3.12 \(\int (f+g x) (d-c^2 d x^2)^{5/2} (a+b \cos ^{-1}(c x)) \, dx\)

Optimal. Leaf size=517 \[ \frac{1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{16} d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{5 d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt{1-c^2 x^2}}-\frac{d^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac{5 b c^3 d^2 f x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}+\frac{25 b c d^2 f x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}-\frac{b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{b c^5 d^2 g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}-\frac{3 b c^3 d^2 g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}+\frac{b c d^2 g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}-\frac{b d^2 g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}} \]

[Out]

-(b*d^2*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2]) + (25*b*c*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 -
c^2*x^2]) + (b*c*d^2*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) - (5*b*c^3*d^2*f*x^4*Sqrt[d - c^2*d*x^2
])/(96*Sqrt[1 - c^2*x^2]) - (3*b*c^3*d^2*g*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*g*x^7*
Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) - (b*d^2*f*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5*d^
2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/16 + (5*d^2*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos
[c*x]))/24 + (d^2*f*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/6 - (d^2*g*(1 - c^2*x^2)^3*Sqrt
[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^2) - (5*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(32*b*c*Sqr
t[1 - c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.402964, antiderivative size = 517, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {4778, 4764, 4650, 4648, 4642, 30, 14, 261, 4678, 194} \[ \frac{1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{16} d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{5 d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt{1-c^2 x^2}}-\frac{d^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac{5 b c^3 d^2 f x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}+\frac{25 b c d^2 f x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}-\frac{b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{b c^5 d^2 g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}-\frac{3 b c^3 d^2 g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}+\frac{b c d^2 g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}-\frac{b d^2 g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]

[Out]

-(b*d^2*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2]) + (25*b*c*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 -
c^2*x^2]) + (b*c*d^2*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) - (5*b*c^3*d^2*f*x^4*Sqrt[d - c^2*d*x^2
])/(96*Sqrt[1 - c^2*x^2]) - (3*b*c^3*d^2*g*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*g*x^7*
Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) - (b*d^2*f*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5*d^
2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/16 + (5*d^2*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos
[c*x]))/24 + (d^2*f*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/6 - (d^2*g*(1 - c^2*x^2)^3*Sqrt
[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^2) - (5*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(32*b*c*Sqr
t[1 - c^2*x^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]

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 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 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 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 30

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

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 261

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 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 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]

Rubi steps

\begin{align*} \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int (f+g x) \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (f \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )+g x \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d^2 f \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (d^2 g \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{d^2 g \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^2}+\frac{\left (5 d^2 f \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}{6 \sqrt{1-c^2 x^2}}+\frac{\left (b c d^2 f \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^2 \, dx}{6 \sqrt{1-c^2 x^2}}-\frac{\left (b d^2 g \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \, dx}{7 c \sqrt{1-c^2 x^2}}\\ &=-\frac{b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{5}{24} d^2 f 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}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{d^2 g \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^2}+\frac{\left (5 d^2 f \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{8 \sqrt{1-c^2 x^2}}+\frac{\left (5 b c d^2 f \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{24 \sqrt{1-c^2 x^2}}-\frac{\left (b d^2 g \sqrt{d-c^2 d x^2}\right ) \int \left (1-3 c^2 x^2+3 c^4 x^4-c^6 x^6\right ) \, dx}{7 c \sqrt{1-c^2 x^2}}\\ &=-\frac{b d^2 g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}}+\frac{b c d^2 g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}-\frac{3 b c^3 d^2 g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}-\frac{b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{24} d^2 f 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}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{d^2 g \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^2}+\frac{\left (5 d^2 f \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{16 \sqrt{1-c^2 x^2}}+\frac{\left (5 b c d^2 f \sqrt{d-c^2 d x^2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{24 \sqrt{1-c^2 x^2}}+\frac{\left (5 b c d^2 f \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt{1-c^2 x^2}}\\ &=-\frac{b d^2 g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}}+\frac{25 b c d^2 f x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}+\frac{b c d^2 g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}-\frac{5 b c^3 d^2 f x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}-\frac{3 b c^3 d^2 g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}-\frac{b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{24} d^2 f 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}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{d^2 g \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^2}-\frac{5 d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 2.90072, size = 526, normalized size = 1.02 \[ \frac{d^2 \left (\sqrt{d-c^2 d x^2} \left (94080 a c^6 f x^5 \sqrt{1-c^2 x^2}-305760 a c^4 f x^3 \sqrt{1-c^2 x^2}+388080 a c^2 f x \sqrt{1-c^2 x^2}+80640 a c^6 g x^6 \sqrt{1-c^2 x^2}-241920 a c^4 g x^4 \sqrt{1-c^2 x^2}+241920 a c^2 g x^2 \sqrt{1-c^2 x^2}-80640 a g \sqrt{1-c^2 x^2}+66150 b c f \cos \left (2 \cos ^{-1}(c x)\right )-6615 b c f \cos \left (4 \cos ^{-1}(c x)\right )+490 b c f \cos \left (6 \cos ^{-1}(c x)\right )-44100 b c g x+8820 b g \cos \left (3 \cos ^{-1}(c x)\right )-1764 b g \cos \left (5 \cos ^{-1}(c x)\right )+180 b g \cos \left (7 \cos ^{-1}(c x)\right )\right )-176400 a c \sqrt{d} f \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+84 b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \left (3496 c^2 g x^2 \sqrt{1-c^2 x^2}-1816 g \sqrt{1-c^2 x^2}-864 g \left (1-c^2 x^2\right )^{3/2} \cos \left (2 \cos ^{-1}(c x)\right )-120 g \left (1-c^2 x^2\right )^{3/2} \cos \left (4 \cos ^{-1}(c x)\right )+1575 c f \sin \left (2 \cos ^{-1}(c x)\right )-315 c f \sin \left (4 \cos ^{-1}(c x)\right )+35 c f \sin \left (6 \cos ^{-1}(c x)\right )-280 g \sin \left (3 \cos ^{-1}(c x)\right )-168 g \sin \left (5 \cos ^{-1}(c x)\right )\right )-88200 b c f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2\right )}{564480 c^2 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]

