3.1.75 \(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {ArcSin}(c x))}{x^{10}} \, dx\) [75]
Optimal. Leaf size=308 \[ -\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {1-c^2 x^2}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {1-c^2 x^2}}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 x^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{315 d x^5}+\frac {8 b c^9 d \sqrt {d-c^2 d x^2} \log (x)}{315 \sqrt {1-c^2 x^2}} \]
[Out]
-1/9*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/d/x^9-4/63*c^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/d/x^7-8/315*
c^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/d/x^5-1/72*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^8/(-c^2*x^2+1)^(1/2)+5/189*
b*c^3*d*(-c^2*d*x^2+d)^(1/2)/x^6/(-c^2*x^2+1)^(1/2)-1/420*b*c^5*d*(-c^2*d*x^2+d)^(1/2)/x^4/(-c^2*x^2+1)^(1/2)-
2/315*b*c^7*d*(-c^2*d*x^2+d)^(1/2)/x^2/(-c^2*x^2+1)^(1/2)+8/315*b*c^9*d*ln(x)*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1
)^(1/2)
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {277, 270, 4779,
12, 1265, 907} \begin {gather*} -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{315 d x^5}-\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {1-c^2 x^2}}+\frac {8 b c^9 d \log (x) \sqrt {d-c^2 d x^2}}{315 \sqrt {1-c^2 x^2}}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 x^2 \sqrt {1-c^2 x^2}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^10,x]
[Out]
-1/72*(b*c*d*Sqrt[d - c^2*d*x^2])/(x^8*Sqrt[1 - c^2*x^2]) + (5*b*c^3*d*Sqrt[d - c^2*d*x^2])/(189*x^6*Sqrt[1 -
c^2*x^2]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(420*x^4*Sqrt[1 - c^2*x^2]) - (2*b*c^7*d*Sqrt[d - c^2*d*x^2])/(315*x
^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(9*d*x^9) - (4*c^2*(d - c^2*d*x^2)^(5/2)*(
a + b*ArcSin[c*x]))/(63*d*x^7) - (8*c^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(315*d*x^5) + (8*b*c^9*d*Sq
rt[d - c^2*d*x^2]*Log[x])/(315*Sqrt[1 - c^2*x^2])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 270
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 277
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 907
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))
Rule 1265
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]
Rule 4779
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[Si
mplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p
- 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
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^{10}} \, dx &=-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (-35-20 c^2 x^2-8 c^4 x^4\right )}{315 x^9} \, 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^{10}} \, dx\\ &=-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{9 d x^9}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (-35-20 c^2 x^2-8 c^4 x^4\right )}{x^9} \, dx}{315 \sqrt {1-c^2 x^2}}+\frac {1}{9} \left (4 c^2 \left (a+b \sin ^{-1}(c x)\right )\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 )}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{63 d x^7}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\left (1-c^2 x\right )^2 \left (-35-20 c^2 x-8 c^4 x^2\right )}{x^5} \, dx,x,x^2\right )}{630 \sqrt {1-c^2 x^2}}+\frac {1}{63} \left (8 c^4 \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 )}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{315 d x^5}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (-\frac {35}{x^5}+\frac {50 c^2}{x^4}-\frac {3 c^4}{x^3}-\frac {4 c^6}{x^2}-\frac {8 c^8}{x}\right ) \, dx,x,x^2\right )}{630 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {1-c^2 x^2}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {1-c^2 x^2}}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 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 )}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{315 d x^5}+\frac {8 b c^9 d \sqrt {d-c^2 d x^2} \log (x)}{315 \sqrt {1-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.14, size = 197, normalized size = 0.64 \begin {gather*} -\frac {d \sqrt {d-c^2 d x^2} \left (840 a \left (-1+c^2 x^2\right )^3 \left (35+20 c^2 x^2+8 c^4 x^4\right )-b c x \sqrt {1-c^2 x^2} \left (3675-7000 c^2 x^2+630 c^4 x^4+1680 c^6 x^6+18264 c^8 x^8\right )+840 b \left (-1+c^2 x^2\right )^3 \left (35+20 c^2 x^2+8 c^4 x^4\right ) \text {ArcSin}(c x)\right )}{264600 x^9 \left (-1+c^2 x^2\right )}+\frac {8 b c^9 d \sqrt {d-c^2 d x^2} \log (x)}{315 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^10,x]
[Out]
-1/264600*(d*Sqrt[d - c^2*d*x^2]*(840*a*(-1 + c^2*x^2)^3*(35 + 20*c^2*x^2 + 8*c^4*x^4) - b*c*x*Sqrt[1 - c^2*x^
2]*(3675 - 7000*c^2*x^2 + 630*c^4*x^4 + 1680*c^6*x^6 + 18264*c^8*x^8) + 840*b*(-1 + c^2*x^2)^3*(35 + 20*c^2*x^
