3.31 \(\int (f+g x)^3 \sqrt{d-c^2 d x^2} (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=669 \[ -\frac{f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^2}+\frac{1}{2} f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{3}{4} f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{3 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{3 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1-c^2 x^2}}+\frac{g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4}-\frac{g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4}-\frac{b c f^2 g x^3 \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}+\frac{b f^2 g x \sqrt{d-c^2 d x^2}}{c \sqrt{1-c^2 x^2}}-\frac{b c f^3 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}-\frac{3 b c f g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{3 b f g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}}-\frac{b c g^3 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}+\frac{b g^3 x^3 \sqrt{d-c^2 d x^2}}{45 c \sqrt{1-c^2 x^2}}+\frac{2 b g^3 x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{1-c^2 x^2}} \]

[Out]

(b*f^2*g*x*Sqrt[d - c^2*d*x^2])/(c*Sqrt[1 - c^2*x^2]) + (2*b*g^3*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[1 - c^2*x
^2]) - (b*c*f^3*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2]) + (3*b*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqr
t[1 - c^2*x^2]) - (b*c*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(3*Sqrt[1 - c^2*x^2]) + (b*g^3*x^3*Sqrt[d - c^2*d*x^2])/
(45*c*Sqrt[1 - c^2*x^2]) - (3*b*c*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (b*c*g^3*x^5*Sqrt[d
- c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) + (f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/2 - (3*f*g^2*x*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c^2) + (3*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/4 - (f^2*g*(1
- c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/c^2 - (g^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSi
n[c*x]))/(3*c^4) + (g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c^4) + (f^3*Sqrt[d - c^2*d
*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2]) + (3*f*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])

________________________________________________________________________________________

Rubi [A]  time = 0.69255, antiderivative size = 669, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387, Rules used = {4777, 4763, 4647, 4641, 30, 4677, 4697, 4707, 266, 43, 4689, 12} \[ -\frac{f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^2}+\frac{1}{2} f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{3}{4} f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{3 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{3 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1-c^2 x^2}}+\frac{g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4}-\frac{g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4}-\frac{b c f^2 g x^3 \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}+\frac{b f^2 g x \sqrt{d-c^2 d x^2}}{c \sqrt{1-c^2 x^2}}-\frac{b c f^3 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}-\frac{3 b c f g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{3 b f g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}}-\frac{b c g^3 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}+\frac{b g^3 x^3 \sqrt{d-c^2 d x^2}}{45 c \sqrt{1-c^2 x^2}}+\frac{2 b g^3 x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*f^2*g*x*Sqrt[d - c^2*d*x^2])/(c*Sqrt[1 - c^2*x^2]) + (2*b*g^3*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[1 - c^2*x
^2]) - (b*c*f^3*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2]) + (3*b*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqr
t[1 - c^2*x^2]) - (b*c*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(3*Sqrt[1 - c^2*x^2]) + (b*g^3*x^3*Sqrt[d - c^2*d*x^2])/
(45*c*Sqrt[1 - c^2*x^2]) - (3*b*c*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (b*c*g^3*x^5*Sqrt[d
- c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) + (f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/2 - (3*f*g^2*x*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c^2) + (3*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/4 - (f^2*g*(1
- c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/c^2 - (g^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSi
n[c*x]))/(3*c^4) + (g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c^4) + (f^3*Sqrt[d - c^2*d
*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2]) + (3*f*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 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]

Rule 4763

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 4647

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[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(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 4641

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

Rule 30

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

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]

Rule 4697

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[Sqrt[d + e*x^2]/((m + 2)*Sqrt[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*Sqrt[d + e*x^2])/(f*(m
+ 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] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 4707

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

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4689

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(1 - c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcSin[c*x]), u, x] - Dist[b*c*d^p, Int[SimplifyIntegrand[u/Sqrt[1 - c
^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2
, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d, 0]

