3.7.21 \(\int \frac {(d+e x^2)^3 (a+b \text {ArcSin}(c x))}{x^3} \, dx\) [621]

Optimal. Leaf size=262 \[ -\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3 b e^2 \left (8 c^2 d+e\right ) \text {ArcSin}(c x)}{32 c^4}-\frac {3}{2} i b d^2 e \text {ArcSin}(c x)^2-\frac {d^3 (a+b \text {ArcSin}(c x))}{2 x^2}+\frac {3}{2} d e^2 x^2 (a+b \text {ArcSin}(c x))+\frac {1}{4} e^3 x^4 (a+b \text {ArcSin}(c x))+3 b d^2 e \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-3 b d^2 e \text {ArcSin}(c x) \log (x)+3 d^2 e (a+b \text {ArcSin}(c x)) \log (x)-\frac {3}{2} i b d^2 e \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right ) \]

[Out]

-3/32*b*e^2*(8*c^2*d+e)*arcsin(c*x)/c^4-3/2*I*b*d^2*e*arcsin(c*x)^2-1/2*d^3*(a+b*arcsin(c*x))/x^2+3/2*d*e^2*x^
2*(a+b*arcsin(c*x))+1/4*e^3*x^4*(a+b*arcsin(c*x))+3*b*d^2*e*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-3*b
*d^2*e*arcsin(c*x)*ln(x)+3*d^2*e*(a+b*arcsin(c*x))*ln(x)-3/2*I*b*d^2*e*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)
-1/2*b*c*d^3*(-c^2*x^2+1)^(1/2)/x+3/32*b*e^2*(8*c^2*d+e)*x*(-c^2*x^2+1)^(1/2)/c^3+1/16*b*e^3*x^3*(-c^2*x^2+1)^
(1/2)/c

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Rubi [A]
time = 0.56, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 16, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {272, 45, 4815, 12, 6874, 1821, 1598, 470, 327, 222, 2363, 4721, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {d^3 (a+b \text {ArcSin}(c x))}{2 x^2}+3 d^2 e \log (x) (a+b \text {ArcSin}(c x))+\frac {3}{2} d e^2 x^2 (a+b \text {ArcSin}(c x))+\frac {1}{4} e^3 x^4 (a+b \text {ArcSin}(c x))-\frac {3 b e^2 \text {ArcSin}(c x) \left (8 c^2 d+e\right )}{32 c^4}-\frac {3}{2} i b d^2 e \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )-\frac {3}{2} i b d^2 e \text {ArcSin}(c x)^2+3 b d^2 e \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-3 b d^2 e \log (x) \text {ArcSin}(c x)-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {3 b e^2 x \sqrt {1-c^2 x^2} \left (8 c^2 d+e\right )}{32 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(b*c*d^3*Sqrt[1 - c^2*x^2])/x + (3*b*e^2*(8*c^2*d + e)*x*Sqrt[1 - c^2*x^2])/(32*c^3) + (b*e^3*x^3*Sqrt[1
- c^2*x^2])/(16*c) - (3*b*e^2*(8*c^2*d + e)*ArcSin[c*x])/(32*c^4) - ((3*I)/2)*b*d^2*e*ArcSin[c*x]^2 - (d^3*(a
+ b*ArcSin[c*x]))/(2*x^2) + (3*d*e^2*x^2*(a + b*ArcSin[c*x]))/2 + (e^3*x^4*(a + b*ArcSin[c*x]))/4 + 3*b*d^2*e*
ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - 3*b*d^2*e*ArcSin[c*x]*Log[x] + 3*d^2*e*(a + b*ArcSin[c*x])*Log[x]
- ((3*I)/2)*b*d^2*e*PolyLog[2, E^((2*I)*ArcSin[c*x])]

Rule 12

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

Rule 45

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 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 272

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 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 470

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

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1821

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2363

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-e, 2]*(x/Sqr
t[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 2438

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]

