Integrand size = 27, antiderivative size = 293 \[ \int \frac {x^6 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b}{6 c^7 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {1-c^2 x^2}}{4 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^5 (a+b \arccos (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 x^3 (a+b \arccos (c x))}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 c^6 d^3}+\frac {5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 b c^7 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^7 d^2 \sqrt {d-c^2 d x^2}} \] Output:
-1/6*b/c^7/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/4*b*x^2*(-c^2*x^2 +1)^(1/2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)+1/3*x^5*(a+b*arccos(c*x))/c^2/d/(-c ^2*d*x^2+d)^(3/2)-5/3*x^3*(a+b*arccos(c*x))/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-5 /2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/c^6/d^3+5/4*(-c^2*x^2+1)^(1/2) *(a+b*arccos(c*x))^2/b/c^7/d^2/(-c^2*d*x^2+d)^(1/2)-7/6*b*(-c^2*x^2+1)^(1/ 2)*ln(-c^2*x^2+1)/c^7/d^2/(-c^2*d*x^2+d)^(1/2)
Time = 0.66 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.86 \[ \int \frac {x^6 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {4 b c \sqrt {d} x \left (15-20 c^2 x^2+3 c^4 x^4\right ) \arccos (c x)+30 b \sqrt {d} \left (1-c^2 x^2\right )^{3/2} \arccos (c x)^2-60 a \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d} \left (b \sqrt {1-c^2 x^2} \left (-7+9 c^2 x^2-6 c^4 x^4\right )+4 a c x \left (15-20 c^2 x^2+3 c^4 x^4\right )-28 b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )\right )}{24 c^7 d^{5/2} \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \] Input:
Integrate[(x^6*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^(5/2),x]
Output:
(4*b*c*Sqrt[d]*x*(15 - 20*c^2*x^2 + 3*c^4*x^4)*ArcCos[c*x] + 30*b*Sqrt[d]* (1 - c^2*x^2)^(3/2)*ArcCos[c*x]^2 - 60*a*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2 ]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d]*(b* Sqrt[1 - c^2*x^2]*(-7 + 9*c^2*x^2 - 6*c^4*x^4) + 4*a*c*x*(15 - 20*c^2*x^2 + 3*c^4*x^4) - 28*b*(1 - c^2*x^2)^(3/2)*Log[1 - c^2*x^2]))/(24*c^7*d^(5/2) *(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])
Time = 1.38 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.22, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {5207, 243, 49, 2009, 5207, 243, 49, 2009, 5211, 15, 5153}
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.
| \(\displaystyle \int \frac {x^6 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5207 |
| \(\displaystyle -\frac {5 \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}+\frac {b \sqrt {1-c^2 x^2} \int \frac {x^5}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^5 (a+b \arccos (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {5 \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}+\frac {b \sqrt {1-c^2 x^2} \int \frac {x^4}{\left (1-c^2 x^2\right )^2}dx^2}{6 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^5 (a+b \arccos (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\frac {5 \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}+\frac {b \sqrt {1-c^2 x^2} \int \left (\frac {1}{c^4}+\frac {2}{c^4 \left (c^2 x^2-1\right )}+\frac {1}{c^4 \left (c^2 x^2-1\right )^2}\right )dx^2}{6 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^5 (a+b \arccos (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5 \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}+\frac {x^5 (a+b \arccos (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5207 |
| \(\displaystyle -\frac {5 \left (-\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {b \sqrt {1-c^2 x^2} \int \frac {x^3}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^5 (a+b \arccos (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {5 \left (-\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {b \sqrt {1-c^2 x^2} \int \frac {x^2}{1-c^2 x^2}dx^2}{2 c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^5 (a+b \arccos (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\frac {5 \left (-\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {b \sqrt {1-c^2 x^2} \int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (c^2 x^2-1\right )}\right )dx^2}{2 c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^5 (a+b \arccos (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5 \left (-\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{2 c d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^5 (a+b \arccos (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle -\frac {5 \left (-\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {b \sqrt {1-c^2 x^2} \int xdx}{2 c \sqrt {d-c^2 d x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 c^2 d}\right )}{c^2 d}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{2 c d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^5 (a+b \arccos (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {5 \left (-\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 c^2 d}-\frac {b x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{2 c d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^5 (a+b \arccos (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5153 |
| \(\displaystyle \frac {x^5 (a+b \arccos (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 \left (\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 c^2 d}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}+\frac {b \sqrt {1-c^2 x^2} \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{2 c d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(x^6*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^(5/2),x]
