Integrand size = 25, antiderivative size = 211 \[ \int x^2 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {52 b^2 d x}{225 c^2}-\frac {26}{675} b^2 d x^3+\frac {2}{125} b^2 c^2 d x^5+\frac {8 b d \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{45 c^3}+\frac {4 b d x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{45 c}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{15 c^3}-\frac {2 b d \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{25 c^3}+\frac {2}{15} d x^3 (a+b \arccos (c x))^2+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2 \] Output:
-52/225*b^2*d*x/c^2-26/675*b^2*d*x^3+2/125*b^2*c^2*d*x^5+8/45*b*d*(-c^2*x^ 2+1)^(1/2)*(a+b*arccos(c*x))/c^3+4/45*b*d*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arcc os(c*x))/c+2/15*b*d*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))/c^3-2/25*b*d*(-c^ 2*x^2+1)^(5/2)*(a+b*arccos(c*x))/c^3+2/15*d*x^3*(a+b*arccos(c*x))^2+1/5*d* x^3*(-c^2*x^2+1)*(a+b*arccos(c*x))^2
Time = 0.19 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.85 \[ \int x^2 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {d \left (225 a^2 c^3 x^3 \left (-5+3 c^2 x^2\right )-30 a b \sqrt {1-c^2 x^2} \left (-26-13 c^2 x^2+9 c^4 x^4\right )+b^2 \left (780 c x+130 c^3 x^3-54 c^5 x^5\right )-30 b \left (b \sqrt {1-c^2 x^2} \left (-26-13 c^2 x^2+9 c^4 x^4\right )+a \left (75 c^3 x^3-45 c^5 x^5\right )\right ) \arccos (c x)+225 b^2 c^3 x^3 \left (-5+3 c^2 x^2\right ) \arccos (c x)^2\right )}{3375 c^3} \] Input:
Integrate[x^2*(d - c^2*d*x^2)*(a + b*ArcCos[c*x])^2,x]
Output:
-1/3375*(d*(225*a^2*c^3*x^3*(-5 + 3*c^2*x^2) - 30*a*b*Sqrt[1 - c^2*x^2]*(- 26 - 13*c^2*x^2 + 9*c^4*x^4) + b^2*(780*c*x + 130*c^3*x^3 - 54*c^5*x^5) - 30*b*(b*Sqrt[1 - c^2*x^2]*(-26 - 13*c^2*x^2 + 9*c^4*x^4) + a*(75*c^3*x^3 - 45*c^5*x^5))*ArcCos[c*x] + 225*b^2*c^3*x^3*(-5 + 3*c^2*x^2)*ArcCos[c*x]^2 ))/c^3
Time = 1.07 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5203, 5139, 5195, 27, 2009, 5211, 15, 5183, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5203 |
| \(\displaystyle \frac {2}{5} b c d \int x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {2}{5} d \int x^2 (a+b \arccos (c x))^2dx+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 5139 |
| \(\displaystyle \frac {2}{5} d \left (\frac {2}{3} b c \int \frac {x^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {2}{5} b c d \int x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 5195 |
| \(\displaystyle \frac {2}{5} d \left (\frac {2}{3} b c \int \frac {x^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {2}{5} b c d \left (b c \int -\frac {-3 c^4 x^4+c^2 x^2+2}{15 c^4}dx+\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}\right )+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} d \left (\frac {2}{3} b c \int \frac {x^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {2}{5} b c d \left (-\frac {b \int \left (-3 c^4 x^4+c^2 x^2+2\right )dx}{15 c^3}+\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}\right )+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{5} d \left (\frac {2}{3} b c \int \frac {x^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{5} b c d \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}-\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle \frac {2}{5} d \left (\frac {2}{3} b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {b \int x^2dx}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{5} b c d \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}-\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {2}{5} d \left (\frac {2}{3} b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{5} b c d \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}-\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle \frac {2}{5} d \left (\frac {2}{3} b c \left (\frac {2 \left (-\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {1}{5} d x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{5} b c d \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}-\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{5} d x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{5} d \left (\frac {2}{3} b c \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}+\frac {2 \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {2}{5} b c d \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}-\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\) |
