3.6 \(\int (f+g x)^3 (d-c^2 d x^2)^{3/2} (a+b \cos ^{-1}(c x)) \, dx\)

Optimal. Leaf size=959 \[ -\frac {b c^3 d g^3 \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f g^2 \sqrt {d-c^2 d x^2} x^6}{12 \sqrt {1-c^2 x^2}}+\frac {8 b c d g^3 \sqrt {d-c^2 d x^2} x^5}{175 \sqrt {1-c^2 x^2}}-\frac {3 b c^3 d f^2 g \sqrt {d-c^2 d x^2} x^5}{25 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f^3 \sqrt {d-c^2 d x^2} x^4}{16 \sqrt {1-c^2 x^2}}+\frac {7 b c d f g^2 \sqrt {d-c^2 d x^2} x^4}{32 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac {1}{2} d f g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3-\frac {b d g^3 \sqrt {d-c^2 d x^2} x^3}{105 c \sqrt {1-c^2 x^2}}+\frac {2 b c d f^2 g \sqrt {d-c^2 d x^2} x^3}{5 \sqrt {1-c^2 x^2}}+\frac {5 b c d f^3 \sqrt {d-c^2 d x^2} x^2}{16 \sqrt {1-c^2 x^2}}-\frac {3 b d f g^2 \sqrt {d-c^2 d x^2} x^2}{32 c \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x}{16 c^2}+\frac {1}{4} d f^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac {2 b d g^3 \sqrt {d-c^2 d x^2} x}{35 c^3 \sqrt {1-c^2 x^2}}-\frac {3 b d f^2 g \sqrt {d-c^2 d x^2} x}{5 c \sqrt {1-c^2 x^2}}-\frac {3 d f^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1-c^2 x^2}}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2} \]

[Out]

