3.7 \(\int (f+g x)^2 (d-c^2 d x^2)^{3/2} (a+b \cos ^{-1}(c x)) \, dx\)
Optimal. Leaf size=680 \[ \frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{4} d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d f^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}-\frac {2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} d g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {d g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1-c^2 x^2}}+\frac {5 b c d f^2 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {2 b d f g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}+\frac {4 b c d f g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}-\frac {b d g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}}+\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d f g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}-\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}} \]
[Out]
3/8*d*f^2*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)-1/16*d*g^2*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/8
*d*g^2*x^3*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)+1/4*d*f^2*x*(-c^2*x^2+1)*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1
/2)+1/6*d*g^2*x^3*(-c^2*x^2+1)*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)-2/5*d*f*g*(-c^2*x^2+1)^2*(a+b*arccos(c*x
))*(-c^2*d*x^2+d)^(1/2)/c^2-2/5*b*d*f*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+5/16*b*c*d*f^2*x^2*(-c^2*d
*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/32*b*d*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+4/15*b*c*d*f*g*x^3
*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/16*b*c^3*d*f^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+7/96*b*c
*d*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2/25*b*c^3*d*f*g*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2
)-1/36*b*c^3*d*g^2*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-3/16*d*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)-1/32*d*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 = 0.74, antiderivative size = 680, normalized size of antiderivative =
1.00, number of steps used = 20, number of rules used = 12, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) =
0.387, Rules used = {4778, 4764, 4650, 4648, 4642, 30, 14, 4678, 194, 4700, 4698, 4708}
\[ \frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{4} d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d f^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}-\frac {2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} d g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{16 c^2}-\frac {d g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {5 b c d f^2 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d f g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {4 b c d f g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}-\frac {2 b d f g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}-\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}+\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {b d g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Int[(f + g*x)^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]),x]
[Out]
(-2*b*d*f*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2]) + (5*b*c*d*f^2*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1
- c^2*x^2]) - (b*d*g^2*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[1 - c^2*x^2]) + (4*b*c*d*f*g*x^3*Sqrt[d - c^2*d*x^2
])/(15*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (7*b*c*d*g^2*x^4*Sq
rt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) - (2*b*c^3*d*f*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) - (
b*c^3*d*g^2*x^6*Sqrt[d - c^2*d*x^2])/(36*Sqrt[1 - c^2*x^2]) + (3*d*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x
]))/8 - (d*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(16*c^2) + (d*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*Arc
Cos[c*x]))/8 + (d*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/4 + (d*g^2*x^3*(1 - c^2*x^2)*Sq
rt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/6 - (2*d*f*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(
5*c^2) - (3*d*f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2]) - (d*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 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 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 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 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)^2 \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)^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 \sqrt {d-c^2 d x^2}\right ) \int \left (f^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )+2 f g x \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )+g^2 x^2 \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^2 \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 (2 d f 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 (d 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 {1}{4} d f^2 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 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 {2 d f 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 {\left (3 d 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}{4 \sqrt {1-c^2 x^2}}+\frac {\left (b c d f^2 \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 (2 b d f 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 (d 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 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{6 \sqrt {1-c^2 x^2}}\\ &=\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{4} d f^2 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 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 {2 d f 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 {\left (3 d 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}{8 \sqrt {1-c^2 x^2}}+\frac {\left (b c d f^2 \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^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (2 b d f 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 (d 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 (b c d 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 g^2 \sqrt {d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{6 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b d f g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}+\frac {5 b c d f^2 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {4 b c d f g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d f g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}-\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{4} d f^2 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 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 {2 d f 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 {3 d f^2 \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 (d 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 (b d g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b d f g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}+\frac {5 b c d f^2 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {b d g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}}+\frac {4 b c d f g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d f g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}-\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{4} d f^2 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 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 {2 d f 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 {3 d f^2 \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 {d 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*}
________________________________________________________________________________________
Mathematica [A] time = 2.27, size = 591, normalized size = 0.87 \[ \frac {-d \sqrt {d-c^2 d x^2} \left (23040 a c f g \sqrt {1-c^2 x^2}+3600 a c g^2 x \sqrt {1-c^2 x^2}+14400 a c^5 f^2 x^3 \sqrt {1-c^2 x^2}+23040 a c^5 f g x^4 \sqrt {1-c^2 x^2}+9600 a c^5 g^2 x^5 \sqrt {1-c^2 x^2}-36000 a c^3 f^2 x \sqrt {1-c^2 x^2}-46080 a c^3 f g x^2 \sqrt {1-c^2 x^2}-16800 a c^3 g^2 x^3 \sqrt {1-c^2 x^2}-450 b \left (16 c^2 f^2+g^2\right ) \cos \left (2 \cos ^{-1}(c x)\right )+450 b c^2 f^2 \cos \left (4 \cos ^{-1}(c x)\right )+14400 b c^2 f g x-2400 b c f g \cos \left (3 \cos ^{-1}(c x)\right )+288 b c f g \cos \left (5 \cos ^{-1}(c x)\right )-225 b g^2 \cos \left (4 \cos ^{-1}(c x)\right )+50 b g^2 \cos \left (6 \cos ^{-1}(c x)\right )\right )-3600 a d^{3/2} \sqrt {1-c^2 x^2} \left (6 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-1800 b d \sqrt {d-c^2 d x^2} \left (6 c^2 f^2+g^2\right ) \cos ^{-1}(c x)^2+60 b d \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \left (15 \left (16 c^2 f^2+g^2\right ) \sin \left (2 \cos ^{-1}(c x)\right )-30 c^2 f^2 \sin \left (4 \cos ^{-1}(c x)\right )-400 c f g \sqrt {1-c^2 x^2}+640 c^3 f g x^2 \sqrt {1-c^2 x^2}-40 c f g \sin \left (3 \cos ^{-1}(c x)\right )-24 c f g \sin \left (5 \cos ^{-1}(c x)\right )+15 g^2 \sin \left (4 \cos ^{-1}(c x)\right )-5 g^2 \sin \left (6 \cos ^{-1}(c x)\right )\right )}{57600 c^3 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(f + g*x)^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]),x]
[Out]
(-1800*b*d*(6*c^2*f^2 + g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 3600*a*d^(3/2)*(6*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))] - d*Sqrt[d - c^2*d*x^2]*(14400*b*c^2*f*g*x +
23040*a*c*f*g*Sqrt[1 - c^2*x^2] - 36000*a*c^3*f^2*x*Sqrt[1 - c^2*x^2] + 3600*a*c*g^2*x*Sqrt[1 - c^2*x^2] - 460
80*a*c^3*f*g*x^2*Sqrt[1 - c^2*x^2] + 14400*a*c^5*f^2*x^3*Sqrt[1 - c^2*x^2] - 16800*a*c^3*g^2*x^3*Sqrt[1 - c^2*
x^2] + 23040*a*c^5*f*g*x^4*Sqrt[1 - c^2*x^2] + 9600*a*c^5*g^2*x^5*Sqrt[1 - c^2*x^2] - 450*b*(16*c^2*f^2 + g^2)
*Cos[2*ArcCos[c*x]] - 2400*b*c*f*g*Cos[3*ArcCos[c*x]] + 450*b*c^2*f^2*Cos[4*ArcCos[c*x]] - 225*b*g^2*Cos[4*Arc
