\(\int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx\) [44]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4861, 4847, 4737, 4767, 8, 4795, 30} \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {f^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{c^2 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{3 c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{3 c^4 \sqrt {d-c^2 d x^2}}+\frac {3 f g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{4 b c^3 \sqrt {d-c^2 d x^2}}+\frac {3 b f^2 g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {3 b f g^2 x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}}+\frac {b g^3 x^3 \sqrt {1-c^2 x^2}}{9 c \sqrt {d-c^2 d x^2}}+\frac {2 b g^3 x \sqrt {1-c^2 x^2}}{3 c^3 \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Rule 8

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

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

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.76 \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-18 b c \sqrt {d} f \left (2 c^2 f^2+3 g^2\right ) \left (-1+c^2 x^2\right ) \arcsin (c x)^2-36 a c f \left (2 c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-\sqrt {d} g \left (-1+c^2 x^2\right ) \left (8 b c x \left (6 g^2+c^2 \left (27 f^2+g^2 x^2\right )\right )-12 a \sqrt {1-c^2 x^2} \left (4 g^2+c^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )\right )-27 b c f g \cos (2 \arcsin (c x))\right )+6 b \sqrt {d} g \left (-1+c^2 x^2\right ) \arcsin (c x) \left (4 \sqrt {1-c^2 x^2} \left (2 g^2+c^2 \left (9 f^2+g^2 x^2\right )\right )+9 c f g \sin (2 \arcsin (c x))\right )}{72 c^4 \sqrt {d} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

(-18*b*c*Sqrt[d]*f*(2*c^2*f^2 + 3*g^2)*(-1 + c^2*x^2)*ArcSin[c*x]^2 - 36*a*c*f*(2*c^2*f^2 + 3*g^2)*Sqrt[1 - c^
2*x^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - Sqrt[d]*g*(-1 + c^2*x^
2)*(8*b*c*x*(6*g^2 + c^2*(27*f^2 + g^2*x^2)) - 12*a*Sqrt[1 - c^2*x^2]*(4*g^2 + c^2*(18*f^2 + 9*f*g*x + 2*g^2*x
^2)) - 27*b*c*f*g*Cos[2*ArcSin[c*x]]) + 6*b*Sqrt[d]*g*(-1 + c^2*x^2)*ArcSin[c*x]*(4*Sqrt[1 - c^2*x^2]*(2*g^2 +
c^2*(9*f^2 + g^2*x^2)) + 9*c*f*g*Sin[2*ArcSin[c*x]]))/(72*c^4*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 = 0.55 (sec) , antiderivative size = 856, normalized size of antiderivative = 1.90

method	result	size
default	\(a \left (\frac {f^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+g^{3} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+3 f \,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 {3 f^{2} g \sqrt {-c^{2} d \,x^{2}+d}}{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} f \left (2 c^{2} f^{2}+3 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 (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g^{3} \left (i+3 \arcsin \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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 ) g \left (4 \arcsin \left (c x \right ) c^{2} f^{2}+4 i c^{2} f^{2}+\arcsin \left (c x \right ) g^{2}+i g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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 ) g \left (4 \arcsin \left (c x \right ) c^{2} f^{2}-4 i c^{2} f^{2}+\arcsin \left (c x \right ) g^{2}-i g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) g^{3} \left (-i+3 \arcsin \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, f \,g^{2}}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,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 )}\, \arcsin \left (c x \right ) g^{3} \cos \left (4 \arcsin \left (c x \right )\right )}{24 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \sin \left (4 \arcsin \left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \cos \left (3 \arcsin \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,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 )\)	\(856\)
parts	\(a \left (\frac {f^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+g^{3} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+3 f \,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 {3 f^{2} g \sqrt {-c^{2} d \,x^{2}+d}}{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} f \left (2 c^{2} f^{2}+3 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 (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g^{3} \left (i+3 \arcsin \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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 ) g \left (4 \arcsin \left (c x \right ) c^{2} f^{2}+4 i c^{2} f^{2}+\arcsin \left (c x \right ) g^{2}+i g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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 ) g \left (4 \arcsin \left (c x \right ) c^{2} f^{2}-4 i c^{2} f^{2}+\arcsin \left (c x \right ) g^{2}-i g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) g^{3} \left (-i+3 \arcsin \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, f \,g^{2}}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,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 )}\, \arcsin \left (c x \right ) g^{3} \cos \left (4 \arcsin \left (c x \right )\right )}{24 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \sin \left (4 \arcsin \left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \cos \left (3 \arcsin \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,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 )\)	\(856\)

[In]

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

[Out]

a*(f^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+g^3*(-1/3*x^2/c^2/d*(-c^2*d*x^2+d)^(1/2)-2/3
/d/c^4*(-c^2*d*x^2+d)^(1/2))+3*f*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)))-3*f^2*g/c^2/d*(-c^2*d*x^2+d)^(1/2))+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*f*(2*c^2*f^2+3*g^2)+1/144*(-d*(c^2*x^2-1))^(1/2)*(2*c^2*x^2-2*I*c*x*(-c^
2*x^2+1)^(1/2)-1)*g^3*(I+3*arcsin(c*x))/c^4/d/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(
1/2)*x*c-1)*g*(4*arcsin(c*x)*c^2*f^2+4*I*c^2*f^2+arcsin(c*x)*g^2+I*g^2)/c^4/d/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1))
^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*g*(4*arcsin(c*x)*c^2*f^2-4*I*c^2*f^2+arcsin(c*x)*g^2-I*g^2)/c^4/d/
(c^2*x^2-1)+1/144*(-d*(c^2*x^2-1))^(1/2)*(2*I*c*x*(-c^2*x^2+1)^(1/2)+2*c^2*x^2-1)*g^3*(-I+3*arcsin(c*x))/c^4/d
/(c^2*x^2-1)+3/16*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*f*g^2+3/8*(-d*(c^2*x^2-1))^(1/2)
/c^2/d/(c^2*x^2-1)*f*g^2*arcsin(c*x)*x-1/24*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^2*x^2-1)*arcsin(c*x)*g^3*cos(4*arc
sin(c*x))+1/72*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^2*x^2-1)*g^3*sin(4*arcsin(c*x))+3/16*(-d*(c^2*x^2-1))^(1/2)/c^3
/d/(c^2*x^2-1)*f*g^2*cos(3*arcsin(c*x))+3/8*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*f*g^2*arcsin(c*x)*sin(3*a
rcsin(c*x)))

Fricas [F]

\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]

[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)/(c^2*d*x^2 - d), x)

Sympy [F(-2)]

Exception generated. \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [F]

\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \]

[In]

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

[Out]

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