3/8*d*f^3*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)-3/16*d*f*g^2*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+3
/8*d*f*g^2*x^3*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)+1/4*d*f^3*x*(-c^2*x^2+1)*(a+b*arccos(c*x))*(-c^2*d*x^2+d
)^(1/2)+1/2*d*f*g^2*x^3*(-c^2*x^2+1)*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)-3/5*d*f^2*g*(-c^2*x^2+1)^2*(a+b*ar
ccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2-1/5*d*g^3*(-c^2*x^2+1)^2*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4+1/7*d*
g^3*(-c^2*x^2+1)^3*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4-3/5*b*d*f^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2
+1)^(1/2)-2/35*b*d*g^3*x*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+5/16*b*c*d*f^3*x^2*(-c^2*d*x^2+d)^(1/2)/(
-c^2*x^2+1)^(1/2)-3/32*b*d*f*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+2/5*b*c*d*f^2*g*x^3*(-c^2*d*x^2
+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/105*b*d*g^3*x^3*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-1/16*b*c^3*d*f^3*x^4*
(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+7/32*b*c*d*f*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-3/25*b*c^
3*d*f^2*g*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+8/175*b*c*d*g^3*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1
/2)-1/12*b*c^3*d*f*g^2*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/49*b*c^3*d*g^3*x^7*(-c^2*d*x^2+d)^(1/2)/(
-c^2*x^2+1)^(1/2)-3/16*d*f^3*(a+b*arccos(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)-3/32*d*f*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 = 0.96, antiderivative size = 959, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 17, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {4778, 4764, 4650, 4648, 4642, 30, 14, 4678, 194, 4700, 4698, 4708, 266, 43, 4690, 12, 373} \[ -\frac {b c^3 d g^3 \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f g^2 \sqrt {d-c^2 d x^2} x^6}{12 \sqrt {1-c^2 x^2}}+\frac {8 b c d g^3 \sqrt {d-c^2 d x^2} x^5}{175 \sqrt {1-c^2 x^2}}-\frac {3 b c^3 d f^2 g \sqrt {d-c^2 d x^2} x^5}{25 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f^3 \sqrt {d-c^2 d x^2} x^4}{16 \sqrt {1-c^2 x^2}}+\frac {7 b c d f g^2 \sqrt {d-c^2 d x^2} x^4}{32 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac {1}{2} d f g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3-\frac {b d g^3 \sqrt {d-c^2 d x^2} x^3}{105 c \sqrt {1-c^2 x^2}}+\frac {2 b c d f^2 g \sqrt {d-c^2 d x^2} x^3}{5 \sqrt {1-c^2 x^2}}+\frac {5 b c d f^3 \sqrt {d-c^2 d x^2} x^2}{16 \sqrt {1-c^2 x^2}}-\frac {3 b d f g^2 \sqrt {d-c^2 d x^2} x^2}{32 c \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x}{16 c^2}+\frac {1}{4} d f^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac {2 b d g^3 \sqrt {d-c^2 d x^2} x}{35 c^3 \sqrt {1-c^2 x^2}}-\frac {3 b d f^2 g \sqrt {d-c^2 d x^2} x}{5 c \sqrt {1-c^2 x^2}}-\frac {3 d f^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1-c^2 x^2}}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*d*f^2*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2]) - (2*b*d*g^3*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[
1 - c^2*x^2]) + (5*b*c*d*f^3*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (3*b*d*f*g^2*x^2*Sqrt[d - c^2*d
*x^2])/(32*c*Sqrt[1 - c^2*x^2]) + (2*b*c*d*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(5*Sqrt[1 - c^2*x^2]) - (b*d*g^3*x^3
*Sqrt[d - c^2*d*x^2])/(105*c*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f^3*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2])
+ (7*b*c*d*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(32*Sqrt[1 - c^2*x^2]) - (3*b*c^3*d*f^2*g*x^5*Sqrt[d - c^2*d*x^2])/
(25*Sqrt[1 - c^2*x^2]) + (8*b*c*d*g^3*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f*g^2*x^6*Sq
rt[d - c^2*d*x^2])/(12*Sqrt[1 - c^2*x^2]) - (b*c^3*d*g^3*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) + (3*
d*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/8 - (3*d*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(16
*c^2) + (3*d*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/8 + (d*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]
*(a + b*ArcCos[c*x]))/4 + (d*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/2 - (3*d*f^2*g*(
1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(5*c^2) - (d*g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(
a + b*ArcCos[c*x]))/(5*c^4) + (d*g^3*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^4) - (3*d*f
^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2]) - (3*d*f*g^2*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCos[c*x])^2)/(32*b*c^3*Sqrt[1 - c^2*x^2])

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

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

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 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]

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 4650

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(x*(d + e*x^2)^p*(
a + b*ArcCos[c*x])^n)/(2*p + 1), x] + (Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n,
x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c
^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Gt
Q[n, 0] && GtQ[p, 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 4690

Int[((a_.) + ArcCos[(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*ArcCos[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 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 4700

