3.71 \(\int \frac {(f+g x)^2 (a+b \sin ^{-1}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\)

Optimal. Leaf size=410 \[ \frac {f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d-c^2 d x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f g x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {b g^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {d-c^2 d x^2}}+\frac {4 b^2 f g \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 g^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt {d-c^2 d x^2}} \]

[Out]

4*b^2*f*g*(-c^2*x^2+1)/c^2/(-c^2*d*x^2+d)^(1/2)+1/4*b^2*g^2*x*(-c^2*x^2+1)/c^2/(-c^2*d*x^2+d)^(1/2)-2*f*g*(-c^
2*x^2+1)*(a+b*arcsin(c*x))^2/c^2/(-c^2*d*x^2+d)^(1/2)-1/2*g^2*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/c^2/(-c^2*d*x
^2+d)^(1/2)-1/4*b^2*g^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c^3/(-c^2*d*x^2+d)^(1/2)+4*b*f*g*x*(a+b*arcsin(c*x))*(-
c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)+1/2*b*g^2*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1
/2)+1/3*f^2*(a+b*arcsin(c*x))^3*(-c^2*x^2+1)^(1/2)/b/c/(-c^2*d*x^2+d)^(1/2)+1/6*g^2*(a+b*arcsin(c*x))^3*(-c^2*
x^2+1)^(1/2)/b/c^3/(-c^2*d*x^2+d)^(1/2)

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Rubi [A]  time = 0.55, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4777, 4773, 3317, 3296, 2638, 3311, 32, 2635, 8} \[ \frac {f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d-c^2 d x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f g x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {b g^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt {d-c^2 d x^2}}+\frac {4 b^2 f g \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 g^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b^2*f*g*(1 - c^2*x^2))/(c^2*Sqrt[d - c^2*d*x^2]) + (b^2*g^2*x*(1 - c^2*x^2))/(4*c^2*Sqrt[d - c^2*d*x^2]) -
(b^2*g^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c^3*Sqrt[d - c^2*d*x^2]) + (4*b*f*g*x*Sqrt[1 - c^2*x^2]*(a + b*ArcS
in[c*x]))/(c*Sqrt[d - c^2*d*x^2]) + (b*g^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c*Sqrt[d - c^2*d*x^2]
) - (2*f*g*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c^2*Sqrt[d - c^2*d*x^2]) - (g^2*x*(1 - c^2*x^2)*(a + b*ArcSin
[c*x])^2)/(2*c^2*Sqrt[d - c^2*d*x^2]) + (f^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d - c^2*d*x^
2]) + (g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^2])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

Rule 4773

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol]
:> Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a,
b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 4777

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a +
b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ
[p - 1/2] &&  !GtQ[d, 0]

