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

Optimal. Leaf size=229 \[ -\frac{1}{2} i b d^2 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+d^2 \log (x) \left (a+b \sin ^{-1}(c x)\right )+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{b d e x \sqrt{1-c^2 x^2}}{2 c}-\frac{b d e \sin ^{-1}(c x)}{2 c^2}+\frac{b e^2 x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{3 b e^2 x \sqrt{1-c^2 x^2}}{32 c^3}-\frac{3 b e^2 \sin ^{-1}(c x)}{32 c^4}-\frac{1}{2} i b d^2 \sin ^{-1}(c x)^2+b d^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^2 \log (x) \sin ^{-1}(c x) \]

[Out]

(b*d*e*x*Sqrt[1 - c^2*x^2])/(2*c) + (3*b*e^2*x*Sqrt[1 - c^2*x^2])/(32*c^3) + (b*e^2*x^3*Sqrt[1 - c^2*x^2])/(16
*c) - (b*d*e*ArcSin[c*x])/(2*c^2) - (3*b*e^2*ArcSin[c*x])/(32*c^4) - (I/2)*b*d^2*ArcSin[c*x]^2 + d*e*x^2*(a +
b*ArcSin[c*x]) + (e^2*x^4*(a + b*ArcSin[c*x]))/4 + b*d^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - b*d^2*Ar
cSin[c*x]*Log[x] + d^2*(a + b*ArcSin[c*x])*Log[x] - (I/2)*b*d^2*PolyLog[2, E^((2*I)*ArcSin[c*x])]

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Rubi [A]  time = 0.334752, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {266, 43, 4731, 6742, 321, 216, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac{1}{2} i b d^2 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+d^2 \log (x) \left (a+b \sin ^{-1}(c x)\right )+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{b d e x \sqrt{1-c^2 x^2}}{2 c}-\frac{b d e \sin ^{-1}(c x)}{2 c^2}+\frac{b e^2 x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{3 b e^2 x \sqrt{1-c^2 x^2}}{32 c^3}-\frac{3 b e^2 \sin ^{-1}(c x)}{32 c^4}-\frac{1}{2} i b d^2 \sin ^{-1}(c x)^2+b d^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^2 \log (x) \sin ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d*e*x*Sqrt[1 - c^2*x^2])/(2*c) + (3*b*e^2*x*Sqrt[1 - c^2*x^2])/(32*c^3) + (b*e^2*x^3*Sqrt[1 - c^2*x^2])/(16
*c) - (b*d*e*ArcSin[c*x])/(2*c^2) - (3*b*e^2*ArcSin[c*x])/(32*c^4) - (I/2)*b*d^2*ArcSin[c*x]^2 + d*e*x^2*(a +
b*ArcSin[c*x]) + (e^2*x^4*(a + b*ArcSin[c*x]))/4 + b*d^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - b*d^2*Ar
cSin[c*x]*Log[x] + d^2*(a + b*ArcSin[c*x])*Log[x] - (I/2)*b*d^2*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 6742