Input:
Int[x^2*(d - c^2*d*x^2)*(a + b*ArcCos[c*x])^2,x]
Output:
(d*x^3*(1 - c^2*x^2)*(a + b*ArcCos[c*x])^2)/5 + (2*b*c*d*(-1/15*(b*(2*x + (c^2*x^3)/3 - (3*c^4*x^5)/5))/c^3 - ((1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x ]))/(3*c^4) + ((1 - c^2*x^2)^(5/2)*(a + b*ArcCos[c*x]))/(5*c^4)))/5 + (2*d *((x^3*(a + b*ArcCos[c*x])^2)/3 + (2*b*c*(-1/9*(b*x^3)/c - (x^2*Sqrt[1 - c ^2*x^2]*(a + b*ArcCos[c*x]))/(3*c^2) + (2*(-((b*x)/c) - (Sqrt[1 - c^2*x^2] *(a + b*ArcCos[c*x]))/c^2))/(3*c^2)))/3))/5
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
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] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(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]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) , x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos [c*x]) u, x] + Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[Sim plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
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*ArcC os[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[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] + Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Time = 0.27 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.32
| method | result | size |
| parts | \(-d \,a^{2} \left (\frac {1}{5} c^{2} x^{5}-\frac {1}{3} x^{3}\right )-\frac {d \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{15}+\frac {4 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{135}+\frac {\arccos \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}\right )}{c^{3}}-\frac {2 d a b \left (\frac {\arccos \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}+\frac {13 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {-c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}\right )}{c^{3}}\) | \(279\) |
| derivativedivides | \(\frac {-d \,a^{2} \left (\frac {1}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{15}+\frac {4 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{135}+\frac {\arccos \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}\right )-2 d a b \left (\frac {\arccos \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}+\frac {13 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {-c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}\right )}{c^{3}}\) | \(280\) |
| default | \(\frac {-d \,a^{2} \left (\frac {1}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{15}+\frac {4 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{135}+\frac {\arccos \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}\right )-2 d a b \left (\frac {\arccos \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}+\frac {13 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {-c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}\right )}{c^{3}}\) | \(280\) |
| orering | \(\frac {\left (1647 c^{8} x^{8}-4862 c^{6} x^{6}-4033 c^{4} x^{4}+7800 c^{2} x^{2}-3120\right ) \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2}}{3375 x \,c^{4} \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (324 c^{6} x^{6}-893 c^{4} x^{4}-2665 c^{2} x^{2}+1950\right ) \left (2 x \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2}-2 x^{3} d \,c^{2} \left (a +b \arccos \left (c x \right )\right )^{2}-\frac {2 x^{2} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{3375 x^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\left (27 c^{4} x^{4}-65 c^{2} x^{2}-390\right ) \left (2 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2}-10 x^{2} d \,c^{2} \left (a +b \arccos \left (c x \right )\right )^{2}-\frac {8 x \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {8 x^{3} d \,c^{3} \left (a +b \arccos \left (c x \right )\right ) b}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 x^{2} \left (-c^{2} d \,x^{2}+d \right ) b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {2 x^{3} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{3375 x \,c^{4}}\) | \(412\) |