Rubi steps

\begin {align*} \int \frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int (a+b x)^2 (c f+g \sin (x))^2 \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \left (c^2 f^2 (a+b x)^2+2 c f g (a+b x)^2 \sin (x)+g^2 (a+b x)^2 \sin ^2(x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \sqrt {d-c^2 d x^2}}\\ &=\frac {f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {\left (2 f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \sqrt {d-c^2 d x^2}}\\ &=\frac {b g^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt {d-c^2 d x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {\left (4 b f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \cos (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \, dx,x,\sin ^{-1}(c x)\right )}{2 c^3 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 g^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^3 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f g x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt {d-c^2 d x^2}}+\frac {b g^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt {d-c^2 d x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 g^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 \sqrt {d-c^2 d x^2}}\\ &=\frac {4 b^2 f g \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 g^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt {d-c^2 d x^2}}+\frac {4 b f g x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt {d-c^2 d x^2}}+\frac {b g^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt {d-c^2 d x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]  time = 1.40, size = 400, normalized size = 0.98 \[ \frac {3 \sqrt {d} g \left (c^2 x^2-1\right ) \left (4 c \left (a^2 \sqrt {1-c^2 x^2} (4 f+g x)-8 a b c f x-8 b^2 f \sqrt {1-c^2 x^2}\right )+2 a b g \cos \left (2 \sin ^{-1}(c x)\right )+b^2 (-g) \sin \left (2 \sin ^{-1}(c x)\right )\right )-12 a^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (2 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+6 b \sqrt {d} \left (c^2 x^2-1\right ) \sin ^{-1}(c x)^2 \left (-2 a \left (2 c^2 f^2+g^2\right )+8 b c f g \sqrt {1-c^2 x^2}+b g^2 \sin \left (2 \sin ^{-1}(c x)\right )\right )+6 b \sqrt {d} g \left (c^2 x^2-1\right ) \sin ^{-1}(c x) \left (16 c f \left (a \sqrt {1-c^2 x^2}-b c x\right )+2 a g \sin \left (2 \sin ^{-1}(c x)\right )+b g \cos \left (2 \sin ^{-1}(c x)\right )\right )-4 b^2 \sqrt {d} \left (c^2 x^2-1\right ) \left (2 c^2 f^2+g^2\right ) \sin ^{-1}(c x)^3}{24 c^3 \sqrt {d} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b^2*Sqrt[d]*(2*c^2*f^2 + g^2)*(-1 + c^2*x^2)*ArcSin[c*x]^3 - 12*a^2*(2*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*Sq
rt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 6*b*Sqrt[d]*g*(-1 + c^2*x^2)*Ar
cSin[c*x]*(16*c*f*(-(b*c*x) + a*Sqrt[1 - c^2*x^2]) + b*g*Cos[2*ArcSin[c*x]] + 2*a*g*Sin[2*ArcSin[c*x]]) + 3*Sq
rt[d]*g*(-1 + c^2*x^2)*(4*c*(-8*a*b*c*f*x - 8*b^2*f*Sqrt[1 - c^2*x^2] + a^2*(4*f + g*x)*Sqrt[1 - c^2*x^2]) + 2
*a*b*g*Cos[2*ArcSin[c*x]] - b^2*g*Sin[2*ArcSin[c*x]]) + 6*b*Sqrt[d]*(-1 + c^2*x^2)*ArcSin[c*x]^2*(-2*a*(2*c^2*
f^2 + g^2) + 8*b*c*f*g*Sqrt[1 - c^2*x^2] + b*g^2*Sin[2*ArcSin[c*x]]))/(24*c^3*Sqrt[d]*Sqrt[1 - c^2*x^2]*Sqrt[d
- c^2*d*x^2])

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fricas [F]  time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} g^{2} x^{2} + 2 \, a^{2} f g x + a^{2} f^{2} + {\left (b^{2} g^{2} x^{2} + 2 \, b^{2} f g x + b^{2} f^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b g^{2} x^{2} + 2 \, a b f g x + a b f^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{2} d x^{2} - d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral(-(a^2*g^2*x^2 + 2*a^2*f*g*x + a^2*f^2 + (b^2*g^2*x^2 + 2*b^2*f*g*x + b^2*f^2)*arcsin(c*x)^2 + 2*(a*b*
g^2*x^2 + 2*a*b*f*g*x + a*b*f^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/(c^2*d*x^2 - d), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((g*x + f)^2*(b*arcsin(c*x) + a)^2/sqrt(-c^2*d*x^2 + d), x)

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maple [B]  time = 1.00, size = 1187, normalized size = 2.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)

[Out]

