3.1.74 \(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {ArcSin}(c x))}{x^8} \, dx\) [74]
Optimal. Leaf size=231 \[ -\frac {b c d \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d \sqrt {d-c^2 d x^2}}{35 x^4 \sqrt {1-c^2 x^2}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{70 x^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{35 d x^5}+\frac {2 b c^7 d \sqrt {d-c^2 d x^2} \log (x)}{35 \sqrt {1-c^2 x^2}} \]
[Out]
-1/7*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/d/x^7-2/35*c^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/d/x^5-1/42*b
*c*d*(-c^2*d*x^2+d)^(1/2)/x^6/(-c^2*x^2+1)^(1/2)+2/35*b*c^3*d*(-c^2*d*x^2+d)^(1/2)/x^4/(-c^2*x^2+1)^(1/2)-1/70
*b*c^5*d*(-c^2*d*x^2+d)^(1/2)/x^2/(-c^2*x^2+1)^(1/2)+2/35*b*c^7*d*ln(x)*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2
)
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Rubi [A]
time = 0.11, antiderivative size = 231, 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, 457, 77} \begin {gather*} -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{35 d x^5}-\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}}-\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}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^8,x]
[Out]
-1/42*(b*c*d*Sqrt[d - c^2*d*x^2])/(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]*
Log[x])/(35*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 77
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])
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 457
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 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^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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.13, size = 173, normalized size = 0.75 \begin {gather*} -\frac {d \sqrt {d-c^2 d x^2} \left (30 a \left (-1+c^2 x^2\right )^3 \left (5+2 c^2 x^2\right )-b c x \sqrt {1-c^2 x^2} \left (25-60 c^2 x^2+15 c^4 x^4+147 c^6 x^6\right )+30 b \left (-1+c^2 x^2\right )^3 \left (5+2 c^2 x^2\right ) \text {ArcSin}(c x)\right )}{1050 x^7 \left (-1+c^2 x^2\right )}+\frac {2 b c^7 d \sqrt {d-c^2 d x^2} \log (x)}{35 \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^8,x]
[Out]
-1/1050*(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]))/(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] Result contains complex when optimal does not.
time = 0.40, size = 3384, normalized size = 14.65
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\(\text {Expression too large to display}\) |
\(3384\) |
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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,method=_RETURNVERBOSE)
[Out]
-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^8-2*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+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+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)*arcsin(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+154*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^2*x^2+1)^(1/2)*c^9+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)*arcsin(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^4-1
05*c^2*x^2+25)/x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^5+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^4-55/14*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^2*x^2+1)^(1/2)*c^3-170/7*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+4*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*d*c^7/(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+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+a*(-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))+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+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)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^9-2*I*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)*x^10/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^17+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)*(-c^2*x^2+1)^(1/2)
*arcsin(c*x)*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+154*c^4*x^4-105*c^2*x^2+2
5)*x^6/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^13-44/5*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+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)*arcsin(c*x)*c^11+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^8-10/7*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)/(c^2*x^2-1)*(-c^2*x^2+1)^
(1/2)*arcsin(c*x)*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+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)*d*...
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Maxima [A]
time = 0.49, size = 151, normalized size = 0.65 \begin {gather*} \frac {1}{210} \, {\left (12 \, c^{6} d^{\frac {3}{2}} \log \left (x\right ) - \frac {3 \, c^{4} d^{\frac {3}{2}} x^{4} - 12 \, c^{2} d^{\frac {3}{2}} x^{2} + 5 \, d^{\frac {3}{2}}}{x^{6}}\right )} b c - \frac {1}{35} \, b {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{5}} + \frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{7}}\right )} \arcsin \left (c x\right ) - \frac {1}{35} \, a {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{5}} + \frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{7}}\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^8,x, algorithm="maxima")
[Out]
1/210*(12*c^6*d^(3/2)*log(x) - (3*c^4*d^(3/2)*x^4 - 12*c^2*d^(3/2)*x^2 + 5*d^(3/2))/x^6)*b*c - 1/35*b*(2*(-c^2
*d*x^2 + d)^(5/2)*c^2/(d*x^5) + 5*(-c^2*d*x^2 + d)^(5/2)/(d*x^7))*arcsin(c*x) - 1/35*a*(2*(-c^2*d*x^2 + d)^(5/
2)*c^2/(d*x^5) + 5*(-c^2*d*x^2 + d)^(5/2)/(d*x^7))
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Fricas [A]
time = 2.41, size = 599, normalized size = 2.59 \begin {gather*} \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 {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^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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{x^{8}}\, dx \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**8,x)
[Out]
Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x))/x**8, x)
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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^8,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
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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^8} \,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^8,x)
[Out]
int(((a + b*asin(c*x))*(d - c^2*d*x^2)^(3/2))/x^8, x)
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