Input:
int(x^2*(-c^2*d*x^2+d)*(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)
Output:
-d*a^2*(1/5*c^2*x^5-1/3*x^3)-d*b^2/c^3*(1/3*arccos(c*x)^2*(c^2*x^2-3)*c*x+ 4/15*c*x+4/15*arccos(c*x)*(-c^2*x^2+1)^(1/2)-2/45*arccos(c*x)*(c^2*x^2-1)* (-c^2*x^2+1)^(1/2)-2/135*(c^2*x^2-3)*c*x+1/15*arccos(c*x)^2*(3*c^4*x^4-10* c^2*x^2+15)*c*x-2/25*arccos(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/2)-2/375*(3 *c^4*x^4-10*c^2*x^2+15)*c*x)-2*d*a*b/c^3*(1/5*arccos(c*x)*c^5*x^5-1/3*c^3* x^3*arccos(c*x)+13/225*c^2*x^2*(-c^2*x^2+1)^(1/2)+26/225*(-c^2*x^2+1)^(1/2 )-1/25*c^4*x^4*(-c^2*x^2+1)^(1/2))
Time = 0.12 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.92 \[ \int x^2 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {27 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{5} d x^{5} - 5 \, {\left (225 \, a^{2} - 26 \, b^{2}\right )} c^{3} d x^{3} + 780 \, b^{2} c d x + 225 \, {\left (3 \, b^{2} c^{5} d x^{5} - 5 \, b^{2} c^{3} d x^{3}\right )} \arccos \left (c x\right )^{2} + 450 \, {\left (3 \, a b c^{5} d x^{5} - 5 \, a b c^{3} d x^{3}\right )} \arccos \left (c x\right ) - 30 \, {\left (9 \, a b c^{4} d x^{4} - 13 \, a b c^{2} d x^{2} - 26 \, a b d + {\left (9 \, b^{2} c^{4} d x^{4} - 13 \, b^{2} c^{2} d x^{2} - 26 \, b^{2} d\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{3375 \, c^{3}} \] Input:
integrate(x^2*(-c^2*d*x^2+d)*(a+b*arccos(c*x))^2,x, algorithm="fricas")
Output:
-1/3375*(27*(25*a^2 - 2*b^2)*c^5*d*x^5 - 5*(225*a^2 - 26*b^2)*c^3*d*x^3 + 780*b^2*c*d*x + 225*(3*b^2*c^5*d*x^5 - 5*b^2*c^3*d*x^3)*arccos(c*x)^2 + 45 0*(3*a*b*c^5*d*x^5 - 5*a*b*c^3*d*x^3)*arccos(c*x) - 30*(9*a*b*c^4*d*x^4 - 13*a*b*c^2*d*x^2 - 26*a*b*d + (9*b^2*c^4*d*x^4 - 13*b^2*c^2*d*x^2 - 26*b^2 *d)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/c^3
Time = 0.50 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.51 \[ \int x^2 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=\begin {cases} - \frac {a^{2} c^{2} d x^{5}}{5} + \frac {a^{2} d x^{3}}{3} - \frac {2 a b c^{2} d x^{5} \operatorname {acos}{\left (c x \right )}}{5} + \frac {2 a b c d x^{4} \sqrt {- c^{2} x^{2} + 1}}{25} + \frac {2 a b d x^{3} \operatorname {acos}{\left (c x \right )}}{3} - \frac {26 a b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{225 c} - \frac {52 a b d \sqrt {- c^{2} x^{2} + 1}}{225 c^{3}} - \frac {b^{2} c^{2} d x^{5} \operatorname {acos}^{2}{\left (c x \right )}}{5} + \frac {2 b^{2} c^{2} d x^{5}}{125} + \frac {2 b^{2} c d x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{25} + \frac {b^{2} d x^{3} \operatorname {acos}^{2}{\left (c x \right )}}{3} - \frac {26 b^{2} d x^{3}}{675} - \frac {26 b^{2} d x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{225 c} - \frac {52 b^{2} d x}{225 c^{2}} - \frac {52 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{225 c^{3}} & \text {for}\: c \neq 0 \\\frac {d x^{3} \left (a + \frac {\pi b}{2}\right )^{2}}{3} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(-c**2*d*x**2+d)*(a+b*acos(c*x))**2,x)
Output:
Piecewise((-a**2*c**2*d*x**5/5 + a**2*d*x**3/3 - 2*a*b*c**2*d*x**5*acos(c* x)/5 + 2*a*b*c*d*x**4*sqrt(-c**2*x**2 + 1)/25 + 2*a*b*d*x**3*acos(c*x)/3 - 26*a*b*d*x**2*sqrt(-c**2*x**2 + 1)/(225*c) - 52*a*b*d*sqrt(-c**2*x**2 + 1 )/(225*c**3) - b**2*c**2*d*x**5*acos(c*x)**2/5 + 2*b**2*c**2*d*x**5/125 + 2*b**2*c*d*x**4*sqrt(-c**2*x**2 + 1)*acos(c*x)/25 + b**2*d*x**3*acos(c*x)* *2/3 - 26*b**2*d*x**3/675 - 26*b**2*d*x**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/ (225*c) - 52*b**2*d*x/(225*c**2) - 52*b**2*d*sqrt(-c**2*x**2 + 1)*acos(c*x )/(225*c**3), Ne(c, 0)), (d*x**3*(a + pi*b/2)**2/3, True))
