Integrand size = 27, antiderivative size = 308 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{x^{10}} \, dx=-\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {1-c^2 x^2}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {1-c^2 x^2}}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 x^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{315 d x^5}+\frac {8 b c^9 d \sqrt {d-c^2 d x^2} \log (x)}{315 \sqrt {1-c^2 x^2}} \] Output:
-1/72*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^8/(-c^2*x^2+1)^(1/2)+5/189*b*c^3*d*(-c^ 2*d*x^2+d)^(1/2)/x^6/(-c^2*x^2+1)^(1/2)-1/420*b*c^5*d*(-c^2*d*x^2+d)^(1/2) /x^4/(-c^2*x^2+1)^(1/2)-2/315*b*c^7*d*(-c^2*d*x^2+d)^(1/2)/x^2/(-c^2*x^2+1 )^(1/2)-1/9*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/d/x^9-4/63*c^2*(-c^2*d* x^2+d)^(5/2)*(a+b*arccos(c*x))/d/x^7-8/315*c^4*(-c^2*d*x^2+d)^(5/2)*(a+b*a rccos(c*x))/d/x^5+8/315*b*c^9*d*(-c^2*d*x^2+d)^(1/2)*ln(x)/(-c^2*x^2+1)^(1 /2)
Time = 0.37 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.77 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{x^{10}} \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (-3675 b c x+7000 b c^3 x^3-630 b c^5 x^5-1680 b c^7 x^7-18264 b c^9 x^9+29400 a \sqrt {1-c^2 x^2}-42000 a c^2 x^2 \sqrt {1-c^2 x^2}+2520 a c^4 x^4 \sqrt {1-c^2 x^2}+3360 a c^6 x^6 \sqrt {1-c^2 x^2}+6720 a c^8 x^8 \sqrt {1-c^2 x^2}+840 b \left (1-c^2 x^2\right )^{5/2} \left (35+20 c^2 x^2+8 c^4 x^4\right ) \arccos (c x)+6720 b c^9 x^9 \log (x)\right )}{264600 x^9 \sqrt {1-c^2 x^2}} \] Input:
Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]))/x^10,x]
Output:
-1/264600*(d*Sqrt[d - c^2*d*x^2]*(-3675*b*c*x + 7000*b*c^3*x^3 - 630*b*c^5 *x^5 - 1680*b*c^7*x^7 - 18264*b*c^9*x^9 + 29400*a*Sqrt[1 - c^2*x^2] - 4200 0*a*c^2*x^2*Sqrt[1 - c^2*x^2] + 2520*a*c^4*x^4*Sqrt[1 - c^2*x^2] + 3360*a* c^6*x^6*Sqrt[1 - c^2*x^2] + 6720*a*c^8*x^8*Sqrt[1 - c^2*x^2] + 840*b*(1 - c^2*x^2)^(5/2)*(35 + 20*c^2*x^2 + 8*c^4*x^4)*ArcCos[c*x] + 6720*b*c^9*x^9* Log[x]))/(x^9*Sqrt[1 - c^2*x^2])
Time = 0.51 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.61, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {5195, 27, 1578, 1195, 2009}
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 5195 |
| \(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d \left (1-c^2 x^2\right )^2 \left (8 c^4 x^4+20 c^2 x^2+35\right )}{315 x^9}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{315 d x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 \left (8 c^4 x^4+20 c^2 x^2+35\right )}{x^9}dx}{315 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{315 d x^5}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 \left (8 c^4 x^4+20 c^2 x^2+35\right )}{x^{10}}dx^2}{630 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{315 d x^5}\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle -\frac {b c d \sqrt {d-c^2 d x^2} \int \left (\frac {8 c^8}{x^2}+\frac {4 c^6}{x^4}+\frac {3 c^4}{x^6}-\frac {50 c^2}{x^8}+\frac {35}{x^{10}}\right )dx^2}{630 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{315 d x^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{315 d x^5}-\frac {b c d \sqrt {d-c^2 d x^2} \left (8 c^8 \log \left (x^2\right )-\frac {4 c^6}{x^2}-\frac {3 c^4}{2 x^4}+\frac {50 c^2}{3 x^6}-\frac {35}{4 x^8}\right )}{630 \sqrt {1-c^2 x^2}}\) |
Input:
Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]))/x^10,x]
Output:
-1/9*((d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]))/(d*x^9) - (4*c^2*(d - c^2 *d*x^2)^(5/2)*(a + b*ArcCos[c*x]))/(63*d*x^7) - (8*c^4*(d - c^2*d*x^2)^(5/ 2)*(a + b*ArcCos[c*x]))/(315*d*x^5) - (b*c*d*Sqrt[d - c^2*d*x^2]*(-35/(4*x ^8) + (50*c^2)/(3*x^6) - (3*c^4)/(2*x^4) - (4*c^6)/x^2 + 8*c^8*Log[x^2]))/ (630*Sqrt[1 - c^2*x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
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])
Result contains complex when optimal does not.
