Integrand size = 26, antiderivative size = 253 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{20}} \, dx=-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac {a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10} \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )} \]
-1/19*a^5*((b*x^3+a)^2)^(1/2)/x^19/(b*x^3+a)-5/16*a^4*b*((b*x^3+a)^2)^(1/2 )/x^16/(b*x^3+a)-10/13*a^3*b^2*((b*x^3+a)^2)^(1/2)/x^13/(b*x^3+a)-a^2*b^3* ((b*x^3+a)^2)^(1/2)/x^10/(b*x^3+a)-5/7*a*b^4*((b*x^3+a)^2)^(1/2)/x^7/(b*x^ 3+a)-1/4*b^5*((b*x^3+a)^2)^(1/2)/x^4/(b*x^3+a)
Time = 1.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.33 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{20}} \, dx=-\frac {\sqrt {\left (a+b x^3\right )^2} \left (1456 a^5+8645 a^4 b x^3+21280 a^3 b^2 x^6+27664 a^2 b^3 x^9+19760 a b^4 x^{12}+6916 b^5 x^{15}\right )}{27664 x^{19} \left (a+b x^3\right )} \]
-1/27664*(Sqrt[(a + b*x^3)^2]*(1456*a^5 + 8645*a^4*b*x^3 + 21280*a^3*b^2*x ^6 + 27664*a^2*b^3*x^9 + 19760*a*b^4*x^12 + 6916*b^5*x^15))/(x^19*(a + b*x ^3))
Time = 0.24 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.39, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1384, 27, 802, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{20}} \, dx\) |
\(\Big \downarrow \) 1384 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {b^5 \left (b x^3+a\right )^5}{x^{20}}dx}{b^5 \left (a+b x^3\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (b x^3+a\right )^5}{x^{20}}dx}{a+b x^3}\) |
\(\Big \downarrow \) 802 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a^5}{x^{20}}+\frac {5 b a^4}{x^{17}}+\frac {10 b^2 a^3}{x^{14}}+\frac {10 b^3 a^2}{x^{11}}+\frac {5 b^4 a}{x^8}+\frac {b^5}{x^5}\right )dx}{a+b x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (-\frac {a^5}{19 x^{19}}-\frac {5 a^4 b}{16 x^{16}}-\frac {10 a^3 b^2}{13 x^{13}}-\frac {a^2 b^3}{x^{10}}-\frac {5 a b^4}{7 x^7}-\frac {b^5}{4 x^4}\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}\) |
((-1/19*a^5/x^19 - (5*a^4*b)/(16*x^16) - (10*a^3*b^2)/(13*x^13) - (a^2*b^3 )/x^10 - (5*a*b^4)/(7*x^7) - b^5/(4*x^4))*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) /(a + b*x^3)
3.1.83.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[p])) Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
Time = 33.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.31
method | result | size |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-\frac {1}{19} a^{5}-\frac {5}{16} a^{4} b \,x^{3}-\frac {10}{13} a^{3} b^{2} x^{6}-a^{2} b^{3} x^{9}-\frac {5}{7} a \,b^{4} x^{12}-\frac {1}{4} b^{5} x^{15}\right )}{\left (b \,x^{3}+a \right ) x^{19}}\) | \(79\) |
gosper | \(-\frac {\left (6916 b^{5} x^{15}+19760 a \,b^{4} x^{12}+27664 a^{2} b^{3} x^{9}+21280 a^{3} b^{2} x^{6}+8645 a^{4} b \,x^{3}+1456 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{27664 x^{19} \left (b \,x^{3}+a \right )^{5}}\) | \(80\) |
default | \(-\frac {\left (6916 b^{5} x^{15}+19760 a \,b^{4} x^{12}+27664 a^{2} b^{3} x^{9}+21280 a^{3} b^{2} x^{6}+8645 a^{4} b \,x^{3}+1456 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{27664 x^{19} \left (b \,x^{3}+a \right )^{5}}\) | \(80\) |
((b*x^3+a)^2)^(1/2)/(b*x^3+a)*(-1/19*a^5-5/16*a^4*b*x^3-10/13*a^3*b^2*x^6- a^2*b^3*x^9-5/7*a*b^4*x^12-1/4*b^5*x^15)/x^19
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{20}} \, dx=-\frac {6916 \, b^{5} x^{15} + 19760 \, a b^{4} x^{12} + 27664 \, a^{2} b^{3} x^{9} + 21280 \, a^{3} b^{2} x^{6} + 8645 \, a^{4} b x^{3} + 1456 \, a^{5}}{27664 \, x^{19}} \]
-1/27664*(6916*b^5*x^15 + 19760*a*b^4*x^12 + 27664*a^2*b^3*x^9 + 21280*a^3 *b^2*x^6 + 8645*a^4*b*x^3 + 1456*a^5)/x^19
\[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{20}} \, dx=\int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{20}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{20}} \, dx=-\frac {6916 \, b^{5} x^{15} + 19760 \, a b^{4} x^{12} + 27664 \, a^{2} b^{3} x^{9} + 21280 \, a^{3} b^{2} x^{6} + 8645 \, a^{4} b x^{3} + 1456 \, a^{5}}{27664 \, x^{19}} \]
-1/27664*(6916*b^5*x^15 + 19760*a*b^4*x^12 + 27664*a^2*b^3*x^9 + 21280*a^3 *b^2*x^6 + 8645*a^4*b*x^3 + 1456*a^5)/x^19
Time = 0.30 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{20}} \, dx=-\frac {6916 \, b^{5} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + 19760 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 27664 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 21280 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 8645 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 1456 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{27664 \, x^{19}} \]
-1/27664*(6916*b^5*x^15*sgn(b*x^3 + a) + 19760*a*b^4*x^12*sgn(b*x^3 + a) + 27664*a^2*b^3*x^9*sgn(b*x^3 + a) + 21280*a^3*b^2*x^6*sgn(b*x^3 + a) + 864 5*a^4*b*x^3*sgn(b*x^3 + a) + 1456*a^5*sgn(b*x^3 + a))/x^19
Time = 8.35 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{20}} \, dx=-\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{19\,x^{19}\,\left (b\,x^3+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{4\,x^4\,\left (b\,x^3+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{7\,x^7\,\left (b\,x^3+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{16\,x^{16}\,\left (b\,x^3+a\right )}-\frac {a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{x^{10}\,\left (b\,x^3+a\right )}-\frac {10\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{13\,x^{13}\,\left (b\,x^3+a\right )} \]
- (a^5*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(19*x^19*(a + b*x^3)) - (b^5*(a^ 2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(4*x^4*(a + b*x^3)) - (5*a*b^4*(a^2 + b^2* x^6 + 2*a*b*x^3)^(1/2))/(7*x^7*(a + b*x^3)) - (5*a^4*b*(a^2 + b^2*x^6 + 2* a*b*x^3)^(1/2))/(16*x^16*(a + b*x^3)) - (a^2*b^3*(a^2 + b^2*x^6 + 2*a*b*x^ 3)^(1/2))/(x^10*(a + b*x^3)) - (10*a^3*b^2*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/ 2))/(13*x^13*(a + b*x^3))