3.220 \(\int x (d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=279 \[ \frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {2 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {16 b^2 d \sqrt {d-c^2 d x^2}}{75 c^2}+\frac {8 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2} \]

[Out]

-1/5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/c^2/d+16/75*b^2*d*(-c^2*d*x^2+d)^(1/2)/c^2+8/225*b^2*d*(-c^2*x^2
+1)*(-c^2*d*x^2+d)^(1/2)/c^2+2/125*b^2*d*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)/c^2+2/5*b*d*x*(a+b*arcsin(c*x))*(
-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-4/15*b*c*d*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1
/2)+2/25*b*c^3*d*x^5*(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 = 0.23, antiderivative size = 279, 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 = {4677, 194, 4645, 12, 1247, 698} \[ \frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}+\frac {2 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {16 b^2 d \sqrt {d-c^2 d x^2}}{75 c^2}+\frac {8 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(16*b^2*d*Sqrt[d - c^2*d*x^2])/(75*c^2) + (8*b^2*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(225*c^2) + (2*b^2*d*(1
- c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(125*c^2) + (2*b*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 -
c^2*x^2]) - (4*b*c*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*x^5*Sqrt
[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(5
*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 194

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 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(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]

Rule 4645

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 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}+\frac {\left (2 b d \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 c \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \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 )^2}{5 c^2 d}-\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {1-c^2 x^2}} \, dx}{5 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \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 )^2}{5 c^2 d}-\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{75 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \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 )^2}{5 c^2 d}-\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{75 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \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 )^2}{5 c^2 d}-\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {8}{\sqrt {1-c^2 x}}+4 \sqrt {1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \sqrt {1-c^2 x^2}}\\ &=\frac {16 b^2 d \sqrt {d-c^2 d x^2}}{75 c^2}+\frac {8 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2}+\frac {2 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \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 )^2}{5 c^2 d}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 159, normalized size = 0.57 \[ \frac {2 b d \sqrt {d-c^2 d x^2} \left (15 a c x \left (3 c^4 x^4-10 c^2 x^2+15\right )+b \sqrt {1-c^2 x^2} \left (9 c^4 x^4-38 c^2 x^2+149\right )+15 b c x \left (3 c^4 x^4-10 c^2 x^2+15\right ) \sin ^{-1}(c x)\right )}{1125 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

-1/5*((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c^2*d) + (2*b*d*Sqrt[d - c^2*d*x^2]*(15*a*c*x*(15 - 10*c^2
*x^2 + 3*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(149 - 38*c^2*x^2 + 9*c^4*x^4) + 15*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4
)*ArcSin[c*x]))/(1125*c^2*Sqrt[1 - c^2*x^2])

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fricas [A]  time = 0.64, size = 295, normalized size = 1.06 \[ -\frac {30 \, {\left (3 \, a b c^{5} d x^{5} - 10 \, a b c^{3} d x^{3} + 15 \, a b c d x + {\left (3 \, b^{2} c^{5} d x^{5} - 10 \, b^{2} c^{3} d x^{3} + 15 \, b^{2} c d x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left (9 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{6} d x^{6} - {\left (675 \, a^{2} - 94 \, b^{2}\right )} c^{4} d x^{4} + {\left (675 \, a^{2} - 374 \, b^{2}\right )} c^{2} d x^{2} + 225 \, {\left (b^{2} c^{6} d x^{6} - 3 \, b^{2} c^{4} d x^{4} + 3 \, b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} - {\left (225 \, a^{2} - 298 \, b^{2}\right )} d + 450 \, {\left (a b c^{6} d x^{6} - 3 \, a b c^{4} d x^{4} + 3 \, a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{1125 \, {\left (c^{4} x^{2} - c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

-1/1125*(30*(3*a*b*c^5*d*x^5 - 10*a*b*c^3*d*x^3 + 15*a*b*c*d*x + (3*b^2*c^5*d*x^5 - 10*b^2*c^3*d*x^3 + 15*b^2*
c*d*x)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + (9*(25*a^2 - 2*b^2)*c^6*d*x^6 - (675*a^2 - 94*b^
2)*c^4*d*x^4 + (675*a^2 - 374*b^2)*c^2*d*x^2 + 225*(b^2*c^6*d*x^6 - 3*b^2*c^4*d*x^4 + 3*b^2*c^2*d*x^2 - b^2*d)
*arcsin(c*x)^2 - (225*a^2 - 298*b^2)*d + 450*(a*b*c^6*d*x^6 - 3*a*b*c^4*d*x^4 + 3*a*b*c^2*d*x^2 - a*b*d)*arcsi
n(c*x))*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [C]  time = 0.39, size = 1151, normalized size = 4.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x)

