3.1.68 \(\int x^4 (d-c^2 d x^2)^{3/2} (a+b \text {ArcSin}(c x)) \, dx\) [68]

Optimal. Leaf size=340 \[ \frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {3 d x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{128 c^4}-\frac {d x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))+\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{256 b c^5 \sqrt {1-c^2 x^2}} \]

[Out]

1/8*x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))-3/128*d*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4-1/64*d*x^3
*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/16*d*x^5*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+3/256*b*d*x^2*(-
c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/256*b*d*x^4*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-1/32*b*c*d*x
^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/64*b*c^3*d*x^8*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+3/256*d*(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 = 0.28, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4787, 4783, 4795, 4737, 30, 14} \begin {gather*} \frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))-\frac {d x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{64 c^2}+\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{256 b c^5 \sqrt {1-c^2 x^2}}-\frac {3 d x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{128 c^4}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}+\frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]

[Out]

(3*b*d*x^2*Sqrt[d - c^2*d*x^2])/(256*c^3*Sqrt[1 - c^2*x^2]) + (b*d*x^4*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^
2*x^2]) - (b*c*d*x^6*Sqrt[d - c^2*d*x^2])/(32*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[
1 - c^2*x^2]) - (3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c^4) - (d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*
ArcSin[c*x]))/(64*c^2) + (d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/16 + (x^5*(d - c^2*d*x^2)^(3/2)*(a +
b*ArcSin[c*x]))/8 + (3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(256*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 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 )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} (3 d) \int x^4 \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^5 \left (1-c^2 x^2\right ) \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} x^5 \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^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (x^5-c^2 x^7\right ) \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (3 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}{64 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{64 c \sqrt {1-c^2 x^2}}\\ &=\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {3 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^4}-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (3 d \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^4 \sqrt {1-c^2 x^2}}+\frac {\left (3 b d \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{128 c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {3 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^4}-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{256 b c^5 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 193, normalized size = 0.57 \begin {gather*} \frac {d \sqrt {d-c^2 d x^2} \left (3 a^2+b^2 c^2 x^2 \left (3+c^2 x^2-8 c^4 x^4+4 c^6 x^6\right )-2 a b c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right )-2 b \left (-3 a+b c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right )\right ) \text {ArcSin}(c x)+3 b^2 \text {ArcSin}(c x)^2\right )}{256 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)^(3/2)*(a + b*ArcSin[c*x]),x]

[Out]

(d*Sqrt[d - c^2*d*x^2]*(3*a^2 + b^2*c^2*x^2*(3 + c^2*x^2 - 8*c^4*x^4 + 4*c^6*x^6) - 2*a*b*c*x*Sqrt[1 - c^2*x^2
]*(3 + 2*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6) - 2*b*(-3*a + b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2 - 24*c^4*x^4
+ 16*c^6*x^6))*ArcSin[c*x] + 3*b^2*ArcSin[c*x]^2))/(256*b*c^5*Sqrt[1 - c^2*x^2])

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Maple [C] Result contains complex when optimal does not.
time = 0.47, size = 770, normalized size = 2.26

method	result	size
default	\(-\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8 c^{2} d}-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{16 c^{4} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{64 c^{4}}+\frac {3 a d x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{4}}+\frac {3 a \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 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}{256 c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-128 i \sqrt {-c^{2} x^{2}+1}\, x^{8} c^{8}+128 c^{9} x^{9}+256 i \sqrt {-c^{2} x^{2}+1}\, x^{6} c^{6}-320 c^{7} x^{7}-160 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+272 c^{5} x^{5}+32 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-88 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+8 c x \right ) \left (i+8 \arcsin \left (c x \right )\right ) d}{16384 c^{5} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (i+4 \arcsin \left (c x \right )\right ) d}{1024 c^{5} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-i+4 \arcsin \left (c x \right )\right ) d}{1024 c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (128 i \sqrt {-c^{2} x^{2}+1}\, x^{8} c^{8}+128 c^{9} x^{9}-256 i \sqrt {-c^{2} x^{2}+1}\, x^{6} c^{6}-320 c^{7} x^{7}+160 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+272 c^{5} x^{5}-32 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-88 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+8 c x \right ) \left (-i+8 \arcsin \left (c x \right )\right ) d}{16384 c^{5} \left (c^{2} x^{2}-1\right )}\right )\)	\(770\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)

[Out]

-1/8*a*x^3*(-c^2*d*x^2+d)^(5/2)/c^2/d-1/16*a/c^4*x*(-c^2*d*x^2+d)^(5/2)/d+1/64*a/c^4*x*(-c^2*d*x^2+d)^(3/2)+3/
128*a/c^4*d*x*(-c^2*d*x^2+d)^(1/2)+3/128*a/c^4*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+
b*(-3/256*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/(c^2*x^2-1)*arcsin(c*x)^2*d-1/16384*(-d*(c^2*x^2-1))^(
1/2)*(-128*I*(-c^2*x^2+1)^(1/2)*x^8*c^8+128*c^9*x^9+256*I*(-c^2*x^2+1)^(1/2)*x^6*c^6-320*c^7*x^7-160*I*(-c^2*x
^2+1)^(1/2)*x^4*c^4+272*c^5*x^5+32*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-88*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+8*c*x)*(I+8*ar
csin(c*x))*d/c^5/(c^2*x^2-1)+1/1024*(-d*(c^2*x^2-1))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c^
2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*(I+4*arcsin(c*x))*d/c^5/(c^2*x^2-1)+1/1024*(-d*(
c^2*x^2-1))^(1/2)*(8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5-8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3+I*(-c^2*
x^2+1)^(1/2)+4*c*x)*(-I+4*arcsin(c*x))*d/c^5/(c^2*x^2-1)-1/16384*(-d*(c^2*x^2-1))^(1/2)*(128*I*(-c^2*x^2+1)^(1
/2)*x^8*c^8+128*c^9*x^9-256*I*(-c^2*x^2+1)^(1/2)*x^6*c^6-320*c^7*x^7+160*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+272*c^5*
x^5-32*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-88*c^3*x^3+I*(-c^2*x^2+1)^(1/2)+8*c*x)*(-I+8*arcsin(c*x))*d/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)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

b*sqrt(d)*integrate(-(c^2*d*x^6 - d*x^4)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1
)), x) - 1/128*(16*(-c^2*d*x^2 + d)^(5/2)*x^3/(c^2*d) - 2*(-c^2*d*x^2 + d)^(3/2)*x/c^4 + 8*(-c^2*d*x^2 + d)^(5
/2)*x/(c^4*d) - 3*sqrt(-c^2*d*x^2 + d)*d*x/c^4 - 3*d^(3/2)*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)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral(-(a*c^2*d*x^6 - a*d*x^4 + (b*c^2*d*x^6 - b*d*x^4)*arcsin(c*x))*sqrt(-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 x^{4} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x)),x)

[Out]

Integral(x**4*(-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x)), 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*(-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^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 )}^{3/2} \,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)^(3/2),x)

[Out]

int(x^4*(a + b*asin(c*x))*(d - c^2*d*x^2)^(3/2), x)

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