3.3.28 \(\int x (d-c^2 d x^2)^{5/2} (a+b \text {ArcSin}(c x))^2 \, dx\) [228]

Optimal. Leaf size=382 \[ \frac {32 b^2 d^2 \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2}+\frac {12 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{49 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {ArcSin}(c x))^2}{7 c^2 d} \]

[Out]

-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arcsin(c*x))^2/c^2/d+32/245*b^2*d^2*(-c^2*d*x^2+d)^(1/2)/c^2+16/735*b^2*d^2*(-c
^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^2+12/1225*b^2*d^2*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)/c^2+2/343*b^2*d^2*(-c^2
*x^2+1)^3*(-c^2*d*x^2+d)^(1/2)/c^2+2/7*b*d^2*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-2/7
*b*c*d^2*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+6/35*b*c^3*d^2*x^5*(a+b*arcsin(c*x))*(-
c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2/49*b*c^5*d^2*x^7*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(
1/2)

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Rubi [A]
time = 0.20, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4767, 200, 4739, 12, 1813, 1864} \begin {gather*} \frac {2 b d^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{7 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {ArcSin}(c x))^2}{7 c^2 d}-\frac {2 b c^5 d^2 x^7 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{49 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{35 \sqrt {1-c^2 x^2}}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {32 b^2 d^2 \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {12 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(32*b^2*d^2*Sqrt[d - c^2*d*x^2])/(245*c^2) + (16*b^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(735*c^2) + (12*b^
2*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1225*c^2) + (2*b^2*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c
^2) + (2*b*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c*Sqrt[1 - c^2*x^2]) - (2*b*c*d^2*x^3*Sqrt[d - c^
2*d*x^2]*(a + b*ArcSin[c*x]))/(7*Sqrt[1 - c^2*x^2]) + (6*b*c^3*d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])
)/(35*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(49*Sqrt[1 - c^2*x^2]) -
((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x])^2)/(7*c^2*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 4739

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*
d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}+\frac {\left (2 b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{7 c \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}-\frac {\left (2 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt {1-c^2 x^2}} \, dx}{7 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}-\frac {\left (2 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt {1-c^2 x^2}} \, dx}{245 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}-\frac {\left (b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}-\frac {\left (b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {16}{\sqrt {1-c^2 x}}+8 \sqrt {1-c^2 x}+6 \left (1-c^2 x\right )^{3/2}+5 \left (1-c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}}\\ &=\frac {32 b^2 d^2 \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2}+\frac {12 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 226, normalized size = 0.59 \begin {gather*} \frac {d^2 \sqrt {d-c^2 d x^2} \left (3675 a^2 \left (-1+c^2 x^2\right )^4+210 a b c x \sqrt {1-c^2 x^2} \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )-2 b^2 \left (2161-2918 c^2 x^2+1108 c^4 x^4-426 c^6 x^6+75 c^8 x^8\right )+210 b \left (35 a \left (-1+c^2 x^2\right )^4+b c x \sqrt {1-c^2 x^2} \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )\right ) \text {ArcSin}(c x)+3675 b^2 \left (-1+c^2 x^2\right )^4 \text {ArcSin}(c x)^2\right )}{25725 c^2 \left (-1+c^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(3675*a^2*(-1 + c^2*x^2)^4 + 210*a*b*c*x*Sqrt[1 - c^2*x^2]*(-35 + 35*c^2*x^2 - 21*c^4
*x^4 + 5*c^6*x^6) - 2*b^2*(2161 - 2918*c^2*x^2 + 1108*c^4*x^4 - 426*c^6*x^6 + 75*c^8*x^8) + 210*b*(35*a*(-1 +
c^2*x^2)^4 + b*c*x*Sqrt[1 - c^2*x^2]*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6))*ArcSin[c*x] + 3675*b^2*(-1 +
c^2*x^2)^4*ArcSin[c*x]^2))/(25725*c^2*(-1 + c^2*x^2))

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Maple [C] Result contains complex when optimal does not.
time = 0.32, size = 1611, normalized size = 4.22

method	result	size
default	\(\text {Expression too large to display}\)	\(1611\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

