Integrand size = 27, antiderivative size = 385 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^{12}} \, dx=-\frac {b c d \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {1-c^2 x^2}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{66 x^8 \sqrt {1-c^2 x^2}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{1386 x^6 \sqrt {1-c^2 x^2}}-\frac {b c^7 d \sqrt {d-c^2 d x^2}}{770 x^4 \sqrt {1-c^2 x^2}}-\frac {4 b c^9 d \sqrt {d-c^2 d x^2}}{1155 x^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{33 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{231 d x^7}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{1155 d x^5}+\frac {16 b c^{11} d \sqrt {d-c^2 d x^2} \log (x)}{1155 \sqrt {1-c^2 x^2}} \]
-1/11*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/d/x^11-2/33*c^2*(-c^2*d*x^2+d )^(5/2)*(a+b*arcsin(c*x))/d/x^9-8/231*c^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin (c*x))/d/x^7-16/1155*c^6*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/d/x^5-1/11 0*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^10/(-c^2*x^2+1)^(1/2)+1/66*b*c^3*d*(-c^2*d* x^2+d)^(1/2)/x^8/(-c^2*x^2+1)^(1/2)-1/1386*b*c^5*d*(-c^2*d*x^2+d)^(1/2)/x^ 6/(-c^2*x^2+1)^(1/2)-1/770*b*c^7*d*(-c^2*d*x^2+d)^(1/2)/x^4/(-c^2*x^2+1)^( 1/2)-4/1155*b*c^9*d*(-c^2*d*x^2+d)^(1/2)/x^2/(-c^2*x^2+1)^(1/2)+16/1155*b* c^11*d*ln(x)*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)
Time = 0.35 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^{12}} \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (6615 b c x-11025 b c^3 x^3+525 b c^5 x^5+945 b c^7 x^7+2520 b c^9 x^9+29524 b c^{11} x^{11}+66150 a \sqrt {1-c^2 x^2}-88200 a c^2 x^2 \sqrt {1-c^2 x^2}+3150 a c^4 x^4 \sqrt {1-c^2 x^2}+3780 a c^6 x^6 \sqrt {1-c^2 x^2}+5040 a c^8 x^8 \sqrt {1-c^2 x^2}+10080 a c^{10} x^{10} \sqrt {1-c^2 x^2}+630 b \left (1-c^2 x^2\right )^{5/2} \left (105+70 c^2 x^2+40 c^4 x^4+16 c^6 x^6\right ) \arcsin (c x)-10080 b c^{11} x^{11} \log (x)\right )}{727650 x^{11} \sqrt {1-c^2 x^2}} \]
-1/727650*(d*Sqrt[d - c^2*d*x^2]*(6615*b*c*x - 11025*b*c^3*x^3 + 525*b*c^5 *x^5 + 945*b*c^7*x^7 + 2520*b*c^9*x^9 + 29524*b*c^11*x^11 + 66150*a*Sqrt[1 - c^2*x^2] - 88200*a*c^2*x^2*Sqrt[1 - c^2*x^2] + 3150*a*c^4*x^4*Sqrt[1 - c^2*x^2] + 3780*a*c^6*x^6*Sqrt[1 - c^2*x^2] + 5040*a*c^8*x^8*Sqrt[1 - c^2* x^2] + 10080*a*c^10*x^10*Sqrt[1 - c^2*x^2] + 630*b*(1 - c^2*x^2)^(5/2)*(10 5 + 70*c^2*x^2 + 40*c^4*x^4 + 16*c^6*x^6)*ArcSin[c*x] - 10080*b*c^11*x^11* Log[x]))/(x^11*Sqrt[1 - c^2*x^2])
Time = 0.66 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.59, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {5194, 27, 2331, 2123, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^{12}} \, dx\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d \left (1-c^2 x^2\right )^2 \left (16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right )}{1155 x^{11}}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{231 d x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 \left (16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right )}{x^{11}}dx}{1155 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{231 d x^7}\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 \left (16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right )}{x^{12}}dx^2}{2310 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{231 d x^7}\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \int \left (\frac {16 c^{10}}{x^2}+\frac {8 c^8}{x^4}+\frac {6 c^6}{x^6}+\frac {5 c^4}{x^8}-\frac {140 c^2}{x^{10}}+\frac {105}{x^{12}}\right )dx^2}{2310 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{231 d x^7}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{231 d x^7}+\frac {b c d \sqrt {d-c^2 d x^2} \left (16 c^{10} \log \left (x^2\right )-\frac {8 c^8}{x^2}-\frac {3 c^6}{x^4}-\frac {5 c^4}{3 x^6}+\frac {35 c^2}{x^8}-\frac {21}{x^{10}}\right )}{2310 \sqrt {1-c^2 x^2}}\) |
-1/11*((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(d*x^11) - (2*c^2*(d - c ^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(33*d*x^9) - (8*c^4*(d - c^2*d*x^2)^( 5/2)*(a + b*ArcSin[c*x]))/(231*d*x^7) - (16*c^6*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(1155*d*x^5) + (b*c*d*Sqrt[d - c^2*d*x^2]*(-21/x^10 + (35 *c^2)/x^8 - (5*c^4)/(3*x^6) - (3*c^6)/x^4 - (8*c^8)/x^2 + 16*c^10*Log[x^2] ))/(2310*Sqrt[1 - c^2*x^2])
3.1.76.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[((a_.) + ArcSin[(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*ArcSin [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 = 0.39 (sec) , antiderivative size = 5886, normalized size of antiderivative = 15.29
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(5886\) |
| parts | \(\text {Expression too large to display}\) | \(5886\) |
Time = 0.35 (sec) , antiderivative size = 743, normalized size of antiderivative = 1.93 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^{12}} \, dx=\left [\frac {48 \, {\left (b c^{13} d x^{13} - b c^{11} d x^{11}\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 (24 \, b c^{9} d x^{9} + 9 \, b c^{7} d x^{7} - {\left (24 \, b c^{9} + 9 \, b c^{7} + 5 \, b c^{5} - 105 \, b c^{3} + 63 \, b c\right )} d x^{11} + 5 \, b c^{5} d x^{5} - 105 \, b c^{3} d x^{3} + 63 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 6 \, {\left (16 \, a c^{12} d x^{12} - 8 \, a c^{10} d x^{10} - 2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 145 \, a c^{4} d x^{4} + 245 \, a c^{2} d x^{2} - 105 \, a d + {\left (16 \, b c^{12} d x^{12} - 8 \, b c^{10} d x^{10} - 2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 145 \, b c^{4} d x^{4} + 245 \, b c^{2} d x^{2} - 105 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{6930 \, {\left (c^{2} x^{13} - x^{11}\right )}}, \frac {96 \, {\left (b c^{13} d x^{13} - b c^{11} d x^{11}\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 (24 \, b c^{9} d x^{9} + 9 \, b c^{7} d x^{7} - {\left (24 \, b c^{9} + 9 \, b c^{7} + 5 \, b c^{5} - 105 \, b c^{3} + 63 \, b c\right )} d x^{11} + 5 \, b c^{5} d x^{5} - 105 \, b c^{3} d x^{3} + 63 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 6 \, {\left (16 \, a c^{12} d x^{12} - 8 \, a c^{10} d x^{10} - 2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 145 \, a c^{4} d x^{4} + 245 \, a c^{2} d x^{2} - 105 \, a d + {\left (16 \, b c^{12} d x^{12} - 8 \, b c^{10} d x^{10} - 2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 145 \, b c^{4} d x^{4} + 245 \, b c^{2} d x^{2} - 105 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{6930 \, {\left (c^{2} x^{13} - x^{11}\right )}}\right ] \]
