\(\int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) [71]

Optimal result

Integrand size = 33, antiderivative size = 410 \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {4 b^2 f g \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 g^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c^3 \sqrt {d-c^2 d x^2}}+\frac {4 b f g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c \sqrt {d-c^2 d x^2}}+\frac {b g^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c \sqrt {d-c^2 d x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}} \]

[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)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4861, 4857, 3398, 3377, 2718, 3392, 32, 2715, 8} \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {f^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {b g^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b^2 g^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c^3 \sqrt {d-c^2 d x^2}}+\frac {4 b^2 f g \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {d-c^2 d x^2}} \]

[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 2715

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] && Integ
erQ[2*n]

Rule 2718

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

Rule 3377

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 3392

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 3398

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 4857

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 4861

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; F
reeQ[{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*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}} \\ & = \frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x)^2 (c f+g \sin (x))^2 \, dx,x,\arcsin (c x)\right )}{c^3 \sqrt {d-c^2 d x^2}} \\ & = \frac {\sqrt {1-c^2 x^2} \text {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,\arcsin (c x)\right )}{c^3 \sqrt {d-c^2 d x^2}} \\ & = \frac {f^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {\left (2 f g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\arcsin (c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\arcsin (c x)\right )}{c^3 \sqrt {d-c^2 d x^2}} \\ & = \frac {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}}-\frac {2 f g \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {\left (4 b f g \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \cos (x) \, dx,x,\arcsin (c x))}{c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \, dx,x,\arcsin (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 ) \text {Subst}\left (\int \sin ^2(x) \, dx,x,\arcsin (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} (a+b \arcsin (c x))}{c \sqrt {d-c^2 d x^2}}+\frac {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}}-\frac {2 f g \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {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}}-\frac {\left (4 b^2 f g \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int \sin (x) \, dx,x,\arcsin (c x))}{c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int 1 \, dx,x,\arcsin (c x))}{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} \arcsin (c x)}{4 c^3 \sqrt {d-c^2 d x^2}}+\frac {4 b f g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c \sqrt {d-c^2 d x^2}}+\frac {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}}-\frac {2 f g \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.98 \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-4 b^2 \sqrt {d} \left (2 c^2 f^2+g^2\right ) \left (-1+c^2 x^2\right ) \arcsin (c x)^3-12 a^2 \left (2 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+6 b \sqrt {d} g \left (-1+c^2 x^2\right ) \arcsin (c x) \left (16 c f \left (-b c x+a \sqrt {1-c^2 x^2}\right )+b g \cos (2 \arcsin (c x))+2 a g \sin (2 \arcsin (c x))\right )+3 \sqrt {d} g \left (-1+c^2 x^2\right ) \left (4 c \left (-8 a b c f x-8 b^2 f \sqrt {1-c^2 x^2}+a^2 (4 f+g x) \sqrt {1-c^2 x^2}\right )+2 a b g \cos (2 \arcsin (c x))-b^2 g \sin (2 \arcsin (c x))\right )+6 b \sqrt {d} \left (-1+c^2 x^2\right ) \arcsin (c x)^2 \left (-2 a \left (2 c^2 f^2+g^2\right )+8 b c f g \sqrt {1-c^2 x^2}+b g^2 \sin (2 \arcsin (c x))\right )}{24 c^3 \sqrt {d} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}} \]

[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])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.70 (sec) , antiderivative size = 928, normalized size of antiderivative = 2.26

method	result	size
default	\(a^{2} \left (\frac {f^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+g^{2} \left (-\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )-\frac {2 f g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} \left (2 c^{2} f^{2}+g^{2}\right )}{6 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) f g \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) f g \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) g^{2}}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \left (2 \arcsin \left (c x \right )^{2}-1\right ) x}{16 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) g^{2} \cos \left (3 \arcsin \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \left (2 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} \left (2 c^{2} f^{2}+g^{2}\right )}{4 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) f g \left (\arcsin \left (c x \right )+i\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) f g \left (\arcsin \left (c x \right )-i\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2}}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \arcsin \left (c x \right ) x}{8 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \cos \left (3 \arcsin \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}\right )\)	\(928\)
parts	\(a^{2} \left (\frac {f^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+g^{2} \left (-\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )-\frac {2 f g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} \left (2 c^{2} f^{2}+g^{2}\right )}{6 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) f g \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) f g \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) g^{2}}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \left (2 \arcsin \left (c x \right )^{2}-1\right ) x}{16 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) g^{2} \cos \left (3 \arcsin \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \left (2 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} \left (2 c^{2} f^{2}+g^{2}\right )}{4 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) f g \left (\arcsin \left (c x \right )+i\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) f g \left (\arcsin \left (c x \right )-i\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2}}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \arcsin \left (c x \right ) x}{8 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \cos \left (3 \arcsin \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}\right )\)	\(928\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]

[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)

Sympy [F(-2)]

Exception generated. \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]

[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 >> Invalid comparison of non-real zoo

Maxima [F]

\[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]

[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)