3.233 \(\int \frac{(d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x))^2}{x^4} \, dx\)

Optimal. Leaf size=591 \[ \frac{7 i b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt{1-c^2 x^2}}-\frac{5 b c^5 d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}+\frac{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b \sqrt{1-c^2 x^2}}+\frac{7 i c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt{1-c^2 x^2}}-\frac{7}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac{14 b c^3 d^2 \sqrt{d-c^2 d x^2} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\frac{7}{12} b^2 c^4 d^2 x \sqrt{d-c^2 d x^2}-\frac{b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{3 x}+\frac{23 b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{12 \sqrt{1-c^2 x^2}} \]

[Out]

(-7*b^2*c^4*d^2*x*Sqrt[d - c^2*d*x^2])/12 - (b^2*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(3*x) + (23*b^2*c^
3*d^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(12*Sqrt[1 - c^2*x^2]) - (5*b*c^5*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*Ar
cSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) - (7*b*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/3
- (b*c*d^2*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*x^2) + (5*c^4*d^2*x*Sqrt[d - c^2*d*
x^2]*(a + b*ArcSin[c*x])^2)/2 + (((7*I)/3)*c^3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2
] + (5*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*x) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2
)/(3*x^3) + (5*c^3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*Sqrt[1 - c^2*x^2]) - (14*b*c^3*d^2*Sqrt
[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])])/(3*Sqrt[1 - c^2*x^2]) + (((7*I)/3)*b^2*c^3
*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]

________________________________________________________________________________________

Rubi [A]  time = 0.883494, antiderivative size = 591, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 15, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.517, Rules used = {4695, 4647, 4641, 4627, 321, 216, 4683, 4625, 3717, 2190, 2279, 2391, 195, 4685, 277} \[ \frac{7 i b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt{1-c^2 x^2}}-\frac{5 b c^5 d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}+\frac{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b \sqrt{1-c^2 x^2}}+\frac{7 i c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt{1-c^2 x^2}}-\frac{7}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac{14 b c^3 d^2 \sqrt{d-c^2 d x^2} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\frac{7}{12} b^2 c^4 d^2 x \sqrt{d-c^2 d x^2}-\frac{b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{3 x}+\frac{23 b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{12 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x^4,x]

[Out]

(-7*b^2*c^4*d^2*x*Sqrt[d - c^2*d*x^2])/12 - (b^2*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(3*x) + (23*b^2*c^
3*d^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(12*Sqrt[1 - c^2*x^2]) - (5*b*c^5*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*Ar
cSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) - (7*b*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/3
- (b*c*d^2*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*x^2) + (5*c^4*d^2*x*Sqrt[d - c^2*d*
x^2]*(a + b*ArcSin[c*x])^2)/2 + (((7*I)/3)*c^3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2
] + (5*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*x) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2
)/(3*x^3) + (5*c^3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*Sqrt[1 - c^2*x^2]) - (14*b*c^3*d^2*Sqrt
[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])])/(3*Sqrt[1 - c^2*x^2]) + (((7*I)/3)*b^2*c^3
*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]

Rule 4695

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/
(f*(m + 1)*(1 - c^2*x^2)^FracPart[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]

Rule 4647

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(x*Sqrt[d + e*x^2]*(
a + b*ArcSin[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 - c^2*x^2]), Int[x*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 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 4683

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.))/(x_), x_Symbol] :> Simp[((d + e*x^2)^p*(a
+ b*ArcSin[c*x]))/(2*p), x] + (Dist[d, Int[((d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x]))/x, x], x] - Dist[(b*c*d^
p)/(2*p), Int[(1 - c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

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]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 4685

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((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]))/(f*(m + 1)), x] + (-Dist[(b*c*d^p)/(f*(m + 1)), Int[(f*x)^(m + 1)*
(1 - c^2*x^2)^(p - 1/2), x], x] - Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*Arc
Sin[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]

