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

Optimal. Leaf size=185 \[ -i b d e \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+2 d e \log (x) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \sqrt{1-c^2 x^2}}{2 x}+\frac{b e^2 x \sqrt{1-c^2 x^2}}{4 c}-\frac{b e^2 \sin ^{-1}(c x)}{4 c^2}-i b d e \sin ^{-1}(c x)^2+2 b d e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-2 b d e \log (x) \sin ^{-1}(c x) \]

[Out]

-(b*c*d^2*Sqrt[1 - c^2*x^2])/(2*x) + (b*e^2*x*Sqrt[1 - c^2*x^2])/(4*c) - (b*e^2*ArcSin[c*x])/(4*c^2) - I*b*d*e
*ArcSin[c*x]^2 - (d^2*(a + b*ArcSin[c*x]))/(2*x^2) + (e^2*x^2*(a + b*ArcSin[c*x]))/2 + 2*b*d*e*ArcSin[c*x]*Log
[1 - E^((2*I)*ArcSin[c*x])] - 2*b*d*e*ArcSin[c*x]*Log[x] + 2*d*e*(a + b*ArcSin[c*x])*Log[x] - I*b*d*e*PolyLog[
2, E^((2*I)*ArcSin[c*x])]

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Rubi [A]  time = 0.338032, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 14, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {266, 43, 4731, 12, 6742, 264, 321, 216, 2326, 4625, 3717, 2190, 2279, 2391} \[ -i b d e \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+2 d e \log (x) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \sqrt{1-c^2 x^2}}{2 x}+\frac{b e^2 x \sqrt{1-c^2 x^2}}{4 c}-\frac{b e^2 \sin ^{-1}(c x)}{4 c^2}-i b d e \sin ^{-1}(c x)^2+2 b d e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-2 b d e \log (x) \sin ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*d^2*Sqrt[1 - c^2*x^2])/(2*x) + (b*e^2*x*Sqrt[1 - c^2*x^2])/(4*c) - (b*e^2*ArcSin[c*x])/(4*c^2) - I*b*d*e
*ArcSin[c*x]^2 - (d^2*(a + b*ArcSin[c*x]))/(2*x^2) + (e^2*x^2*(a + b*ArcSin[c*x]))/2 + 2*b*d*e*ArcSin[c*x]*Log
[1 - E^((2*I)*ArcSin[c*x])] - 2*b*d*e*ArcSin[c*x]*Log[x] + 2*d*e*(a + b*ArcSin[c*x])*Log[x] - I*b*d*e*PolyLog[
2, E^((2*I)*ArcSin[c*x])]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] ||
(IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 264

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

Rule 321

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 216

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 2326

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(ArcSin[(Rt[-e, 2]*x)/S
qrt[d]]*(a + b*Log[c*x^n]))/Rt[-e, 2], x] - Dist[(b*n)/Rt[-e, 2], Int[ArcSin[(Rt[-e, 2]*x)/Sqrt[d]]/x, x], x]
/; FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && NegQ[e]

