3.2.29 \(\int \frac {x^6 (a+b \text {ArcSin}(c x))}{(d-c^2 d x^2)^{5/2}} \, dx\) [129]
Optimal. Leaf size=293 \[ -\frac {b}{6 c^7 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {1-c^2 x^2}}{4 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^5 (a+b \text {ArcSin}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 x^3 (a+b \text {ArcSin}(c x))}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{2 c^6 d^3}+\frac {5 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{4 b c^7 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^7 d^2 \sqrt {d-c^2 d x^2}} \]
[Out]
1/3*x^5*(a+b*arcsin(c*x))/c^2/d/(-c^2*d*x^2+d)^(3/2)-5/3*x^3*(a+b*arcsin(c*x))/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/
6*b/c^7/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/4*b*x^2*(-c^2*x^2+1)^(1/2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)+
5/4*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/b/c^7/d^2/(-c^2*d*x^2+d)^(1/2)-7/6*b*ln(-c^2*x^2+1)*(-c^2*x^2+1)^(1
/2)/c^7/d^2/(-c^2*d*x^2+d)^(1/2)-5/2*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^6/d^3
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Rubi [A]
time = 0.30, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4791, 4795,
4737, 30, 272, 45} \begin {gather*} \frac {x^5 (a+b \text {ArcSin}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{4 b c^7 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{2 c^6 d^3}-\frac {5 x^3 (a+b \text {ArcSin}(c x))}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b}{6 c^7 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {7 b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^7 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {1-c^2 x^2}}{4 c^5 d^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x^6*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]
[Out]
-1/6*b/(c^7*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (b*x^2*Sqrt[1 - c^2*x^2])/(4*c^5*d^2*Sqrt[d - c^2*d*x
^2]) + (x^5*(a + b*ArcSin[c*x]))/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (5*x^3*(a + b*ArcSin[c*x]))/(3*c^4*d^2*Sqrt
[d - c^2*d*x^2]) - (5*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*c^6*d^3) + (5*Sqrt[1 - c^2*x^2]*(a + b*Arc
Sin[c*x])^2)/(4*b*c^7*d^2*Sqrt[d - c^2*d*x^2]) - (7*b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2])/(6*c^7*d^2*Sqrt[d -
c^2*d*x^2])
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rule 45
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rule 272
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rule 4737
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]
Rule 4791
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + (-Dist[f^2*((m - 1)/(2*e*(p + 1
))), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(2*c*(p + 1)))*Simp[(d
+ e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]
Rule 4795
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
Rubi steps
\begin {align*} \int \frac {x^6 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {x^5 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {x^5}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x^5 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}} \, dx}{c^4 d^2}+\frac {\left (5 b \sqrt {1-c^2 x^2}\right ) \int \frac {x^3}{1-c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1-c^2 x\right )^2} \, dx,x,x^2\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x^5 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^6 d^3}+\frac {5 \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {d-c^2 d x^2}} \, dx}{2 c^6 d^2}+\frac {\left (5 b \sqrt {1-c^2 x^2}\right ) \int x \, dx}{2 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^4}+\frac {1}{c^4 \left (-1+c^2 x\right )^2}+\frac {2}{c^4 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b}{6 c^7 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {13 b x^2 \sqrt {1-c^2 x^2}}{12 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^5 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^6 d^3}-\frac {b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{3 c^7 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b}{6 c^7 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {1-c^2 x^2}}{4 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^5 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^6 d^3}+\frac {5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^7 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^7 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 253, normalized size = 0.86 \begin {gather*} \frac {4 b c \sqrt {d} x \left (15-20 c^2 x^2+3 c^4 x^4\right ) \text {ArcSin}(c x)-30 b \sqrt {d} \left (1-c^2 x^2\right )^{3/2} \text {ArcSin}(c x)^2-60 a \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d} \left (4 a c x \left (15-20 c^2 x^2+3 c^4 x^4\right )+b \sqrt {1-c^2 x^2} \left (7-9 c^2 x^2+6 c^4 x^4\right )+28 b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )\right )}{24 c^7 d^{5/2} \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x^6*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]
[Out]
(4*b*c*Sqrt[d]*x*(15 - 20*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x] - 30*b*Sqrt[d]*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]^2 -
60*a*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d]*(
4*a*c*x*(15 - 20*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(7 - 9*c^2*x^2 + 6*c^4*x^4) + 28*b*(1 - c^2*x^2)^(
3/2)*Log[1 - c^2*x^2]))/(24*c^7*d^(5/2)*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])
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Maple [C] Result contains complex when optimal does not.
