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d^{2}}{1024 c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(907\)

input

int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
 

output

-1/8*a*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/48*a/c^2*x*(-c^2*d*x^2+d)^(5/2)+5/19 
2*a/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/128*a/c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/ 
128*a/c^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b 
*(-5/256*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin( 
c*x)^2*d^2+1/16384*(-d*(c^2*x^2-1))^(1/2)*(-128*I*(-c^2*x^2+1)^(1/2)*x^8*c 
^8+128*c^9*x^9+256*I*(-c^2*x^2+1)^(1/2)*x^6*c^6-320*c^7*x^7-160*I*(-c^2*x^ 
2+1)^(1/2)*x^4*c^4+272*c^5*x^5+32*I*(-c^2*x^2+1)^(1/2)*c^2*x^2-88*c^3*x^3- 
I*(-c^2*x^2+1)^(1/2)+8*c*x)*(8*arcsin(c*x)+I)*d^2/c^3/(c^2*x^2-1)+1/256*(- 
d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2 
+1)^(1/2)-2*c*x)*(-I+2*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)+1/147456*(-d*(c^2* 
x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(73*I+312*arcsin(c*x))* 
cos(7*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)-1/147456*(-d*(c^2*x^2-1))^(1/2)*(I* 
(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(55*I+456*arcsin(c*x))*sin(7*arcsin(c*x) 
)*d^2/c^3/(c^2*x^2-1)+1/9216*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x 
^2+1)^(1/2)-I)*(13*I+12*arcsin(c*x))*cos(5*arcsin(c*x))*d^2/c^3/(c^2*x^2-1 
)-5/9216*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(I+12 
*arcsin(c*x))*sin(5*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)-3/1024*(-d*(c^2*x^2-1 
))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(4*arcsin(c*x)+I)*cos(3*arcs 
in(c*x))*d^2/c^3/(c^2*x^2-1)+1/1024*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1) 
^(1/2)*x*c+c^2*x^2-1)*(5*I+4*arcsin(c*x))*sin(3*arcsin(c*x))*d^2/c^3/(c...
 

3.1.85.5 Fricas [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{2} \,d x } \]

input

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="fricas" 
)
 

output

integral((a*c^4*d^2*x^6 - 2*a*c^2*d^2*x^4 + a*d^2*x^2 + (b*c^4*d^2*x^6 - 2 
*b*c^2*d^2*x^4 + b*d^2*x^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)
 

3.1.85.6 Sympy [F(-1)]

Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \]

input

integrate(x**2*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x)),x)
 

output

Timed out
 

3.1.85.7 Maxima [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{2} \,d x } \]

input

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="maxima" 
)
 

output

b*sqrt(d)*integrate((c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2)*sqrt(c*x + 1)* 
sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + 1/384*(8*( 
-c^2*d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d) + 10*(-c 
^2*d*x^2 + d)^(3/2)*d*x/c^2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/ 
2)*arcsin(c*x)/c^3)*a
 

3.1.85.8 Giac [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{2} \,d x } \]

input

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="giac")
 

output

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)*x^2, x)
 

3.1.85.9 Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

input

int(x^2*(a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2),x)
 

output

int(x^2*(a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2), x)