3.2.57 \(\int x^3 (d-c^2 d x^2) (a+b \text {ArcSin}(c x))^2 \, dx\) [157]

Optimal. Leaf size=202 \[ -\frac {b^2 d x^2}{24 c^2}-\frac {1}{72} b^2 d x^4+\frac {1}{108} b^2 c^2 d x^6+\frac {b d x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{12 c^3}+\frac {b d x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{18 c}-\frac {1}{18} b c d x^5 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {d (a+b \text {ArcSin}(c x))^2}{24 c^4}+\frac {1}{12} d x^4 (a+b \text {ArcSin}(c x))^2+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2 \]

[Out]

-1/24*b^2*d*x^2/c^2-1/72*b^2*d*x^4+1/108*b^2*c^2*d*x^6-1/24*d*(a+b*arcsin(c*x))^2/c^4+1/12*d*x^4*(a+b*arcsin(c
*x))^2+1/6*d*x^4*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2+1/12*b*d*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+1/18*b*d
*x^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c-1/18*b*c*d*x^5*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.38, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4787, 4723, 4795, 4737, 30, 4783} \begin {gather*} -\frac {d (a+b \text {ArcSin}(c x))^2}{24 c^4}-\frac {1}{18} b c d x^5 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2+\frac {b d x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{18 c}+\frac {b d x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{12 c^3}+\frac {1}{12} d x^4 (a+b \text {ArcSin}(c x))^2+\frac {1}{108} b^2 c^2 d x^6-\frac {b^2 d x^2}{24 c^2}-\frac {1}{72} b^2 d x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*(b^2*d*x^2)/c^2 - (b^2*d*x^4)/72 + (b^2*c^2*d*x^6)/108 + (b*d*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(
12*c^3) + (b*d*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(18*c) - (b*c*d*x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[
c*x]))/18 - (d*(a + b*ArcSin[c*x])^2)/(24*c^4) + (d*x^4*(a + b*ArcSin[c*x])^2)/12 + (d*x^4*(1 - c^2*x^2)*(a +
b*ArcSin[c*x])^2)/6

Rule 30

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

Rule 4723

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

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^3 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} d \int x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{3} (b c d) \int x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {1}{18} b c d x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{12} d x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{18} (b c d) \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{6} (b c d) \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{18} \left (b^2 c^2 d\right ) \int x^5 \, dx\\ &=\frac {1}{108} b^2 c^2 d x^6+\frac {b d x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {1}{18} b c d x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{12} d x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{72} \left (b^2 d\right ) \int x^3 \, dx-\frac {1}{24} \left (b^2 d\right ) \int x^3 \, dx-\frac {(b d) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{24 c}-\frac {(b d) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{8 c}\\ &=-\frac {1}{72} b^2 d x^4+\frac {1}{108} b^2 c^2 d x^6+\frac {b d x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 c^3}+\frac {b d x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {1}{18} b c d x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{12} d x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {(b d) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{48 c^3}-\frac {(b d) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 c^3}-\frac {\left (b^2 d\right ) \int x \, dx}{48 c^2}-\frac {\left (b^2 d\right ) \int x \, dx}{16 c^2}\\ &=-\frac {b^2 d x^2}{24 c^2}-\frac {1}{72} b^2 d x^4+\frac {1}{108} b^2 c^2 d x^6+\frac {b d x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 c^3}+\frac {b d x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {1}{18} b c d x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {d \left (a+b \sin ^{-1}(c x)\right )^2}{24 c^4}+\frac {1}{12} d x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 192, normalized size = 0.95 \begin {gather*} -\frac {d \left (b^2 c^2 x^2 \left (9+3 c^2 x^2-2 c^4 x^4\right )+6 a b c x \sqrt {1-c^2 x^2} \left (-3-2 c^2 x^2+2 c^4 x^4\right )+9 a^2 \left (1-6 c^4 x^4+4 c^6 x^6\right )+6 b \left (b c x \sqrt {1-c^2 x^2} \left (-3-2 c^2 x^2+2 c^4 x^4\right )+3 a \left (1-6 c^4 x^4+4 c^6 x^6\right )\right ) \text {ArcSin}(c x)+9 b^2 \left (1-6 c^4 x^4+4 c^6 x^6\right ) \text {ArcSin}(c x)^2\right )}{216 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/216*(d*(b^2*c^2*x^2*(9 + 3*c^2*x^2 - 2*c^4*x^4) + 6*a*b*c*x*Sqrt[1 - c^2*x^2]*(-3 - 2*c^2*x^2 + 2*c^4*x^4)
+ 9*a^2*(1 - 6*c^4*x^4 + 4*c^6*x^6) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(-3 - 2*c^2*x^2 + 2*c^4*x^4) + 3*a*(1 - 6*c
^4*x^4 + 4*c^6*x^6))*ArcSin[c*x] + 9*b^2*(1 - 6*c^4*x^4 + 4*c^6*x^6)*ArcSin[c*x]^2))/c^4

