3.1.72 \(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {ArcSin}(c x))}{x^4} \, dx\) [72]

Optimal. Leaf size=191 \[ -\frac {b c d \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}+\frac {c^2 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{2 b \sqrt {1-c^2 x^2}}-\frac {4 b c^3 d \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {1-c^2 x^2}} \]

[Out]

-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))/x^3+c^2*d*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/x-1/6*b*c*d*(-c^2
*d*x^2+d)^(1/2)/x^2/(-c^2*x^2+1)^(1/2)+1/2*c^3*d*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/(-c^2*x^2+1)^(1/2)
-4/3*b*c^3*d*ln(x)*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {4785, 4781, 29, 4737, 14} \begin {gather*} \frac {c^2 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{2 b \sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {4 b c^3 d \log (x) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(b*c*d*Sqrt[d - c^2*d*x^2])/(x^2*Sqrt[1 - c^2*x^2]) + (c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x -
((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*x^3) + (c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*
Sqrt[1 - c^2*x^2]) - (4*b*c^3*d*Sqrt[d - c^2*d*x^2]*Log[x])/(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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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 4781

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 + 1))), x] + (-Dist[b*c*(n/(f*(m + 1)))*Simp[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] + Dist[(c^2/(f^2*(m + 1)))*S
imp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(f*x)^(m + 2)*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x]) /
; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]

Rule 4785

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 + 1))), x] + (-Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x)^
(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\left (c^2 d\right ) \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {1-c^2 x^2}{x^3} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {1}{x^3}-\frac {c^2}{x}\right ) \, dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (b c^3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{x} \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (c^4 d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b \sqrt {1-c^2 x^2}}-\frac {4 b c^3 d \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.48, size = 211, normalized size = 1.10 \begin {gather*} \frac {b d \left (-1+4 c^2 x^2\right ) \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{3 x^3}+\frac {b c^3 d \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)^2}{2 \sqrt {1-c^2 x^2}}-a c^3 d^{3/2} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-\frac {d \sqrt {d-c^2 d x^2} \left (b c x+2 a \left (1-4 c^2 x^2\right ) \sqrt {1-c^2 x^2}+8 b c^3 x^3 \log (c x)\right )}{6 x^3 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d*(-1 + 4*c^2*x^2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(3*x^3) + (b*c^3*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/(
2*Sqrt[1 - c^2*x^2]) - a*c^3*d^(3/2)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - (d*Sqrt[d -
c^2*d*x^2]*(b*c*x + 2*a*(1 - 4*c^2*x^2)*Sqrt[1 - c^2*x^2] + 8*b*c^3*x^3*Log[c*x]))/(6*x^3*Sqrt[1 - c^2*x^2])

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Maple [C] Result contains complex when optimal does not.
time = 0.28, size = 1289, normalized size = 6.75

method	result	size
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d \,x^{3}}+\frac {2 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d x}+\frac {2 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+a \,c^{4} d x \sqrt {-c^{2} d \,x^{2}+d}+\frac {a \,c^{4} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d \,c^{3}}{2 \left (c^{2} x^{2}-1\right )}+\frac {4 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{3}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {10 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{3} c^{6}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {32 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{5} \arcsin \left (c x \right ) c^{8}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {8 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{5} c^{8}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {8 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{3} \left (-c^{2} x^{2}+1\right ) c^{6}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d x \,c^{4}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {52 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{3} \arcsin \left (c x \right ) c^{6}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {32 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{4} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{7}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {12 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{5}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{2} \sqrt {-c^{2} x^{2}+1}\, c^{5}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {8 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) d \,c^{3}}{3 c^{2} x^{2}-3}+\frac {73 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d x \arcsin \left (c x \right ) c^{4}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d x \left (-c^{2} x^{2}+1\right ) c^{4}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \sqrt {-c^{2} x^{2}+1}\, c^{3}}{2 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {14 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) c^{2}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) x \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \sqrt {-c^{2} x^{2}+1}\, c}{6 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) x^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right )}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) x^{3} \left (c^{2} x^{2}-1\right )}+\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) d \,c^{3}}{3 \left (c^{2} x^{2}-1\right )}\)	\(1289\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a/d/x^3*(-c^2*d*x^2+d)^(5/2)+2/3*a*c^2/d/x*(-c^2*d*x^2+d)^(5/2)+2/3*a*c^4*x*(-c^2*d*x^2+d)^(3/2)+a*c^4*d*
x*(-c^2*d*x^2+d)^(1/2)+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))-1/2*b*(-d*(c^2*x^2
-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arcsin(c*x)^2*d*c^3+4/3*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c
^2*x^2+1)/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3+10/3*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x
^2+1)*x^3/(c^2*x^2-1)*c^6+32*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c^2*x^2-1)*arcsin(c*x)*c
^8-8/3*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c^2*x^2-1)*c^8-8/3*I*b*(-d*(c^2*x^2-1))^(1/2
)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c^6-2/3*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c
^2*x^2+1)*x/(c^2*x^2-1)*c^4-52*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)
*c^6+32*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^4/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c
^7-12*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^5
+4*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^5-8*I*b*(-d*(c^2*x
^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*d*c^3/(3*c^2*x^2-3)+73/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*
c^2*x^2+1)*x/(c^2*x^2-1)*arcsin(c*x)*c^4+2/3*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x/(c^2*x^2-
1)*(-c^2*x^2+1)*c^4-3/2*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^3
-14/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x/(c^2*x^2-1)*arcsin(c*x)*c^2+1/6*b*(-d*(c^2*x^2-1))
^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c+1/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*
x^4-9*c^2*x^2+1)/x^3/(c^2*x^2-1)*arcsin(c*x)+4/3*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*ln((I
*c*x+(-c^2*x^2+1)^(1/2))^2-1)*d*c^3

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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((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))/x^4,x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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 )}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))/x^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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