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(
f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(
f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(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] && GtQ[p, 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)^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int (f+g x)^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (f^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )+3 f^2 g x \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )+3 f g^2 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )+g^3 x^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d f^2 g \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d g^3 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}+\frac {\left (3 d f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (b c d f^3 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (3 b d f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \, dx}{5 c \sqrt {1-c^2 x^2}}+\frac {\left (3 d f 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}{2 \sqrt {1-c^2 x^2}}+\frac {\left (b c d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (b c d g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2}{35 c^4} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}+\frac {\left (3 d f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cos ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (b c d f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (3 b c d f^3 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (3 b d f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt {1-c^2 x^2}}+\frac {\left (3 d f 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}{8 \sqrt {1-c^2 x^2}}+\frac {\left (3 b c d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (b c d f g^2 \sqrt {d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (b d g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2 \, dx}{35 c^3 \sqrt {1-c^2 x^2}}\\ &=-\frac {3 b d f^2 g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}+\frac {5 b c d f^3 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {2 b c d f^2 g x^3 \sqrt {d-c^2 d x^2}}{5 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f^3 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {7 b c d f g^2 x^4 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}-\frac {3 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2}}{12 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}-\frac {3 d f^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}+\frac {\left (3 d f 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}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (3 b d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt {1-c^2 x^2}}+\frac {\left (b d g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+8 c^4 x^4-5 c^6 x^6\right ) \, dx}{35 c^3 \sqrt {1-c^2 x^2}}\\ &=-\frac {3 b d f^2 g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}-\frac {2 b d g^3 x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {5 b c d f^3 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {3 b d f g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}}+\frac {2 b c d f^2 g x^3 \sqrt {d-c^2 d x^2}}{5 \sqrt {1-c^2 x^2}}-\frac {b d g^3 x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {1-c^2 x^2}}-\frac {b c^3 d f^3 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {7 b c d f g^2 x^4 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}-\frac {3 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {8 b c d g^3 x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2}}{12 \sqrt {1-c^2 x^2}}-\frac {b c^3 d g^3 x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}-\frac {3 d f^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 4.84, size = 910, normalized size = 0.95 \[ \frac {-88200 b c d f \left (2 c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)^2+140 b d \sqrt {d-c^2 d x^2} \left (6720 f^2 g x^2 \sqrt {1-c^2 x^2} c^4+1680 f^3 \sin \left (2 \cos ^{-1}(c x)\right ) c^3-210 f^3 \sin \left (4 \cos ^{-1}(c x)\right ) c^3-420 f^2 g \sin \left (3 \cos ^{-1}(c x)\right ) c^2-252 f^2 g \sin \left (5 \cos ^{-1}(c x)\right ) c^2-1256 g^3 x^2 \sqrt {1-c^2 x^2} c^2-4200 f^2 g \sqrt {1-c^2 x^2} c^2+315 f g^2 \sin \left (2 \cos ^{-1}(c x)\right ) c+315 f g^2 \sin \left (4 \cos ^{-1}(c x)\right ) c-105 f g^2 \sin \left (6 \cos ^{-1}(c x)\right ) c+864 g^3 \left (1-c^2 x^2\right )^{3/2} \cos \left (2 \cos ^{-1}(c x)\right )+120 g^3 \left (1-c^2 x^2\right )^{3/2} \cos \left (4 \cos ^{-1}(c x)\right )+140 g^3 \sin \left (3 \cos ^{-1}(c x)\right )+84 g^3 \sin \left (5 \cos ^{-1}(c x)\right )+416 g^3 \sqrt {1-c^2 x^2}\right ) \cos ^{-1}(c x)-176400 a c d^{3/2} f \left (2 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-d \sqrt {d-c^2 d x^2} \left (134400 a g^3 x^6 \sqrt {1-c^2 x^2} c^6+470400 a f g^2 x^5 \sqrt {1-c^2 x^2} c^6+564480 a f^2 g x^4 \sqrt {1-c^2 x^2} c^6+235200 a f^3 x^3 \sqrt {1-c^2 x^2} c^6-215040 a g^3 x^4 \sqrt {1-c^2 x^2} c^4-823200 a f g^2 x^3 \sqrt {1-c^2 x^2} c^4-1128960 a f^2 g x^2 \sqrt {1-c^2 x^2} c^4-588000 a f^3 x \sqrt {1-c^2 x^2} c^4+352800 b f^2 g x c^3+7350 b f^3 \cos \left (4 \cos ^{-1}(c x)\right ) c^3+7056 b f^2 g \cos \left (5 \cos ^{-1}(c x)\right ) c^2+26880 a g^3 x^2 \sqrt {1-c^2 x^2} c^2+564480 a f^2 g \sqrt {1-c^2 x^2} c^2+176400 a f g^2 x \sqrt {1-c^2 x^2} c^2+44100 b g^3 x c-7350 b f \left (16 c^2 f^2+3 g^2\right ) \cos \left (2 \cos ^{-1}(c x)\right ) c-11025 b f g^2 \cos \left (4 \cos ^{-1}(c x)\right ) c+2450 b f g^2 \cos \left (6 \cos ^{-1}(c x)\right ) c-4900 b g \left (12 c^2 f^2+g^2\right ) \cos \left (3 \cos ^{-1}(c x)\right )-588 b g^3 \cos \left (5 \cos ^{-1}(c x)\right )+300 b g^3 \cos \left (7 \cos ^{-1}(c x)\right )+53760 a g^3 \sqrt {1-c^2 x^2}\right )}{940800 c^4 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-88200*b*c*d*f*(2*c^2*f^2 + g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 176400*a*c*d^(3/2)*f*(2*c^2*f^2 + g^2)*S
qrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - d*Sqrt[d - c^2*d*x^2]*(352800*b*
c^3*f^2*g*x + 44100*b*c*g^3*x + 564480*a*c^2*f^2*g*Sqrt[1 - c^2*x^2] + 53760*a*g^3*Sqrt[1 - c^2*x^2] - 588000*
a*c^4*f^3*x*Sqrt[1 - c^2*x^2] + 176400*a*c^2*f*g^2*x*Sqrt[1 - c^2*x^2] - 1128960*a*c^4*f^2*g*x^2*Sqrt[1 - c^2*
x^2] + 26880*a*c^2*g^3*x^2*Sqrt[1 - c^2*x^2] + 235200*a*c^6*f^3*x^3*Sqrt[1 - c^2*x^2] - 823200*a*c^4*f*g^2*x^3
*Sqrt[1 - c^2*x^2] + 564480*a*c^6*f^2*g*x^4*Sqrt[1 - c^2*x^2] - 215040*a*c^4*g^3*x^4*Sqrt[1 - c^2*x^2] + 47040
0*a*c^6*f*g^2*x^5*Sqrt[1 - c^2*x^2] + 134400*a*c^6*g^3*x^6*Sqrt[1 - c^2*x^2] - 7350*b*c*f*(16*c^2*f^2 + 3*g^2)
*Cos[2*ArcCos[c*x]] - 4900*b*g*(12*c^2*f^2 + g^2)*Cos[3*ArcCos[c*x]] + 7350*b*c^3*f^3*Cos[4*ArcCos[c*x]] - 110
25*b*c*f*g^2*Cos[4*ArcCos[c*x]] + 7056*b*c^2*f^2*g*Cos[5*ArcCos[c*x]] - 588*b*g^3*Cos[5*ArcCos[c*x]] + 2450*b*
c*f*g^2*Cos[6*ArcCos[c*x]] + 300*b*g^3*Cos[7*ArcCos[c*x]]) + 140*b*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-4200*c^
2*f^2*g*Sqrt[1 - c^2*x^2] + 416*g^3*Sqrt[1 - c^2*x^2] + 6720*c^4*f^2*g*x^2*Sqrt[1 - c^2*x^2] - 1256*c^2*g^3*x^
2*Sqrt[1 - c^2*x^2] + 864*g^3*(1 - c^2*x^2)^(3/2)*Cos[2*ArcCos[c*x]] + 120*g^3*(1 - c^2*x^2)^(3/2)*Cos[4*ArcCo
s[c*x]] + 1680*c^3*f^3*Sin[2*ArcCos[c*x]] + 315*c*f*g^2*Sin[2*ArcCos[c*x]] - 420*c^2*f^2*g*Sin[3*ArcCos[c*x]]
+ 140*g^3*Sin[3*ArcCos[c*x]] - 210*c^3*f^3*Sin[4*ArcCos[c*x]] + 315*c*f*g^2*Sin[4*ArcCos[c*x]] - 252*c^2*f^2*g
*Sin[5*ArcCos[c*x]] + 84*g^3*Sin[5*ArcCos[c*x]] - 105*c*f*g^2*Sin[6*ArcCos[c*x]]))/(940800*c^4*Sqrt[1 - c^2*x^
2])

