\(\int \frac {(f+g x) (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) [72]

Optimal result

Integrand size = 31, antiderivative size = 171 \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {2 b^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {2 b g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c \sqrt {d-c^2 d x^2}}-\frac {g \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.21, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4861, 4847, 4737, 4767, 4715, 267} \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}}-\frac {g \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {2 a b g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 g x \sqrt {1-c^2 x^2} \arcsin (c x)}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Rule 267

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

Rule 4715

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

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

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.69 \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c}+\frac {f (a+b \arcsin (c x))^3}{3 b}+\frac {2 b g \left (a c x+b \sqrt {1-c^2 x^2}+b c x \arcsin (c x)\right )}{c}\right )}{c \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.69

method	result	size
default	\(\frac {a^{2} f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {a^{2} g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} f}{3 c d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f}{2 c d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arcsin \left (c x \right )+i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arcsin \left (c x \right )-i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\)	\(460\)
parts	\(\frac {a^{2} f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {a^{2} g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} f}{3 c d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f}{2 c d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arcsin \left (c x \right )+i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arcsin \left (c x \right )-i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\)	\(460\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

integrate((g*x+f)*(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 [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.08 \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {b^{2} f \arcsin \left (c x\right )^{3}}{3 \, c \sqrt {d}} + 2 \, b^{2} g {\left (\frac {x \arcsin \left (c x\right )}{c \sqrt {d}} + \frac {\sqrt {-c^{2} x^{2} + 1}}{c^{2} \sqrt {d}}\right )} + \frac {a b f \arcsin \left (c x\right )^{2}}{c \sqrt {d}} + \frac {2 \, a b g x}{c \sqrt {d}} + \frac {a^{2} f \arcsin \left (c x\right )}{c \sqrt {d}} - \frac {\sqrt {-c^{2} d x^{2} + d} b^{2} g \arcsin \left (c x\right )^{2}}{c^{2} d} - \frac {2 \, \sqrt {-c^{2} d x^{2} + d} a b g \arcsin \left (c x\right )}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} + d} a^{2} g}{c^{2} d} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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