Output:
(x^5*(a + b*ArcCos[c*x]))/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) + (b*Sqrt[1 - c^ 2*x^2]*(x^2/c^4 + 1/(c^6*(1 - c^2*x^2)) + (2*Log[1 - c^2*x^2])/c^6))/(6*c* d^2*Sqrt[d - c^2*d*x^2]) - (5*((x^3*(a + b*ArcCos[c*x]))/(c^2*d*Sqrt[d - c ^2*d*x^2]) - (3*(-1/4*(b*x^2*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(2*c^2*d) - (Sqrt[1 - c^2*x^2] *(a + b*ArcCos[c*x])^2)/(4*b*c^3*Sqrt[d - c^2*d*x^2])))/(c^2*d) + (b*Sqrt[ 1 - c^2*x^2]*(-(x^2/c^2) - Log[1 - c^2*x^2]/c^4))/(2*c*d*Sqrt[d - c^2*d*x^ 2])))/(3*c^2*d)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> 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] && EqQ[c^ 2*d + e, 0] && NeQ[n, -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/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Simp [b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[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]
Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.46
| method | result | size |
| default | \(-\frac {a \,x^{5}}{2 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,x^{3}}{6 c^{4} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {5 a x}{2 c^{6} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {5 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{6} d^{2} \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-12 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x^{5} c^{5}-6 c^{6} x^{6}-30 \arccos \left (c x \right )^{2} x^{4} c^{4}+112 i \arccos \left (c x \right ) c^{2} x^{2}+56 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{4} c^{4}+80 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x^{3} c^{3}+15 c^{4} x^{4}+60 \arccos \left (c x \right )^{2} x^{2} c^{2}-56 i \arccos \left (c x \right ) x^{4} c^{4}-112 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{2} c^{2}-60 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c -16 c^{2} x^{2}-30 \arccos \left (c x \right )^{2}-56 i \arccos \left (c x \right )+56 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right )+7\right )}{24 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) c^{7}}\) | \(429\) |
| parts | \(-\frac {a \,x^{5}}{2 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,x^{3}}{6 c^{4} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {5 a x}{2 c^{6} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {5 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{6} d^{2} \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-12 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x^{5} c^{5}-6 c^{6} x^{6}-30 \arccos \left (c x \right )^{2} x^{4} c^{4}+112 i \arccos \left (c x \right ) c^{2} x^{2}+56 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{4} c^{4}+80 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x^{3} c^{3}+15 c^{4} x^{4}+60 \arccos \left (c x \right )^{2} x^{2} c^{2}-56 i \arccos \left (c x \right ) x^{4} c^{4}-112 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{2} c^{2}-60 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c -16 c^{2} x^{2}-30 \arccos \left (c x \right )^{2}-56 i \arccos \left (c x \right )+56 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right )+7\right )}{24 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) c^{7}}\) | \(429\) |
Input:
int(x^6*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/2*a*x^5/c^2/d/(-c^2*d*x^2+d)^(3/2)+5/6*a/c^4*x^3/d/(-c^2*d*x^2+d)^(3/2) -5/2*a/c^6/d^2*x/(-c^2*d*x^2+d)^(1/2)+5/2*a/c^6/d^2/(c^2*d)^(1/2)*arctan(( c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/24*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x ^2+1)^(1/2)*(-12*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x^5*c^5-6*c^6*x^6-30*arcco s(c*x)^2*x^4*c^4+112*I*arccos(c*x)*x^2*c^2+56*ln((c*x+I*(-c^2*x^2+1)^(1/2) )^2-1)*x^4*c^4+80*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x^3*c^3+15*c^4*x^4+60*arc cos(c*x)^2*x^2*c^2-56*I*arccos(c*x)*x^4*c^4-112*ln((c*x+I*(-c^2*x^2+1)^(1/ 2))^2-1)*x^2*c^2-60*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x*c-16*c^2*x^2-30*arcco s(c*x)^2-56*I*arccos(c*x)+56*ln((c*x+I*(-c^2*x^2+1)^(1/2))^2-1)+7)/d^3/(c^ 6*x^6-3*c^4*x^4+3*c^2*x^2-1)/c^7
\[ \int \frac {x^6 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{6}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^6*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas" )
Output:
integral(-(b*x^6*arccos(c*x) + a*x^6)*sqrt(-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
\[ \int \frac {x^6 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{6} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x**6*(a+b*acos(c*x))/(-c**2*d*x**2+d)**(5/2),x)
Output:
Integral(x**6*(a + b*acos(c*x))/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
\[ \int \frac {x^6 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{6}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^6*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima" )
Output:
-1/6*a*(3*x^5/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 5*x*(3*x^2/((-c^2*d*x^2 + d )^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c^4*d))/c^2 + 5*x/(sqrt(-c^2*d* x^2 + d)*c^6*d^2) - 15*arcsin(c*x)/(c^7*d^(5/2))) + b*integrate(x^6*arctan 2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)* sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d)
Leaf count of result is larger than twice the leaf count of optimal. 1418 vs. \(2 (257) = 514\).