2 + 8*c^4*x^4)*ArcSin[c*x]))/(x^9*(-1 + c^2*x^2)) + (8*b*c^9*d*Sqrt[d - c^2*d*x^2]*Log[x])/(315*Sqrt[1 - c^2*x
^2])
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.46, size = 4563, normalized size = 14.81
| | |
| method |
result |
size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(4563\) |
| | |
|
|
|
|
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^10,x,method=_RETURNVERBOSE)
[Out]
-24/5*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2
*x^2+1225)*x^8/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^17-1104/7*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x
^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*
arcsin(c*x)*c^13+120*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c
^4*x^4-4725*c^2*x^2+1225)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^11-64/3*I*b*(-d*(c^2*x^2-1))^(1/2)*
d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^12/(c^2*x^2-1)*(-c^2
*x^2+1)^(1/2)*arcsin(c*x)*c^21+208/3*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-273
0*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^6/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^15+24*I*b*(-d*(c^2*
x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^10/(c^
2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^19+922/945*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10
+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^11/(c^2*x^2-1)*c^20-2906/945*I*b*(-d*(c^2*x^2-1))^
(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^9/(c^2*x^2-1)*
c^18-2069/189*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-
4725*c^2*x^2+1225)*x^7/(c^2*x^2-1)*c^16+4639/189*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189
*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^5/(c^2*x^2-1)*c^14-455/27*I*b*(-d*(c^2*x^2-1))^(1/2)*d
/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^3/(c^2*x^2-1)*c^12+35
/9*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^
2+1225)*x/(c^2*x^2-1)*c^10+a*(-1/9/d/x^9*(-c^2*d*x^2+d)^(5/2)+4/9*c^2*(-1/7/d/x^7*(-c^2*d*x^2+d)^(5/2)-2/35*c^
2/d/x^5*(-c^2*d*x^2+d)^(5/2)))+16/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c
^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^10/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^19-212/15*b*(-d*(c^2*x^2-1))^(1/2
)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^9/(c^2*x^2-1)*arcs
in(c*x)*c^18-4*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4
725*c^2*x^2+1225)*x^8/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^17+3151/15*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-94
5*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^7/(c^2*x^2-1)*arcsin(c*x)*c^16-4189/180
*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+12
25)*x^6/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^15-60632/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+
189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^5/(c^2*x^2-1)*arcsin(c*x)*c^14+1187/60*b*(-d*(c^2*x
^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^4/(c^2*
x^2-1)*(-c^2*x^2+1)^(1/2)*c^13+59884/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2
730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^3/(c^2*x^2-1)*arcsin(c*x)*c^12+829/56*b*(-d*(c^2*x^2-1))^(1/2)*d
/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^2/(c^2*x^2-1)*(-c^2*x
^2+1)^(1/2)*c^11-43264/63*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+621
0*c^4*x^4-4725*c^2*x^2+1225)*x/(c^2*x^2-1)*arcsin(c*x)*c^10+113594/63*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^1
2-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x/(c^2*x^2-1)*arcsin(c*x)*c^8-25915/1
26*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+
1225)/x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^7-174520/63*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10
+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^3/(c^2*x^2-1)*arcsin(c*x)*c^6+1285/6*b*(-d*(c^2*x^
2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^4/(c^2*x
^2-1)*(-c^2*x^2+1)^(1/2)*c^5+19540/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*
c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^5/(c^2*x^2-1)*arcsin(c*x)*c^4-21175/216*b*(-d*(c^2*x^2-1))^(1/2)*d/(
840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^6/(c^2*x^2-1)*(-c^2*x^2
+1)^(1/2)*c^3-7700/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4
*x^4-4725*c^2*x^2+1225)/x^7/(c^2*x^2-1)*arcsin(c*x)*c^2+1225/72*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*
c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^8/(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)*arcsin(c*x)*d*c^9/(315*c^2*x^2-315)-128/315*I*b*(-d*(c^2*x^2-1))^(1
/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-...
