\(\int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx\) [32]

Optimal result

Integrand size = 31, antiderivative size = 450 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {1-c^2 x^2}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}}+\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 b c^3 \sqrt {1-c^2 x^2}} \]

[Out]

1/2*f^2*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)-1/8*g^2*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/4*g^2*
x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)-2/3*f*g*(-c^2*x^2+1)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+2/3
*b*f*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-1/4*b*c*f^2*x^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/1
6*b*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-2/9*b*c*f*g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-
1/16*b*c*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/4*f^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/
(-c^2*x^2+1)^(1/2)+1/16*g^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4861, 4847, 4741, 4737, 30, 4767, 4783, 4795} \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2}-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 b c^3 \sqrt {1-c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {1-c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}} \]

[In]

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

[Out]

(2*b*f*g*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[1 - c^2*x^2]) - (b*c*f^2*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^
2]) + (b*g^2*x^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqrt[1 - c^2*x^2]) - (2*b*c*f*g*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[1
- c^2*x^2]) - (b*c*g^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*Ar
cSin[c*x]))/2 - (g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c^2) + (g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*
ArcSin[c*x]))/4 - (2*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c^2) + (f^2*Sqrt[d - c^2*d*
x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2]) + (g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*
c^3*Sqrt[1 - c^2*x^2])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 4741

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

Rule 4783

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

Rule 4847

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

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 {d-c^2 d x^2} \int (f+g x)^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\sqrt {d-c^2 d x^2} \int \left (f^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+2 f g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+g^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right ) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (f^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (2 f g \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {\left (f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c f^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b f g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 c \sqrt {1-c^2 x^2}}+\frac {\left (g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (b c g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt {1-c^2 x^2}} \\ & = \frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}}+\frac {\left (g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{8 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt {1-c^2 x^2}} \\ & = \frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {1-c^2 x^2}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}}+\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 b c^3 \sqrt {1-c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.53 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\frac {\sqrt {d-c^2 d x^2} \left (-36 b c f^2 x^2-9 b c g^2 x^4-\frac {32 b f g x \left (-3+c^2 x^2\right )}{c}+72 f^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+36 g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {96 f g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{c^2}+\frac {36 f^2 (a+b \arcsin (c x))^2}{b c}+\frac {9 g^2 \left (b c^2 x^2-2 c x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{b}\right )}{c^3}\right )}{144 \sqrt {1-c^2 x^2}} \]

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*(-36*b*c*f^2*x^2 - 9*b*c*g^2*x^4 - (32*b*f*g*x*(-3 + c^2*x^2))/c + 72*f^2*x*Sqrt[1 - c^2*
x^2]*(a + b*ArcSin[c*x]) + 36*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - (96*f*g*(1 - c^2*x^2)^(3/2)*(a +
b*ArcSin[c*x]))/c^2 + (36*f^2*(a + b*ArcSin[c*x])^2)/(b*c) + (9*g^2*(b*c^2*x^2 - 2*c*x*Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x]) + (a + b*ArcSin[c*x])^2/b))/c^3))/(144*Sqrt[1 - c^2*x^2])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.50 (sec) , antiderivative size = 973, normalized size of antiderivative = 2.16

method	result	size
default	\(a \left (f^{2} \left (\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )-\frac {2 f g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}\right )+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 (4 c^{2} f^{2}+g^{2}\right )}{16 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) g^{2} \left (4 \arcsin \left (c x \right )+i\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) f g \left (i+3 \arcsin \left (c x \right )\right )}{36 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f^{2} \left (i+2 \arcsin \left (c x \right )\right )}{16 c \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 )}{4 c^{2} \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 )}{4 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f^{2} \left (-i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+4 c^{4} x^{4}-3 i c x \sqrt {-c^{2} x^{2}+1}-5 c^{2} x^{2}+1\right ) f g \left (-i+3 \arcsin \left (c x \right )\right )}{36 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) g^{2} \left (-i+4 \arcsin \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(973\)
parts	\(a \left (f^{2} \left (\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )-\frac {2 f g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}\right )+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 (4 c^{2} f^{2}+g^{2}\right )}{16 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) g^{2} \left (4 \arcsin \left (c x \right )+i\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) f g \left (i+3 \arcsin \left (c x \right )\right )}{36 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f^{2} \left (i+2 \arcsin \left (c x \right )\right )}{16 c \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 )}{4 c^{2} \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 )}{4 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f^{2} \left (-i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+4 c^{4} x^{4}-3 i c x \sqrt {-c^{2} x^{2}+1}-5 c^{2} x^{2}+1\right ) f g \left (-i+3 \arcsin \left (c x \right )\right )}{36 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) g^{2} \left (-i+4 \arcsin \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(973\)

[In]

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

[Out]

a*(f^2*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2)))+g^2*(-1/4
*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/4/c^2*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(
-c^2*d*x^2+d)^(1/2))))-2/3*f*g*(-c^2*d*x^2+d)^(3/2)/c^2/d)+b*(-1/16*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/
c^3/(c^2*x^2-1)*arcsin(c*x)^2*(4*c^2*f^2+g^2)+1/256*(-d*(c^2*x^2-1))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*
c^5*x^5+8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*g^2*(4*arcsin(c*x)+I)/c^3/(c^2*x
^2-1)+1/36*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+3*I*(-c^2*x^2+1)^(1/2)*x
*c+1)*f*g*(I+3*arcsin(c*x))/c^2/(c^2*x^2-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3
*x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*f^2*(I+2*arcsin(c*x))/c/(c^2*x^2-1)-1/4*(-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/(c^2*x^2-1)-1/4*(-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/(c^2*x^2-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2
*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*f^2*(-I+2*arcsin(c*x))/c/(c^2*x^2-1)+1/36*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3
*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*f*g*(-I+3*arcsin(c*x))/c^2/(c^2*x^2-
1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(8*I*(-c^2*x^2+1)^(1/2)*c^4*x^4+8*c^5*x^5-8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^
3*x^3+I*(-c^2*x^2+1)^(1/2)+4*c*x)*g^2*(-I+4*arcsin(c*x))/c^3/(c^2*x^2-1))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )^{2}\, dx \]

[In]

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

[Out]

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))*(f + g*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a*f^2 + 1/8*a*g^2*(sqrt(-c^2*d*x^2 + d)*x/c^2 - 2*(-c^2*d
*x^2 + d)^(3/2)*x/(c^2*d) + sqrt(d)*arcsin(c*x)/c^3) - 2/3*(-c^2*d*x^2 + d)^(3/2)*a*f*g/(c^2*d) + sqrt(d)*inte
grate((b*g^2*x^2 + 2*b*f*g*x + b*f^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)),
x)

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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