-1/7*a^2/c^2/d*(-c^2*d*x^2+d)^(7/2)+b^2*(1/43904*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6-64*I*(-c^2*x^2
+1)^(1/2)*x^7*c^7+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*c^2*x^2-56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*
(-c^2*x^2+1)^(1/2)*x*c+1)*(14*I*arcsin(c*x)+49*arcsin(c*x)^2-2)*d^2/c^2/(c^2*x^2-1)-1/3200*(-d*(c^2*x^2-1))^(1
/2)*(16*c^6*x^6-28*c^4*x^4-16*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-5*I*(-c^
2*x^2+1)^(1/2)*x*c-1)*(10*I*arcsin(c*x)+25*arcsin(c*x)^2-2)*d^2/c^2/(c^2*x^2-1)-5/128*(-d*(c^2*x^2-1))^(1/2)*(
c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)^2-2+2*I*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-5/128*(-d*(c^2*x^2-1
))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)^2-2-2*I*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)+1/384*(-d*
(c^2*x^2-1))^(1/2)*(4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*(-6*I*arc
sin(c*x)+9*arcsin(c*x)^2-2)*d^2/c^2/(c^2*x^2-1)+1/43904*(-d*(c^2*x^2-1))^(1/2)*(64*I*(-c^2*x^2+1)^(1/2)*x^7*c^
7+64*c^8*x^8-112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-144*c^6*x^6+56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+104*c^4*x^4-7*I*(-c^
2*x^2+1)^(1/2)*x*c-25*c^2*x^2+1)*(-14*I*arcsin(c*x)+49*arcsin(c*x)^2-2)*d^2/c^2/(c^2*x^2-1)-1/2400*(-d*(c^2*x^
2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(30*I*arcsin(c*x)+75*arcsin(c*x)^2-14)*cos(4*arcsin(c*x))*d^2
/c^2/(c^2*x^2-1)-1/4800*(-d*(c^2*x^2-1))^(1/2)*(I*x^2*c^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(90*I*arcsin(c*x)+75*arcsi
n(c*x)^2-22)*sin(4*arcsin(c*x))*d^2/c^2/(c^2*x^2-1))+2*a*b*(1/6272*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*
x^6-64*I*(-c^2*x^2+1)^(1/2)*x^7*c^7+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*c^2*x^2-56*I*(-c^2*x^2+1)^
(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(I+7*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-5/128*(-d*(c^2*x^2-1))^(1/2)
*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)+I)*d^2/c^2/(c^2*x^2-1)-5/128*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^
2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)-I)*d^2/c^2/(c^2*x^2-1)+1/128*(-d*(c^2*x^2-1))^(1/2)*(4*I*(-c^2*x^2+
1)^(1/2)*x^3*c^3+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*(-I+3*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-1/78
40*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(11*I+70*arcsin(c*x))*cos(6*arcsin(c*x))*d^2/c^
2/(c^2*x^2-1)-3/15680*(-d*(c^2*x^2-1))^(1/2)*(I*x^2*c^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(9*I+35*arcsin(c*x))*sin(6*a
rcsin(c*x))*d^2/c^2/(c^2*x^2-1)-1/160*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(I+5*arcsin(
c*x))*cos(4*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-1/320*(-d*(c^2*x^2-1))^(1/2)*(I*x^2*c^2-c*x*(-c^2*x^2+1)^(1/2)-I)
*(3*I+5*arcsin(c*x))*sin(4*arcsin(c*x))*d^2/c^2/(c^2*x^2-1))

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Maxima [A]
time = 0.51, size = 281, normalized size = 0.74 \begin {gather*} -\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} b^{2} \arcsin \left (c x\right )^{2}}{7 \, c^{2} d} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a b \arcsin \left (c x\right )}{7 \, c^{2} d} - \frac {2}{25725} \, b^{2} {\left (\frac {75 \, \sqrt {-c^{2} x^{2} + 1} c^{4} d^{\frac {7}{2}} x^{6} - 351 \, \sqrt {-c^{2} x^{2} + 1} c^{2} d^{\frac {7}{2}} x^{4} + 757 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {7}{2}} x^{2} - \frac {2161 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {7}{2}}}{c^{2}}}{d} + \frac {105 \, {\left (5 \, c^{6} d^{\frac {7}{2}} x^{7} - 21 \, c^{4} d^{\frac {7}{2}} x^{5} + 35 \, c^{2} d^{\frac {7}{2}} x^{3} - 35 \, d^{\frac {7}{2}} x\right )} \arcsin \left (c x\right )}{c d}\right )} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a^{2}}{7 \, c^{2} d} - \frac {2 \, {\left (5 \, c^{6} d^{\frac {7}{2}} x^{7} - 21 \, c^{4} d^{\frac {7}{2}} x^{5} + 35 \, c^{2} d^{\frac {7}{2}} x^{3} - 35 \, d^{\frac {7}{2}} x\right )} a b}{245 \, c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