Rule 277

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

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\frac{1}{3} \left (5 c^2 d\right ) \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx+\frac{\left (2 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx}{3 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\left (5 c^4 d^2\right ) \int \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\frac{\left (b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x^2} \, dx}{3 \sqrt{1-c^2 x^2}}-\frac{\left (4 b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx}{3 \sqrt{1-c^2 x^2}}-\frac{\left (10 b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx}{3 \sqrt{1-c^2 x^2}}\\ &=-\frac{b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{3 x}-\frac{7}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\frac{\left (4 b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x} \, dx}{3 \sqrt{1-c^2 x^2}}-\frac{\left (10 b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x} \, dx}{3 \sqrt{1-c^2 x^2}}+\frac{\left (5 c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \, dx}{3 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \, dx}{3 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c^5 d^2 \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{2}{3} b^2 c^4 d^2 x \sqrt{d-c^2 d x^2}-\frac{b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{3 x}-\frac{5 b c^5 d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{7}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{5 c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b \sqrt{1-c^2 x^2}}-\frac{\left (4 b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}-\frac{\left (10 b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{3 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{6 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^6 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}\\ &=-\frac{7}{12} b^2 c^4 d^2 x \sqrt{d-c^2 d x^2}-\frac{b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{3 x}+\frac{2 b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt{1-c^2 x^2}}-\frac{5 b c^5 d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{7}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{7 i c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt{1-c^2 x^2}}+\frac{5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{5 c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b \sqrt{1-c^2 x^2}}+\frac{\left (8 i b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{\left (20 i b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{4 \sqrt{1-c^2 x^2}}\\ &=-\frac{7}{12} b^2 c^4 d^2 x \sqrt{d-c^2 d x^2}-\frac{b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{3 x}+\frac{23 b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{12 \sqrt{1-c^2 x^2}}-\frac{5 b c^5 d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{7}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{7 i c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt{1-c^2 x^2}}+\frac{5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{5 c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b \sqrt{1-c^2 x^2}}-\frac{14 b c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt{1-c^2 x^2}}+\frac{\left (4 b^2 c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{\left (10 b^2 c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}\\ &=-\frac{7}{12} b^2 c^4 d^2 x \sqrt{d-c^2 d x^2}-\frac{b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{3 x}+\frac{23 b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{12 \sqrt{1-c^2 x^2}}-\frac{5 b c^5 d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{7}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{7 i c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt{1-c^2 x^2}}+\frac{5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{5 c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b \sqrt{1-c^2 x^2}}-\frac{14 b c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt{1-c^2 x^2}}-\frac{\left (2 i b^2 c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt{1-c^2 x^2}}-\frac{\left (5 i b^2 c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt{1-c^2 x^2}}\\ &=-\frac{7}{12} b^2 c^4 d^2 x \sqrt{d-c^2 d x^2}-\frac{b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{3 x}+\frac{23 b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{12 \sqrt{1-c^2 x^2}}-\frac{5 b c^5 d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{7}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{7 i c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt{1-c^2 x^2}}+\frac{5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{5 c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b \sqrt{1-c^2 x^2}}-\frac{14 b c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt{1-c^2 x^2}}+\frac{7 i b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 3.56288, size = 690, normalized size = 1.17 \[ \frac{d^2 \left (28 i b^2 c^3 x^3 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+6 a^2 c^4 x^4 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}+28 a^2 c^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}-4 a^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}-30 a^2 c^3 \sqrt{d} x^3 \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-6 a b c^5 x^5 \sqrt{d-c^2 d x^2}+3 a b c^3 x^3 \sqrt{d-c^2 d x^2}-4 a b c x \sqrt{d-c^2 d x^2}-56 a b c^3 x^3 \sqrt{d-c^2 d x^2} \log (c x)+b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \left (30 a c^3 x^3+4 b \left (7 i c^3 x^3+7 c^2 x^2 \sqrt{1-c^2 x^2}-\sqrt{1-c^2 x^2}\right )+3 b c^3 x^3 \sin \left (2 \sin ^{-1}(c x)\right )\right )+b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \left (48 a c^2 x^2 \sqrt{1-c^2 x^2}-6 a \sqrt{1-c^2 x^2}+6 a c^3 x^3 \sin \left (2 \sin ^{-1}(c x)\right )-2 a \cos \left (3 \sin ^{-1}(c x)\right )-56 b c^3 x^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+3 b c^3 x^3 \cos \left (2 \sin ^{-1}(c x)\right )-4 b c x\right )-3 b^2 c^4 x^4 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}-4 b^2 c^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}+10 b^2 c^3 x^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^3\right )}{12 x^3 \sqrt{1-c^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x^4,x]