Rule 12

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

Rubi steps

\begin{align*} \int (f+g x)^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{\sqrt{d-c^2 d x^2} \int (f+g x)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{d-c^2 d x^2} \int \left (f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+3 f^2 g x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+3 f g^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+g^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (f^3 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 f^2 g \sqrt{d-c^2 d x^2}\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (g^3 \sqrt{d-c^2 d x^2}\right ) \int x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{2} f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{4} f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^2}-\frac{g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4}+\frac{g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4}+\frac{\left (f^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}-\frac{\left (b c f^3 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt{1-c^2 x^2}}+\frac{\left (b f^2 g \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{c \sqrt{1-c^2 x^2}}+\frac{\left (3 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{4 \sqrt{1-c^2 x^2}}-\frac{\left (3 b c f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt{1-c^2 x^2}}-\frac{\left (b c g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{b f^2 g x \sqrt{d-c^2 d x^2}}{c \sqrt{1-c^2 x^2}}-\frac{b c f^3 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}-\frac{b c f^2 g x^3 \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}-\frac{3 b c f g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{1}{2} f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{3 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{3}{4} f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^2}-\frac{g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4}+\frac{g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4}+\frac{f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{\left (3 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{8 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (3 b f g^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt{1-c^2 x^2}}-\frac{\left (b g^3 \sqrt{d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{15 c^3 \sqrt{1-c^2 x^2}}\\ &=\frac{b f^2 g x \sqrt{d-c^2 d x^2}}{c \sqrt{1-c^2 x^2}}+\frac{2 b g^3 x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{1-c^2 x^2}}-\frac{b c f^3 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}+\frac{3 b f g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}}-\frac{b c f^2 g x^3 \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}+\frac{b g^3 x^3 \sqrt{d-c^2 d x^2}}{45 c \sqrt{1-c^2 x^2}}-\frac{3 b c f g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{b c g^3 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}+\frac{1}{2} f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{3 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{3}{4} f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^2}-\frac{g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4}+\frac{g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4}+\frac{f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{3 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.443516, size = 356, normalized size = 0.53 \[ \frac{\sqrt{d-c^2 d x^2} \left (225 a^2 \left (4 c^3 f^3+3 c f g^2\right )+30 a b \sqrt{1-c^2 x^2} \left (6 c^4 x \left (20 f^2 g x+10 f^3+15 f g^2 x^2+4 g^3 x^3\right )-c^2 g \left (120 f^2+45 f g x+8 g^2 x^2\right )-16 g^3\right )+30 b \sin ^{-1}(c x) \left (15 a \left (4 c^3 f^3+3 c f g^2\right )+b \sqrt{1-c^2 x^2} \left (6 c^4 x \left (20 f^2 g x+10 f^3+15 f g^2 x^2+4 g^3 x^3\right )-c^2 g \left (120 f^2+45 f g x+8 g^2 x^2\right )-16 g^3\right )\right )+b^2 c x \left (-3 c^4 x \left (400 f^2 g x+300 f^3+225 f g^2 x^2+48 g^3 x^3\right )+5 c^2 g \left (720 f^2+135 f g x+16 g^2 x^2\right )+480 g^3\right )+225 b^2 c f \left (4 c^2 f^2+3 g^2\right ) \sin ^{-1}(c x)^2\right )}{3600 b c^4 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*(225*a^2*(4*c^3*f^3 + 3*c*f*g^2) + 30*a*b*Sqrt[1 - c^2*x^2]*(-16*g^3 - c^2*g*(120*f^2 + 4
5*f*g*x + 8*g^2*x^2) + 6*c^4*x*(10*f^3 + 20*f^2*g*x + 15*f*g^2*x^2 + 4*g^3*x^3)) + b^2*c*x*(480*g^3 + 5*c^2*g*
(720*f^2 + 135*f*g*x + 16*g^2*x^2) - 3*c^4*x*(300*f^3 + 400*f^2*g*x + 225*f*g^2*x^2 + 48*g^3*x^3)) + 30*b*(15*
a*(4*c^3*f^3 + 3*c*f*g^2) + b*Sqrt[1 - c^2*x^2]*(-16*g^3 - c^2*g*(120*f^2 + 45*f*g*x + 8*g^2*x^2) + 6*c^4*x*(1
0*f^3 + 20*f^2*g*x + 15*f*g^2*x^2 + 4*g^3*x^3)))*ArcSin[c*x] + 225*b^2*c*f*(4*c^2*f^2 + 3*g^2)*ArcSin[c*x]^2))
/(3600*b*c^4*Sqrt[1 - c^2*x^2])