[Out]

(d^2*(-88200*b*c*f*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 176400*a*c*Sqrt[d]*f*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt
[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d - c^2*d*x^2]*(-44100*b*c*g*x - 80640*a*g*Sqrt[1 - c^2*x^2]
+ 388080*a*c^2*f*x*Sqrt[1 - c^2*x^2] + 241920*a*c^2*g*x^2*Sqrt[1 - c^2*x^2] - 305760*a*c^4*f*x^3*Sqrt[1 - c^2
*x^2] - 241920*a*c^4*g*x^4*Sqrt[1 - c^2*x^2] + 94080*a*c^6*f*x^5*Sqrt[1 - c^2*x^2] + 80640*a*c^6*g*x^6*Sqrt[1
- c^2*x^2] + 66150*b*c*f*Cos[2*ArcCos[c*x]] + 8820*b*g*Cos[3*ArcCos[c*x]] - 6615*b*c*f*Cos[4*ArcCos[c*x]] - 17
64*b*g*Cos[5*ArcCos[c*x]] + 490*b*c*f*Cos[6*ArcCos[c*x]] + 180*b*g*Cos[7*ArcCos[c*x]]) + 84*b*Sqrt[d - c^2*d*x
^2]*ArcCos[c*x]*(-1816*g*Sqrt[1 - c^2*x^2] + 3496*c^2*g*x^2*Sqrt[1 - c^2*x^2] - 864*g*(1 - c^2*x^2)^(3/2)*Cos[
2*ArcCos[c*x]] - 120*g*(1 - c^2*x^2)^(3/2)*Cos[4*ArcCos[c*x]] + 1575*c*f*Sin[2*ArcCos[c*x]] - 280*g*Sin[3*ArcC
os[c*x]] - 315*c*f*Sin[4*ArcCos[c*x]] - 168*g*Sin[5*ArcCos[c*x]] + 35*c*f*Sin[6*ArcCos[c*x]])))/(564480*c^2*Sq
rt[1 - c^2*x^2])

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Maple [B]  time = 0.602, size = 931, normalized size = 1.8 \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)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x)

[Out]

-1/7*a*g*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/6*a*f*x*(-c^2*d*x^2+d)^(5/2)+5/24*a*f*d*x*(-c^2*d*x^2+d)^(3/2)+5/16*a*f*
d^2*x*(-c^2*d*x^2+d)^(1/2)+5/16*a*f*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+5/32*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*d^2-1/36*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^
5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^6+13/96*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x
^4-11/32*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-1/49*b*(-d*(c^2*x^2-1))^(1/2)*g*d
^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^7+3/35*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/
2)*x^5-1/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3+1/7*b*(-d*(c^2*x^2-1))^(1/2)*g*
d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+1/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/c^2/(c^2*x^2-1)*arccos(c*x)+1/7*b*(-
d*(c^2*x^2-1))^(1/2)*g*d^2*c^6/(c^2*x^2-1)*arccos(c*x)*x^8-4/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^4/(c^2*x^2-1)*
arccos(c*x)*x^6+6/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^2/(c^2*x^2-1)*arccos(c*x)*x^4-4/7*b*(-d*(c^2*x^2-1))^(1/2
)*g*d^2/(c^2*x^2-1)*arccos(c*x)*x^2+1/6*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^6/(c^2*x^2-1)*arccos(c*x)*x^7-17/24*b
*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^4/(c^2*x^2-1)*arccos(c*x)*x^5+59/48*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^2/(c^2*x^
2-1)*arccos(c*x)*x^3-11/16*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2/(c^2*x^2-1)*arccos(c*x)*x+299/2304*b*(-d*(c^2*x^2-1)
)^(1/2)*f*d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(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)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),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 c^{4} d^{2} g x^{5} + a c^{4} d^{2} f x^{4} - 2 \, a c^{2} d^{2} g x^{3} - 2 \, a c^{2} d^{2} f x^{2} + a d^{2} g x + a d^{2} f +{\left (b c^{4} d^{2} g x^{5} + b c^{4} d^{2} f x^{4} - 2 \, b c^{2} d^{2} g x^{3} - 2 \, b c^{2} d^{2} f x^{2} + b d^{2} g x + b d^{2} f\right )} \arccos \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)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (g x + f\right )}{\left (b \arccos \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)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(g*x + f)*(b*arccos(c*x) + a), x)