[Out]

(d^2*(-4*a*b*c*x*Sqrt[d - c^2*d*x^2] + 3*a*b*c^3*x^3*Sqrt[d - c^2*d*x^2] - 6*a*b*c^5*x^5*Sqrt[d - c^2*d*x^2] -
4*a^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + 28*a^2*c^2*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] - 4*b^2*c^
2*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + 6*a^2*c^4*x^4*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] - 3*b^2*c^4*
x^4*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + 10*b^2*c^3*x^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^3 - 30*a^2*c^3*Sqrt
[d]*x^3*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 56*a*b*c^3*x^3*Sqrt[d -
c^2*d*x^2]*Log[c*x] + (28*I)*b^2*c^3*x^3*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])] + b*Sqrt[d - c
^2*d*x^2]*ArcSin[c*x]*(-4*b*c*x - 6*a*Sqrt[1 - c^2*x^2] + 48*a*c^2*x^2*Sqrt[1 - c^2*x^2] + 3*b*c^3*x^3*Cos[2*A
rcSin[c*x]] - 2*a*Cos[3*ArcSin[c*x]] - 56*b*c^3*x^3*Log[1 - E^((2*I)*ArcSin[c*x])] + 6*a*c^3*x^3*Sin[2*ArcSin[
c*x]]) + b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*(30*a*c^3*x^3 + 4*b*((7*I)*c^3*x^3 - Sqrt[1 - c^2*x^2] + 7*c^2*x^
2*Sqrt[1 - c^2*x^2]) + 3*b*c^3*x^3*Sin[2*ArcSin[c*x]])))/(12*x^3*Sqrt[1 - c^2*x^2])

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Maple [B]  time = 0.465, size = 3855, normalized size = 6.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/x^4,x)

[Out]