Time = 1.14 (sec) , antiderivative size = 4563, normalized size of antiderivative = 14.81
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(4563\) |
| parts | \(\text {Expression too large to display}\) | \(4563\) |
Input:
int((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/x^10,x,method=_RETURNVERBOSE)
Output:
a*(-1/9/d/x^9*(-c^2*d*x^2+d)^(5/2)+4/9*c^2*(-1/7/d/x^7*(-c^2*d*x^2+d)^(5/2 )-2/35*c^2/d/x^5*(-c^2*d*x^2+d)^(5/2)))-24*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(8 40*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2* x^2+1225)*x^10/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^19+24/5*I*b*(- d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x ^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^8/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcc os(c*x)*c^17-208/3*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^ 10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^6/(c^2*x^2-1 )*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^15+1104/7*I*b*(-d*(c^2*x^2-1))^(1/2)*d/ (840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^ 2*x^2+1225)*x^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^13-120*I*b*(- d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x ^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcc os(c*x)*c^11+64/3*I*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^1 0+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^12/(c^2*x^2-1 )*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^21+8/315*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2 *x^2+1)^(1/2)/(c^2*x^2-1)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)*d*c^9+1225/9* b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c ^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^9/(c^2*x^2-1)*arccos(c*x)-30055/5 04*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-...
Time = 0.23 (sec) , antiderivative size = 672, normalized size of antiderivative = 2.18 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{x^{10}} \, dx=\left [\frac {96 \, {\left (b c^{11} d x^{11} - b c^{9} d x^{9}\right )} \sqrt {d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {d} - d}{c^{2} x^{4} - x^{2}}\right ) - {\left (48 \, b c^{7} d x^{7} + 18 \, b c^{5} d x^{5} - {\left (48 \, b c^{7} + 18 \, b c^{5} - 200 \, b c^{3} + 105 \, b c\right )} d x^{9} - 200 \, b c^{3} d x^{3} + 105 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 24 \, {\left (8 \, a c^{10} d x^{10} - 4 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 53 \, a c^{4} d x^{4} + 85 \, a c^{2} d x^{2} - 35 \, a d + {\left (8 \, b c^{10} d x^{10} - 4 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 53 \, b c^{4} d x^{4} + 85 \, b c^{2} d x^{2} - 35 \, b d\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{7560 \, {\left (c^{2} x^{11} - x^{9}\right )}}, -\frac {192 \, {\left (b c^{11} d x^{11} - b c^{9} d x^{9}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{2} - 1\right )} \sqrt {-d}}{c^{2} d x^{4} + {\left (c^{2} - 1\right )} d x^{2} - d}\right ) + {\left (48 \, b c^{7} d x^{7} + 18 \, b c^{5} d x^{5} - {\left (48 \, b c^{7} + 18 \, b c^{5} - 200 \, b c^{3} + 105 \, b c\right )} d x^{9} - 200 \, b c^{3} d x^{3} + 105 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 24 \, {\left (8 \, a c^{10} d x^{10} - 4 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 53 \, a c^{4} d x^{4} + 85 \, a c^{2} d x^{2} - 35 \, a d + {\left (8 \, b c^{10} d x^{10} - 4 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 53 \, b c^{4} d x^{4} + 85 \, b c^{2} d x^{2} - 35 \, b d\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{7560 \, {\left (c^{2} x^{11} - x^{9}\right )}}\right ] \] Input:
integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/x^10,x, algorithm="fricas ")
Output:
