3.61 \(\int \frac{\sqrt{d-c^2 d x^2} (a+b \sin ^{-1}(c x))}{x^8} \, dx\)

Optimal. Leaf size=263 \[ -\frac{8 c^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 d x^3}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{35 d x^5}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d x^7}+\frac{2 b c^5 \sqrt{d-c^2 d x^2}}{105 x^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{140 x^4 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{1-c^2 x^2}}-\frac{8 b c^7 \log (x) \sqrt{d-c^2 d x^2}}{105 \sqrt{1-c^2 x^2}} \]

[Out]

-(b*c*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[1 - c^2*x^2]) + (b*c^3*Sqrt[d - c^2*d*x^2])/(140*x^4*Sqrt[1 - c^2*x^2]
) + (2*b*c^5*Sqrt[d - c^2*d*x^2])/(105*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(7
*d*x^7) - (4*c^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(35*d*x^5) - (8*c^4*(d - c^2*d*x^2)^(3/2)*(a + b*A
rcSin[c*x]))/(105*d*x^3) - (8*b*c^7*Sqrt[d - c^2*d*x^2]*Log[x])/(105*Sqrt[1 - c^2*x^2])

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Rubi [A]  time = 0.168816, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {271, 264, 4691, 12, 14} \[ -\frac{8 c^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 d x^3}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{35 d x^5}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d x^7}+\frac{2 b c^5 \sqrt{d-c^2 d x^2}}{105 x^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{140 x^4 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{1-c^2 x^2}}-\frac{8 b c^7 \log (x) \sqrt{d-c^2 d x^2}}{105 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x^8,x]

[Out]

-(b*c*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[1 - c^2*x^2]) + (b*c^3*Sqrt[d - c^2*d*x^2])/(140*x^4*Sqrt[1 - c^2*x^2]
) + (2*b*c^5*Sqrt[d - c^2*d*x^2])/(105*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(7
*d*x^7) - (4*c^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(35*d*x^5) - (8*c^4*(d - c^2*d*x^2)^(3/2)*(a + b*A
rcSin[c*x]))/(105*d*x^3) - (8*b*c^7*Sqrt[d - c^2*d*x^2]*Log[x])/(105*Sqrt[1 - c^2*x^2])

Rule 271

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

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 4691

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(1 - c^2*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], Int[x^m*(d + e*x^2)^p, x], x] - Dist[(b*c*d^(p - 1/2)*Sqrt[d +
e*x^2])/Sqrt[1 - c^2*x^2], Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x
] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 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 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]]

Rubi steps

\begin{align*} \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^8} \, dx &=-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \frac{-15+3 c^2 x^2+4 c^4 x^4+8 c^6 x^6}{105 x^7} \, dx}{\sqrt{1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int \frac{\sqrt{d-c^2 d x^2}}{x^8} \, dx\\ &=-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d x^7}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \frac{-15+3 c^2 x^2+4 c^4 x^4+8 c^6 x^6}{x^7} \, dx}{105 \sqrt{1-c^2 x^2}}+\frac{1}{7} \left (4 c^2 \left (a+b \sin ^{-1}(c x)\right )\right ) \int \frac{\sqrt{d-c^2 d x^2}}{x^6} \, dx\\ &=-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d x^7}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{35 d x^5}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \left (-\frac{15}{x^7}+\frac{3 c^2}{x^5}+\frac{4 c^4}{x^3}+\frac{8 c^6}{x}\right ) \, dx}{105 \sqrt{1-c^2 x^2}}+\frac{1}{35} \left (8 c^4 \left (a+b \sin ^{-1}(c x)\right )\right ) \int \frac{\sqrt{d-c^2 d x^2}}{x^4} \, dx\\ &=-\frac{b c \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{1-c^2 x^2}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{140 x^4 \sqrt{1-c^2 x^2}}+\frac{2 b c^5 \sqrt{d-c^2 d x^2}}{105 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d x^7}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{35 d x^5}-\frac{8 c^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 d x^3}-\frac{8 b c^7 \sqrt{d-c^2 d x^2} \log (x)}{105 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.158904, size = 187, normalized size = 0.71 \[ \frac{\sqrt{d-c^2 d x^2} \left (20 a \left (8 c^4 x^4+12 c^2 x^2+15\right ) \left (c^2 x^2-1\right )^2-b c x \sqrt{1-c^2 x^2} \left (392 c^6 x^6+40 c^4 x^4+15 c^2 x^2-50\right )+20 b \left (8 c^4 x^4+12 c^2 x^2+15\right ) \left (c^2 x^2-1\right )^2 \sin ^{-1}(c x)\right )}{2100 x^7 \left (c^2 x^2-1\right )}-\frac{8 b c^7 \log (x) \sqrt{d-c^2 d x^2}}{105 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x^8,x]

