3.3.35 \(\int \frac {x^4 (a+b \text {ArcSin}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) [235]

Optimal. Leaf size=337 \[ \frac {15 b^2 x \left (1-c^2 x^2\right )}{64 c^4 \sqrt {d-c^2 d x^2}}+\frac {b^2 x^3 \left (1-c^2 x^2\right )}{32 c^2 \sqrt {d-c^2 d x^2}}-\frac {15 b^2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{64 c^5 \sqrt {d-c^2 d x^2}}+\frac {3 b x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{8 c^3 \sqrt {d-c^2 d x^2}}+\frac {b x^4 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{4 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{8 b c^5 \sqrt {d-c^2 d x^2}} \]

[Out]

15/64*b^2*x*(-c^2*x^2+1)/c^4/(-c^2*d*x^2+d)^(1/2)+1/32*b^2*x^3*(-c^2*x^2+1)/c^2/(-c^2*d*x^2+d)^(1/2)-15/64*b^2
*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c^5/(-c^2*d*x^2+d)^(1/2)+3/8*b*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3/(-
c^2*d*x^2+d)^(1/2)+1/8*b*x^4*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)+1/8*(a+b*arcsin(c*x))
^3*(-c^2*x^2+1)^(1/2)/b/c^5/(-c^2*d*x^2+d)^(1/2)-3/8*x*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^4/d-1/4*x^3*
(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2/d

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Rubi [A]
time = 0.30, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {4795, 4737, 4723, 327, 222} \begin {gather*} \frac {b x^4 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{8 c \sqrt {d-c^2 d x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{4 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{8 b c^5 \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{8 c^4 d}+\frac {3 b x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {15 b^2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{64 c^5 \sqrt {d-c^2 d x^2}}+\frac {b^2 x^3 \left (1-c^2 x^2\right )}{32 c^2 \sqrt {d-c^2 d x^2}}+\frac {15 b^2 x \left (1-c^2 x^2\right )}{64 c^4 \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(15*b^2*x*(1 - c^2*x^2))/(64*c^4*Sqrt[d - c^2*d*x^2]) + (b^2*x^3*(1 - c^2*x^2))/(32*c^2*Sqrt[d - c^2*d*x^2]) -
(15*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(64*c^5*Sqrt[d - c^2*d*x^2]) + (3*b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSi
n[c*x]))/(8*c^3*Sqrt[d - c^2*d*x^2]) + (b*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c*Sqrt[d - c^2*d*x^2])
- (3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8*c^4*d) - (x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)
/(4*c^2*d) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(8*b*c^5*Sqrt[d - c^2*d*x^2])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 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 \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx &=-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d}+\frac {3 \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx}{4 c^2}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 c \sqrt {d-c^2 d x^2}}\\ &=\frac {b x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d}+\frac {3 \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx}{8 c^4}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b \sqrt {1-c^2 x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 c^3 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 x^3 \left (1-c^2 x^2\right )}{32 c^2 \sqrt {d-c^2 d x^2}}+\frac {3 b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^3 \sqrt {d-c^2 d x^2}}+\frac {b x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d}+\frac {\left (3 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{8 c^4 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{32 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 c^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {15 b^2 x \left (1-c^2 x^2\right )}{64 c^4 \sqrt {d-c^2 d x^2}}+\frac {b^2 x^3 \left (1-c^2 x^2\right )}{32 c^2 \sqrt {d-c^2 d x^2}}+\frac {3 b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^3 \sqrt {d-c^2 d x^2}}+\frac {b x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d}+\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^5 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 c^4 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 c^4 \sqrt {d-c^2 d x^2}}\\ &=\frac {15 b^2 x \left (1-c^2 x^2\right )}{64 c^4 \sqrt {d-c^2 d x^2}}+\frac {b^2 x^3 \left (1-c^2 x^2\right )}{32 c^2 \sqrt {d-c^2 d x^2}}-\frac {15 b^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{64 c^5 \sqrt {d-c^2 d x^2}}+\frac {3 b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^3 \sqrt {d-c^2 d x^2}}+\frac {b x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d}+\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^5 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.92, size = 283, normalized size = 0.84 \begin {gather*} \frac {32 a^2 c \sqrt {d} x \left (-1+c^2 x^2\right ) \left (3+2 c^2 x^2\right )-96 a^2 \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+b^2 \sqrt {d} \sqrt {1-c^2 x^2} \left (32 \text {ArcSin}(c x)^3+4 \text {ArcSin}(c x) (-16 \cos (2 \text {ArcSin}(c x))+\cos (4 \text {ArcSin}(c x)))+32 \sin (2 \text {ArcSin}(c x))-\sin (4 \text {ArcSin}(c x))+8 \text {ArcSin}(c x)^2 (-8 \sin (2 \text {ArcSin}(c x))+\sin (4 \text {ArcSin}(c x)))\right )-4 a b \sqrt {d} \sqrt {1-c^2 x^2} (16 \cos (2 \text {ArcSin}(c x))-\cos (4 \text {ArcSin}(c x))-4 \text {ArcSin}(c x) (6 \text {ArcSin}(c x)-8 \sin (2 \text {ArcSin}(c x))+\sin (4 \text {ArcSin}(c x))))}{256 c^5 \sqrt {d} \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(32*a^2*c*Sqrt[d]*x*(-1 + c^2*x^2)*(3 + 2*c^2*x^2) - 96*a^2*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2
])/(Sqrt[d]*(-1 + c^2*x^2))] + b^2*Sqrt[d]*Sqrt[1 - c^2*x^2]*(32*ArcSin[c*x]^3 + 4*ArcSin[c*x]*(-16*Cos[2*ArcS
in[c*x]] + Cos[4*ArcSin[c*x]]) + 32*Sin[2*ArcSin[c*x]] - Sin[4*ArcSin[c*x]] + 8*ArcSin[c*x]^2*(-8*Sin[2*ArcSin
[c*x]] + Sin[4*ArcSin[c*x]])) - 4*a*b*Sqrt[d]*Sqrt[1 - c^2*x^2]*(16*Cos[2*ArcSin[c*x]] - Cos[4*ArcSin[c*x]] -
4*ArcSin[c*x]*(6*ArcSin[c*x] - 8*Sin[2*ArcSin[c*x]] + Sin[4*ArcSin[c*x]])))/(256*c^5*Sqrt[d]*Sqrt[d - c^2*d*x^
2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(721\) vs. \(2(297)=594\).
time = 0.60, size = 722, normalized size = 2.14

