3.74 \(\int \frac{(d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x))}{x^8} \, dx\)
Optimal. Leaf size=231 \[ -\frac{2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 d x^5}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d x^7}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{70 x^2 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d \sqrt{d-c^2 d x^2}}{35 x^4 \sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{1-c^2 x^2}}+\frac{2 b c^7 d \log (x) \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}} \]
[Out]
-(b*c*d*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*Sqrt[d - c^2*d*x^2])/(35*x^4*Sqrt[1 - c^2
*x^2]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(70*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])
)/(7*d*x^7) - (2*c^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(35*d*x^5) + (2*b*c^7*d*Sqrt[d - c^2*d*x^2]*Lo
g[x])/(35*Sqrt[1 - c^2*x^2])
________________________________________________________________________________________
Rubi [A] time = 0.16388, antiderivative size = 231, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used =
{271, 264, 4691, 12, 446, 76} \[ -\frac{2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 d x^5}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d x^7}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{70 x^2 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d \sqrt{d-c^2 d x^2}}{35 x^4 \sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{1-c^2 x^2}}+\frac{2 b c^7 d \log (x) \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^8,x]
[Out]
-(b*c*d*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*Sqrt[d - c^2*d*x^2])/(35*x^4*Sqrt[1 - c^2
*x^2]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(70*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])
)/(7*d*x^7) - (2*c^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(35*d*x^5) + (2*b*c^7*d*Sqrt[d - c^2*d*x^2]*Lo
g[x])/(35*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 446
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rule 76
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x^8} \, dx &=-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-5-2 c^2 x^2\right ) \left (1-c^2 x^2\right )^2}{35 x^7} \, dx}{\sqrt{1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int \frac{\left (d-c^2 d x^2\right )^{3/2}}{x^8} \, dx\\ &=-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d x^7}-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-5-2 c^2 x^2\right ) \left (1-c^2 x^2\right )^2}{x^7} \, dx}{35 \sqrt{1-c^2 x^2}}+\frac{1}{7} \left (2 c^2 \left (a+b \sin ^{-1}(c x)\right )\right ) \int \frac{\left (d-c^2 d x^2\right )^{3/2}}{x^6} \, dx\\ &=-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d x^7}-\frac{2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 d x^5}-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (-5-2 c^2 x\right ) \left (1-c^2 x\right )^2}{x^4} \, dx,x,x^2\right )}{70 \sqrt{1-c^2 x^2}}\\ &=-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d x^7}-\frac{2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 d x^5}-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{5}{x^4}+\frac{8 c^2}{x^3}-\frac{c^4}{x^2}-\frac{2 c^6}{x}\right ) \, dx,x,x^2\right )}{70 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d \sqrt{d-c^2 d x^2}}{35 x^4 \sqrt{1-c^2 x^2}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{70 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d x^7}-\frac{2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 d x^5}+\frac{2 b c^7 d \sqrt{d-c^2 d x^2} \log (x)}{35 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.18152, size = 173, normalized size = 0.75 \[ \frac{2 b c^7 d \log (x) \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}-\frac{d \sqrt{d-c^2 d x^2} \left (30 a \left (2 c^2 x^2+5\right ) \left (c^2 x^2-1\right )^3-b c x \sqrt{1-c^2 x^2} \left (147 c^6 x^6+15 c^4 x^4-60 c^2 x^2+25\right )+30 b \left (2 c^2 x^2+5\right ) \left (c^2 x^2-1\right )^3 \sin ^{-1}(c x)\right )}{1050 x^7 \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^8,x]
[Out]
-(d*Sqrt[d - c^2*d*x^2]*(30*a*(-1 + c^2*x^2)^3*(5 + 2*c^2*x^2) - b*c*x*Sqrt[1 - c^2*x^2]*(25 - 60*c^2*x^2 + 15
*c^4*x^4 + 147*c^6*x^6) + 30*b*(-1 + c^2*x^2)^3*(5 + 2*c^2*x^2)*ArcSin[c*x]))/(1050*x^7*(-1 + c^2*x^2)) + (2*b
*c^7*d*Sqrt[d - c^2*d*x^2]*Log[x])/(35*Sqrt[1 - c^2*x^2])
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Maple [C] time = 0.361, size = 3383, normalized size = 14.7 \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)^(3/2)*(a+b*arcsin(c*x))/x^8,x)
[Out]
-2/35*a*c^2/d/x^5*(-c^2*d*x^2+d)^(5/2)-44/5*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+1
54*c^4*x^4-105*c^2*x^2+25)*x^4/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^11+6*I*b*(-d*(c^2*x^2-1))^(1/2)*d/
(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)
*c^9+2*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^8/(c^2*x
^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^15+4*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+1
54*c^4*x^4-105*c^2*x^2+25)*x^6/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^13-2*I*b*(-d*(c^2*x^2-1))^(1/2)*d/
(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^10/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2
)*c^17-55/14*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)/x^4/(c
^2*x^2-1)*c^3*(-c^2*x^2+1)^(1/2)-170/7*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*
x^4-105*c^2*x^2+25)/x^5/(c^2*x^2-1)*arcsin(c*x)*c^2+25/42*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-
70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)/x^6/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c+1/2*b*(-d*(c^2*x^2-1))^(1/2)*d/(35
*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^8/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^15+9/35*I*b*
(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^11/(c^2*x^2-1)*c^18