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 210, normalized size = 0.68 \begin {gather*} \frac {1}{7560} \, {\left (192 \, c^{8} d^{\frac {3}{2}} \log \left (x\right ) - \frac {48 \, c^{6} d^{\frac {3}{2}} x^{6} + 18 \, c^{4} d^{\frac {3}{2}} x^{4} - 200 \, c^{2} d^{\frac {3}{2}} x^{2} + 105 \, d^{\frac {3}{2}}}{x^{8}}\right )} b c - \frac {1}{315} \, b {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{5}} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{7}} + \frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{9}}\right )} \arcsin \left (c x\right ) - \frac {1}{315} \, a {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{5}} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{7}} + \frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{9}}\right )} \end {gather*}
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^10,x, algorithm="maxima")
[Out]
1/7560*(192*c^8*d^(3/2)*log(x) - (48*c^6*d^(3/2)*x^6 + 18*c^4*d^(3/2)*x^4 - 200*c^2*d^(3/2)*x^2 + 105*d^(3/2))
/x^8)*b*c - 1/315*b*(8*(-c^2*d*x^2 + d)^(5/2)*c^4/(d*x^5) + 20*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x^7) + 35*(-c^2*d
*x^2 + d)^(5/2)/(d*x^9))*arcsin(c*x) - 1/315*a*(8*(-c^2*d*x^2 + d)^(5/2)*c^4/(d*x^5) + 20*(-c^2*d*x^2 + d)^(5/
2)*c^2/(d*x^7) + 35*(-c^2*d*x^2 + d)^(5/2)/(d*x^9))
________________________________________________________________________________________
Fricas [A]
time = 2.53, size = 671, normalized size = 2.18 \begin {gather*} \left [\frac {96 \, {\left (b c^{11} d x^{11} - b c^{9} d x^{9}\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 (48 \, b c^{7} d x^{7} + 18 \, b c^{5} d x^{5} - {\left (48 \, b c^{7} + 18 \, b c^{5} - 200 \, b c^{3} + 105 \, b c\right )} d x^{9} - 200 \, b c^{3} d x^{3} + 105 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 24 \, {\left (8 \, a c^{10} d x^{10} - 4 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 53 \, a c^{4} d x^{4} + 85 \, a c^{2} d x^{2} - 35 \, a d + {\left (8 \, b c^{10} d x^{10} - 4 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 53 \, b c^{4} d x^{4} + 85 \, b c^{2} d x^{2} - 35 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{7560 \, {\left (c^{2} x^{11} - x^{9}\right )}}, \frac {192 \, {\left (b c^{11} d x^{11} - b c^{9} d x^{9}\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 (48 \, b c^{7} d x^{7} + 18 \, b c^{5} d x^{5} - {\left (48 \, b c^{7} + 18 \, b c^{5} - 200 \, b c^{3} + 105 \, b c\right )} d x^{9} - 200 \, b c^{3} d x^{3} + 105 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 24 \, {\left (8 \, a c^{10} d x^{10} - 4 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 53 \, a c^{4} d x^{4} + 85 \, a c^{2} d x^{2} - 35 \, a d + {\left (8 \, b c^{10} d x^{10} - 4 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 53 \, b c^{4} d x^{4} + 85 \, b c^{2} d x^{2} - 35 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{7560 \, {\left (c^{2} x^{11} - x^{9}\right )}}\right ] \end {gather*}
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^10,x, algorithm="fricas")
[Out]
[1/7560*(96*(b*c^11*d*x^11 - b*c^9*d*x^9)*sqrt(d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 - sqrt(-c^2*d*x^2 + d)*sq
rt(-c^2*x^2 + 1)*(x^4 - 1)*sqrt(d) - d)/(c^2*x^4 - x^2)) + (48*b*c^7*d*x^7 + 18*b*c^5*d*x^5 - (48*b*c^7 + 18*b
*c^5 - 200*b*c^3 + 105*b*c)*d*x^9 - 200*b*c^3*d*x^3 + 105*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 2
4*(8*a*c^10*d*x^10 - 4*a*c^8*d*x^8 - a*c^6*d*x^6 - 53*a*c^4*d*x^4 + 85*a*c^2*d*x^2 - 35*a*d + (8*b*c^10*d*x^10
- 4*b*c^8*d*x^8 - b*c^6*d*x^6 - 53*b*c^4*d*x^4 + 85*b*c^2*d*x^2 - 35*b*d)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/
(c^2*x^11 - x^9), 1/7560*(192*(b*c^11*d*x^11 - b*c^9*d*x^9)*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)) + (48*b*c^7*d*x^7 + 18*b*c^5*d*x^5 - (48*b*c^7 + 1
8*b*c^5 - 200*b*c^3 + 105*b*c)*d*x^9 - 200*b*c^3*d*x^3 + 105*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)
- 24*(8*a*c^10*d*x^10 - 4*a*c^8*d*x^8 - a*c^6*d*x^6 - 53*a*c^4*d*x^4 + 85*a*c^2*d*x^2 - 35*a*d + (8*b*c^10*d*x
^10 - 4*b*c^8*d*x^8 - b*c^6*d*x^6 - 53*b*c^4*d*x^4 + 85*b*c^2*d*x^2 - 35*b*d)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d
))/(c^2*x^11 - x^9)]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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**10,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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^10,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
________________________________________________________________________________________
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 (d-c^2\,d\,x^2\right )}^{3/2}}{x^{10}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((a + b*asin(c*x))*(d - c^2*d*x^2)^(3/2))/x^10,x)
[Out]
int(((a + b*asin(c*x))*(d - c^2*d*x^2)^(3/2))/x^10, x)
________________________________________________________________________________________