Integrand size = 27, antiderivative size = 203 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^8} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d^2 \sqrt {d-c^2 d x^2}}{28 x^4 \sqrt {1-c^2 x^2}}-\frac {3 b c^5 d^2 \sqrt {d-c^2 d x^2}}{14 x^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 d x^7}-\frac {b c^7 d^2 \sqrt {d-c^2 d x^2} \log (x)}{7 \sqrt {1-c^2 x^2}} \]
-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arcsin(c*x))/d/x^7-1/42*b*c*d^2*(-c^2*d*x^2 +d)^(1/2)/x^6/(-c^2*x^2+1)^(1/2)+3/28*b*c^3*d^2*(-c^2*d*x^2+d)^(1/2)/x^4/( -c^2*x^2+1)^(1/2)-3/14*b*c^5*d^2*(-c^2*d*x^2+d)^(1/2)/x^2/(-c^2*x^2+1)^(1/ 2)-1/7*b*c^7*d^2*ln(x)*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)
Time = 0.31 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.94 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^8} \, dx=-\frac {d^2 \sqrt {d-c^2 d x^2} \left (10 b c x-45 b c^3 x^3+90 b c^5 x^5-147 b c^7 x^7+60 a \sqrt {1-c^2 x^2}-180 a c^2 x^2 \sqrt {1-c^2 x^2}+180 a c^4 x^4 \sqrt {1-c^2 x^2}-60 a c^6 x^6 \sqrt {1-c^2 x^2}+60 b \left (1-c^2 x^2\right )^{7/2} \arcsin (c x)+60 b c^7 x^7 \log (x)\right )}{420 x^7 \sqrt {1-c^2 x^2}} \]
-1/420*(d^2*Sqrt[d - c^2*d*x^2]*(10*b*c*x - 45*b*c^3*x^3 + 90*b*c^5*x^5 - 147*b*c^7*x^7 + 60*a*Sqrt[1 - c^2*x^2] - 180*a*c^2*x^2*Sqrt[1 - c^2*x^2] + 180*a*c^4*x^4*Sqrt[1 - c^2*x^2] - 60*a*c^6*x^6*Sqrt[1 - c^2*x^2] + 60*b*( 1 - c^2*x^2)^(7/2)*ArcSin[c*x] + 60*b*c^7*x^7*Log[x]))/(x^7*Sqrt[1 - c^2*x ^2])
Time = 0.32 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.53, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5186, 243, 49, 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.
\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^8} \, dx\) |
\(\Big \downarrow \) 5186 |
\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^3}{x^7}dx}{7 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 d x^7}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^3}{x^8}dx^2}{14 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 d x^7}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (-\frac {c^6}{x^2}+\frac {3 c^4}{x^4}-\frac {3 c^2}{x^6}+\frac {1}{x^8}\right )dx^2}{14 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 d x^7}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^6 \left (-\log \left (x^2\right )\right )-\frac {3 c^4}{x^2}+\frac {3 c^2}{2 x^4}-\frac {1}{3 x^6}\right )}{14 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 d x^7}\) |
-1/7*((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(d*x^7) + (b*c*d^2*Sqrt[d - c^2*d*x^2]*(-1/3*1/x^6 + (3*c^2)/(2*x^4) - (3*c^4)/x^2 - c^6*Log[x^2])) /(14*Sqrt[1 - c^2*x^2])
3.1.90.3.1 Defintions of rubi rules used
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_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 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*A rcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 4031, normalized size of antiderivative = 19.86
method | result | size |
default | \(\text {Expression too large to display}\) | \(4031\) |
parts | \(\text {Expression too large to display}\) | \(4031\) |
I*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6 *x^6+21*c^4*x^4-7*c^2*x^2+1)*x^12/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c* x)*c^19-5*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8* x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^6/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)* arcsin(c*x)*c^13+3*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^1 0+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^4/(c^2*x^2-1)*(-c^2*x^2+ 1)^(1/2)*arcsin(c*x)*c^11-I*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c ^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c ^2*x^2+1)^(1/2)*arcsin(c*x)*c^9-3*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x ^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^10/(c^2*x ^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^17+5*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2 /(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x ^8/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^15-2*I*b*(-d*(c^2*x^2-1))^ (1/2)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^7*d^2/(7*c^2*x^2-7)-3/14*I*b*(-d*(c ^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^ 4*x^4-7*c^2*x^2+1)*x^13/(c^2*x^2-1)*c^20+27/28*I*b*(-d*(c^2*x^2-1))^(1/2)* d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1 )*x^11/(c^2*x^2-1)*c^18-7*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^1 0*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^11/(c^2*x^2-1)*arcs in(c*x)*c^18-1/42*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x...
