3.85 \(\int x^2 (d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x)) \, dx\)

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

[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 = 0.47, 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 = {4699, 4697, 4707, 4641, 30, 14, 266, 43} \[ \frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^2}+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}} \]

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 43

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 266

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 4641

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

Rule 4697

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[Sqrt[d + e*x^2]/((m + 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] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m
+ 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] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 4699

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(
f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[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]
 && (RationalQ[m] || EqQ[n, 1])

Rule 4707

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

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 ) \operatorname {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 ) \operatorname {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 = 0.23, size = 196, normalized size = 0.56 \[ \frac {d^2 \sqrt {d-c^2 d x^2} \left (45 a^2+6 a b c x \sqrt {1-c^2 x^2} \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )+6 b \sin ^{-1}(c x) \left (15 a+b c x \sqrt {1-c^2 x^2} \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )\right )+b^2 c^2 x^2 \left (-36 c^6 x^6+136 c^4 x^4-177 c^2 x^2+45\right )+45 b^2 \sin ^{-1}(c x)^2\right )}{2304 b c^3 \sqrt {1-c^2 x^2}} \]

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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fricas [F]  time = 1.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{4} d^{2} x^{6} - 2 \, a c^{2} d^{2} x^{4} + a d^{2} x^{2} + {\left (b c^{4} d^{2} x^{6} - 2 \, b c^{2} d^{2} x^{4} + b d^{2} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{2}\,{d x} \]

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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maple [C]  time = 0.38, size = 2828, normalized size = 8.06 \[ \text {output too large to display} \]

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)

[Out]

-7/256*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*arcsin(c*x)*x^3+73/147456*b*(-d*(c^2*x^2-1))^(1/2)*cos(7*arcsi
n(c*x))*d^2/c^3/(c^2*x^2-1)+13/9216*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)-3/1024*b*(
-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)-63/16384*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c^3/(c^2*x^
2-1)*(-c^2*x^2+1)^(1/2)-27/2048*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*x^3+5/768*b*(-d*(c^2*x^2-1))^(1/2)*
sin(5*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)*arcsin(c*x)-1/256*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))*d^2/c^3/(
c^2*x^2-1)*arcsin(c*x)+1/16*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^6/(c^2*x^2-1)*arcsin(c*x)*x^9-5/32*b*(-d*(c^2*x^2-1
))^(1/2)*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)*x^7+17/128*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^2/(c^2*x^2-1)*arcsin(c*x)*x
^5-3/256*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c^2/(c^2*x^2-1)*arcsin(c*x)*x-5/256*b*(-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+3/1024*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d^2/c/(c^2*x^2-1)
*x^2+1/128*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^8-1/64*b*(-d*(c^2*x^2-1))^(1/2)*d
^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^6+5/512*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*
x^4+3/512*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-73/147456*b*(-d*(c^2*x^2-1))^(1/2)
*cos(7*arcsin(c*x))*d^2/c/(c^2*x^2-1)*x^2-13/9216*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d^2/c/(c^2*x^2-1
)*x^2+1/48*a/c^2*x*(-c^2*d*x^2+d)^(5/2)+5/128*a/c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+55/147456*I*b*(-d*(c^2*x^2-1))^
(1/2)*sin(7*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)+5/9216*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d^2/c^3/(c^2
*x^2-1)-5/1024*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)+1/128*I*b*(-d*(c^2*x^2-1))^(1
/2)*d^2*c^6/(c^2*x^2-1)*x^9-5/256*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^4/(c^2*x^2-1)*x^7+17/1024*I*b*(-d*(c^2*x^2-
1))^(1/2)*d^2*c^2/(c^2*x^2-1)*x^5+17/2048*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c^2/(c^2*x^2-1)*x+19/6144*b*(-d*(c^2*
x^2-1))^(1/2)*sin(7*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)*arcsin(c*x)+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))+5/192*a/c^2*d*x*(-c^2*d*x^2+d)^(3/2)-1/8*a*x*(-c^2*d*x^2+d)^(7/2)/c^2/d-1/768*I*
b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)*arcsin(c*x)-5/9216*I*b*(-d*(c^2*x^2-1))^(1/2)*
sin(5*arcsin(c*x))*d^2/c/(c^2*x^2-1)*x^2+5/9216*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d^2/c^2/(c^2*x^2-1
)*(-c^2*x^2+1)^(1/2)*x-5/1024*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/
2)*x+55/147456*b*(-d*(c^2*x^2-1))^(1/2)*sin(7*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+1/256*b*(-
d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))*d^2/c/(c^2*x^2-1)*arcsin(c*x)*x^2-19/6144*b*(-d*(c^2*x^2-1))^(1/2)*sin
(7*arcsin(c*x))*d^2/c/(c^2*x^2-1)*arcsin(c*x)*x^2-5/768*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d^2/c/(c^2
*x^2-1)*arcsin(c*x)*x^2+3/256*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)*arcsin(c*x)+5/
1024*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))*d^2/c/(c^2*x^2-1)*x^2-17/2048*I*b*(-d*(c^2*x^2-1))^(1/2)*d^
2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)-13/6144*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(7*arcsin(c*x))*d^2/c^3
/(c^2*x^2-1)*arcsin(c*x)-55/147456*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(7*arcsin(c*x))*d^2/c/(c^2*x^2-1)*x^2-1/768*b
*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x+3/256*b*(-d*(c
^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x-13/6144*b*(-d*(c^2*x^
2-1))^(1/2)*cos(7*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x-13/9216*I*b*(-d*(c^2*x^2-1
))^(1/2)*cos(5*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-3/256*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*ar
csin(c*x))*d^2/c/(c^2*x^2-1)*arcsin(c*x)*x^2+3/1024*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d^2/c^2/(c^2
*x^2-1)*(-c^2*x^2+1)^(1/2)*x-1/16*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x
)*x^8+1/8*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^6-5/64*I*b*(-d*(c^2*
x^2-1))^(1/2)*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^4+1/32*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c/(c^2*
x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2+13/6144*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(7*arcsin(c*x))*d^2/c/(c^2*x^2
-1)*arcsin(c*x)*x^2-73/147456*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(7*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(
1/2)*x+1/768*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d^2/c/(c^2*x^2-1)*arcsin(c*x)*x^2-19/6144*I*b*(-d*(
c^2*x^2-1))^(1/2)*sin(7*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x-5/768*I*b*(-d*(c^2*x
^2-1))^(1/2)*sin(5*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x+1/256*I*b*(-d*(c^2*x^2-1)
)^(1/2)*sin(3*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b \sqrt {d} \int {\left (c^{4} d^{2} x^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\,{d x} + \frac {1}{384} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2}} - \frac {48 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x}{c^{2} d} + \frac {10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x}{c^{2}} + \frac {15 \, \sqrt {-c^{2} d x^{2} + d} d^{2} x}{c^{2}} + \frac {15 \, d^{\frac {5}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} a \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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