[Out]

(Sqrt[d - c^2*d*x^2]*(20*a*(-1 + c^2*x^2)^2*(15 + 12*c^2*x^2 + 8*c^4*x^4) - b*c*x*Sqrt[1 - c^2*x^2]*(-50 + 15*
c^2*x^2 + 40*c^4*x^4 + 392*c^6*x^6) + 20*b*(-1 + c^2*x^2)^2*(15 + 12*c^2*x^2 + 8*c^4*x^4)*ArcSin[c*x]))/(2100*
x^7*(-1 + c^2*x^2)) - (8*b*c^7*Sqrt[d - c^2*d*x^2]*Log[x])/(105*Sqrt[1 - c^2*x^2])

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Maple [C]  time = 0.361, size = 2748, normalized size = 10.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^8,x)

[Out]

73/20*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/(c^2*x^2-1)*c^7*(-c^2*x^2+
1)^(1/2)+8/105*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^7+
225/7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^7/(c^2*x^2-1)*arcsin(c*x
)-4/35*a*c^2/d/x^5*(-c^2*d*x^2+d)^(3/2)+20/7*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-31
5*c^2*x^2+225)*x/(c^2*x^2-1)*(-c^2*x^2+1)*c^8+120/7*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4
*x^4-315*c^2*x^2+225)/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^7+128/105*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c
^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^11/(c^2*x^2-1)*(-c^2*x^2+1)*c^18+16/15*I*b*(-d*(c^2*x^2-1))^(
1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^9/(c^2*x^2-1)*(-c^2*x^2+1)*c^16-88/105*I*b*(-d*(c^
2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^7/(c^2*x^2-1)*(-c^2*x^2+1)*c^14-302/105
*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^5/(c^2*x^2-1)*(-c^2*x^2+1)*
c^12-10/7*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^3/(c^2*x^2-1)*(-c^
2*x^2+1)*c^10-8/105*a*c^4/d/x^3*(-c^2*d*x^2+d)^(3/2)+64/3*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21
*c^4*x^4-315*c^2*x^2+225)*x^9/(c^2*x^2-1)*arcsin(c*x)*c^16-56/3*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*
x^6-21*c^4*x^4-315*c^2*x^2+225)*x^7/(c^2*x^2-1)*arcsin(c*x)*c^14-16/3*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-10
5*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^6/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^13-4/15*b*(-d*(c^2*x^2-1))^(1/2)/(2
80*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^5/(c^2*x^2-1)*arcsin(c*x)*c^12-351/5*b*(-d*(c^2*x^2-1))^(
1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^3/(c^2*x^2-1)*arcsin(c*x)*c^10+469/60*b*(-d*(c^2*x
^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^2/(c^2*x^2-1)*c^9*(-c^2*x^2+1)^(1/2)+3057/
35*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x/(c^2*x^2-1)*arcsin(c*x)*c^8
-594/35*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x/(c^2*x^2-1)*arcsin(c*x
)*c^6-71/28*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^2/(c^2*x^2-1)*c^5*
(-c^2*x^2+1)^(1/2)+342/7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^3/(c^
2*x^2-1)*arcsin(c*x)*c^4-255/28*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/
x^4/(c^2*x^2-1)*c^3*(-c^2*x^2+1)^(1/2)-585/7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*
c^2*x^2+225)/x^5/(c^2*x^2-1)*arcsin(c*x)*c^2+75/14*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^
4-315*c^2*x^2+225)/x^6/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c-16*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*arcsi
n(c*x)*c^7/(105*c^2*x^2-105)+128/105*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^
2+225)*x^13/(c^2*x^2-1)*c^20-16/105*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2
+225)*x^11/(c^2*x^2-1)*c^18-40/21*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+2
25)*x^9/(c^2*x^2-1)*c^16-214/105*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+22
5)*x^7/(c^2*x^2-1)*c^14+152/105*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225
)*x^5/(c^2*x^2-1)*c^12+30/7*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^
3/(c^2*x^2-1)*c^10-20/7*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x/(c^2
*x^2-1)*c^8+64/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^8/(c^2*x^2-
1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^15-8*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^
2*x^2+225)*x^6/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^13-8/5*I*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105
*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^4/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^11-24*I*b*(-d*(c^2*x^2-1
))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c
^9-1/7*a/d/x^7*(-c^2*d*x^2+d)^(3/2)