method	result	size
default	\(-\frac {a^{2} x^{3} \sqrt {-c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{4} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{8 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{8 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arcsin \left (c x \right )^{2}-1\right ) x}{16 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (5 \arcsin \left (c x \right )\right )}{128 c^{5} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (5 \arcsin \left (c x \right )\right )}{512 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{128 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (56 \arcsin \left (c x \right )^{2}-31\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{512 c^{5} d \left (c^{2} x^{2}-1\right )}\right )+2 a 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}}{16 c^{5} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}}{16 c^{5} \sqrt {-d \left (c^{2} x^{2}-1\right )}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (5 \arcsin \left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (5 \arcsin \left (c x \right )\right )}{64 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (3 \arcsin \left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {7 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{64 c^{5} d \left (c^{2} x^{2}-1\right )}\right )\)	\(722\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/4*a^2*x^3/c^2/d*(-c^2*d*x^2+d)^(1/2)-3/8*a^2/c^4*x/d*(-c^2*d*x^2+d)^(1/2)+3/8*a^2/c^4/(c^2*d)^(1/2)*arctan(
(c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(-1/8*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/d/(c^2*x^2-1)*ar
csin(c*x)^3+1/8*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/d/(c^2*x^2-1)*arcsin(c*x)+1/16*(-d*(c^2*x^2-1))^
(1/2)/c^4/d/(c^2*x^2-1)*(2*arcsin(c*x)^2-1)*x-1/128*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*arcsin(c*x)*cos(5
*arcsin(c*x))-1/512*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*(8*arcsin(c*x)^2-1)*sin(5*arcsin(c*x))+15/128*(-d
*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*arcsin(c*x)*cos(3*arcsin(c*x))+1/512*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x
^2-1)*(56*arcsin(c*x)^2-31)*sin(3*arcsin(c*x)))+2*a*b*(-3/16*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/d/(
c^2*x^2-1)*arcsin(c*x)^2-1/16/c^5/(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)+1/8*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(
c^2*x^2-1)*arcsin(c*x)*x-1/256*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*cos(5*arcsin(c*x))-1/64*(-d*(c^2*x^2-1
))^(1/2)/c^5/d/(c^2*x^2-1)*arcsin(c*x)*sin(5*arcsin(c*x))+15/256*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*cos(
3*arcsin(c*x))+7/64*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*arcsin(c*x)*sin(3*arcsin(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^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

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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*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(x**4*(a + b*asin(c*x))**2/sqrt(-d*(c*x - 1)*(c*x + 1)), x)

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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*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^2*x^4/sqrt(-c^2*d*x^2 + d), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(1/2),x)

[Out]

int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(1/2), x)

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