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Maple [A]
time = 0.06, size = 320, normalized size = 1.58

method	result	size
derivativedivides	\(\frac {-d \,a^{2} \left (\frac {1}{6} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\arcsin \left (c x \right ) \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{16}-\frac {\arcsin \left (c x \right )^{2}}{24}+\frac {\left (2 c^{2} x^{2}+3\right )^{2}}{128}+\frac {\arcsin \left (c x \right )^{2} c^{6} x^{6}}{6}-\frac {\arcsin \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-10 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-15 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{144}-\frac {c^{6} x^{6}}{108}-\frac {5 c^{4} x^{4}}{288}-\frac {5 c^{2} x^{2}}{96}\right )-2 d a b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{24}+\frac {\arcsin \left (c x \right )}{24}\right )}{c^{4}}\)	\(320\)
default	\(\frac {-d \,a^{2} \left (\frac {1}{6} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\arcsin \left (c x \right ) \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{16}-\frac {\arcsin \left (c x \right )^{2}}{24}+\frac {\left (2 c^{2} x^{2}+3\right )^{2}}{128}+\frac {\arcsin \left (c x \right )^{2} c^{6} x^{6}}{6}-\frac {\arcsin \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-10 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-15 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{144}-\frac {c^{6} x^{6}}{108}-\frac {5 c^{4} x^{4}}{288}-\frac {5 c^{2} x^{2}}{96}\right )-2 d a b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{24}+\frac {\arcsin \left (c x \right )}{24}\right )}{c^{4}}\)	\(320\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(-d*a^2*(1/6*c^6*x^6-1/4*c^4*x^4)-d*b^2*(-1/4*arcsin(c*x)^2*c^4*x^4+1/16*arcsin(c*x)*(-2*c^3*x^3*(-c^2*x
^2+1)^(1/2)-3*c*x*(-c^2*x^2+1)^(1/2)+3*arcsin(c*x))-1/24*arcsin(c*x)^2+1/128*(2*c^2*x^2+3)^2+1/6*arcsin(c*x)^2
*c^6*x^6-1/144*arcsin(c*x)*(-8*c^5*x^5*(-c^2*x^2+1)^(1/2)-10*c^3*x^3*(-c^2*x^2+1)^(1/2)-15*c*x*(-c^2*x^2+1)^(1
/2)+15*arcsin(c*x))-1/108*c^6*x^6-5/288*c^4*x^4-5/96*c^2*x^2)-2*d*a*b*(1/6*arcsin(c*x)*c^6*x^6-1/4*c^4*x^4*arc
sin(c*x)+1/36*c^5*x^5*(-c^2*x^2+1)^(1/2)-1/36*c^3*x^3*(-c^2*x^2+1)^(1/2)-1/24*c*x*(-c^2*x^2+1)^(1/2)+1/24*arcs
in(c*x)))

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

[Out]

-1/6*a^2*c^2*d*x^6 + 1/4*a^2*d*x^4 - 1/144*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*
x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*a*b*c^2*d + 1/16*(8*x^4*arcsin(c*x) +
(2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*a*b*d - 1/12*(2*b^2*c^2*d*x
^6 - 3*b^2*d*x^4)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 - integrate(1/6*(2*b^2*c^3*d*x^6 - 3*b^2*c*d*x^
4)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*x^2 - 1), x)

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Fricas [A]
time = 2.28, size = 211, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (18 \, a^{2} - b^{2}\right )} c^{6} d x^{6} - 3 \, {\left (18 \, a^{2} - b^{2}\right )} c^{4} d x^{4} + 9 \, b^{2} c^{2} d x^{2} + 9 \, {\left (4 \, b^{2} c^{6} d x^{6} - 6 \, b^{2} c^{4} d x^{4} + b^{2} d\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (4 \, a b c^{6} d x^{6} - 6 \, a b c^{4} d x^{4} + a b d\right )} \arcsin \left (c x\right ) + 6 \, {\left (2 \, a b c^{5} d x^{5} - 2 \, a b c^{3} d x^{3} - 3 \, a b c d x + {\left (2 \, b^{2} c^{5} d x^{5} - 2 \, b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{216 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/216*(2*(18*a^2 - b^2)*c^6*d*x^6 - 3*(18*a^2 - b^2)*c^4*d*x^4 + 9*b^2*c^2*d*x^2 + 9*(4*b^2*c^6*d*x^6 - 6*b^2
*c^4*d*x^4 + b^2*d)*arcsin(c*x)^2 + 18*(4*a*b*c^6*d*x^6 - 6*a*b*c^4*d*x^4 + a*b*d)*arcsin(c*x) + 6*(2*a*b*c^5*
d*x^5 - 2*a*b*c^3*d*x^3 - 3*a*b*c*d*x + (2*b^2*c^5*d*x^5 - 2*b^2*c^3*d*x^3 - 3*b^2*c*d*x)*arcsin(c*x))*sqrt(-c
^2*x^2 + 1))/c^4