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

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} \, dx &=d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(b c) \int \frac{d e x^2+\frac{e^2 x^4}{4}+d^2 \log (x)}{\sqrt{1-c^2 x^2}} \, dx\\ &=d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(b c) \int \left (\frac{d e x^2}{\sqrt{1-c^2 x^2}}+\frac{e^2 x^4}{4 \sqrt{1-c^2 x^2}}+\frac{d^2 \log (x)}{\sqrt{1-c^2 x^2}}\right ) \, dx\\ &=d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (b c d^2\right ) \int \frac{\log (x)}{\sqrt{1-c^2 x^2}} \, dx-(b c d e) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{4} \left (b c e^2\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b d e x \sqrt{1-c^2 x^2}}{2 c}+\frac{b e^2 x^3 \sqrt{1-c^2 x^2}}{16 c}+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-b d^2 \sin ^{-1}(c x) \log (x)+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\left (b d^2\right ) \int \frac{\sin ^{-1}(c x)}{x} \, dx-\frac{(b d e) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{2 c}-\frac{\left (3 b e^2\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{16 c}\\ &=\frac{b d e x \sqrt{1-c^2 x^2}}{2 c}+\frac{3 b e^2 x \sqrt{1-c^2 x^2}}{32 c^3}+\frac{b e^2 x^3 \sqrt{1-c^2 x^2}}{16 c}-\frac{b d e \sin ^{-1}(c x)}{2 c^2}+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-b d^2 \sin ^{-1}(c x) \log (x)+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\left (b d^2\right ) \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac{\left (3 b e^2\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{32 c^3}\\ &=\frac{b d e x \sqrt{1-c^2 x^2}}{2 c}+\frac{3 b e^2 x \sqrt{1-c^2 x^2}}{32 c^3}+\frac{b e^2 x^3 \sqrt{1-c^2 x^2}}{16 c}-\frac{b d e \sin ^{-1}(c x)}{2 c^2}-\frac{3 b e^2 \sin ^{-1}(c x)}{32 c^4}-\frac{1}{2} i b d^2 \sin ^{-1}(c x)^2+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-b d^2 \sin ^{-1}(c x) \log (x)+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (2 i b d^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac{b d e x \sqrt{1-c^2 x^2}}{2 c}+\frac{3 b e^2 x \sqrt{1-c^2 x^2}}{32 c^3}+\frac{b e^2 x^3 \sqrt{1-c^2 x^2}}{16 c}-\frac{b d e \sin ^{-1}(c x)}{2 c^2}-\frac{3 b e^2 \sin ^{-1}(c x)}{32 c^4}-\frac{1}{2} i b d^2 \sin ^{-1}(c x)^2+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+b d^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^2 \sin ^{-1}(c x) \log (x)+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (b d^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac{b d e x \sqrt{1-c^2 x^2}}{2 c}+\frac{3 b e^2 x \sqrt{1-c^2 x^2}}{32 c^3}+\frac{b e^2 x^3 \sqrt{1-c^2 x^2}}{16 c}-\frac{b d e \sin ^{-1}(c x)}{2 c^2}-\frac{3 b e^2 \sin ^{-1}(c x)}{32 c^4}-\frac{1}{2} i b d^2 \sin ^{-1}(c x)^2+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+b d^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^2 \sin ^{-1}(c x) \log (x)+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\frac{1}{2} \left (i b d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=\frac{b d e x \sqrt{1-c^2 x^2}}{2 c}+\frac{3 b e^2 x \sqrt{1-c^2 x^2}}{32 c^3}+\frac{b e^2 x^3 \sqrt{1-c^2 x^2}}{16 c}-\frac{b d e \sin ^{-1}(c x)}{2 c^2}-\frac{3 b e^2 \sin ^{-1}(c x)}{32 c^4}-\frac{1}{2} i b d^2 \sin ^{-1}(c x)^2+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+b d^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^2 \sin ^{-1}(c x) \log (x)+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac{1}{2} i b d^2 \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.465204, size = 184, normalized size = 0.8 \[ -\frac{1}{2} i b d^2 \left (\sin ^{-1}(c x)^2+\text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )\right )+a d^2 \log (x)+a d e x^2+\frac{1}{4} a e^2 x^4+\frac{b d e \left (c x \sqrt{1-c^2 x^2}-\sin ^{-1}(c x)\right )}{2 c^2}+\frac{b e^2 \left (c x \sqrt{1-c^2 x^2} \left (2 c^2 x^2+3\right )-3 \sin ^{-1}(c x)\right )}{32 c^4}+b d^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b d e x^2 \sin ^{-1}(c x)+\frac{1}{4} b e^2 x^4 \sin ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*d*e*x^2 + (a*e^2*x^4)/4 + (b*e^2*(c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) - 3*ArcSin[c*x]))/(32*c^4) + (b*d*e*
(c*x*Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(2*c^2) + b*d*e*x^2*ArcSin[c*x] + (b*e^2*x^4*ArcSin[c*x])/4 + b*d^2*Arc
Sin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + a*d^2*Log[x] - (I/2)*b*d^2*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcS
in[c*x])])

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Maple [A]  time = 0.208, size = 272, normalized size = 1.2 \begin{align*}{\frac{a{e}^{2}{x}^{4}}{4}}+aed{x}^{2}+{d}^{2}a\ln \left ( cx \right ) +{\frac{b\arcsin \left ( cx \right ){e}^{2}{x}^{4}}{4}}+b\arcsin \left ( cx \right ) ed{x}^{2}+{\frac{bedx}{2\,c}\sqrt{-{c}^{2}{x}^{2}+1}}+{d}^{2}b\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +{d}^{2}b\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -i{d}^{2}b{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -{\frac{3\,b\arcsin \left ( cx \right ){e}^{2}}{32\,{c}^{4}}}+{\frac{b{e}^{2}{x}^{3}}{16\,c}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{3\,b{e}^{2}x}{32\,{c}^{3}}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arcsin \left ( cx \right ) ed}{2\,{c}^{2}}}-{\frac{i}{2}}b{d}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}-i{d}^{2}b{\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,x)

[Out]

1/4*a*e^2*x^4+a*e*d*x^2+d^2*a*ln(c*x)+1/4*b*arcsin(c*x)*e^2*x^4+b*arcsin(c*x)*e*d*x^2+1/2*b*d*e*x*(-c^2*x^2+1)
^(1/2)/c+d^2*b*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+d^2*b*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-I*d
^2*b*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-3/32*b*e^2*arcsin(c*x)/c^4+1/16*b*e^2*x^3*(-c^2*x^2+1)^(1/2)/c+3/32*
b*e^2*x*(-c^2*x^2+1)^(1/2)/c^3-1/2*b*d*e*arcsin(c*x)/c^2-1/2*I*b*d^2*arcsin(c*x)^2-I*d^2*b*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}{4} \, a e^{2} x^{4} + a d e x^{2} + a d^{2} \log \left (x\right ) + \int \frac{{\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\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,x, algorithm="maxima")

[Out]

1/4*a*e^2*x^4 + a*d*e*x^2 + a*d^2*log(x) + integrate((b*e^2*x^4 + 2*b*d*e*x^2 + b*d^2)*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}, 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,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, 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}\, 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,x)

[Out]

Integral((a + b*asin(c*x))*(d + e*x**2)**2/x, 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}\,{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,x, algorithm="giac")

[Out]

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