4/3*a^2*c^2/d/x*(-c^2*d*x^2+d)^(7/2)+5/3*a^2*c^4*d*x*(-c^2*d*x^2+d)^(3/2)+5/2*a^2*c^4*d^3/(c^2*d)^(1/2)*arctan
((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+147*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^4/(c^2
*x^2-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^7-49/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x
^3/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)*c^6-35*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^2/
(c^2*x^2-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^5+7/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1
)*x/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)*c^4-1/4*a*b*(-d*(c^2*x^2-1))^(1/2)*c^3*d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(
1/2)-1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*c^3*d^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+1/2*b^2*(-d*(c^2*x^2-1)
)^(1/2)*c^6*d^2/(c^2*x^2-1)*arcsin(c*x)^2*x^3-1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*c^4*d^2/(c^2*x^2-1)*arcsin(c*x)^2
*x-56/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^5/(c^2*x^2-1)*c^8+71/3*b^2*(-d*(c^2*x^2-1))
^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^3/(c^2*x^2-1)*c^6-16/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^
2*x^2+1)*x/(c^2*x^2-1)*c^4+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)/x/(c^2*x^2-1)*c^2+1/3*
b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)/x^3/(c^2*x^2-1)*arcsin(c*x)^2-5/6*b^2*(-d*(c^2*x^2-1)
)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arcsin(c*x)^3*c^3*d^2+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15
*c^2*x^2+1)/x^3/(c^2*x^2-1)*arcsin(c*x)+14/3*a*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)*c^3*d^2-5*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)/(c^2*x^2-1)*(-c
^2*x^2+1)^(1/2)*c^3+a*b*(-d*(c^2*x^2-1))^(1/2)*c^6*d^2/(c^2*x^2-1)*arcsin(c*x)*x^3-a*b*(-d*(c^2*x^2-1))^(1/2)*
c^4*d^2/(c^2*x^2-1)*arcsin(c*x)*x+14*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^3*d^2/(3*c^2*x^2-3)*arcsi
n(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*c^5*d^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2
+1)^(1/2)*x^2+7/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c^6-1/
3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^3-14*I*b^2*(-c^2
*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^3*d^2/(3*c^2*x^2-3)*arcsin(c*x)^2-14*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*
x^2-1))^(1/2)*c^3*d^2/(3*c^2*x^2-3)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-14*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*
x^2-1))^(1/2)*c^3*d^2/(3*c^2*x^2-3)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+1/2*a*b*(-d*(c^2*x^2-1))^(1/2)*c^5*d^2
/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-5/2*a*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*c^3*d^2-5*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2
)*c^3-23/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)/x/(c^2*x^2-1)*arcsin(c*x)^2*c^2+147*b^2*(-
d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^5/(c^2*x^2-1)*arcsin(c*x)^2*c^8-203*b^2*(-d*(c^2*x^2-1))^
(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)^2*c^6+190/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63
*c^4*x^4-15*c^2*x^2+1)*x/(c^2*x^2-1)*arcsin(c*x)^2*c^4+14*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^3*d^
2/(3*c^2*x^2-3)*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-49/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-1
5*c^2*x^2+1)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c^6+7/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x
/(c^2*x^2-1)*(-c^2*x^2+1)*c^4+14/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)/(c^2*x^2-1)*arcs
in(c*x)*(-c^2*x^2+1)^(1/2)*c^3+294*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^4/(c^2*x^2-1)*
arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^7-70*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^2/(c^2*x^2-
1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^5+5/2*a^2*c^4*d^2*x*(-c^2*d*x^2+d)^(1/2)-1/3*a^2/d/x^3*(-c^2*d*x^2+d)^(7/2
)+4/3*a^2*c^4*x*(-c^2*d*x^2+d)^(5/2)+294*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^5/(c^2*x^2
-1)*arcsin(c*x)*c^8-406*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)*c
^6+21*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^5+380/3*a*
b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x/(c^2*x^2-1)*arcsin(c*x)*c^4-46/3*a*b*(-d*(c^2*x^2-1))
^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)/x/(c^2*x^2-1)*arcsin(c*x)*c^2+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*
x^4-15*c^2*x^2+1)/x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c+21*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^
2+1)*x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^5-7/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^
2*x^2+1)*x/(c^2*x^2-1)*arcsin(c*x)*c^4+7/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)/(c^2*x^2
-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^3-49/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^5/(
c^2*x^2-1)*arcsin(c*x)*c^8-21*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^4/(c^2*x^2-1)*(-c^2
*x^2+1)^(1/2)*c^7+56/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)*
c^6+5*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^5+1/3*b^
2*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)/x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c-28*I*a
*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*arcsin(c*x)*c^3*d^2/(3*c^2*x^2-3)-49/3*I*a*b*(-d*(c^2*x^2-1))^(1/
2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x^5/(c^2*x^2-1)*c^8+56/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*
x^2+1)*x^3/(c^2*x^2-1)*c^6-7/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(63*c^4*x^4-15*c^2*x^2+1)*x/(c^2*x^2-1)*c^4-1/
4*b^2*(-d*(c^2*x^2-1))^(1/2)*c^6*d^2/(c^2*x^2-1)*x^3+1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*c^4*d^2/(c^2*x^2-1)*x

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**2/x**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/x^4,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)^2/x^4, x)