[1/6930*(48*(b*c^13*d*x^13 - b*c^11*d*x^11)*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)) + (24*b*c^9*d*x^9 + 9*b*c^7*d*x^7 - (24*b*c^9 + 9*b*c ^7 + 5*b*c^5 - 105*b*c^3 + 63*b*c)*d*x^11 + 5*b*c^5*d*x^5 - 105*b*c^3*d*x^ 3 + 63*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 6*(16*a*c^12*d*x ^12 - 8*a*c^10*d*x^10 - 2*a*c^8*d*x^8 - a*c^6*d*x^6 - 145*a*c^4*d*x^4 + 24 5*a*c^2*d*x^2 - 105*a*d + (16*b*c^12*d*x^12 - 8*b*c^10*d*x^10 - 2*b*c^8*d* x^8 - b*c^6*d*x^6 - 145*b*c^4*d*x^4 + 245*b*c^2*d*x^2 - 105*b*d)*arcsin(c* x))*sqrt(-c^2*d*x^2 + d))/(c^2*x^13 - x^11), 1/6930*(96*(b*c^13*d*x^13 - b *c^11*d*x^11)*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)) + (24*b*c^9*d*x^9 + 9*b* c^7*d*x^7 - (24*b*c^9 + 9*b*c^7 + 5*b*c^5 - 105*b*c^3 + 63*b*c)*d*x^11 + 5 *b*c^5*d*x^5 - 105*b*c^3*d*x^3 + 63*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^ 2*x^2 + 1) - 6*(16*a*c^12*d*x^12 - 8*a*c^10*d*x^10 - 2*a*c^8*d*x^8 - a*c^6 *d*x^6 - 145*a*c^4*d*x^4 + 245*a*c^2*d*x^2 - 105*a*d + (16*b*c^12*d*x^12 - 8*b*c^10*d*x^10 - 2*b*c^8*d*x^8 - b*c^6*d*x^6 - 145*b*c^4*d*x^4 + 245*b*c ^2*d*x^2 - 105*b*d)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^2*x^13 - x^11)]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^{12}} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.70 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^{12}} \, dx=\frac {1}{6930} \, {\left (96 \, c^{10} d^{\frac {3}{2}} \log \left (x\right ) - \frac {24 \, c^{8} d^{\frac {3}{2}} x^{8} + 9 \, c^{6} d^{\frac {3}{2}} x^{6} + 5 \, c^{4} d^{\frac {3}{2}} x^{4} - 105 \, c^{2} d^{\frac {3}{2}} x^{2} + 63 \, d^{\frac {3}{2}}}{x^{10}}\right )} b c - \frac {1}{1155} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{6}}{d x^{5}} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{7}} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{9}} + \frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{11}}\right )} b \arcsin \left (c x\right ) - \frac {1}{1155} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{6}}{d x^{5}} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{7}} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{9}} + \frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{11}}\right )} a \]
1/6930*(96*c^10*d^(3/2)*log(x) - (24*c^8*d^(3/2)*x^8 + 9*c^6*d^(3/2)*x^6 + 5*c^4*d^(3/2)*x^4 - 105*c^2*d^(3/2)*x^2 + 63*d^(3/2))/x^10)*b*c - 1/1155* (16*(-c^2*d*x^2 + d)^(5/2)*c^6/(d*x^5) + 40*(-c^2*d*x^2 + d)^(5/2)*c^4/(d* x^7) + 70*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x^9) + 105*(-c^2*d*x^2 + d)^(5/2)/ (d*x^11))*b*arcsin(c*x) - 1/1155*(16*(-c^2*d*x^2 + d)^(5/2)*c^6/(d*x^5) + 40*(-c^2*d*x^2 + d)^(5/2)*c^4/(d*x^7) + 70*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x ^9) + 105*(-c^2*d*x^2 + d)^(5/2)/(d*x^11))*a
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^{12}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^{12}} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^{12}} \,d x \]