time = 0.39, size = 2245, normalized size = 7.66
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\(2245\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^6*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
[Out]
1177/8*b*(-d*(c^2*x^2-1))^(1/2)*x^5/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)/c^2*arcsin(c*x)-30
1/24*b*(-d*(c^2*x^2-1))^(1/2)*x^3/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)/c^4*arcsin(c*x)-5/4*
b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^7/d^3/(c^2*x^2-1)*arcsin(c*x)^2+49/12*I*b*(-d*(c^2*x^2-1))^(1/2)
*x/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)/c^6+5/24*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*
c^2*x^2+1)/c^6*x-629/8*b*(-d*(c^2*x^2-1))^(1/2)*x^7/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)*ar
csin(c*x)+1/16*b*(-d*(c^2*x^2-1))^(1/2)/c^7/d^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-11/48*b*(-d*(c^2*x^2-1))^(1/2)/
d^3/(c^4*x^4-2*c^2*x^2+1)/c^7*(-c^2*x^2+1)^(1/2)-1/4*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*x^7+
17/2*I*b*(-d*(c^2*x^2-1))^(1/2)*x^7/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)-10*I*b*(-d*(c^2*x^
2-1))^(1/2)*x^5/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)/c^2*(-c^2*x^2+1)+10*I*b*(-d*(c^2*x^2-1
))^(1/2)*x^3/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)/c^4*(-c^2*x^2+1)-14/3*I*b*(-c^2*x^2+1)^(1
/2)*(-d*(c^2*x^2-1))^(1/2)/c^7/d^3/(c^2*x^2-1)*arcsin(c*x)-7/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x
^2+1)/c^7*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-1/4*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/c^2*(-c^2*x^
2+1)*x^5+3/8*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/c^4*(-c^2*x^2+1)*x^3-1/8*I*b*(-d*(c^2*x^2-1)
)^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/c^6*(-c^2*x^2+1)*x-5/2*a/c^6/d^2*x/(-c^2*d*x^2+d)^(1/2)+5/2*a/c^6/d^2/(c^2*d
)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+5/6*a/c^4*x^3/d/(-c^2*d*x^2+d)^(3/2)-1/2*a*x^5/c^2/d/(-c^
2*d*x^2+d)^(3/2)+5/8*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/c^2*x^5-2/3*I*b*(-d*(c^2*x^2-1))^(1/
2)/d^3/(c^4*x^4-2*c^2*x^2+1)/c^4*x^3-65/4*I*b*(-d*(c^2*x^2-1))^(1/2)*x^5/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*
x^4-64*c^2*x^2-49)/c^2-14/3*I*b*(-d*(c^2*x^2-1))^(1/2)*x^3/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2
-49)/c^4-37/2*b*(-d*(c^2*x^2-1))^(1/2)*x^6/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)/c*(-c^2*x^2
+1)^(1/2)+27*b*(-d*(c^2*x^2-1))^(1/2)*x^4/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)/c^3*(-c^2*x^
2+1)^(1/2)+49/6*b*(-d*(c^2*x^2-1))^(1/2)*x^2/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)/c^5*(-c^2
*x^2+1)^(1/2)-1/4*b*(-d*(c^2*x^2-1))^(1/2)/c^5/d^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2+11/48*b*(-d*(c^2*x^2-1))
^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/c^5*(-c^2*x^2+1)^(1/2)*x^2-343/24*b*(-d*(c^2*x^2-1))^(1/2)*x/d^3/(111*c^8*x^8
-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)/c^6*arcsin(c*x)+7/3*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^7/d^
3/(c^2*x^2-1)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*b*(-d*(c^2*x^2-1))^(1/2)/c^4/d^3/(c^2*x^2-1)*arcsin(c*x)*
x^3+3/8*b*(-d*(c^2*x^2-1))^(1/2)/c^6/d^3/(c^2*x^2-1)*arcsin(c*x)*x-29/12*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4
-2*c^2*x^2+1)/c^6*arcsin(c*x)*x+19/6*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/c^4*arcsin(c*x)*x^3+19
/6*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/c^5*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2-185/2*I*b*(-d*(
c^2*x^2-1))^(1/2)*x^6/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)/c*(-c^2*x^2+1)^(1/2)*arcsin(c*x)
+135*I*b*(-d*(c^2*x^2-1))^(1/2)*x^4/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)/c^3*(-c^2*x^2+1)^(
1/2)*arcsin(c*x)+245/6*I*b*(-d*(c^2*x^2-1))^(1/2)*x^2/d^3/(111*c^8*x^8-384*c^6*x^6+386*c^4*x^4-64*c^2*x^2-49)/
c^5*(-c^2*x^2+1)^(1/2)*arcsin(c*x)
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^6*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")
[Out]
-1/6*a*(3*x^5/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 5*x*(3*x^2/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)
^(3/2)*c^4*d))/c^2 + 5*x/(sqrt(-c^2*d*x^2 + d)*c^6*d^2) - 15*arcsin(c*x)/(c^7*d^(5/2))) + b*integrate(x^6*arct
an2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/
sqrt(d)
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^6*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")
[Out]
integral(-(b*x^6*arcsin(c*x) + a*x^6)*sqrt(-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3)
, x)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**6*(a+b*asin(c*x))/(-c**2*d*x**2+d)**(5/2),x)
[Out]
Integral(x**6*(a + b*asin(c*x))/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^6*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")
[Out]
integrate((b*arcsin(c*x) + a)*x^6/(-c^2*d*x^2 + d)^(5/2), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^6\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^6*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(5/2),x)
[Out]
int((x^6*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(5/2), x)
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