Rule 3798

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 4721

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

Rule 4815

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 6874

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

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(b c) \int \frac {-2 d^3+6 d e^2 x^4+e^3 x^6+12 d^2 e x^2 \log (x)}{4 x^2 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{4} (b c) \int \frac {-2 d^3+6 d e^2 x^4+e^3 x^6+12 d^2 e x^2 \log (x)}{x^2 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{4} (b c) \int \left (\frac {-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt {1-c^2 x^2}}+\frac {12 d^2 e \log (x)}{\sqrt {1-c^2 x^2}}\right ) \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{4} (b c) \int \frac {-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt {1-c^2 x^2}} \, dx-\left (3 b c d^2 e\right ) \int \frac {\log (x)}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )-3 b d^2 e \sin ^{-1}(c x) \log (x)+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\frac {1}{4} (b c) \int \frac {-6 d e^2 x^3-e^3 x^5}{x \sqrt {1-c^2 x^2}} \, dx+\left (3 b d^2 e\right ) \int \frac {\sin ^{-1}(c x)}{x} \, dx\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )-3 b d^2 e \sin ^{-1}(c x) \log (x)+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\frac {1}{4} (b c) \int \frac {x^2 \left (-6 d e^2-e^3 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx+\left (3 b d^2 e\right ) \text {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3}{2} i b d^2 e \sin ^{-1}(c x)^2-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )-3 b d^2 e \sin ^{-1}(c x) \log (x)+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (6 i b d^2 e\right ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )-\frac {\left (3 b e^2 \left (8 c^2 d+e\right )\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 c}\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3}{2} i b d^2 e \sin ^{-1}(c x)^2-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 b d^2 e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-3 b d^2 e \sin ^{-1}(c x) \log (x)+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (3 b d^2 e\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\frac {\left (3 b e^2 \left (8 c^2 d+e\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^3}\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3 b e^2 \left (8 c^2 d+e\right ) \sin ^{-1}(c x)}{32 c^4}-\frac {3}{2} i b d^2 e \sin ^{-1}(c x)^2-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 b d^2 e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-3 b d^2 e \sin ^{-1}(c x) \log (x)+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\frac {1}{2} \left (3 i b d^2 e\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3 b e^2 \left (8 c^2 d+e\right ) \sin ^{-1}(c x)}{32 c^4}-\frac {3}{2} i b d^2 e \sin ^{-1}(c x)^2-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 b d^2 e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-3 b d^2 e \sin ^{-1}(c x) \log (x)+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {3}{2} i b d^2 e \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 262, normalized size = 1.00 \begin {gather*} \frac {1}{32} \left (-\frac {16 a d^3}{x^2}+48 a d e^2 x^2+8 a e^3 x^4-\frac {16 b d^3 \left (c x \sqrt {1-c^2 x^2}+\text {ArcSin}(c x)\right )}{x^2}+\frac {b e^3 \left (c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )+8 c^4 x^4 \text {ArcSin}(c x)-6 \text {ArcTan}\left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )}{c^4}+\frac {24 b d e^2 \left (c x \sqrt {1-c^2 x^2}+2 c^2 x^2 \text {ArcSin}(c x)-2 \text {ArcTan}\left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )}{c^2}+96 a d^2 e \log (x)+96 b d^2 e \left (\text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-\frac {1}{2} i \left (\text {ArcSin}(c x)^2+\text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-16*a*d^3)/x^2 + 48*a*d*e^2*x^2 + 8*a*e^3*x^4 - (16*b*d^3*(c*x*Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/x^2 + (b*e^
3*(c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) + 8*c^4*x^4*ArcSin[c*x] - 6*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])]))/
c^4 + (24*b*d*e^2*(c*x*Sqrt[1 - c^2*x^2] + 2*c^2*x^2*ArcSin[c*x] - 2*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])]))/
c^2 + 96*a*d^2*e*Log[x] + 96*b*d^2*e*(ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - (I/2)*(ArcSin[c*x]^2 + Poly
Log[2, E^((2*I)*ArcSin[c*x])])))/32

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Maple [A]
time = 0.49, size = 396, normalized size = 1.51

