3.1.84 \(\int x^4 (d-c^2 d x^2)^{5/2} (a+b \text {ArcSin}(c x)) \, dx\) [84]

Optimal. Leaf size=430 \[ \frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {1-c^2 x^2}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {1-c^2 x^2}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {1-c^2 x^2}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))+\frac {3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{512 b c^5 \sqrt {1-c^2 x^2}} \]

[Out]

1/16*d*x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))+1/10*x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))-3/256*d^2*x*
(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4-1/128*d^2*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/32*d^2*x
^5*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+3/512*b*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/512*b*
d^2*x^4*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-31/960*b*c*d^2*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+2
1/640*b*c^3*d^2*x^8*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/100*b*c^5*d^2*x^10*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^
2+1)^(1/2)+3/512*d^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c^5/(-c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.38, antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 14, 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 {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{128 c^2}+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))+\frac {3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{512 b c^5 \sqrt {1-c^2 x^2}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{256 c^4}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {1-c^2 x^2}}+\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {1-c^2 x^2}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]

[Out]

(3*b*d^2*x^2*Sqrt[d - c^2*d*x^2])/(512*c^3*Sqrt[1 - c^2*x^2]) + (b*d^2*x^4*Sqrt[d - c^2*d*x^2])/(512*c*Sqrt[1
- c^2*x^2]) - (31*b*c*d^2*x^6*Sqrt[d - c^2*d*x^2])/(960*Sqrt[1 - c^2*x^2]) + (21*b*c^3*d^2*x^8*Sqrt[d - c^2*d*
x^2])/(640*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^10*Sqrt[d - c^2*d*x^2])/(100*Sqrt[1 - c^2*x^2]) - (3*d^2*x*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[c*x]))/(256*c^4) - (d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c^2) + (
d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/32 + (d*x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/16 + (
x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/10 + (3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(512*b*c
^5*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^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d \int x^4 \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^5 \left (1-c^2 x^2\right )^2 \, dx}{10 \sqrt {1-c^2 x^2}}\\ &=\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{16} \left (3 d^2\right ) \int x^4 \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^2 \left (1-c^2 x\right )^2 \, dx,x,x^2\right )}{20 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \left (1-c^2 x^2\right ) \, dx}{16 \sqrt {1-c^2 x^2}}\\ &=\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{32 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{32 \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^2-2 c^2 x^3+c^4 x^4\right ) \, dx,x,x^2\right )}{20 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (x^5-c^2 x^7\right ) \, dx}{16 \sqrt {1-c^2 x^2}}\\ &=-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {1-c^2 x^2}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {1-c^2 x^2}}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (3 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}{128 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{128 c \sqrt {1-c^2 x^2}}\\ &=\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {1-c^2 x^2}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {1-c^2 x^2}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {1-c^2 x^2}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (3 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}{256 c^4 \sqrt {1-c^2 x^2}}+\frac {\left (3 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{256 c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {1-c^2 x^2}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {1-c^2 x^2}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {1-c^2 x^2}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{512 b c^5 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 220, normalized size = 0.51 \begin {gather*} \frac {d^2 \sqrt {d-c^2 d x^2} \left (225 a^2+b^2 c^2 x^2 \left (225+75 c^2 x^2-1240 c^4 x^4+1260 c^6 x^6-384 c^8 x^8\right )+30 a b c x \sqrt {1-c^2 x^2} \left (-15-10 c^2 x^2+248 c^4 x^4-336 c^6 x^6+128 c^8 x^8\right )+30 b \left (15 a+b c x \sqrt {1-c^2 x^2} \left (-15-10 c^2 x^2+248 c^4 x^4-336 c^6 x^6+128 c^8 x^8\right )\right ) \text {ArcSin}(c x)+225 b^2 \text {ArcSin}(c x)^2\right )}{38400 b c^5 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(225*a^2 + b^2*c^2*x^2*(225 + 75*c^2*x^2 - 1240*c^4*x^4 + 1260*c^6*x^6 - 384*c^8*x^8)
+ 30*a*b*c*x*Sqrt[1 - c^2*x^2]*(-15 - 10*c^2*x^2 + 248*c^4*x^4 - 336*c^6*x^6 + 128*c^8*x^8) + 30*b*(15*a + b*
c*x*Sqrt[1 - c^2*x^2]*(-15 - 10*c^2*x^2 + 248*c^4*x^4 - 336*c^6*x^6 + 128*c^8*x^8))*ArcSin[c*x] + 225*b^2*ArcS
in[c*x]^2))/(38400*b*c^5*Sqrt[1 - c^2*x^2])