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c^{2} d g^{3} x^{5} + 3 \, a c^{2} d f g^{2} x^{4} - 3 \, a d f^{2} g x - a d f^{3} + {\left (3 \, a c^{2} d f^{2} g - a d g^{3}\right )} x^{3} + {\left (a c^{2} d f^{3} - 3 \, a d f g^{2}\right )} x^{2} + {\left (b c^{2} d g^{3} x^{5} + 3 \, b c^{2} d f g^{2} x^{4} - 3 \, b d f^{2} g x - b d f^{3} + {\left (3 \, b c^{2} d f^{2} g - b d g^{3}\right )} x^{3} + {\left (b c^{2} d f^{3} - 3 \, b d f g^{2}\right )} x^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a*c^2*d*g^3*x^5 + 3*a*c^2*d*f*g^2*x^4 - 3*a*d*f^2*g*x - a*d*f^3 + (3*a*c^2*d*f^2*g - a*d*g^3)*x^3 +
(a*c^2*d*f^3 - 3*a*d*f*g^2)*x^2 + (b*c^2*d*g^3*x^5 + 3*b*c^2*d*f*g^2*x^4 - 3*b*d*f^2*g*x - b*d*f^3 + (3*b*c^2
*d*f^2*g - b*d*g^3)*x^3 + (b*c^2*d*f^3 - 3*b*d*f*g^2)*x^2)*arccos(c*x))*sqrt(-c^2*d*x^2 + d), x)