Rule 4625

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(b c) \int \frac{-\frac{d^2}{x^2}+e^2 x^2+4 d e \log (x)}{2 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac{1}{2} (b c) \int \frac{-\frac{d^2}{x^2}+e^2 x^2+4 d e \log (x)}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac{1}{2} (b c) \int \left (-\frac{d^2}{x^2 \sqrt{1-c^2 x^2}}+\frac{e^2 x^2}{\sqrt{1-c^2 x^2}}+\frac{4 d e \log (x)}{\sqrt{1-c^2 x^2}}\right ) \, dx\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\frac{1}{2} \left (b c d^2\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^2}} \, dx-(2 b c d e) \int \frac{\log (x)}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{2} \left (b c e^2\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{b c d^2 \sqrt{1-c^2 x^2}}{2 x}+\frac{b e^2 x \sqrt{1-c^2 x^2}}{4 c}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )-2 b d e \sin ^{-1}(c x) \log (x)+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+(2 b d e) \int \frac{\sin ^{-1}(c x)}{x} \, dx-\frac{\left (b e^2\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{4 c}\\ &=-\frac{b c d^2 \sqrt{1-c^2 x^2}}{2 x}+\frac{b e^2 x \sqrt{1-c^2 x^2}}{4 c}-\frac{b e^2 \sin ^{-1}(c x)}{4 c^2}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )-2 b d e \sin ^{-1}(c x) \log (x)+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+(2 b d e) \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{b c d^2 \sqrt{1-c^2 x^2}}{2 x}+\frac{b e^2 x \sqrt{1-c^2 x^2}}{4 c}-\frac{b e^2 \sin ^{-1}(c x)}{4 c^2}-i b d e \sin ^{-1}(c x)^2-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )-2 b d e \sin ^{-1}(c x) \log (x)+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(4 i b d e) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{b c d^2 \sqrt{1-c^2 x^2}}{2 x}+\frac{b e^2 x \sqrt{1-c^2 x^2}}{4 c}-\frac{b e^2 \sin ^{-1}(c x)}{4 c^2}-i b d e \sin ^{-1}(c x)^2-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 b d e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-2 b d e \sin ^{-1}(c x) \log (x)+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(2 b d e) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{b c d^2 \sqrt{1-c^2 x^2}}{2 x}+\frac{b e^2 x \sqrt{1-c^2 x^2}}{4 c}-\frac{b e^2 \sin ^{-1}(c x)}{4 c^2}-i b d e \sin ^{-1}(c x)^2-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 b d e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-2 b d e \sin ^{-1}(c x) \log (x)+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+(i b d e) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac{b c d^2 \sqrt{1-c^2 x^2}}{2 x}+\frac{b e^2 x \sqrt{1-c^2 x^2}}{4 c}-\frac{b e^2 \sin ^{-1}(c x)}{4 c^2}-i b d e \sin ^{-1}(c x)^2-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 b d e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-2 b d e \sin ^{-1}(c x) \log (x)+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-i b d e \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.333032, size = 159, normalized size = 0.86 \[ \frac{1}{4} \left (-4 i b d e \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )-\frac{2 a d^2}{x^2}+8 a d e \log (x)+2 a e^2 x^2+b \sin ^{-1}(c x) \left (-\frac{e^2}{c^2}+8 d e \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac{2 d^2}{x^2}+2 e^2 x^2\right )-\frac{2 b c d^2 \sqrt{1-c^2 x^2}}{x}+\frac{b e^2 x \sqrt{1-c^2 x^2}}{c}-4 i b d e \sin ^{-1}(c x)^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a*d^2)/x^2 + 2*a*e^2*x^2 - (2*b*c*d^2*Sqrt[1 - c^2*x^2])/x + (b*e^2*x*Sqrt[1 - c^2*x^2])/c - (4*I)*b*d*e*
ArcSin[c*x]^2 + b*ArcSin[c*x]*(-(e^2/c^2) - (2*d^2)/x^2 + 2*e^2*x^2 + 8*d*e*Log[1 - E^((2*I)*ArcSin[c*x])]) +
8*a*d*e*Log[x] - (4*I)*b*d*e*PolyLog[2, E^((2*I)*ArcSin[c*x])])/4

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Maple [A]  time = 0.371, size = 248, normalized size = 1.3 \begin{align*}{\frac{a{x}^{2}{e}^{2}}{2}}-{\frac{a{d}^{2}}{2\,{x}^{2}}}+2\,aed\ln \left ( cx \right ) -ibde \left ( \arcsin \left ( cx \right ) \right ) ^{2}+{\frac{b{e}^{2}x}{4\,c}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b\arcsin \left ( cx \right ){x}^{2}{e}^{2}}{2}}-{\frac{b{e}^{2}\arcsin \left ( cx \right ) }{4\,{c}^{2}}}+{\frac{i}{2}}{c}^{2}b{d}^{2}-{\frac{bc{d}^{2}}{2\,x}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b{d}^{2}\arcsin \left ( cx \right ) }{2\,{x}^{2}}}+2\,bed\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +2\,bed\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -2\,ibed{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -2\,ibed{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2*e^2-1/2*a*d^2/x^2+2*a*e*d*ln(c*x)-I*b*d*e*arcsin(c*x)^2+1/4*b*e^2*x*(-c^2*x^2+1)^(1/2)/c+1/2*b*arcsi
n(c*x)*x^2*e^2-1/4*b*e^2*arcsin(c*x)/c^2+1/2*I*c^2*b*d^2-1/2*b*c*d^2*(-c^2*x^2+1)^(1/2)/x-1/2*b*arcsin(c*x)*d^
2/x^2+2*b*e*d*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+2*b*e*d*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*
I*b*e*d*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*b*e*d*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a e^{2} x^{2} - \frac{1}{2} \, b d^{2}{\left (\frac{\sqrt{-c^{2} x^{2} + 1} c}{x} + \frac{\arcsin \left (c x\right )}{x^{2}}\right )} + 2 \, a d e \log \left (x\right ) - \frac{a d^{2}}{2 \, x^{2}} + \int \frac{{\left (b e^{2} x^{2} + 2 \, b d e\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*e^2*x^2 - 1/2*b*d^2*(sqrt(-c^2*x^2 + 1)*c/x + arcsin(c*x)/x^2) + 2*a*d*e*log(x) - 1/2*a*d^2/x^2 + integr
ate((b*e^2*x^2 + 2*b*d*e)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} +{\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \arcsin \left (c x\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*e^2*x^4 + 2*a*d*e*x^2 + a*d^2 + (b*e^2*x^4 + 2*b*d*e*x^2 + b*d^2)*arcsin(c*x))/x^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{2}{\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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