\(\int (f+g x) (d-c^2 d x^2)^{5/2} (a+b \arccos (c x)) \, dx\) [12]

Optimal result

Integrand size = 29, antiderivative size = 517 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=-\frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {4862, 4848, 4744, 4742, 4738, 30, 14, 267, 4768, 200} \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}+\frac {25 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}-\frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}} \]

[In]

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

[Out]

-1/7*(b*d^2*g*x*Sqrt[d - c^2*d*x^2])/(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*ArcC
os[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*Sq
rt[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*S
qrt[1 - c^2*x^2])

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 200

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 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 4738

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

Rule 4742

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[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcCos[c*x])^n/S
qrt[1 - c^2*x^2], x], x] + Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^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 4744

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/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcC
os[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]

Rule 4768

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/(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*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 4848

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 4862

Int[((a_.) + ArcCos[(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*ArcCos[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 {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (f+g x) \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \, 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} (a+b \arccos (c x))+g x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))\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} (a+b \arccos (c x)) \, 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} (a+b \arccos (c x)) \, 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} (a+b \arccos (c x))-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{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} (a+b \arccos (c x)) \, 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} (a+b \arccos (c x))+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \, 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} (a+b \arccos (c x))+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arccos (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} (a+b \arccos (c x))+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.47 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.02 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {d^2 \left (-88200 b c f \sqrt {d-c^2 d x^2} \arccos (c x)^2-176400 a c \sqrt {d} f \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d-c^2 d x^2} \left (-44100 b c g x-80640 a g \sqrt {1-c^2 x^2}+388080 a c^2 f x \sqrt {1-c^2 x^2}+241920 a c^2 g x^2 \sqrt {1-c^2 x^2}-305760 a c^4 f x^3 \sqrt {1-c^2 x^2}-241920 a c^4 g x^4 \sqrt {1-c^2 x^2}+94080 a c^6 f x^5 \sqrt {1-c^2 x^2}+80640 a c^6 g x^6 \sqrt {1-c^2 x^2}+66150 b c f \cos (2 \arccos (c x))+8820 b g \cos (3 \arccos (c x))-6615 b c f \cos (4 \arccos (c x))-1764 b g \cos (5 \arccos (c x))+490 b c f \cos (6 \arccos (c x))+180 b g \cos (7 \arccos (c x))\right )+84 b \sqrt {d-c^2 d x^2} \arccos (c x) \left (-1816 g \sqrt {1-c^2 x^2}+3496 c^2 g x^2 \sqrt {1-c^2 x^2}-864 g \left (1-c^2 x^2\right )^{3/2} \cos (2 \arccos (c x))-120 g \left (1-c^2 x^2\right )^{3/2} \cos (4 \arccos (c x))+1575 c f \sin (2 \arccos (c x))-280 g \sin (3 \arccos (c x))-315 c f \sin (4 \arccos (c x))-168 g \sin (5 \arccos (c x))+35 c f \sin (6 \arccos (c x))\right )\right )}{564480 c^2 \sqrt {1-c^2 x^2}} \]

[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])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.42 (sec) , antiderivative size = 1419, normalized size of antiderivative = 2.74

method	result	size
default	\(\text {Expression too large to display}\)	\(1419\)
parts	\(\text {Expression too large to display}\)	\(1419\)

[In]

int((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)

[Out]

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))-1/7*a*g*(-c^2*d*x^2+d)^(7/2)/c^2/d+b*(5/32*(-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+1/6272*(-d*(c^2*x^2-1))^(1/2)*(64*I*c^
7*x^7*(-c^2*x^2+1)^(1/2)+64*c^8*x^8-112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-144*c^6*x^6+56*I*(-c^2*x^2+1)^(1/2)*x^3*c
^3+104*c^4*x^4-7*I*(-c^2*x^2+1)^(1/2)*x*c-25*c^2*x^2+1)*g*(I+7*arccos(c*x))*d^2/c^2/(c^2*x^2-1)+1/2304*(-d*(c^
2*x^2-1))^(1/2)*(32*I*(-c^2*x^2+1)^(1/2)*c^6*x^6+32*c^7*x^7-48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-64*c^5*x^5+18*I*(-
c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-6*c*x)*f*(I+6*arccos(c*x))*d^2/c/(c^2*x^2-1)-5/128*(-
d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*g*(arccos(c*x)+I)*d^2/c^2/(c^2*x^2-1)-5/128*(-d*(c^2
*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*g*(arccos(c*x)-I)*d^2/c^2/(c^2*x^2-1)+15/256*(-d*(c^2*x^2-
1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*f*(-I+2*arccos(c*x))*d^2/c/(c
^2*x^2-1)+1/128*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+3*I*(-c^2*x^2+1)^(1
/2)*x*c+1)*g*(-I+3*arccos(c*x))*d^2/c^2/(c^2*x^2-1)+1/7840*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2
)*x*c-1)*g*(11*I+70*arccos(c*x))*cos(6*arccos(c*x))*d^2/c^2/(c^2*x^2-1)+3/15680*(-d*(c^2*x^2-1))^(1/2)*(c*x*(-
c^2*x^2+1)^(1/2)+I*c^2*x^2-I)*g*(9*I+35*arccos(c*x))*sin(6*arccos(c*x))*d^2/c^2/(c^2*x^2-1)+5/4608*(-d*(c^2*x^
2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*f*(5*I+24*arccos(c*x))*cos(5*arccos(c*x))*d^2/c/(c^2*x^2-1)+1
/4608*(-d*(c^2*x^2-1))^(1/2)*(c*x*(-c^2*x^2+1)^(1/2)+I*c^2*x^2-I)*f*(29*I+96*arccos(c*x))*sin(5*arccos(c*x))*d
^2/c/(c^2*x^2-1)-1/160*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*g*(I+5*arccos(c*x))*cos(4*a
rccos(c*x))*d^2/c^2/(c^2*x^2-1)-1/320*(-d*(c^2*x^2-1))^(1/2)*(c*x*(-c^2*x^2+1)^(1/2)+I*c^2*x^2-I)*g*(3*I+5*arc
cos(c*x))*sin(4*arccos(c*x))*d^2/c^2/(c^2*x^2-1)-9/512*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*
c-1)*f*(3*I+8*arccos(c*x))*cos(3*arccos(c*x))*d^2/c/(c^2*x^2-1)-3/512*(-d*(c^2*x^2-1))^(1/2)*(c*x*(-c^2*x^2+1)
^(1/2)+I*c^2*x^2-I)*f*(11*I+16*arccos(c*x))*sin(3*arccos(c*x))*d^2/c/(c^2*x^2-1))

Fricas [F]

\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]

[In]

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

[Out]

1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt(-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*
arcsin(c*x)/c)*a*f - 1/7*(-c^2*d*x^2 + d)^(7/2)*a*g/(c^2*d) + sqrt(d)*integrate((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)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(
c*x + 1)*sqrt(-c*x + 1), c*x), x)

Giac [F(-2)]

Exception generated. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(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

Mupad [F(-1)]

Timed out. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

[In]

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

[Out]

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