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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)^(1/2)*(a+b*arcsin(c*x))/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.31668, size = 1218, normalized size = 4.63 \begin{align*} \left [\frac{16 \,{\left (b c^{9} x^{9} - b c^{7} x^{7}\right )} \sqrt{d} \log \left (\frac{c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1}{\left (x^{4} - 1\right )} \sqrt{d} - d}{c^{2} x^{4} - x^{2}}\right ) -{\left (8 \, b c^{5} x^{5} -{\left (8 \, b c^{5} + 3 \, b c^{3} - 10 \, b c\right )} x^{7} + 3 \, b c^{3} x^{3} - 10 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} + 4 \,{\left (8 \, a c^{8} x^{8} - 4 \, a c^{6} x^{6} - a c^{4} x^{4} - 18 \, a c^{2} x^{2} +{\left (8 \, b c^{8} x^{8} - 4 \, b c^{6} x^{6} - b c^{4} x^{4} - 18 \, b c^{2} x^{2} + 15 \, b\right )} \arcsin \left (c x\right ) + 15 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{420 \,{\left (c^{2} x^{9} - x^{7}\right )}}, -\frac{32 \,{\left (b c^{9} x^{9} - b c^{7} x^{7}\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1}{\left (x^{2} + 1\right )} \sqrt{-d}}{c^{2} d x^{4} -{\left (c^{2} + 1\right )} d x^{2} + d}\right ) +{\left (8 \, b c^{5} x^{5} -{\left (8 \, b c^{5} + 3 \, b c^{3} - 10 \, b c\right )} x^{7} + 3 \, b c^{3} x^{3} - 10 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} - 4 \,{\left (8 \, a c^{8} x^{8} - 4 \, a c^{6} x^{6} - a c^{4} x^{4} - 18 \, a c^{2} x^{2} +{\left (8 \, b c^{8} x^{8} - 4 \, b c^{6} x^{6} - b c^{4} x^{4} - 18 \, b c^{2} x^{2} + 15 \, b\right )} \arcsin \left (c x\right ) + 15 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{420 \,{\left (c^{2} x^{9} - x^{7}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^8,x, algorithm="fricas")

[Out]

[1/420*(16*(b*c^9*x^9 - b*c^7*x^7)*sqrt(d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 + sqrt(-c^2*d*x^2 + d)*sqrt(-c^2
*x^2 + 1)*(x^4 - 1)*sqrt(d) - d)/(c^2*x^4 - x^2)) - (8*b*c^5*x^5 - (8*b*c^5 + 3*b*c^3 - 10*b*c)*x^7 + 3*b*c^3*
x^3 - 10*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 4*(8*a*c^8*x^8 - 4*a*c^6*x^6 - a*c^4*x^4 - 18*a*c^2*
x^2 + (8*b*c^8*x^8 - 4*b*c^6*x^6 - b*c^4*x^4 - 18*b*c^2*x^2 + 15*b)*arcsin(c*x) + 15*a)*sqrt(-c^2*d*x^2 + d))/
(c^2*x^9 - x^7), -1/420*(32*(b*c^9*x^9 - b*c^7*x^7)*sqrt(-d)*arctan(sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*(x
^2 + 1)*sqrt(-d)/(c^2*d*x^4 - (c^2 + 1)*d*x^2 + d)) + (8*b*c^5*x^5 - (8*b*c^5 + 3*b*c^3 - 10*b*c)*x^7 + 3*b*c^
3*x^3 - 10*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 4*(8*a*c^8*x^8 - 4*a*c^6*x^6 - a*c^4*x^4 - 18*a*c^
2*x^2 + (8*b*c^8*x^8 - 4*b*c^6*x^6 - b*c^4*x^4 - 18*b*c^2*x^2 + 15*b)*arcsin(c*x) + 15*a)*sqrt(-c^2*d*x^2 + d)
)/(c^2*x^9 - x^7)]

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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)**(1/2)*(a+b*asin(c*x))/x**8,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}}{x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^8,x, algorithm="giac")

[Out]

integrate(sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/x^8, x)