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Maple [C] Result contains complex when optimal does not.
time = 0.51, size = 1106, normalized size = 2.57

method	result	size
default	\(-\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{10 c^{2} d}-\frac {3 a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{80 c^{4} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{160 c^{4}}+\frac {a d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{128 c^{4}}+\frac {3 a \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{256 c^{4}}+\frac {3 a \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{256 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d^{2}}{512 c^{5} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-512 i \sqrt {-c^{2} x^{2}+1}\, x^{10} c^{10}+512 c^{11} x^{11}+1280 i \sqrt {-c^{2} x^{2}+1}\, x^{8} c^{8}-1536 c^{9} x^{9}-1120 i \sqrt {-c^{2} x^{2}+1}\, x^{6} c^{6}+1696 c^{7} x^{7}+400 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}-832 c^{5} x^{5}-50 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+170 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-10 c x \right ) \left (i+10 \arcsin \left (c x \right )\right ) d^{2}}{102400 c^{5} \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}}{2048 c^{5} \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 (11 i+40 \arcsin \left (c x \right )\right ) \cos \left (9 \arcsin \left (c x \right )\right ) d^{2}}{819200 c^{5} \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 (17 i+280 \arcsin \left (c x \right )\right ) \sin \left (9 \arcsin \left (c x \right )\right ) d^{2}}{819200 c^{5} \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 (5 i+72 \arcsin \left (c x \right )\right ) \cos \left (7 \arcsin \left (c x \right )\right ) d^{2}}{98304 c^{5} \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 (11 i+24 \arcsin \left (c x \right )\right ) \sin \left (7 \arcsin \left (c x \right )\right ) d^{2}}{98304 c^{5} \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 (7 i+18 \arcsin \left (c x \right )\right ) \cos \left (5 \arcsin \left (c x \right )\right ) d^{2}}{12288 c^{5} \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+6 \arcsin \left (c x \right )\right ) \sin \left (5 \arcsin \left (c x \right )\right ) d^{2}}{12288 c^{5} \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 ) \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right ) d^{2}}{1024 c^{5} \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 (\arcsin \left (c x \right )-i\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d^{2}}{1024 c^{5} \left (c^{2} x^{2}-1\right )}\right )\)	\(1106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)

[Out]

-1/10*a*x^3*(-c^2*d*x^2+d)^(7/2)/c^2/d-3/80*a/c^4*x*(-c^2*d*x^2+d)^(7/2)/d+1/160*a/c^4*x*(-c^2*d*x^2+d)^(5/2)+
1/128*a/c^4*d*x*(-c^2*d*x^2+d)^(3/2)+3/256*a/c^4*d^2*x*(-c^2*d*x^2+d)^(1/2)+3/256*a/c^4*d^3/(c^2*d)^(1/2)*arct
an((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-3/512*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/(c^2*x^2-1)*a
rcsin(c*x)^2*d^2+1/102400*(-d*(c^2*x^2-1))^(1/2)*(-512*I*(-c^2*x^2+1)^(1/2)*x^10*c^10+512*c^11*x^11+1280*I*(-c
^2*x^2+1)^(1/2)*x^8*c^8-1536*c^9*x^9-1120*I*(-c^2*x^2+1)^(1/2)*x^6*c^6+1696*c^7*x^7+400*I*(-c^2*x^2+1)^(1/2)*x
^4*c^4-832*c^5*x^5-50*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+170*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-10*c*x)*(I+10*arcsin(c*x))
*d^2/c^5/(c^2*x^2-1)+1/2048*(-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^5/(c^2*x^2-1)-3/819200*(-d*(c^2*x^2-1))^(1/2)*(I*x^2*c^2-c*x*(-c^2*x^2+1)^
(1/2)-I)*(11*I+40*arcsin(c*x))*cos(9*arcsin(c*x))*d^2/c^5/(c^2*x^2-1)+1/819200*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2
*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(17*I+280*arcsin(c*x))*sin(9*arcsin(c*x))*d^2/c^5/(c^2*x^2-1)+1/98304*(-d*(c^2*x^
2-1))^(1/2)*(I*x^2*c^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(5*I+72*arcsin(c*x))*cos(7*arcsin(c*x))*d^2/c^5/(c^2*x^2-1)-1
/98304*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(11*I+24*arcsin(c*x))*sin(7*arcsin(c*x))*d^
2/c^5/(c^2*x^2-1)+1/12288*(-d*(c^2*x^2-1))^(1/2)*(I*x^2*c^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(7*I+18*arcsin(c*x))*cos
(5*arcsin(c*x))*d^2/c^5/(c^2*x^2-1)-5/12288*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(I+6*a
rcsin(c*x))*sin(5*arcsin(c*x))*d^2/c^5/(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)*arcsin(c*x)*cos(3*arcsin(c*x))*d^2/c^5/(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)*(arcsin(c*x)-I)*sin(3*arcsin(c*x))*d^2/c^5/(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^4*(-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^8 - 2*c^2*d^2*x^6 + d^2*x^4)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x
+ 1)*sqrt(-c*x + 1)), x) - 1/1280*(128*(-c^2*d*x^2 + d)^(7/2)*x^3/(c^2*d) - 8*(-c^2*d*x^2 + d)^(5/2)*x/c^4 +
48*(-c^2*d*x^2 + d)^(7/2)*x/(c^4*d) - 10*(-c^2*d*x^2 + d)^(3/2)*d*x/c^4 - 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^4 -
15*d^(5/2)*arcsin(c*x)/c^5)*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^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*x^8 - 2*a*c^2*d^2*x^6 + a*d^2*x^4 + (b*c^4*d^2*x^8 - 2*b*c^2*d^2*x^6 + b*d^2*x^4)*arcsin(c
*x))*sqrt(-c^2*d*x^2 + d), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3876 deep

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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^4*(-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^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,\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^4*(a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2),x)

[Out]

int(x^4*(a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2), x)

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