Time = 0.31 (sec) , antiderivative size = 655, normalized size of antiderivative = 3.23 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^8} \, dx=\left [\frac {6 \, {\left (b c^{9} d^{2} x^{9} - b c^{7} d^{2} x^{7}\right )} \sqrt {d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {d} - d}{c^{2} x^{4} - x^{2}}\right ) + {\left (18 \, b c^{5} d^{2} x^{5} - {\left (18 \, b c^{5} - 9 \, b c^{3} + 2 \, b c\right )} d^{2} x^{7} - 9 \, b c^{3} d^{2} x^{3} + 2 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 12 \, {\left (a c^{8} d^{2} x^{8} - 4 \, a c^{6} d^{2} x^{6} + 6 \, a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + a d^{2} + {\left (b c^{8} d^{2} x^{8} - 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{84 \, {\left (c^{2} x^{9} - x^{7}\right )}}, -\frac {12 \, {\left (b c^{9} d^{2} x^{9} - b c^{7} d^{2} x^{7}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{2} + 1\right )} \sqrt {-d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - {\left (18 \, b c^{5} d^{2} x^{5} - {\left (18 \, b c^{5} - 9 \, b c^{3} + 2 \, b c\right )} d^{2} x^{7} - 9 \, b c^{3} d^{2} x^{3} + 2 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 12 \, {\left (a c^{8} d^{2} x^{8} - 4 \, a c^{6} d^{2} x^{6} + 6 \, a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + a d^{2} + {\left (b c^{8} d^{2} x^{8} - 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{84 \, {\left (c^{2} x^{9} - x^{7}\right )}}\right ] \]
[1/84*(6*(b*c^9*d^2*x^9 - b*c^7*d^2*x^7)*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)) + (18*b*c^5*d^2*x^5 - (18*b*c^5 - 9*b*c^3 + 2*b*c)*d^2*x ^7 - 9*b*c^3*d^2*x^3 + 2*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1 ) + 12*(a*c^8*d^2*x^8 - 4*a*c^6*d^2*x^6 + 6*a*c^4*d^2*x^4 - 4*a*c^2*d^2*x^ 2 + a*d^2 + (b*c^8*d^2*x^8 - 4*b*c^6*d^2*x^6 + 6*b*c^4*d^2*x^4 - 4*b*c^2*d ^2*x^2 + b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7), -1/84* (12*(b*c^9*d^2*x^9 - b*c^7*d^2*x^7)*sqrt(-d)*arctan(sqrt(-c^2*d*x^2 + d)*s qrt(-c^2*x^2 + 1)*(x^2 + 1)*sqrt(-d)/(c^2*d*x^4 - (c^2 + 1)*d*x^2 + d)) - (18*b*c^5*d^2*x^5 - (18*b*c^5 - 9*b*c^3 + 2*b*c)*d^2*x^7 - 9*b*c^3*d^2*x^3 + 2*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 12*(a*c^8*d^2*x^ 8 - 4*a*c^6*d^2*x^6 + 6*a*c^4*d^2*x^4 - 4*a*c^2*d^2*x^2 + a*d^2 + (b*c^8*d ^2*x^8 - 4*b*c^6*d^2*x^6 + 6*b*c^4*d^2*x^4 - 4*b*c^2*d^2*x^2 + b*d^2)*arcs in(c*x))*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7)]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^8} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^8} \, dx=\frac {{\left (6 \, \left (-1\right )^{-2 \, c^{2} d x^{2} + 2 \, d} c^{6} d^{\frac {7}{2}} \log \left (-2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right ) + 6 \, c^{6} d^{\frac {7}{2}} \log \left (x^{2} - \frac {1}{c^{2}}\right ) - \frac {11 \, \sqrt {c^{4} d x^{4} - 2 \, c^{2} d x^{2} + d} c^{4} d^{3}}{x^{2}} + \frac {7 \, \sqrt {c^{4} d x^{4} - 2 \, c^{2} d x^{2} + d} c^{2} d^{3}}{x^{4}} - \frac {2 \, \sqrt {c^{4} d x^{4} - 2 \, c^{2} d x^{2} + d} d^{3}}{x^{6}}\right )} b c}{84 \, d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} b \arcsin \left (c x\right )}{7 \, d x^{7}} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a}{7 \, d x^{7}} \]
1/84*(6*(-1)^(-2*c^2*d*x^2 + 2*d)*c^6*d^(7/2)*log(-2*c^2*d + 2*d/x^2) + 6* c^6*d^(7/2)*log(x^2 - 1/c^2) - 11*sqrt(c^4*d*x^4 - 2*c^2*d*x^2 + d)*c^4*d^ 3/x^2 + 7*sqrt(c^4*d*x^4 - 2*c^2*d*x^2 + d)*c^2*d^3/x^4 - 2*sqrt(c^4*d*x^4 - 2*c^2*d*x^2 + d)*d^3/x^6)*b*c/d - 1/7*(-c^2*d*x^2 + d)^(7/2)*b*arcsin(c *x)/(d*x^7) - 1/7*(-c^2*d*x^2 + d)^(7/2)*a/(d*x^7)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^8} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^8} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^8} \,d x \]