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Sympy [A]
time = 0.84, size = 332, normalized size = 1.64 \begin {gather*} \begin {cases} - \frac {a^{2} c^{2} d x^{6}}{6} + \frac {a^{2} d x^{4}}{4} - \frac {a b c^{2} d x^{6} \operatorname {asin}{\left (c x \right )}}{3} - \frac {a b c d x^{5} \sqrt {- c^{2} x^{2} + 1}}{18} + \frac {a b d x^{4} \operatorname {asin}{\left (c x \right )}}{2} + \frac {a b d x^{3} \sqrt {- c^{2} x^{2} + 1}}{18 c} + \frac {a b d x \sqrt {- c^{2} x^{2} + 1}}{12 c^{3}} - \frac {a b d \operatorname {asin}{\left (c x \right )}}{12 c^{4}} - \frac {b^{2} c^{2} d x^{6} \operatorname {asin}^{2}{\left (c x \right )}}{6} + \frac {b^{2} c^{2} d x^{6}}{108} - \frac {b^{2} c d x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{18} + \frac {b^{2} d x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{4} - \frac {b^{2} d x^{4}}{72} + \frac {b^{2} d x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{18 c} - \frac {b^{2} d x^{2}}{24 c^{2}} + \frac {b^{2} d x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{12 c^{3}} - \frac {b^{2} d \operatorname {asin}^{2}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{4}}{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*c**2*d*x**6/6 + a**2*d*x**4/4 - a*b*c**2*d*x**6*asin(c*x)/3 - a*b*c*d*x**5*sqrt(-c**2*x**2 +
1)/18 + a*b*d*x**4*asin(c*x)/2 + a*b*d*x**3*sqrt(-c**2*x**2 + 1)/(18*c) + a*b*d*x*sqrt(-c**2*x**2 + 1)/(12*c**
3) - a*b*d*asin(c*x)/(12*c**4) - b**2*c**2*d*x**6*asin(c*x)**2/6 + b**2*c**2*d*x**6/108 - b**2*c*d*x**5*sqrt(-
c**2*x**2 + 1)*asin(c*x)/18 + b**2*d*x**4*asin(c*x)**2/4 - b**2*d*x**4/72 + b**2*d*x**3*sqrt(-c**2*x**2 + 1)*a
sin(c*x)/(18*c) - b**2*d*x**2/(24*c**2) + b**2*d*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(12*c**3) - b**2*d*asin(c*x)
**2/(24*c**4), Ne(c, 0)), (a**2*d*x**4/4, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (177) = 354\).
time = 0.44, size = 377, normalized size = 1.87 \begin {gather*} -\frac {1}{6} \, a^{2} c^{2} d x^{6} + \frac {1}{4} \, a^{2} d x^{4} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d x \arcsin \left (c x\right )}{18 \, c^{3}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d \arcsin \left (c x\right )^{2}}{6 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d x}{18 \, c^{3}} + \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d x \arcsin \left (c x\right )}{18 \, c^{3}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} a b d \arcsin \left (c x\right )}{3 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d \arcsin \left (c x\right )^{2}}{4 \, c^{4}} + \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d x}{18 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} d x \arcsin \left (c x\right )}{12 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d}{108 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} a b d \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} a b d x}{12 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d}{72 \, c^{4}} + \frac {b^{2} d \arcsin \left (c x\right )^{2}}{24 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d}{24 \, c^{4}} + \frac {a b d \arcsin \left (c x\right )}{12 \, c^{4}} - \frac {5 \, b^{2} d}{216 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*a^2*c^2*d*x^6 + 1/4*a^2*d*x^4 - 1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d*x*arcsin(c*x)/c^3 - 1/6*(c^
2*x^2 - 1)^3*b^2*d*arcsin(c*x)^2/c^4 - 1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d*x/c^3 + 1/18*(-c^2*x^2 +
1)^(3/2)*b^2*d*x*arcsin(c*x)/c^3 - 1/3*(c^2*x^2 - 1)^3*a*b*d*arcsin(c*x)/c^4 - 1/4*(c^2*x^2 - 1)^2*b^2*d*arcsi
n(c*x)^2/c^4 + 1/18*(-c^2*x^2 + 1)^(3/2)*a*b*d*x/c^3 + 1/12*sqrt(-c^2*x^2 + 1)*b^2*d*x*arcsin(c*x)/c^3 + 1/108
*(c^2*x^2 - 1)^3*b^2*d/c^4 - 1/2*(c^2*x^2 - 1)^2*a*b*d*arcsin(c*x)/c^4 + 1/12*sqrt(-c^2*x^2 + 1)*a*b*d*x/c^3 +
1/72*(c^2*x^2 - 1)^2*b^2*d/c^4 + 1/24*b^2*d*arcsin(c*x)^2/c^4 - 1/24*(c^2*x^2 - 1)*b^2*d/c^4 + 1/12*a*b*d*arc
sin(c*x)/c^4 - 5/216*b^2*d/c^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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