Cos[c*x]] + 288*b*c*f*g*Cos[5*ArcCos[c*x]] + 50*b*g^2*Cos[6*ArcCos[c*x]]) + 60*b*d*Sqrt[d - c^2*d*x^2]*ArcCos[
c*x]*(-400*c*f*g*Sqrt[1 - c^2*x^2] + 640*c^3*f*g*x^2*Sqrt[1 - c^2*x^2] + 15*(16*c^2*f^2 + g^2)*Sin[2*ArcCos[c*
x]] - 40*c*f*g*Sin[3*ArcCos[c*x]] - 30*c^2*f^2*Sin[4*ArcCos[c*x]] + 15*g^2*Sin[4*ArcCos[c*x]] - 24*c*f*g*Sin[5
*ArcCos[c*x]] - 5*g^2*Sin[6*ArcCos[c*x]]))/(57600*c^3*Sqrt[1 - c^2*x^2])
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c^{2} d g^{2} x^{4} + 2 \, a c^{2} d f g x^{3} - 2 \, a d f g x - a d f^{2} + {\left (a c^{2} d f^{2} - a d g^{2}\right )} x^{2} + {\left (b c^{2} d g^{2} x^{4} + 2 \, b c^{2} d f g x^{3} - 2 \, b d f g x - b d f^{2} + {\left (b c^{2} d f^{2} - b d g^{2}\right )} x^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x)),x, algorithm="fricas")
[Out]
integral(-(a*c^2*d*g^2*x^4 + 2*a*c^2*d*f*g*x^3 - 2*a*d*f*g*x - a*d*f^2 + (a*c^2*d*f^2 - a*d*g^2)*x^2 + (b*c^2*
d*g^2*x^4 + 2*b*c^2*d*f*g*x^3 - 2*b*d*f*g*x - b*d*f^2 + (b*c^2*d*f^2 - b*d*g^2)*x^2)*arccos(c*x))*sqrt(-c^2*d*
x^2 + d), 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*(-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
________________________________________________________________________________________
maple [C] time = 1.34, size = 3643, normalized size = 5.36 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x)),x)
[Out]
-5/16*b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*arccos(c*x)*x*f^2-1/32*I*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d/(c^2*x^2-
1)*x^3+7/64*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*x*f^2-17/256*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arccos(c*x))*
d/c/(c^2*x^2-1)*f^2-1/512*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arccos(c*x))*d/c^3/(c^2*x^2-1)*g^2+5/4608*b*(-d*(c^2*
x^2-1))^(1/2)*g^2*d/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+17/256*b*(-d*(c^2*x^2-1))^(1/2)*d/c/(c^2*x^2-1)*(-c^2*x
^2+1)^(1/2)*f^2-53/192*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d/(c^2*x^2-1)*arccos(c*x)*x^3-1/512*I*b*(-d*(c^2*x^2-1))^(
1/2)*sin(3*arccos(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x*g^2+9/64*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcco
s(c*x))*d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x*f^2-1/64*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arccos(c*x))*d/
c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x*g^2-1/6*a*g^2*x*(-c^2*d*x^2+d)^(5/2)/c^2/d+1/16*a*g^2/c^2*d*x
*(-c^2*d*x^2+d)^(1/2)+1/16*a*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))-2/5*a*f*g/
c^2/d*(-c^2*d*x^2+d)^(5/2)+1/64*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arccos(c*x))*d/c^3/(c^2*x^2-1)*arccos(c*x)*g^
2-17/256*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arccos(c*x))*d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x*f^2+3/40*b*(-d*(c^2*
x^2-1))^(1/2)*f*g*cos(4*arccos(c*x))*d/c^2/(c^2*x^2-1)*arccos(c*x)-9/64*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arccos(
c*x))*d*c/(c^2*x^2-1)*arccos(c*x)*x^2*f^2-7/64*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arccos(c*x))*d/(c^2*x^2-1)*(-c^2
*x^2+1)^(1/2)*arccos(c*x)*x*f^2+1/25*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^5-2/1
5*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3+13/40*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d/c/(
c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-3/512*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arccos(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+
1)^(1/2)*x*g^2-3/40*b*(-d*(c^2*x^2-1))^(1/2)*f*g*cos(4*arccos(c*x))*d/(c^2*x^2-1)*arccos(c*x)*x^2-1/5*b*(-d*(c
^2*x^2-1))^(1/2)*f*g*d*c^4/(c^2*x^2-1)*arccos(c*x)*x^6+3/5*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d*c^2/(c^2*x^2-1)*arcc
os(c*x)*x^4+1/4*a*f^2*x*(-c^2*d*x^2+d)^(3/2)+1/8*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(
1/2)*arccos(c*x)*x-7/300*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g*sin(4*arccos(c*x))*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*
x-1/20*b*(-d*(c^2*x^2-1))^(1/2)*f*g*sin(4*arccos(c*x))*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x-1/5*I*
b*(-d*(c^2*x^2-1))^(1/2)*f*g*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x^5+1/24*a*g^2/c^2*x*(-c^2*d*x^2
+d)^(3/2)-11/600*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g*cos(4*arccos(c*x))*d/(c^2*x^2-1)*x^2+1/64*I*b*(-d*(c^2*x^2-1))
^(1/2)*g^2*d/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)-1/25*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d*c^4/(c^2*x^2
-1)*x^6-1/75*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d*c^2/(c^2*x^2-1)*x^4+7/64*I*b*(-d*(c^2*x^2-1))^(1/2)*d/c/(c^2*x^2
-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)*f^2+11/600*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g*cos(4*arccos(c*x))*d/c^2/(c^2*x^2
-1)-15/256*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arccos(c*x))*d*c/(c^2*x^2-1)*x^2*f^2-3/512*I*b*(-d*(c^2*x^2-1))^(1