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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)^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x)),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

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maple [C]  time = 1.94, size = 3314, normalized size = 3.46 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c^2*x^2-1)*c^4*arccos(c*x)*x^8+13/35*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c^2*
x^2-1)*c^2*arccos(c*x)*x^6-1/35*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c^2*x^2-1)/c^2*arccos(c*x)*x^2+3/5*b*(-d*(c^2*
x^2-1))^(1/2)*g*d/(c^2*x^2-1)/c^2*f^2*arccos(c*x)-1/8*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d*c^4/(c^2*x^2-1)*arccos(c*
x)*x^5+7/16*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d*c^2/(c^2*x^2-1)*arccos(c*x)*x^3+9/64*b*(-d*(c^2*x^2-1))^(1/2)*f^3*c
os(3*arccos(c*x))*d/c/(c^2*x^2-1)*arccos(c*x)+3/512*b*(-d*(c^2*x^2-1))^(1/2)*f*sin(3*arccos(c*x))*d/c/(c^2*x^2
-1)*x^2*g^2+1/12*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^6-11/64*b*(-d*(c^2*x^2-
1))^(1/2)*f*g^2*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4+9/128*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/c/(c^2*x^2-1)*(-
c^2*x^2+1)^(1/2)*x^2+3/25*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c^2*x^2-1)*c^3*(-c^2*x^2+1)^(1/2)*x^5*f^2+9/5*b*(-d*(c
^2*x^2-1))^(1/2)*g*d/(c^2*x^2-1)*c^2*arccos(c*x)*x^4*f^2-7/64*b*(-d*(c^2*x^2-1))^(1/2)*f^3*sin(3*arccos(c*x))*
d/(c^2*x^2-1)*arccos(c*x)*(-c^2*x^2+1)^(1/2)*x+3/32*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*f*d*g^2-1/2*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d*c^4/(c^2*x^2-1)*arccos(c*x)*x^7+19/16*b*(-d*(c^2*
x^2-1))^(1/2)*f*g^2*d*c^2/(c^2*x^2-1)*arccos(c*x)*x^5+9/64*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/c^2/(c^2*x^2-1)*ar
ccos(c*x)*x-3/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c^2*x^2-1)*c^4*arccos(c*x)*x^6*f^2-9/64*b*(-d*(c^2*x^2-1))^(1/2)
*f^3*cos(3*arccos(c*x))*d*c/(c^2*x^2-1)*arccos(c*x)*x^2+3/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c^2*x^2-1)/c*(-c^2*x
^2+1)^(1/2)*x*f^2-17/256*I*b*(-d*(c^2*x^2-1))^(1/2)*f^3*sin(3*arccos(c*x))*d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+
3/64*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d*c^2/(c^2*x^2-1)*x^5-9/512*b*(-d*(c^2*x^2-1))^(1/2)*f*cos(3*arccos(c*x)
)*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x*g^2+17/256*b*(-d*(c^2*x^2-1))^(1/2)*f^3*sin(3*arccos(c*x))*d*c/(c^2*x
^2-1)*x^2-3/512*b*(-d*(c^2*x^2-1))^(1/2)*f*sin(3*arccos(c*x))*d/c^3/(c^2*x^2-1)*g^2+3/8*a*f^3*d*x*(-c^2*d*x^2+
d)^(1/2)+1/32*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4-5/32*b*(-d*(c^2*x^2-1))^(1
/2)*f^3*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-15/256*b*(-d*(c^2*x^2-1))^(1/2)*f^3*cos(3*arccos(c*x))*d/(c^2*x
^2-1)*(-c^2*x^2+1)^(1/2)*x+1/49*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c^2*x^2-1)*c^3*(-c^2*x^2+1)^(1/2)*x^7-8/175*b*
(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c^2*x^2-1)*c*(-c^2*x^2+1)^(1/2)*x^5+1/105*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c^2*x^
2-1)/c*(-c^2*x^2+1)^(1/2)*x^3+2/35*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c^2*x^2-1)/c^3*(-c^2*x^2+1)^(1/2)*x+5/1536*
b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-3/32*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/(c
^2*x^2-1)*x^3-1/32*I*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d*c^4/(c^2*x^2-1)*x^5-5/64*I*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d*
c^2/(c^2*x^2-1)*x^3+15/256*I*b*(-d*(c^2*x^2-1))^(1/2)*f^3*cos(3*arccos(c*x))*d/c/(c^2*x^2-1)-53/64*b*(-d*(c^2*
x^2-1))^(1/2)*f*g^2*d/(c^2*x^2-1)*arccos(c*x)*x^3-9/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c^2*x^2-1)*arccos(c*x)*x^2
*f^2+3/16*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^3*d-1/8*I*b*(-d*(c^2*x^2-1
))^(1/2)*f^3*d*c^3/(c^2*x^2-1)*arccos(c*x)*(-c^2*x^2+1)^(1/2)*x^4-17/256*b*(-d*(c^2*x^2-1))^(1/2)*f^3*sin(3*ar