Time = 0.13 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.68 \[ \int x^2 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {1}{5} \, b^{2} c^{2} d x^{5} \arccos \left (c x\right )^{2} - \frac {1}{5} \, a^{2} c^{2} d x^{5} + \frac {1}{3} \, b^{2} d x^{3} \arccos \left (c x\right )^{2} - \frac {2}{75} \, {\left (15 \, x^{5} \arccos \left (c x\right ) - {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{2} d + \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arccos \left (c x\right ) + \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{2} d + \frac {1}{3} \, a^{2} d x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right ) + \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d \] Input:
integrate(x^2*(-c^2*d*x^2+d)*(a+b*arccos(c*x))^2,x, algorithm="maxima")
Output:
-1/5*b^2*c^2*d*x^5*arccos(c*x)^2 - 1/5*a^2*c^2*d*x^5 + 1/3*b^2*d*x^3*arcco s(c*x)^2 - 2/75*(15*x^5*arccos(c*x) - (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sq rt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*a*b*c^2*d + 2/1125 *(15*(3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt (-c^2*x^2 + 1)/c^6)*c*arccos(c*x) + (9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4)* b^2*c^2*d + 1/3*a^2*d*x^3 + 2/9*(3*x^3*arccos(c*x) - c*(sqrt(-c^2*x^2 + 1) *x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*d - 2/27*(3*c*(sqrt(-c^2*x^2 + 1 )*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arccos(c*x) + (c^2*x^3 + 6*x)/c^2)*b ^2*d
Time = 0.15 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.21 \[ \int x^2 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {1}{5} \, b^{2} c^{2} d x^{5} \arccos \left (c x\right )^{2} - \frac {2}{5} \, a b c^{2} d x^{5} \arccos \left (c x\right ) - \frac {1}{5} \, a^{2} c^{2} d x^{5} + \frac {2}{125} \, b^{2} c^{2} d x^{5} + \frac {2}{25} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c d x^{4} \arccos \left (c x\right ) + \frac {2}{25} \, \sqrt {-c^{2} x^{2} + 1} a b c d x^{4} + \frac {1}{3} \, b^{2} d x^{3} \arccos \left (c x\right )^{2} + \frac {2}{3} \, a b d x^{3} \arccos \left (c x\right ) + \frac {1}{3} \, a^{2} d x^{3} - \frac {26}{675} \, b^{2} d x^{3} - \frac {26 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d x^{2} \arccos \left (c x\right )}{225 \, c} - \frac {26 \, \sqrt {-c^{2} x^{2} + 1} a b d x^{2}}{225 \, c} - \frac {52 \, b^{2} d x}{225 \, c^{2}} - \frac {52 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arccos \left (c x\right )}{225 \, c^{3}} - \frac {52 \, \sqrt {-c^{2} x^{2} + 1} a b d}{225 \, c^{3}} \] Input:
integrate(x^2*(-c^2*d*x^2+d)*(a+b*arccos(c*x))^2,x, algorithm="giac")
Output:
-1/5*b^2*c^2*d*x^5*arccos(c*x)^2 - 2/5*a*b*c^2*d*x^5*arccos(c*x) - 1/5*a^2 *c^2*d*x^5 + 2/125*b^2*c^2*d*x^5 + 2/25*sqrt(-c^2*x^2 + 1)*b^2*c*d*x^4*arc cos(c*x) + 2/25*sqrt(-c^2*x^2 + 1)*a*b*c*d*x^4 + 1/3*b^2*d*x^3*arccos(c*x) ^2 + 2/3*a*b*d*x^3*arccos(c*x) + 1/3*a^2*d*x^3 - 26/675*b^2*d*x^3 - 26/225 *sqrt(-c^2*x^2 + 1)*b^2*d*x^2*arccos(c*x)/c - 26/225*sqrt(-c^2*x^2 + 1)*a* b*d*x^2/c - 52/225*b^2*d*x/c^2 - 52/225*sqrt(-c^2*x^2 + 1)*b^2*d*arccos(c* x)/c^3 - 52/225*sqrt(-c^2*x^2 + 1)*a*b*d/c^3
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,\left (d-c^2\,d\,x^2\right ) \,d x \] Input:
int(x^2*(a + b*acos(c*x))^2*(d - c^2*d*x^2),x)
Output:
int(x^2*(a + b*acos(c*x))^2*(d - c^2*d*x^2), x)
\[ \int x^2 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=\frac {d \left (-90 \mathit {acos} \left (c x \right ) a b \,c^{5} x^{5}+150 \mathit {acos} \left (c x \right ) a b \,c^{3} x^{3}+18 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{4} x^{4}-26 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} x^{2}-52 \sqrt {-c^{2} x^{2}+1}\, a b -225 \left (\int \mathit {acos} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+225 \left (\int \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}-45 a^{2} c^{5} x^{5}+75 a^{2} c^{3} x^{3}\right )}{225 c^{3}} \] Input:
int(x^2*(-c^2*d*x^2+d)*(a+b*acos(c*x))^2,x)
Output:
(d*( - 90*acos(c*x)*a*b*c**5*x**5 + 150*acos(c*x)*a*b*c**3*x**3 + 18*sqrt( - c**2*x**2 + 1)*a*b*c**4*x**4 - 26*sqrt( - c**2*x**2 + 1)*a*b*c**2*x**2 - 52*sqrt( - c**2*x**2 + 1)*a*b - 225*int(acos(c*x)**2*x**4,x)*b**2*c**5 + 225*int(acos(c*x)**2*x**2,x)*b**2*c**3 - 45*a**2*c**5*x**5 + 75*a**2*c**3 *x**3))/(225*c**3)