-1/7*(-c^2*d*x^2 + d)^(7/2)*b^2*arcsin(c*x)^2/(c^2*d) - 2/7*(-c^2*d*x^2 + d)^(7/2)*a*b*arcsin(c*x)/(c^2*d) - 2
/25725*b^2*((75*sqrt(-c^2*x^2 + 1)*c^4*d^(7/2)*x^6 - 351*sqrt(-c^2*x^2 + 1)*c^2*d^(7/2)*x^4 + 757*sqrt(-c^2*x^
2 + 1)*d^(7/2)*x^2 - 2161*sqrt(-c^2*x^2 + 1)*d^(7/2)/c^2)/d + 105*(5*c^6*d^(7/2)*x^7 - 21*c^4*d^(7/2)*x^5 + 35
*c^2*d^(7/2)*x^3 - 35*d^(7/2)*x)*arcsin(c*x)/(c*d)) - 1/7*(-c^2*d*x^2 + d)^(7/2)*a^2/(c^2*d) - 2/245*(5*c^6*d^
(7/2)*x^7 - 21*c^4*d^(7/2)*x^5 + 35*c^2*d^(7/2)*x^3 - 35*d^(7/2)*x)*a*b/(c*d)

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Fricas [A]
time = 1.82, size = 405, normalized size = 1.06 \begin {gather*} \frac {210 \, {\left (5 \, a b c^{7} d^{2} x^{7} - 21 \, a b c^{5} d^{2} x^{5} + 35 \, a b c^{3} d^{2} x^{3} - 35 \, a b c d^{2} x + {\left (5 \, b^{2} c^{7} d^{2} x^{7} - 21 \, b^{2} c^{5} d^{2} x^{5} + 35 \, b^{2} c^{3} d^{2} x^{3} - 35 \, b^{2} c d^{2} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left (75 \, {\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{8} d^{2} x^{8} - 12 \, {\left (1225 \, a^{2} - 71 \, b^{2}\right )} c^{6} d^{2} x^{6} + 2 \, {\left (11025 \, a^{2} - 1108 \, b^{2}\right )} c^{4} d^{2} x^{4} - 4 \, {\left (3675 \, a^{2} - 1459 \, b^{2}\right )} c^{2} d^{2} x^{2} + {\left (3675 \, a^{2} - 4322 \, b^{2}\right )} d^{2} + 3675 \, {\left (b^{2} c^{8} d^{2} x^{8} - 4 \, b^{2} c^{6} d^{2} x^{6} + 6 \, b^{2} c^{4} d^{2} x^{4} - 4 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 7350 \, {\left (a b c^{8} d^{2} x^{8} - 4 \, a b c^{6} d^{2} x^{6} + 6 \, a b c^{4} d^{2} x^{4} - 4 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{25725 \, {\left (c^{4} x^{2} - c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

1/25725*(210*(5*a*b*c^7*d^2*x^7 - 21*a*b*c^5*d^2*x^5 + 35*a*b*c^3*d^2*x^3 - 35*a*b*c*d^2*x + (5*b^2*c^7*d^2*x^
7 - 21*b^2*c^5*d^2*x^5 + 35*b^2*c^3*d^2*x^3 - 35*b^2*c*d^2*x)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2
+ 1) + (75*(49*a^2 - 2*b^2)*c^8*d^2*x^8 - 12*(1225*a^2 - 71*b^2)*c^6*d^2*x^6 + 2*(11025*a^2 - 1108*b^2)*c^4*d^
2*x^4 - 4*(3675*a^2 - 1459*b^2)*c^2*d^2*x^2 + (3675*a^2 - 4322*b^2)*d^2 + 3675*(b^2*c^8*d^2*x^8 - 4*b^2*c^6*d^
2*x^6 + 6*b^2*c^4*d^2*x^4 - 4*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsin(c*x)^2 + 7350*(a*b*c^8*d^2*x^8 - 4*a*b*c^6*d^2
*x^6 + 6*a*b*c^4*d^2*x^4 - 4*a*b*c^2*d^2*x^2 + a*b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**2,x)

[Out]

Integral(x*(-d*(c*x - 1)*(c*x + 1))**(5/2)*(a + b*asin(c*x))**2, 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(x*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,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 x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\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*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2),x)

[Out]

int(x*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2), x)

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