+1/21*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^9/(c^2*x^
2-1)*c^16-142/105*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)
*x^7/(c^2*x^2-1)*c^14+72/35*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c
^2*x^2+25)*x^5/(c^2*x^2-1)*c^12-25/21*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4
*x^4-105*c^2*x^2+25)*x^3/(c^2*x^2-1)*c^10+5/21*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^
6+154*c^4*x^4-105*c^2*x^2+25)*x/(c^2*x^2-1)*c^8+4*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^
7*d/(35*c^2*x^2-35)-2/35*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*
x^2+25)*x^13/(c^2*x^2-1)*c^20-2*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105
*c^2*x^2+25)*x^11/(c^2*x^2-1)*arcsin(c*x)*c^18+3*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^
6+154*c^4*x^4-105*c^2*x^2+25)*x^9/(c^2*x^2-1)*arcsin(c*x)*c^16+12*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*
c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^7/(c^2*x^2-1)*arcsin(c*x)*c^14-5/2*b*(-d*(c^2*x^2-1))^(1/2)*d
/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^6/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^13-164/5
*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^5/(c^2*x^2-1)*ar
csin(c*x)*c^12+11/6*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)
*x^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^11+52/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+1
54*c^4*x^4-105*c^2*x^2+25)*x^3/(c^2*x^2-1)*arcsin(c*x)*c^10+161/30*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35
*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^2/(c^2*x^2-1)*c^9*(-c^2*x^2+1)^(1/2)+1966/35*b*(-d*(c^2*x^2-
1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x/(c^2*x^2-1)*arcsin(c*x)*c^8-3272
/35*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)/x/(c^2*x^2-1)*a
rcsin(c*x)*c^6+421/42*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+2
5)/x^2/(c^2*x^2-1)*c^5*(-c^2*x^2+1)^(1/2)+472/7*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6
+154*c^4*x^4-105*c^2*x^2+25)/x^3/(c^2*x^2-1)*arcsin(c*x)*c^4-10/7*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-3
5*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^7-2/35*I*b*(-d*(
c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^11/(c^2*x^2-1)*(-c^2*x^2
+1)*c^18+1/5*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^9/
(c^2*x^2-1)*(-c^2*x^2+1)*c^16+26/105*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*
x^4-105*c^2*x^2+25)*x^7/(c^2*x^2-1)*(-c^2*x^2+1)*c^14-116/105*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^
8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^5/(c^2*x^2-1)*(-c^2*x^2+1)*c^12+20/21*I*b*(-d*(c^2*x^2-1))^(1/2
)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c^10-5/21*I*b
*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x/(c^2*x^2-1)*(-c^2*
x^2+1)*c^8-1/7*a/d/x^7*(-c^2*d*x^2+d)^(5/2)+25/7*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^
6+154*c^4*x^4-105*c^2*x^2+25)/x^7/(c^2*x^2-1)*arcsin(c*x)-2/35*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*d-359/30*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-7
0*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)/(c^2*x^2-1)*c^7*(-c^2*x^2+1)^(1/2)
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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)^(3/2)*(a+b*arcsin(c*x))/x^8,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.57653, size = 1296, normalized size = 5.61 \begin{align*} \left [\frac{6 \,{\left (b c^{9} d x^{9} - b c^{7} d 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 (3 \, b c^{5} d x^{5} -{\left (3 \, b c^{5} - 12 \, b c^{3} + 5 \, b c\right )} d x^{7} - 12 \, b c^{3} d x^{3} + 5 \, b c d x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} - 6 \,{\left (2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 9 \, a c^{4} d x^{4} + 13 \, a c^{2} d x^{2} - 5 \, a d +{\left (2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 9 \, b c^{4} d x^{4} + 13 \, b c^{2} d x^{2} - 5 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{210 \,{\left (c^{2} x^{9} - x^{7}\right )}}, \frac{12 \,{\left (b c^{9} d x^{9} - b c^{7} d 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 (3 \, b c^{5} d x^{5} -{\left (3 \, b c^{5} - 12 \, b c^{3} + 5 \, b c\right )} d x^{7} - 12 \, b c^{3} d x^{3} + 5 \, b c d x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} - 6 \,{\left (2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 9 \, a c^{4} d x^{4} + 13 \, a c^{2} d x^{2} - 5 \, a d +{\left (2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 9 \, b c^{4} d x^{4} + 13 \, b c^{2} d x^{2} - 5 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{210 \,{\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)^(3/2)*(a+b*arcsin(c*x))/x^8,x, algorithm="fricas")
[Out]
[1/210*(6*(b*c^9*d*x^9 - b*c^7*d*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)) + (3*b*c^5*d*x^5 - (3*b*c^5 - 12*b*c^3 + 5*b*c)*d*x^7 - 1
2*b*c^3*d*x^3 + 5*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 6*(2*a*c^8*d*x^8 - a*c^6*d*x^6 - 9*a*c^4*
d*x^4 + 13*a*c^2*d*x^2 - 5*a*d + (2*b*c^8*d*x^8 - b*c^6*d*x^6 - 9*b*c^4*d*x^4 + 13*b*c^2*d*x^2 - 5*b*d)*arcsin
(c*x))*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7), 1/210*(12*(b*c^9*d*x^9 - b*c^7*d*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)) + (3*b*c^5*d*x^5 - (3*b*c^
5 - 12*b*c^3 + 5*b*c)*d*x^7 - 12*b*c^3*d*x^3 + 5*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 6*(2*a*c^8
*d*x^8 - a*c^6*d*x^6 - 9*a*c^4*d*x^4 + 13*a*c^2*d*x^2 - 5*a*d + (2*b*c^8*d*x^8 - b*c^6*d*x^6 - 9*b*c^4*d*x^4 +
13*b*c^2*d*x^2 - 5*b*d)*arcsin(c*x))*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)**(3/2)*(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{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\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)^(3/2)*(a+b*arcsin(c*x))/x^8,x, algorithm="giac")
[Out]
integrate((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)/x^8, x)