Integrand size = 27, antiderivative size = 277 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^6} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {1-c^2 x^2}}+\frac {11 b c^3 d^2 \sqrt {d-c^2 d x^2}}{30 x^2 \sqrt {1-c^2 x^2}}-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x}+\frac {c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 x^3}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 x^5}-\frac {c^5 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b \sqrt {1-c^2 x^2}}+\frac {23 b c^5 d^2 \sqrt {d-c^2 d x^2} \log (x)}{15 \sqrt {1-c^2 x^2}} \]
1/3*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))/x^3-1/5*(-c^2*d*x^2+d)^(5 /2)*(a+b*arcsin(c*x))/x^5-c^4*d^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/x -1/20*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^4/(-c^2*x^2+1)^(1/2)+11/30*b*c^3*d^2* (-c^2*d*x^2+d)^(1/2)/x^2/(-c^2*x^2+1)^(1/2)-1/2*c^5*d^2*(a+b*arcsin(c*x))^ 2*(-c^2*d*x^2+d)^(1/2)/b/(-c^2*x^2+1)^(1/2)+23/15*b*c^5*d^2*ln(x)*(-c^2*d* x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)
Time = 1.10 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.84 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^6} \, dx=\frac {1}{60} d^2 \left (-\frac {4 b \sqrt {d-c^2 d x^2} \left (3-11 c^2 x^2+23 c^4 x^4\right ) \arcsin (c x)}{x^5}-\frac {30 b c^5 \sqrt {d-c^2 d x^2} \arcsin (c x)^2}{\sqrt {1-c^2 x^2}}+60 a c^5 \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {\sqrt {d-c^2 d x^2} \left (b c x \left (-3+22 c^2 x^2\right )-4 a \sqrt {1-c^2 x^2} \left (3-11 c^2 x^2+23 c^4 x^4\right )+92 b c^5 x^5 \log (c x)\right )}{x^5 \sqrt {1-c^2 x^2}}\right ) \]
(d^2*((-4*b*Sqrt[d - c^2*d*x^2]*(3 - 11*c^2*x^2 + 23*c^4*x^4)*ArcSin[c*x]) /x^5 - (30*b*c^5*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] + 60 *a*c^5*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (Sqrt[d - c^2*d*x^2]*(b*c*x*(-3 + 22*c^2*x^2) - 4*a*Sqrt[1 - c^2*x^2]*(3 - 11*c^2*x^2 + 23*c^4*x^4) + 92*b*c^5*x^5*Log[c*x]))/(x^5*Sqrt[1 - c^2*x^ 2])))/60
Time = 0.95 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {5200, 243, 49, 2009, 5200, 244, 2009, 5196, 14, 5152}
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^6} \, dx\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle -c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^4}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2}{x^5}dx}{5 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 x^5}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^4}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2}{x^6}dx^2}{10 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 x^5}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^4}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (\frac {c^4}{x^2}-\frac {2 c^2}{x^4}+\frac {1}{x^6}\right )dx^2}{10 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 x^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^4}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle -c^2 d \left (c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^2}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {1-c^2 x^2}{x^3}dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 x^3}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -c^2 d \left (c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^2}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \left (\frac {1}{x^3}-\frac {c^2}{x}\right )dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 x^3}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -c^2 d \left (c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 x^3}+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5196 |
| \(\displaystyle -c^2 d \left (c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \int \frac {1}{x}dx}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 x^3}+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle -c^2 d \left (c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 x^3}+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 x^5}-c^2 d \left (c^2 (-d) \left (-\frac {c \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 x^3}+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\) |
-1/5*((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^5 - c^2*d*(-1/3*((d - c ^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^3 + (b*c*d*Sqrt[d - c^2*d*x^2]*(-1/ 2*1/x^2 - c^2*Log[x]))/(3*Sqrt[1 - c^2*x^2]) - c^2*d*(-((Sqrt[d - c^2*d*x^ 2]*(a + b*ArcSin[c*x]))/x) - (c*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2) /(2*b*Sqrt[1 - c^2*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[x])/Sqrt[1 - c^2*x ^2])) + (b*c*d^2*Sqrt[d - c^2*d*x^2]*(-1/2*1/x^4 + (2*c^2)/x^2 + c^4*Log[x ^2]))/(10*Sqrt[1 - c^2*x^2])
3.1.89.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcSin[(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*ArcS in[c*x])^n/(f*(m + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*Simp[Sqrt[d + e*x^ 2]/Sqrt[1 - c^2*x^2]] Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x] + Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int [(f*x)^(m + 2)*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[ {a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
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*((a + b*ArcS in[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], 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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f} , x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 2615, normalized size of antiderivative = 9.44
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2615\) |
| parts | \(\text {Expression too large to display}\) | \(2615\) |
-1/5*a/d/x^5*(-c^2*d*x^2+d)^(7/2)-8/15*a*c^6*x*(-c^2*d*x^2+d)^(5/2)+1173*I *b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2 *x^2+9)*x^6/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^11-1495/3*I*b*(-d *(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9 )*x^4/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^9+115*I*b*(-d*(c^2*x^2- 1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^2/(c^2 *x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^7-1587*I*b*(-d*(c^2*x^2-1))^(1/2) *d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^8/(c^2*x^2-1)*a rcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^13+2/15*a*c^2/d/x^3*(-c^2*d*x^2+d)^(7/2)-8 /15*a*c^4/d/x*(-c^2*d*x^2+d)^(7/2)-2/3*a*c^6*d*x*(-c^2*d*x^2+d)^(3/2)-a*c^ 6*d^2*x*(-c^2*d*x^2+d)^(1/2)-a*c^6*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)* x/(-c^2*d*x^2+d)^(1/2))+9/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765 *c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/x^5/(c^2*x^2-1)*arcsin(c*x)+1/2*b*(-d*( c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arcsin(c*x)^2*c^5*d^2+175 /4*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c ^2*x^2+9)/(c^2*x^2-1)*c^5*(-c^2*x^2+1)^(1/2)-23/15*b*(-d*(c^2*x^2-1))^(1/2 )*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*ln((I*c*x+(-c^2*x^2+1)^(1/2))^2-1)*c^5*d^ 2-759/2*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4 -75*c^2*x^2+9)*x^6/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^11-9595/3*b*(-d*(c^2*x ^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^...
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^6} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{6}} \,d x } \]
integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c ^2*d^2*x^2 + b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/x^6, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^6} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{x^{6}}\, dx \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^6} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{6}} \,d x } \]
b*sqrt(d)*integrate((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(c*x + 1)*sqrt (-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x^6, x) - 1/15*(10*( -c^2*d*x^2 + d)^(3/2)*c^6*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^6*d^2*x + 15*c^5 *d^(5/2)*arcsin(c*x) + 8*(-c^2*d*x^2 + d)^(5/2)*c^4/x - 2*(-c^2*d*x^2 + d) ^(7/2)*c^2/(d*x^3) + 3*(-c^2*d*x^2 + d)^(7/2)/(d*x^5))*a
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^6} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^6} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^6} \,d x \]