[Out]

-1/5*a^2/c^2/d*(-c^2*d*x^2+d)^(5/2)+b^2*(-1/4000*(-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/c^2/(c^2*x^2-1)+1/288*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*(-c^2*x^2+1)^(1/2)*x
^3*c^3+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)*d/c^2/(c^2*x^2-1)-1/16*(-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/c^2/(c^2*x^2-1)-1/16*(-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/c^2/(c^2*x^2-1)+1/288
*(-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*arcsin(c*x)+9*arcsin(c*x)^2-2)*d/c^2/(c^2*x^2-1)-1/4000*(-d*(c^2*x^2-1))^(1/2)*(16*I*(-c^2*x^2+1)^(1/2)*x^5*
c^5+16*c^6*x^6-20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-28*c^4*x^4+5*I*(-c^2*x^2+1)^(1/2)*x*c+13*c^2*x^2-1)*(-10*I*arcs
in(c*x)+25*arcsin(c*x)^2-2)*d/c^2/(c^2*x^2-1))+2*a*b*(-1/800*(-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)*(I+5*arc
sin(c*x))*d/c^2/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(I+arcsin(c*x))*d
/c^2/(c^2*x^2-1)-1/16*(-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/c^2/(c^2*x
^2-1)+1/96*(-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/c^2/(c^2*x^2-1)-1/1200*(-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+45*arcsin(c*x))*cos(4*arcsin(c*x))*d/c^2/(c^2*x^2-1)-1/600*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x
^2+1)^(1/2)-I)*(7*I+15*arcsin(c*x))*sin(4*arcsin(c*x))*d/c^2/(c^2*x^2-1))

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maxima [A]  time = 0.94, size = 236, normalized size = 0.85 \[ -\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b^{2} \arcsin \left (c x\right )^{2}}{5 \, c^{2} d} + \frac {2}{1125} \, b^{2} {\left (\frac {9 \, \sqrt {-c^{2} x^{2} + 1} c^{2} d^{\frac {5}{2}} x^{4} - 38 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {5}{2}} x^{2} + \frac {149 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {5}{2}}}{c^{2}}}{d} + \frac {15 \, {\left (3 \, c^{4} d^{\frac {5}{2}} x^{5} - 10 \, c^{2} d^{\frac {5}{2}} x^{3} + 15 \, d^{\frac {5}{2}} x\right )} \arcsin \left (c x\right )}{c d}\right )} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a b \arcsin \left (c x\right )}{5 \, c^{2} d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a^{2}}{5 \, c^{2} d} + \frac {2 \, {\left (3 \, c^{4} d^{\frac {5}{2}} x^{5} - 10 \, c^{2} d^{\frac {5}{2}} x^{3} + 15 \, d^{\frac {5}{2}} x\right )} a b}{75 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

-1/5*(-c^2*d*x^2 + d)^(5/2)*b^2*arcsin(c*x)^2/(c^2*d) + 2/1125*b^2*((9*sqrt(-c^2*x^2 + 1)*c^2*d^(5/2)*x^4 - 38
*sqrt(-c^2*x^2 + 1)*d^(5/2)*x^2 + 149*sqrt(-c^2*x^2 + 1)*d^(5/2)/c^2)/d + 15*(3*c^4*d^(5/2)*x^5 - 10*c^2*d^(5/
2)*x^3 + 15*d^(5/2)*x)*arcsin(c*x)/(c*d)) - 2/5*(-c^2*d*x^2 + d)^(5/2)*a*b*arcsin(c*x)/(c^2*d) - 1/5*(-c^2*d*x
^2 + d)^(5/2)*a^2/(c^2*d) + 2/75*(3*c^4*d^(5/2)*x^5 - 10*c^2*d^(5/2)*x^3 + 15*d^(5/2)*x)*a*b/(c*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

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)^(3/2),x)

[Out]

int(x*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x))**2,x)

[Out]

Integral(x*(-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x))**2, x)

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