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Maple [B]  time = 0.792, size = 1286, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^4/(c^2*x^2-1)*arcsin(c*x)-4/15*b*(-d*(c^2*x^2-1))^(1/2)*g^3/(c^2*x^2-1)*ar
csin(c*x)*x^4-1/8*b*(-d*(c^2*x^2-1))^(1/2)*f^3/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-1/2*b*(-d*(c^2*x^2-1))^(1/2)*f
^3/(c^2*x^2-1)*arcsin(c*x)*x-1/5*a*g^3*x^2*(-c^2*d*x^2+d)^(3/2)/c^2/d+3/8*a*f*g^2/c^2*x*(-c^2*d*x^2+d)^(1/2)-a
*f^2*g/c^2/d*(-c^2*d*x^2+d)^(3/2)-2/15*a*g^3/d/c^4*(-c^2*d*x^2+d)^(3/2)-3/16*b*(-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*f*g^2+3/4*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*c^2/(c^2*x^2-1)*arcsin(c*x)*
x^5+3/8*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c^2/(c^2*x^2-1)*arcsin(c*x)*x+b*(-d*(c^2*x^2-1))^(1/2)*g*c^2/(c^2*x^2-1
)*arcsin(c*x)*x^4*f^2+1/3*b*(-d*(c^2*x^2-1))^(1/2)*g*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3*f^2-b*(-d*(c^2*x^2-1
))^(1/2)*g/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x*f^2+3/16*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*c/(c^2*x^2-1)*(-c^2*x^2+
1)^(1/2)*x^4-3/16*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-3/4*a*f*g^2*x*(-c^2*d*x^
2+d)^(3/2)/c^2/d+3/8*a*f*g^2/c^2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/25*b*(-d*(c^2*
x^2-1))^(1/2)*g^3*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^5-1/45*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c/(c^2*x^2-1)*(-c^2*x
^2+1)^(1/2)*x^3-2/15*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-9/8*b*(-d*(c^2*x^2-1))^
(1/2)*f*g^2/(c^2*x^2-1)*arcsin(c*x)*x^3+3/128*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2
)-2*b*(-d*(c^2*x^2-1))^(1/2)*g/(c^2*x^2-1)*arcsin(c*x)*x^2*f^2+1/4*b*(-d*(c^2*x^2-1))^(1/2)*f^3*c/(c^2*x^2-1)*
(-c^2*x^2+1)^(1/2)*x^2-1/4*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arcsin(c*x)^2*f^3+1/5*b*(
-d*(c^2*x^2-1))^(1/2)*g^3*c^2/(c^2*x^2-1)*arcsin(c*x)*x^6-1/15*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^2/(c^2*x^2-1)*ar
csin(c*x)*x^2+b*(-d*(c^2*x^2-1))^(1/2)*g/c^2/(c^2*x^2-1)*arcsin(c*x)*f^2+1/2*b*(-d*(c^2*x^2-1))^(1/2)*f^3*c^2/
(c^2*x^2-1)*arcsin(c*x)*x^3+1/2*a*f^3*x*(-c^2*d*x^2+d)^(1/2)+1/2*a*f^3*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/
(-c^2*d*x^2+d)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a g^{3} x^{3} + 3 \, a f g^{2} x^{2} + 3 \, a f^{2} g x + a f^{3} +{\left (b g^{3} x^{3} + 3 \, b f g^{2} x^{2} + 3 \, b f^{2} g x + b f^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral((a*g^3*x^3 + 3*a*f*g^2*x^2 + 3*a*f^2*g*x + a*f^3 + (b*g^3*x^3 + 3*b*f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*
arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-c^{2} d x^{2} + d}{\left (g x + f\right )}^{3}{\left (b \arcsin \left (c x\right ) + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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