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

Optimal. Leaf size=351 \[ \frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))+\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}} \]

[Out]

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)
+5/256*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)

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Rubi [A]
time = 0.32, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {4787, 4783, 4795, 4737, 30, 14, 272, 45} \begin {gather*} -\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))+\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}}+\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 {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

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

Rule 45

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] && NeQ[b*c - a*d, 0] && 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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[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 4783

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*ArcSin[c*x])^n/(f*(m + 2))), x] + (Dist[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/S
qrt[1 - c^2*x^2]], Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x] - Dist[b*c*(n/(f*(m + 2)))*Si
mp[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 4787

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*ArcSin[c*x])^n/(f*(m + 2*p + 1))), x] + (Dist[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] - Dist[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 4795

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] + (Dist[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] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(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]

Rubi steps

\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} (5 d) \int x^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^2 \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{16} \left (5 d^2\right ) \int x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x \left (1-c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{48 \sqrt {1-c^2 x^2}}\\ &=\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (x-2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{48 \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} \left (a+b \sin ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (5 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt {1-c^2 x^2}}\\ &=\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} \left (a+b \sin ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 196, normalized size = 0.56 \begin {gather*} \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 ) \text {ArcSin}(c x)+45 b^2 \text {ArcSin}(c x)^2\right )}{2304 b c^3 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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))/(2304*b*c^3*Sqrt[1 - c^2*x^2])

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Maple [C] Result contains complex when optimal does not.
time = 0.28, size = 907, normalized size = 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}\, x^{6} c^{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 (i+8 \arcsin \left (c x \right )\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 x^{2} c^{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 \sqrt {-c^{2} x^{2}+1}\, x c +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 x^{2} c^{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 \sqrt {-c^{2} x^{2}+1}\, x c +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 x^{2} c^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (i+4 \arcsin \left (c x \right )\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 \sqrt {-c^{2} x^{2}+1}\, x c +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 )\)	\(907\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)+5/192*a/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/1
28*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)*x^2*c^2-88*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+8*c*x)*(I+8
*arcsin(c*x))*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*x^2*c^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*x^2*c^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*x^2*c^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(I+4*arcsin(c*x)
)*cos(3*arcsin(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^2*x^2-1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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