Integrand size = 27, antiderivative size = 354 \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {8 b d^2 x \sqrt {d-c^2 d x^2}}{693 c^5 \sqrt {1-c^2 x^2}}+\frac {4 b d^2 x^3 \sqrt {d-c^2 d x^2}}{2079 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^5 \sqrt {d-c^2 d x^2}}{1155 c \sqrt {1-c^2 x^2}}-\frac {113 b c d^2 x^7 \sqrt {d-c^2 d x^2}}{4851 \sqrt {1-c^2 x^2}}+\frac {23 b c^3 d^2 x^9 \sqrt {d-c^2 d x^2}}{891 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{11} \sqrt {d-c^2 d x^2}}{121 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6 d^3} \]
-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arcsin(c*x))/c^6/d+2/9*(-c^2*d*x^2+d)^(9/2) *(a+b*arcsin(c*x))/c^6/d^2-1/11*(-c^2*d*x^2+d)^(11/2)*(a+b*arcsin(c*x))/c^ 6/d^3+8/693*b*d^2*x*(-c^2*d*x^2+d)^(1/2)/c^5/(-c^2*x^2+1)^(1/2)+4/2079*b*d ^2*x^3*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/1155*b*d^2*x^5*(-c^2* d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-113/4851*b*c*d^2*x^7*(-c^2*d*x^2+d)^(1 /2)/(-c^2*x^2+1)^(1/2)+23/891*b*c^3*d^2*x^9*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2 +1)^(1/2)-1/121*b*c^5*d^2*x^11*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)
Time = 0.13 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.45 \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=-\frac {d^2 \sqrt {d-c^2 d x^2} \left (3465 a \left (1-c^2 x^2\right )^{7/2} \left (8+28 c^2 x^2+63 c^4 x^4\right )+b c x \left (-27720-4620 c^2 x^2-2079 c^4 x^4+55935 c^6 x^6-61985 c^8 x^8+19845 c^{10} x^{10}\right )+3465 b \left (1-c^2 x^2\right )^{7/2} \left (8+28 c^2 x^2+63 c^4 x^4\right ) \arcsin (c x)\right )}{2401245 c^6 \sqrt {1-c^2 x^2}} \]
-1/2401245*(d^2*Sqrt[d - c^2*d*x^2]*(3465*a*(1 - c^2*x^2)^(7/2)*(8 + 28*c^ 2*x^2 + 63*c^4*x^4) + b*c*x*(-27720 - 4620*c^2*x^2 - 2079*c^4*x^4 + 55935* c^6*x^6 - 61985*c^8*x^8 + 19845*c^10*x^10) + 3465*b*(1 - c^2*x^2)^(7/2)*(8 + 28*c^2*x^2 + 63*c^4*x^4)*ArcSin[c*x]))/(c^6*Sqrt[1 - c^2*x^2])
Time = 0.49 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.55, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5194, 27, 1467, 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.
\(\displaystyle \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx\) |
\(\Big \downarrow \) 5194 |
\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d^2 \left (1-c^2 x^2\right )^3 \left (63 c^4 x^4+28 c^2 x^2+8\right )}{693 c^6}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b d^2 \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )^3 \left (63 c^4 x^4+28 c^2 x^2+8\right )dx}{693 c^5 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d}\) |
\(\Big \downarrow \) 1467 |
\(\displaystyle \frac {b d^2 \sqrt {d-c^2 d x^2} \int \left (-63 c^{10} x^{10}+161 c^8 x^8-113 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8\right )dx}{693 c^5 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d}+\frac {b d^2 \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right ) \sqrt {d-c^2 d x^2}}{693 c^5 \sqrt {1-c^2 x^2}}\) |
(b*d^2*Sqrt[d - c^2*d*x^2]*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (113*c^6 *x^7)/7 + (161*c^8*x^9)/9 - (63*c^10*x^11)/11))/(693*c^5*Sqrt[1 - c^2*x^2] ) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^6*d) + (2*(d - c^2*d* x^2)^(9/2)*(a + b*ArcSin[c*x]))/(9*c^6*d^2) - ((d - c^2*d*x^2)^(11/2)*(a + b*ArcSin[c*x]))/(11*c^6*d^3)
3.1.93.3.1 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[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -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 = 0.31 (sec) , antiderivative size = 1644, normalized size of antiderivative = 4.64
method | result | size |
default | \(\text {Expression too large to display}\) | \(1644\) |
parts | \(\text {Expression too large to display}\) | \(1644\) |
