3.75 \(\int \frac{(d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x))}{x^{10}} \, dx\)
Optimal. Leaf size=308 \[ -\frac{8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{315 d x^5}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{63 d x^7}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{9 d x^9}-\frac{2 b c^7 d \sqrt{d-c^2 d x^2}}{315 x^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{420 x^4 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{1-c^2 x^2}}+\frac{8 b c^9 d \log (x) \sqrt{d-c^2 d x^2}}{315 \sqrt{1-c^2 x^2}} \]
[Out]
-(b*c*d*Sqrt[d - c^2*d*x^2])/(72*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*Sqrt
[d - c^2*d*x^2]*Log[x])/(315*Sqrt[1 - c^2*x^2])
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Rubi [A] time = 0.213051, antiderivative size = 308, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used =
{271, 264, 4691, 12, 1251, 893} \[ -\frac{8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{315 d x^5}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{63 d x^7}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{9 d x^9}-\frac{2 b c^7 d \sqrt{d-c^2 d x^2}}{315 x^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{420 x^4 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{1-c^2 x^2}}+\frac{8 b c^9 d \log (x) \sqrt{d-c^2 d x^2}}{315 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^10,x]
[Out]
-(b*c*d*Sqrt[d - c^2*d*x^2])/(72*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*Sqrt
[d - c^2*d*x^2]*Log[x])/(315*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 1251
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 893
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]))
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 ) \operatorname{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 ) \operatorname{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 = 0.257498, size = 197, normalized size = 0.64 \[ \frac{8 b c^9 d \log (x) \sqrt{d-c^2 d x^2}}{315 \sqrt{1-c^2 x^2}}-\frac{d \sqrt{d-c^2 d x^2} \left (840 a \left (8 c^4 x^4+20 c^2 x^2+35\right ) \left (c^2 x^2-1\right )^3-b c x \sqrt{1-c^2 x^2} \left (18264 c^8 x^8+1680 c^6 x^6+630 c^4 x^4-7000 c^2 x^2+3675\right )+840 b \left (8 c^4 x^4+20 c^2 x^2+35\right ) \left (c^2 x^2-1\right )^3 \sin ^{-1}(c x)\right )}{264600 x^9 \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^10,x]
[Out]
-(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]))/(264600*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
])
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Maple [C] time = 0.457, size = 4560, normalized size = 14.8 \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^10,x)
[Out]
-4/63*a*c^2/d/x^7*(-c^2*d*x^2+d)^(5/2)+113594/63*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/(c^2*x^2-1)*arcsin(c*x)*c^8-25915/126*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^7*(-c^2*x^2+1)^(1/2)-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^5*(-c^2*x^2+1
)^(1/2)-64/3*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^4-472
5*c^2*x^2+1225)*x^13/(c^2*x^2-1)*arcsin(c*x)*c^22+104/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^1
0+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)*arcsin(c*x)*c^20+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+1
89*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+16*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2
*x^2+1)^(1/2)*arcsin(c*x)*c^9*d/(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^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^17/(c^2*x^2-1)*c^26+16/315*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^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^15/(c^2*x^2
-1)*c^24+344/189*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^13/(c^2*x^2-1)*c^22+19540/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+18
9*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^3*(-c^2*x^2+1)^(1/2)-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+16/3*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^4-47
25*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)*arcsin(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^4-4725*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^12-945*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^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+1225)*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^8-2730*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^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+6210*c^4*x^4-4725*c^2*x^2
+1225)*x/(c^2*x^2-1)*arcsin(c*x)*c^10-8/315*a*c^4/d/x^5*(-c^2*d*x^2+d)^(5/2)+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)*arcsin(c
*x)*(-c^2*x^2+1)^(1/2)*c^19-24/5*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)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^17-1/9*a/d/x^9*(-c^2*d*
x^2+d)^(5/2)+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^8-2730*c^6*x^6+6210*c^4
*x^4-4725*c^2*x^2+1225)*x^6/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^15-1104/7*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)*arcsin
(c*x)*(-c^2*x^2+1)^(1/2)*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)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*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)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^21-2189/189*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^2*x^2+1)*c^14+350/27*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^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^2*x^2+1)*c^10-280/9*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)/(c^2*x^2-1)*arcsin(c*x)*(-c^2
*x^2+1)^(1/2)*c^9+1225/9*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^9/(c^2*x^2-1)*arcsin(c*x)+30055/504*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)/(c^2*x^2-1)*c^9*(-c^2*x^2+1)^(1/2)-8/315*
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^9*d-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^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*
x^15/(c^2*x^2-1)*(-c^2*x^2+1)*c^24-16/45*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^13/(c^2*x^2-1)*(-c^2*x^2+1)*c^22+1384/945*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^2*x^2+1)*c^20+2306/945*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^2*x^2+1)*c^18-40/63*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^2*x^2+1)*c^
16
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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^10,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.65776, size = 1517, normalized size = 4.93 \begin{align*} \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{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^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)]
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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**10,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^{10}}\,{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^10,x, algorithm="giac")
[Out]
integrate((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)/x^10, x)