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

Optimal. Leaf size=450 \[ \frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {f^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b c \sqrt {1-c^2 x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^2}-\frac {g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c^3 \sqrt {1-c^2 x^2}}+\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}+\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {1-c^2 x^2}}+\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}} \]

[Out]

1/2*f^2*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)-1/8*g^2*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/4*g^2*
x^3*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)-2/3*f*g*(-c^2*x^2+1)*(a+b*arccos(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*arccos(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*arccos(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.53, antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4778, 4764, 4648, 4642, 30, 4678, 4698, 4708} \[ \frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {f^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b c \sqrt {1-c^2 x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 c^2}-\frac {g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c^3 \sqrt {1-c^2 x^2}}+\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}+\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[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*A
rcCos[c*x]))/2 - (g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(8*c^2) + (g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b
*ArcCos[c*x]))/4 - (2*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(3*c^2) - (f^2*Sqrt[d - c^2*d
*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2]) - (g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[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 4642

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> -Simp[(a + b*ArcCos[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 4648

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

Rule 4678

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcCos[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*ArcCos[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 4698

Int[((a_.) + ArcCos[(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*ArcCos[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*ArcCos[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*ArcCos[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 4708

Int[(((a_.) + ArcCos[(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*ArcCos[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcCos[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*ArcCos[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 4764

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcCos[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 4778

Int[((a_.) + ArcCos[(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*ArcCos[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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 1.40, size = 320, normalized size = 0.71 \[ \frac {48 a c \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (12 c^2 f^2 x+16 f g \left (c^2 x^2-1\right )+3 g^2 x \left (2 c^2 x^2-1\right )\right )-144 a \sqrt {d} \sqrt {1-c^2 x^2} \left (4 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+144 b c^2 f^2 \sqrt {d-c^2 d x^2} \left (\cos \left (2 \cos ^{-1}(c x)\right )+2 \cos ^{-1}(c x) \left (\sin \left (2 \cos ^{-1}(c x)\right )-\cos ^{-1}(c x)\right )\right )-64 b c f g \sqrt {d-c^2 d x^2} \left (12 \left (1-c^2 x^2\right )^{3/2} \cos ^{-1}(c x)+9 c x-\cos \left (3 \cos ^{-1}(c x)\right )\right )+9 b g^2 \sqrt {d-c^2 d x^2} \left (-8 \cos ^{-1}(c x)^2+\cos \left (4 \cos ^{-1}(c x)\right )+4 \cos ^{-1}(c x) \sin \left (4 \cos ^{-1}(c x)\right )\right )}{1152 c^3 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(48*a*c*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(12*c^2*f^2*x + 16*f*g*(-1 + c^2*x^2) + 3*g^2*x*(-1 + 2*c^2*x^2)
) - 144*a*Sqrt[d]*(4*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2)
)] - 64*b*c*f*g*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*(1 - c^2*x^2)^(3/2)*ArcCos[c*x] - Cos[3*ArcCos[c*x]]) + 144*b*
c^2*f^2*Sqrt[d - c^2*d*x^2]*(Cos[2*ArcCos[c*x]] + 2*ArcCos[c*x]*(-ArcCos[c*x] + Sin[2*ArcCos[c*x]])) + 9*b*g^2
*Sqrt[d - c^2*d*x^2]*(-8*ArcCos[c*x]^2 + Cos[4*ArcCos[c*x]] + 4*ArcCos[c*x]*Sin[4*ArcCos[c*x]]))/(1152*c^3*Sqr
t[1 - c^2*x^2])

________________________________________________________________________________________

fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-c^{2} d x^{2} + d} {\left (a g^{2} x^{2} + 2 \, a f g x + a f^{2} + {\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \arccos \left (c x\right )\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arccos(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)*arccos(c*x)), x
)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arccos(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

