3.69 \(\int x^2 (d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=265 \[ -\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^2}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}} \]

[Out]

1/6*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))-1/16*d*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/8*d*x^3*(
a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+1/32*b*d*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-7/96*b*c*d*x^4*(-
c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/36*b*c^3*d*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/32*d*(a+b*arc
sin(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 = 265, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4699, 4697, 4707, 4641, 30, 14} \[ \frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^2}+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {b d x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[1 - c^2*x^2]) - (7*b*c*d*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^
2]) + (b*c^3*d*x^6*Sqrt[d - c^2*d*x^2])/(36*Sqrt[1 - c^2*x^2]) - (d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))
/(16*c^2) + (d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/8 + (x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])
)/6 + (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(32*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 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 )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{6 \sqrt {1-c^2 x^2}}\\ &=\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (d \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}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{6 \sqrt {1-c^2 x^2}}\\ &=-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt {1-c^2 x^2}}\\ &=\frac {b d x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 170, normalized size = 0.64 \[ \frac {d \sqrt {d-c^2 d x^2} \left (9 a^2-6 a b c x \sqrt {1-c^2 x^2} \left (8 c^4 x^4-14 c^2 x^2+3\right )+6 b \sin ^{-1}(c x) \left (3 a+b c x \sqrt {1-c^2 x^2} \left (-8 c^4 x^4+14 c^2 x^2-3\right )\right )+b^2 c^2 x^2 \left (8 c^4 x^4-21 c^2 x^2+9\right )+9 b^2 \sin ^{-1}(c x)^2\right )}{288 b c^3 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Sqrt[d - c^2*d*x^2]*(9*a^2 + b^2*c^2*x^2*(9 - 21*c^2*x^2 + 8*c^4*x^4) - 6*a*b*c*x*Sqrt[1 - c^2*x^2]*(3 - 14
*c^2*x^2 + 8*c^4*x^4) + 6*b*(3*a + b*c*x*Sqrt[1 - c^2*x^2]*(-3 + 14*c^2*x^2 - 8*c^4*x^4))*ArcSin[c*x] + 9*b^2*
ArcSin[c*x]^2))/(288*b*c^3*Sqrt[1 - c^2*x^2])

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c^{2} d x^{4} - a d x^{2} + {\left (b c^{2} d x^{4} - b d x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a*c^2*d*x^4 - a*d*x^2 + (b*c^2*d*x^4 - b*d*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 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)*x^2, x)

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maple [C]  time = 0.35, size = 1725, normalized size = 6.51 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/c/(c^2*x^2-1)*arcsin(c*x)*x^2+1/512*I*b*(-d*(c^2*x^2-1))
^(1/2)*cos(3*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+1/12*I*b*(-d*(c^2*x^2-1))^(1/2)*d*c^3/(c^2*x^
2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^6-1/8*I*b*(-d*(c^2*x^2-1))^(1/2)*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcs
in(c*x)*x^4+1/16*I*b*(-d*(c^2*x^2-1))^(1/2)*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2-1/192*b*(-d*(c^
2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x+1/64*b*(-d*(c^2*x^2-1))^
(1/2)*cos(3*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x+1/192*I*b*(-d*(c^2*x^2-1))^(1/2)*c
os(5*arcsin(c*x))*d/c/(c^2*x^2-1)*arcsin(c*x)*x^2-11/4608*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c^2/
(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-1/12*b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*arcsin(c*x)*x^3-1/288*b*(-d*(c^2*
x^2-1))^(1/2)*d/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+11/4608*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c^3/(
c^2*x^2-1)-1/512*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/c^3/(c^2*x^2-1)-7/288*I*b*(-d*(c^2*x^2-1))^(1/2
)*d/(c^2*x^2-1)*x^3+1/16*a/c^2*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/6*a*x*(-c^2*d*
x^2+d)^(5/2)/c^2/d+1/16*a/c^2*d*x*(-c^2*d*x^2+d)^(1/2)-1/96*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c^
2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x+7/4608*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c^2/(c^2*x
^2-1)*(-c^2*x^2+1)^(1/2)*x-1/96*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c/(c^2*x^2-1)*arcsin(c*x)*x^2-3/
512*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-1/96*I*b*(-d*(c^2*x^2-1
))^(1/2)*d/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)-1/192*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*
d/c^3/(c^2*x^2-1)*arcsin(c*x)-7/4608*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c/(c^2*x^2-1)*x^2+1/64*I*
b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/c^3/(c^2*x^2-1)*arcsin(c*x)+3/512*I*b*(-d*(c^2*x^2-1))^(1/2)*sin
(3*arcsin(c*x))*d/c/(c^2*x^2-1)*x^2-1/12*b*(-d*(c^2*x^2-1))^(1/2)*d*c^4/(c^2*x^2-1)*arcsin(c*x)*x^7+1/6*b*(-d*
(c^2*x^2-1))^(1/2)*d*c^2/(c^2*x^2-1)*arcsin(c*x)*x^5+1/96*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c^3/(c
^2*x^2-1)*arcsin(c*x)-1/32*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-1/72*b*
(-d*(c^2*x^2-1))^(1/2)*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^6+1/48*b*(-d*(c^2*x^2-1))^(1/2)*d*c/(c^2*x^2-1)*
(-c^2*x^2+1)^(1/2)*x^4-11/4608*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c/(c^2*x^2-1)*x^2+1/512*b*(-d*(c^
2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/c/(c^2*x^2-1)*x^2-1/72*I*b*(-d*(c^2*x^2-1))^(1/2)*d*c^4/(c^2*x^2-1)*x^7+1
/36*I*b*(-d*(c^2*x^2-1))^(1/2)*d*c^2/(c^2*x^2-1)*x^5+1/96*I*b*(-d*(c^2*x^2-1))^(1/2)*d/c^2/(c^2*x^2-1)*x+7/460
8*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c^3/(c^2*x^2-1)-3/512*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsi
n(c*x))*d/c^3/(c^2*x^2-1)+1/24*a/c^2*x*(-c^2*d*x^2+d)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

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)^(3/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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