method	result	size
derivativedivides	\(c^{2} \left (\frac {3 a d \,e^{2} x^{2}}{2 c^{2}}+\frac {a \,e^{3} x^{4}}{4 c^{2}}-\frac {a \,d^{3}}{2 c^{2} x^{2}}+\frac {3 a \,d^{2} e \ln \left (c x \right )}{c^{2}}+\frac {3 b d \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{4 c^{3}}-\frac {3 i b \,d^{2} e \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}-\frac {b \,d^{3} \sqrt {-c^{2} x^{2}+1}}{2 c x}+\frac {b \,e^{3} x \sqrt {-c^{2} x^{2}+1}}{8 c^{5}}-\frac {3 b d \,e^{2} \arcsin \left (c x \right )}{4 c^{4}}-\frac {b \sin \left (4 \arcsin \left (c x \right )\right ) e^{3}}{128 c^{6}}+\frac {3 b \,d^{2} e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}+\frac {3 b \,d^{2} e \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}-\frac {3 i b \,d^{2} e \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}+\frac {b \arcsin \left (c x \right ) e^{3} x^{2}}{4 c^{4}}-\frac {b \,e^{3} \arcsin \left (c x \right )}{8 c^{6}}-\frac {b \arcsin \left (c x \right ) d^{3}}{2 c^{2} x^{2}}+\frac {3 b \arcsin \left (c x \right ) d \,e^{2} x^{2}}{2 c^{2}}+\frac {b \cos \left (4 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) e^{3}}{32 c^{6}}-\frac {3 i b \arcsin \left (c x \right )^{2} d^{2} e}{2 c^{2}}+\frac {i d^{3} b}{2}\right )\)	\(396\)
default	\(c^{2} \left (\frac {3 a d \,e^{2} x^{2}}{2 c^{2}}+\frac {a \,e^{3} x^{4}}{4 c^{2}}-\frac {a \,d^{3}}{2 c^{2} x^{2}}+\frac {3 a \,d^{2} e \ln \left (c x \right )}{c^{2}}+\frac {3 b d \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{4 c^{3}}-\frac {3 i b \,d^{2} e \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}-\frac {b \,d^{3} \sqrt {-c^{2} x^{2}+1}}{2 c x}+\frac {b \,e^{3} x \sqrt {-c^{2} x^{2}+1}}{8 c^{5}}-\frac {3 b d \,e^{2} \arcsin \left (c x \right )}{4 c^{4}}-\frac {b \sin \left (4 \arcsin \left (c x \right )\right ) e^{3}}{128 c^{6}}+\frac {3 b \,d^{2} e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}+\frac {3 b \,d^{2} e \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}-\frac {3 i b \,d^{2} e \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}+\frac {b \arcsin \left (c x \right ) e^{3} x^{2}}{4 c^{4}}-\frac {b \,e^{3} \arcsin \left (c x \right )}{8 c^{6}}-\frac {b \arcsin \left (c x \right ) d^{3}}{2 c^{2} x^{2}}+\frac {3 b \arcsin \left (c x \right ) d \,e^{2} x^{2}}{2 c^{2}}+\frac {b \cos \left (4 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) e^{3}}{32 c^{6}}-\frac {3 i b \arcsin \left (c x \right )^{2} d^{2} e}{2 c^{2}}+\frac {i d^{3} b}{2}\right )\)	\(396\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2*(3/2*a/c^2*d*e^2*x^2+1/4*a/c^2*e^3*x^4-1/2*a*d^3/c^2/x^2+3*a/c^2*d^2*e*ln(c*x)+3/4*b*d*e^2*x*(-c^2*x^2+1)^
(1/2)/c^3-3/2*I*b/c^2*arcsin(c*x)^2*d^2*e-1/2*b*d^3/c/x*(-c^2*x^2+1)^(1/2)+1/8*b*e^3*x*(-c^2*x^2+1)^(1/2)/c^5-
3/4*b*d*e^2*arcsin(c*x)/c^4-1/128*b/c^6*sin(4*arcsin(c*x))*e^3+3*b/c^2*d^2*e*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+
1)^(1/2))+3*b/c^2*d^2*e*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+1/2*I*d^3*b+1/4*b/c^4*arcsin(c*x)*e^3*x^2-1
/8*b*e^3*arcsin(c*x)/c^6-1/2*b*arcsin(c*x)*d^3/c^2/x^2+3/2*b/c^2*arcsin(c*x)*d*e^2*x^2+1/32*b/c^6*cos(4*arcsin
(c*x))*arcsin(c*x)*e^3-3*I*b/c^2*d^2*e*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-3*I*b/c^2*d^2*e*polylog(2,I*c*x+(-
c^2*x^2+1)^(1/2)))

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

[Out]

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

[Out]

integral((a*x^6*e^3 + 3*a*d*x^4*e^2 + 3*a*d^2*x^2*e + a*d^3 + (b*x^6*e^3 + 3*b*d*x^4*e^2 + 3*b*d^2*x^2*e + b*d
^3)*arcsin(c*x))/x^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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