________________________________________________________________________________________

maple [B]  time = 1.32, size = 912, normalized size = 2.03 \[ -\frac {a \,g^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a \,g^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a \,g^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{2} \sqrt {c^{2} d}}-\frac {2 a f g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+\frac {a \,f^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a \,f^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f^{2} \sqrt {-c^{2} x^{2}+1}}{8 c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f^{2} \arccos \left (c x \right ) x}{2 \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f^{2}}{4 c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} g^{2}}{16 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} c^{2} \arccos \left (c x \right ) x^{5}}{4 c^{2} x^{2}-4}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \arccos \left (c x \right ) x^{3}}{8 \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \sqrt {-c^{2} x^{2}+1}}{128 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \arccos \left (c x \right ) x}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \arccos \left (c x \right )}{3 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f^{2} c^{2} \arccos \left (c x \right ) x^{3}}{2 c^{2} x^{2}-2}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f^{2} c \sqrt {-c^{2} x^{2}+1}\, x^{2}}{4 \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} c \sqrt {-c^{2} x^{2}+1}\, x^{4}}{16 \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2}}{16 c \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \,c^{2} \arccos \left (c x \right ) x^{4}}{3 \left (c^{2} x^{2}-1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \arccos \left (c x \right ) x^{2}}{3 \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g c \sqrt {-c^{2} x^{2}+1}\, x^{3}}{9 \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \sqrt {-c^{2} x^{2}+1}\, x}{3 c \left (c^{2} x^{2}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a*g^2*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/8*a*g^2/c^2*x*(-c^2*d*x^2+d)^(1/2)+1/8*a*g^2/c^2*d/(c^2*d)^(1/2)*arc
tan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-2/3*a*f*g/c^2/d*(-c^2*d*x^2+d)^(3/2)+1/2*a*f^2*x*(-c^2*d*x^2+d)^(1/2
)+1/2*a*f^2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/8*b*(-d*(c^2*x^2-1))^(1/2)*f^2/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^2/(c^2*x^2-1)*arccos(c*x)*x+1/4*b*(-d*(c^2*x^2-1))
^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arccos(c*x)^2*f^2+1/16*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3
/(c^2*x^2-1)*arccos(c*x)^2*g^2+1/4*b*(-d*(c^2*x^2-1))^(1/2)*g^2*c^2/(c^2*x^2-1)*arccos(c*x)*x^5-3/8*b*(-d*(c^2
*x^2-1))^(1/2)*g^2/(c^2*x^2-1)*arccos(c*x)*x^3-1/128*b*(-d*(c^2*x^2-1))^(1/2)*g^2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)
^(1/2)+1/8*b*(-d*(c^2*x^2-1))^(1/2)*g^2/c^2/(c^2*x^2-1)*arccos(c*x)*x+2/3*b*(-d*(c^2*x^2-1))^(1/2)*f*g/c^2/(c^
2*x^2-1)*arccos(c*x)+1/2*b*(-d*(c^2*x^2-1))^(1/2)*f^2*c^2/(c^2*x^2-1)*arccos(c*x)*x^3-1/4*b*(-d*(c^2*x^2-1))^(
1/2)*f^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-1/16*b*(-d*(c^2*x^2-1))^(1/2)*g^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/
2)*x^4+1/16*b*(-d*(c^2*x^2-1))^(1/2)*g^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2+2/3*b*(-d*(c^2*x^2-1))^(1/2)*f*g
*c^2/(c^2*x^2-1)*arccos(c*x)*x^4-4/3*b*(-d*(c^2*x^2-1))^(1/2)*f*g/(c^2*x^2-1)*arccos(c*x)*x^2-2/9*b*(-d*(c^2*x
^2-1))^(1/2)*f*g*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3+2/3*b*(-d*(c^2*x^2-1))^(1/2)*f*g/c/(c^2*x^2-1)*(-c^2*x^2
+1)^(1/2)*x

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (\sqrt {-c^{2} d x^{2} + d} x + \frac {\sqrt {d} \arcsin \left (c x\right )}{c}\right )} a f^{2} + \frac {1}{8} \, a g^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x}{c^{2}} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{2} d} + \frac {\sqrt {d} \arcsin \left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a f g}{3 \, c^{2} d} + \sqrt {d} \int {\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arccos(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(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x),
x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acos}{\left (c x \right )}\right ) \left (f + g x\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________