1/4*a*b*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*g^2*arcsin(c*x)*sin(3*arcsin(c*x))-a*b*(-d*(c^2*x^2-1))^(1/2)
*(-c^2*x^2+1)^(1/2)/c/d/(c^2*x^2-1)*arcsin(c*x)^2*f^2-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d/(c
^2*x^2-1)*arcsin(c*x)^3*f^2-1/6*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arcsin(c*x)^3*
g^2+2*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g/c^2/d/(c^2*x^2-1)*arcsin(c*x)^2+1/8*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+
1)^(1/2)/c^3/d/(c^2*x^2-1)*g^2+1/8*a*b*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*g^2*cos(3*arcsin(c*x))-2*b^2*(
-d*(c^2*x^2-1))^(1/2)*f*g/d/(c^2*x^2-1)*arcsin(c*x)^2*x^2+1/8*b^2*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*g^2
*sin(3*arcsin(c*x))*arcsin(c*x)^2+1/8*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*g^2*arcs
in(c*x)+1/8*b^2*(-d*(c^2*x^2-1))^(1/2)/c^2/d/(c^2*x^2-1)*g^2*x*arcsin(c*x)^2+1/8*b^2*(-d*(c^2*x^2-1))^(1/2)/c^
3/d/(c^2*x^2-1)*g^2*arcsin(c*x)*cos(3*arcsin(c*x))+1/2*a^2*g^2/c^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*
d*x^2+d)^(1/2))-2*a^2*f*g/c^2/d*(-c^2*d*x^2+d)^(1/2)-1/2*a^2*g^2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)-1/2*a*b*(-d*(c^2
*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arcsin(c*x)^2*g^2-4*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g/d/(c^2*
x^2-1)*arcsin(c*x)*x^2+4*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g/c^2/d/(c^2*x^2-1)*arcsin(c*x)+1/4*a*b*(-d*(c^2*x^2-1))
^(1/2)/c^2/d/(c^2*x^2-1)*g^2*arcsin(c*x)*x-1/16*b^2*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*g^2*sin(3*arcsin(
c*x))-4*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g/c^2/d/(c^2*x^2-1)+4*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g/d/(c^2*x^2-1)*x^2-1/
16*b^2*(-d*(c^2*x^2-1))^(1/2)/c^2/d/(c^2*x^2-1)*g^2*x+a^2*f^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2
+d)^(1/2))-4*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g/c/d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x-4*a*b*(-d*(c^2*x^
2-1))^(1/2)*f*g/c/d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, a^{2} g^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x}{c^{2} d} - \frac {\arcsin \left (c x\right )}{c^{3} \sqrt {d}}\right )} + \frac {a b f^{2} \arcsin \left (c x\right )^{2}}{c \sqrt {d}} + \frac {4 \, a b f g x}{c \sqrt {d}} + \frac {a^{2} f^{2} \arcsin \left (c x\right )}{c \sqrt {d}} - \frac {4 \, \sqrt {-c^{2} d x^{2} + d} a b f g \arcsin \left (c x\right )}{c^{2} d} - \frac {2 \, \sqrt {-c^{2} d x^{2} + d} a^{2} f g}{c^{2} d} - \sqrt {d} \int \frac {{\left (2 \, a b g^{2} x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + {\left (b^{2} g^{2} x^{2} + 2 \, b^{2} f g x + b^{2} f^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{2} d x^{2} - d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

-1/2*a^2*g^2*(sqrt(-c^2*d*x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3*sqrt(d))) + a*b*f^2*arcsin(c*x)^2/(c*sqrt(d))
+ 4*a*b*f*g*x/(c*sqrt(d)) + a^2*f^2*arcsin(c*x)/(c*sqrt(d)) - 4*sqrt(-c^2*d*x^2 + d)*a*b*f*g*arcsin(c*x)/(c^2*
d) - 2*sqrt(-c^2*d*x^2 + d)*a^2*f*g/(c^2*d) - sqrt(d)*integrate((2*a*b*g^2*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt
(-c*x + 1)) + (b^2*g^2*x^2 + 2*b^2*f*g*x + b^2*f^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2)*sqrt(c*x + 1
)*sqrt(-c*x + 1)/(c^2*d*x^2 - d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^2*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(1/2),x)

[Out]

int(((f + g*x)^2*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(1/2), x)

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sympy [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((g*x+f)**2*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(1/2),x)

[Out]

Exception raised: TypeError

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