Time = 1.45 (sec) , antiderivative size = 1418, normalized size of antiderivative = 4.84 \[ \int \frac {x^6 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(x^6*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")
Output:
1/24*c*(8*b*arccos(c*x)/((-c^2*x^2 + 1)^(3/2)*c^5*d^(5/2)/x^3 + (c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*c^3*d^(5/2)/x^5) + 8*a/((-c^2*x^2 + 1)^(3/2)*c^5* d^(5/2)/x^3 + (c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*c^3*d^(5/2)/x^5) + 4*sqrt (-c^2*x^2 + 1)*b/(((-c^2*x^2 + 1)^(3/2)*c^5*d^(5/2)/x^3 + (c^2*x^2 - 1)^2* sqrt(-c^2*x^2 + 1)*c^3*d^(5/2)/x^5)*c*x) + 40*(c^2*x^2 - 1)*b*arccos(c*x)/ (((-c^2*x^2 + 1)^(3/2)*c^5*d^(5/2)/x^3 + (c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1 )*c^3*d^(5/2)/x^5)*c^2*x^2) - 30*(-c^2*x^2 + 1)^(3/2)*b*arccos(c*x)^2/(((- c^2*x^2 + 1)^(3/2)*c^5*d^(5/2)/x^3 + (c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*c^ 3*d^(5/2)/x^5)*c^3*x^3) + 40*(c^2*x^2 - 1)*a/(((-c^2*x^2 + 1)^(3/2)*c^5*d^ (5/2)/x^3 + (c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*c^3*d^(5/2)/x^5)*c^2*x^2) - 60*(-c^2*x^2 + 1)^(3/2)*a*arccos(c*x)/(((-c^2*x^2 + 1)^(3/2)*c^5*d^(5/2)/ x^3 + (c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*c^3*d^(5/2)/x^5)*c^3*x^3) + 56*(- c^2*x^2 + 1)^(3/2)*b*log(2)/(((-c^2*x^2 + 1)^(3/2)*c^5*d^(5/2)/x^3 + (c^2* x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*c^3*d^(5/2)/x^5)*c^3*x^3) + 28*(-c^2*x^2 + 1 )^(3/2)*b*log(abs(-c^2*x^2 + 1))/(((-c^2*x^2 + 1)^(3/2)*c^5*d^(5/2)/x^3 + (c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*c^3*d^(5/2)/x^5)*c^3*x^3) + 5*(-c^2*x^2 + 1)^(3/2)*b/(((-c^2*x^2 + 1)^(3/2)*c^5*d^(5/2)/x^3 + (c^2*x^2 - 1)^2*sqr t(-c^2*x^2 + 1)*c^3*d^(5/2)/x^5)*c^3*x^3) - 60*(c^2*x^2 - 1)^2*b*arccos(c* x)/(((-c^2*x^2 + 1)^(3/2)*c^5*d^(5/2)/x^3 + (c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*c^3*d^(5/2)/x^5)*c^4*x^4) - 30*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*...
Timed out. \[ \int \frac {x^6 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^6\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:
int((x^6*(a + b*acos(c*x)))/(d - c^2*d*x^2)^(5/2),x)
Output:
int((x^6*(a + b*acos(c*x)))/(d - c^2*d*x^2)^(5/2), x)
\[ \int \frac {x^6 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {15 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a \,c^{2} x^{2}-15 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a +6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right ) x^{6}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{9} x^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right ) x^{6}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{7}+3 a \,c^{5} x^{5}-20 a \,c^{3} x^{3}+15 a c x}{6 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{7} d^{2} \left (c^{2} x^{2}-1\right )} \] Input:
int(x^6*(a+b*acos(c*x))/(-c^2*d*x^2+d)^(5/2),x)
Output:
(15*sqrt( - c**2*x**2 + 1)*asin(c*x)*a*c**2*x**2 - 15*sqrt( - c**2*x**2 + 1)*asin(c*x)*a + 6*sqrt( - c**2*x**2 + 1)*int((acos(c*x)*x**6)/(sqrt( - c* *2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2 *x**2 + 1)),x)*b*c**9*x**2 - 6*sqrt( - c**2*x**2 + 1)*int((acos(c*x)*x**6) /(sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*b*c**7 + 3*a*c**5*x**5 - 20*a*c**3*x**3 + 15*a* c*x)/(6*sqrt(d)*sqrt( - c**2*x**2 + 1)*c**7*d**2*(c**2*x**2 - 1))