/2)*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)*sin(3*arccos(c*x))*d/c/(c^2*x^2
-1)*arccos(c*x)*f^2-11/600*b*(-d*(c^2*x^2-1))^(1/2)*f*g*cos(4*arccos(c*x))*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*
x-1/8*I*b*(-d*(c^2*x^2-1))^(1/2)*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x^4*f^2-1/8*I*b*(-d*(c^2*x^2
-1))^(1/2)*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x^2*f^2-1/20*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g*sin(4*ar
ccos(c*x))*d/(c^2*x^2-1)*arccos(c*x)*x^2+1/16*I*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2
)*arccos(c*x)*x^4-5/64*I*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x^2+1/20*
I*b*(-d*(c^2*x^2-1))^(1/2)*f*g*sin(4*arccos(c*x))*d/c^2/(c^2*x^2-1)*arccos(c*x)-7/64*I*b*(-d*(c^2*x^2-1))^(1/2
)*sin(3*arccos(c*x))*d*c/(c^2*x^2-1)*arccos(c*x)*x^2*f^2-1/64*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arccos(c*x))*d/
c/(c^2*x^2-1)*arccos(c*x)*x^2*g^2+3/8*a*f^2*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a*f^2*d^2/(c^2*d)^(1/2)*arctan((c^2*d
)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+3/40*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g*cos(4*arccos(c*x))*d/c/(c^2*x^2-1)*(-c^2*x
^2+1)^(1/2)*arccos(c*x)*x-1/32*I*b*(-d*(c^2*x^2-1))^(1/2)*d*c^4/(c^2*x^2-1)*x^5*f^2-5/64*I*b*(-d*(c^2*x^2-1))^
(1/2)*d*c^2/(c^2*x^2-1)*x^3*f^2-29/40*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d/(c^2*x^2-1)*arccos(c*x)*x^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*d*f^2+1/32*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*d*g^2-1/6*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d*c^4/(c^2*x^2-1)*arccos(c*x)*x^
7+19/48*b*(-d*(c^2*x^2-1))^(1/2)*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)*cos(3*arc
cos(c*x))*d/c/(c^2*x^2-1)*f^2*arccos(c*x)+3/64*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d/c^2/(c^2*x^2-1)*arccos(c*x)*x+13
/40*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d/c^2/(c^2*x^2-1)*arccos(c*x)-1/8*b*(-d*(c^2*x^2-1))^(1/2)*d*c^4/(c^2*x^2-1)*
arccos(c*x)*x^5*f^2+7/16*b*(-d*(c^2*x^2-1))^(1/2)*d*c^2/(c^2*x^2-1)*arccos(c*x)*x^3*f^2+1/36*b*(-d*(c^2*x^2-1)
)^(1/2)*g^2*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^6-11/192*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d*c/(c^2*x^2-1)*(-c^2
*x^2+1)^(1/2)*x^4+3/128*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2+1/32*b*(-d*(c^2*x^
2-1))^(1/2)*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4*f^2-5/32*b*(-d*(c^2*x^2-1))^(1/2)*d*c/(c^2*x^2-1)*(-c^2*x
^2+1)^(1/2)*x^2*f^2-7/300*b*(-d*(c^2*x^2-1))^(1/2)*f*g*sin(4*arccos(c*x))*d/c^2/(c^2*x^2-1)+17/256*b*(-d*(c^2*
x^2-1))^(1/2)*sin(3*arccos(c*x))*d*c/(c^2*x^2-1)*x^2*f^2+1/512*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arccos(c*x))*d/c
/(c^2*x^2-1)*x^2*g^2+7/300*b*(-d*(c^2*x^2-1))^(1/2)*f*g*sin(4*arccos(c*x))*d/(c^2*x^2-1)*x^2-15/256*b*(-d*(c^2
*x^2-1))^(1/2)*cos(3*arccos(c*x))*d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x*f^2+43/600*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g
*d/(c^2*x^2-1)*x^2+1/64*I*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d*c^2/(c^2*x^2-1)*x^5+15/256*I*b*(-d*(c^2*x^2-1))^(1/2)
*cos(3*arccos(c*x))*d/c/(c^2*x^2-1)*f^2+3/512*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arccos(c*x))*d/c^3/(c^2*x^2-1)*
g^2+1/64*I*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d/c^2/(c^2*x^2-1)*x-11/600*I*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d/c^2/(c^2*x
^2-1)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, {\left (2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x + 3 \, \sqrt {-c^{2} d x^{2} + d} d x + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c}\right )} a f^{2} + \frac {1}{48} \, a g^{2} {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{2}} - \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2} d} + \frac {3 \, \sqrt {-c^{2} d x^{2} + d} d x}{c^{2}} + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a f g}{5 \, c^{2} d} + \sqrt {d} \int -{\left (b c^{2} d g^{2} x^{4} + 2 \, b c^{2} d f g x^{3} - 2 \, b d f g x - b d f^{2} + {\left (b c^{2} d f^{2} - b d g^{2}\right )} x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)^2*(-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^2 + 1/48*a*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) - 2/5*(-c^2*d*x^2 + d)^(5/2)*a*f*g/(c^2*d) + sqrt(d)*integrate(-(b*c^2*d*g^2*x^4 + 2*b*c^2*d
*f*g*x^3 - 2*b*d*f*g*x - b*d*f^2 + (b*c^2*d*f^2 - b*d*g^2)*x^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 \[ \int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f + g*x)^2*(a + b*acos(c*x))*(d - c^2*d*x^2)^(3/2),x)
[Out]
int((f + g*x)^2*(a + b*acos(c*x))*(d - c^2*d*x^2)^(3/2), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right ) \left (f + g x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)**2*(-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)**2, x)
________________________________________________________________________________________