ccos(c*x))*d/c/(c^2*x^2-1)+17/256*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+2/35*b*(-d*(
c^2*x^2-1))^(1/2)*g^3*d/(c^2*x^2-1)/c^4*arccos(c*x)-9/35*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c^2*x^2-1)*arccos(c*x
)*x^4-5/16*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d/(c^2*x^2-1)*arccos(c*x)*x+7/64*I*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d/(c^2
*x^2-1)*x-1/2*a*f*g^2*x*(-c^2*d*x^2+d)^(5/2)/c^2/d+3/16*a*f*g^2/c^2*d*x*(-c^2*d*x^2+d)^(1/2)+3/16*a*f*g^2/c^2*
d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+3/64*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/c^2/(c^
2*x^2-1)*x-2/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c^2*x^2-1)*c*(-c^2*x^2+1)^(1/2)*x^3*f^2-15/256*I*b*(-d*(c^2*x^2-1
))^(1/2)*f^3*cos(3*arccos(c*x))*d*c/(c^2*x^2-1)*x^2-1/8*I*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d*c/(c^2*x^2-1)*arccos(
c*x)*(-c^2*x^2+1)^(1/2)*x^2-9/512*I*b*(-d*(c^2*x^2-1))^(1/2)*f*cos(3*arccos(c*x))*d/c/(c^2*x^2-1)*x^2*g^2-7/64
*I*b*(-d*(c^2*x^2-1))^(1/2)*f^3*sin(3*arccos(c*x))*d*c/(c^2*x^2-1)*arccos(c*x)*x^2+9/64*I*b*(-d*(c^2*x^2-1))^(
1/2)*f^3*cos(3*arccos(c*x))*d/(c^2*x^2-1)*arccos(c*x)*(-c^2*x^2+1)^(1/2)*x+3/64*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g
^2*d/c^3/(c^2*x^2-1)*arccos(c*x)*(-c^2*x^2+1)^(1/2)+3/64*I*b*(-d*(c^2*x^2-1))^(1/2)*f*sin(3*arccos(c*x))*d/c^3
/(c^2*x^2-1)*arccos(c*x)*g^2-3/512*I*b*(-d*(c^2*x^2-1))^(1/2)*f*sin(3*arccos(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2
+1)^(1/2)*x*g^2-15/64*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/c/(c^2*x^2-1)*arccos(c*x)*(-c^2*x^2+1)^(1/2)*x^2+9/51
2*I*b*(-d*(c^2*x^2-1))^(1/2)*f*cos(3*arccos(c*x))*d/c^3/(c^2*x^2-1)*g^2+7/64*I*b*(-d*(c^2*x^2-1))^(1/2)*f^3*si
n(3*arccos(c*x))*d/c/(c^2*x^2-1)*arccos(c*x)+7/64*I*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d/c/(c^2*x^2-1)*arccos(c*x)*(
-c^2*x^2+1)^(1/2)+3/8*a*f^3*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-2/35*a*g^3/d/c^4*(-
c^2*d*x^2+d)^(5/2)-3/64*I*b*(-d*(c^2*x^2-1))^(1/2)*f*sin(3*arccos(c*x))*d/c/(c^2*x^2-1)*arccos(c*x)*x^2*g^2-3/
64*b*(-d*(c^2*x^2-1))^(1/2)*f*sin(3*arccos(c*x))*d/c^2/(c^2*x^2-1)*arccos(c*x)*(-c^2*x^2+1)^(1/2)*x*g^2+3/16*I
*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d*c/(c^2*x^2-1)*arccos(c*x)*(-c^2*x^2+1)^(1/2)*x^4+1/4*a*f^3*x*(-c^2*d*x^2+d)^
(3/2)-1/7*a*g^3*x^2*(-c^2*d*x^2+d)^(5/2)/c^2/d+1/8*a*f*g^2/c^2*x*(-c^2*d*x^2+d)^(3/2)-3/5*a*f^2*g/c^2/d*(-c^2*
d*x^2+d)^(5/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, {\left (2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x + 3 \, \sqrt {-c^{2} d x^{2} + d} d x + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c}\right )} a f^{3} - \frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a g^{3} + \frac {1}{16} \, a f g^{2} {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{2}} - \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2} d} + \frac {3 \, \sqrt {-c^{2} d x^{2} + d} d x}{c^{2}} + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} - \frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a f^{2} g}{5 \, c^{2} d} + \sqrt {d} \int -{\left (b c^{2} d g^{3} x^{5} + 3 \, b c^{2} d f g^{2} x^{4} - 3 \, b d f^{2} g x - b d f^{3} + {\left (3 \, b c^{2} d f^{2} g - b d g^{3}\right )} x^{3} + {\left (b c^{2} d f^{3} - 3 \, b d f g^{2}\right )} x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*arcsin(c*x)/c)*a*f^3 - 1/35*(5*(-c^2*
d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(5/2)/(c^4*d))*a*g^3 + 1/16*a*f*g^2*(2*(-c^2*d*x^2 + d)^(3/2
)*x/c^2 - 8*(-c^2*d*x^2 + d)^(5/2)*x/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*d*x/c^2 + 3*d^(3/2)*arcsin(c*x)/c^3) - 3
/5*(-c^2*d*x^2 + d)^(5/2)*a*f^2*g/(c^2*d) + sqrt(d)*integrate(-(b*c^2*d*g^3*x^5 + 3*b*c^2*d*f*g^2*x^4 - 3*b*d*
f^2*g*x - b*d*f^3 + (3*b*c^2*d*f^2*g - b*d*g^3)*x^3 + (b*c^2*d*f^3 - 3*b*d*f*g^2)*x^2)*sqrt(c*x + 1)*sqrt(-c*x
+ 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right ) \left (f + g x\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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