[1/7560*(96*(b*c^11*d*x^11 - b*c^9*d*x^9)*sqrt(d)*log((c^2*d*x^6 + c^2*d*x ^2 - d*x^4 + sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*(x^4 - 1)*sqrt(d) - d )/(c^2*x^4 - x^2)) - (48*b*c^7*d*x^7 + 18*b*c^5*d*x^5 - (48*b*c^7 + 18*b*c ^5 - 200*b*c^3 + 105*b*c)*d*x^9 - 200*b*c^3*d*x^3 + 105*b*c*d*x)*sqrt(-c^2 *d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 24*(8*a*c^10*d*x^10 - 4*a*c^8*d*x^8 - a*c ^6*d*x^6 - 53*a*c^4*d*x^4 + 85*a*c^2*d*x^2 - 35*a*d + (8*b*c^10*d*x^10 - 4 *b*c^8*d*x^8 - b*c^6*d*x^6 - 53*b*c^4*d*x^4 + 85*b*c^2*d*x^2 - 35*b*d)*arc cos(c*x))*sqrt(-c^2*d*x^2 + d))/(c^2*x^11 - x^9), -1/7560*(192*(b*c^11*d*x ^11 - b*c^9*d*x^9)*sqrt(-d)*arctan(sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) *(x^2 - 1)*sqrt(-d)/(c^2*d*x^4 + (c^2 - 1)*d*x^2 - d)) + (48*b*c^7*d*x^7 + 18*b*c^5*d*x^5 - (48*b*c^7 + 18*b*c^5 - 200*b*c^3 + 105*b*c)*d*x^9 - 200* b*c^3*d*x^3 + 105*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 24*(8 *a*c^10*d*x^10 - 4*a*c^8*d*x^8 - a*c^6*d*x^6 - 53*a*c^4*d*x^4 + 85*a*c^2*d *x^2 - 35*a*d + (8*b*c^10*d*x^10 - 4*b*c^8*d*x^8 - b*c^6*d*x^6 - 53*b*c^4* d*x^4 + 85*b*c^2*d*x^2 - 35*b*d)*arccos(c*x))*sqrt(-c^2*d*x^2 + d))/(c^2*x ^11 - x^9)]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{x^{10}} \, dx=\text {Timed out} \] Input:
integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acos(c*x))/x**10,x)
Output:
Timed out
Time = 0.14 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.68 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{x^{10}} \, dx=-\frac {1}{7560} \, {\left (192 \, c^{8} d^{\frac {3}{2}} \log \left (x\right ) - \frac {48 \, c^{6} d^{\frac {3}{2}} x^{6} + 18 \, c^{4} d^{\frac {3}{2}} x^{4} - 200 \, c^{2} d^{\frac {3}{2}} x^{2} + 105 \, d^{\frac {3}{2}}}{x^{8}}\right )} b c - \frac {1}{315} \, b {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{5}} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{7}} + \frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{9}}\right )} \arccos \left (c x\right ) - \frac {1}{315} \, a {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{5}} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{7}} + \frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{9}}\right )} \] Input:
integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/x^10,x, algorithm="maxima ")
Output:
-1/7560*(192*c^8*d^(3/2)*log(x) - (48*c^6*d^(3/2)*x^6 + 18*c^4*d^(3/2)*x^4 - 200*c^2*d^(3/2)*x^2 + 105*d^(3/2))/x^8)*b*c - 1/315*b*(8*(-c^2*d*x^2 + d)^(5/2)*c^4/(d*x^5) + 20*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x^7) + 35*(-c^2*d* x^2 + d)^(5/2)/(d*x^9))*arccos(c*x) - 1/315*a*(8*(-c^2*d*x^2 + d)^(5/2)*c^ 4/(d*x^5) + 20*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x^7) + 35*(-c^2*d*x^2 + d)^(5 /2)/(d*x^9))
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{x^{10}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/x^10,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{x^{10}} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^{10}} \,d x \] Input:
int(((a + b*acos(c*x))*(d - c^2*d*x^2)^(3/2))/x^10,x)
Output:
int(((a + b*acos(c*x))*(d - c^2*d*x^2)^(3/2))/x^10, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{x^{10}} \, dx=\frac {\sqrt {d}\, d \left (-8 \sqrt {-c^{2} x^{2}+1}\, a \,c^{8} x^{8}-4 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}-3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}+50 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-35 \sqrt {-c^{2} x^{2}+1}\, a +315 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{x^{10}}d x \right ) b \,x^{9}-315 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{x^{8}}d x \right ) b \,c^{2} x^{9}\right )}{315 x^{9}} \] Input:
int((-c^2*d*x^2+d)^(3/2)*(a+b*acos(c*x))/x^10,x)
Output:
(sqrt(d)*d*( - 8*sqrt( - c**2*x**2 + 1)*a*c**8*x**8 - 4*sqrt( - c**2*x**2 + 1)*a*c**6*x**6 - 3*sqrt( - c**2*x**2 + 1)*a*c**4*x**4 + 50*sqrt( - c**2* x**2 + 1)*a*c**2*x**2 - 35*sqrt( - c**2*x**2 + 1)*a + 315*int((sqrt( - c** 2*x**2 + 1)*acos(c*x))/x**10,x)*b*x**9 - 315*int((sqrt( - c**2*x**2 + 1)*a cos(c*x))/x**8,x)*b*c**2*x**9))/(315*x**9)