a*(-1/11*x^4*(-c^2*d*x^2+d)^(7/2)/c^2/d+4/11/c^2*(-1/9*x^2*(-c^2*d*x^2+d)^ (7/2)/c^2/d-2/63/d/c^4*(-c^2*d*x^2+d)^(7/2)))+b*(1/247808*(-d*(c^2*x^2-1)) ^(1/2)*(1+11*I*(-c^2*x^2+1)^(1/2)*x*c+1024*c^12*x^12+2816*I*(-c^2*x^2+1)^( 1/2)*x^9*c^9-61*c^2*x^2-2352*c^6*x^6+620*c^4*x^4-2816*I*(-c^2*x^2+1)^(1/2) *x^7*c^7-3328*c^10*x^10+4096*c^8*x^8+1232*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-102 4*I*(-c^2*x^2+1)^(1/2)*x^11*c^11-220*I*(-c^2*x^2+1)^(1/2)*x^3*c^3)*(I+11*a rcsin(c*x))*d^2/c^6/(c^2*x^2-1)-1/165888*(-d*(c^2*x^2-1))^(1/2)*(256*c^10* x^10-704*c^8*x^8-256*I*(-c^2*x^2+1)^(1/2)*x^9*c^9+688*c^6*x^6+576*I*(-c^2* x^2+1)^(1/2)*x^7*c^7-280*c^4*x^4-432*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+41*c^2*x ^2+120*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-9*I*(-c^2*x^2+1)^(1/2)*x*c-1)*(I+9*arc sin(c*x))*d^2/c^6/(c^2*x^2-1)-5/100352*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8- 144*c^6*x^6-64*I*c^7*x^7*(-c^2*x^2+1)^(1/2)+104*c^4*x^4+112*I*(-c^2*x^2+1) ^(1/2)*x^5*c^5-25*c^2*x^2-56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1) ^(1/2)*x*c+1)*(I+7*arcsin(c*x))*d^2/c^6/(c^2*x^2-1)+5/9216*(-d*(c^2*x^2-1) )^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+3*I*(-c^2*x^2+ 1)^(1/2)*x*c+1)*(I+3*arcsin(c*x))*d^2/c^6/(c^2*x^2-1)-5/1024*(-d*(c^2*x^2- 1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)+I)*d^2/c^6/(c^ 2*x^2-1)-5/1024*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1 )*(arcsin(c*x)-I)*d^2/c^6/(c^2*x^2-1)+5/9216*(-d*(c^2*x^2-1))^(1/2)*(4*I*c ^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^...
Time = 0.27 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.82 \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {{\left (19845 \, b c^{11} d^{2} x^{11} - 61985 \, b c^{9} d^{2} x^{9} + 55935 \, b c^{7} d^{2} x^{7} - 2079 \, b c^{5} d^{2} x^{5} - 4620 \, b c^{3} d^{2} x^{3} - 27720 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 3465 \, {\left (63 \, a c^{12} d^{2} x^{12} - 224 \, a c^{10} d^{2} x^{10} + 274 \, a c^{8} d^{2} x^{8} - 116 \, a c^{6} d^{2} x^{6} - a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + 8 \, a d^{2} + {\left (63 \, b c^{12} d^{2} x^{12} - 224 \, b c^{10} d^{2} x^{10} + 274 \, b c^{8} d^{2} x^{8} - 116 \, b c^{6} d^{2} x^{6} - b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + 8 \, b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{2401245 \, {\left (c^{8} x^{2} - c^{6}\right )}} \]
1/2401245*((19845*b*c^11*d^2*x^11 - 61985*b*c^9*d^2*x^9 + 55935*b*c^7*d^2* x^7 - 2079*b*c^5*d^2*x^5 - 4620*b*c^3*d^2*x^3 - 27720*b*c*d^2*x)*sqrt(-c^2 *d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 3465*(63*a*c^12*d^2*x^12 - 224*a*c^10*d^2 *x^10 + 274*a*c^8*d^2*x^8 - 116*a*c^6*d^2*x^6 - a*c^4*d^2*x^4 - 4*a*c^2*d^ 2*x^2 + 8*a*d^2 + (63*b*c^12*d^2*x^12 - 224*b*c^10*d^2*x^10 + 274*b*c^8*d^ 2*x^8 - 116*b*c^6*d^2*x^6 - b*c^4*d^2*x^4 - 4*b*c^2*d^2*x^2 + 8*b*d^2)*arc sin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^8*x^2 - c^6)
Timed out. \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.62 \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=-\frac {1}{693} \, {\left (\frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{4}}{c^{2} d} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{6} d}\right )} b \arcsin \left (c x\right ) - \frac {1}{693} \, {\left (\frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{4}}{c^{2} d} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{6} d}\right )} a - \frac {{\left (19845 \, c^{10} d^{\frac {5}{2}} x^{11} - 61985 \, c^{8} d^{\frac {5}{2}} x^{9} + 55935 \, c^{6} d^{\frac {5}{2}} x^{7} - 2079 \, c^{4} d^{\frac {5}{2}} x^{5} - 4620 \, c^{2} d^{\frac {5}{2}} x^{3} - 27720 \, d^{\frac {5}{2}} x\right )} b}{2401245 \, c^{5}} \]
-1/693*(63*(-c^2*d*x^2 + d)^(7/2)*x^4/(c^2*d) + 28*(-c^2*d*x^2 + d)^(7/2)* x^2/(c^4*d) + 8*(-c^2*d*x^2 + d)^(7/2)/(c^6*d))*b*arcsin(c*x) - 1/693*(63* (-c^2*d*x^2 + d)^(7/2)*x^4/(c^2*d) + 28*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^4*d) + 8*(-c^2*d*x^2 + d)^(7/2)/(c^6*d))*a - 1/2401245*(19845*c^10*d^(5/2)*x^1 1 - 61985*c^8*d^(5/2)*x^9 + 55935*c^6*d^(5/2)*x^7 - 2079*c^4*d^(5/2)*x^5 - 4620*c^2*d^(5/2)*x^3 - 27720*d^(5/2)*x)*b/c^5
Exception generated. \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int x^5\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]