Integrand size = 26, antiderivative size = 253 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{14}} \, dx=-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac {a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^5 \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )} \] Output:
-1/13*a^5*((b*x^2+a)^2)^(1/2)/x^13/(b*x^2+a)-5/11*a^4*b*((b*x^2+a)^2)^(1/2 )/x^11/(b*x^2+a)-10/9*a^3*b^2*((b*x^2+a)^2)^(1/2)/x^9/(b*x^2+a)-10/7*a^2*b ^3*((b*x^2+a)^2)^(1/2)/x^7/(b*x^2+a)-a*b^4*((b*x^2+a)^2)^(1/2)/x^5/(b*x^2+ a)-1/3*b^5*((b*x^2+a)^2)^(1/2)/x^3/(b*x^2+a)
Time = 1.01 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.33 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{14}} \, dx=-\frac {\sqrt {\left (a+b x^2\right )^2} \left (693 a^5+4095 a^4 b x^2+10010 a^3 b^2 x^4+12870 a^2 b^3 x^6+9009 a b^4 x^8+3003 b^5 x^{10}\right )}{9009 x^{13} \left (a+b x^2\right )} \] Input:
Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^14,x]
Output:
-1/9009*(Sqrt[(a + b*x^2)^2]*(693*a^5 + 4095*a^4*b*x^2 + 10010*a^3*b^2*x^4 + 12870*a^2*b^3*x^6 + 9009*a*b^4*x^8 + 3003*b^5*x^10))/(x^13*(a + b*x^2))
Time = 0.40 (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, 244, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{14}} \, dx\) |
\(\Big \downarrow \) 1384 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {b^5 \left (b x^2+a\right )^5}{x^{14}}dx}{b^5 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (b x^2+a\right )^5}{x^{14}}dx}{a+b x^2}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (\frac {a^5}{x^{14}}+\frac {5 b a^4}{x^{12}}+\frac {10 b^2 a^3}{x^{10}}+\frac {10 b^3 a^2}{x^8}+\frac {5 b^4 a}{x^6}+\frac {b^5}{x^4}\right )dx}{a+b x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (-\frac {a^5}{13 x^{13}}-\frac {5 a^4 b}{11 x^{11}}-\frac {10 a^3 b^2}{9 x^9}-\frac {10 a^2 b^3}{7 x^7}-\frac {a b^4}{x^5}-\frac {b^5}{3 x^3}\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}\) |
Input:
Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^14,x]
Output:
((-1/13*a^5/x^13 - (5*a^4*b)/(11*x^11) - (10*a^3*b^2)/(9*x^9) - (10*a^2*b^ 3)/(7*x^7) - (a*b^4)/x^5 - b^5/(3*x^3))*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/( a + b*x^2)
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_)^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[(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 = 7.86 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.31
method | result | size |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {1}{3} x^{10} b^{5}-a \,x^{8} b^{4}-\frac {10}{7} a^{2} x^{6} b^{3}-\frac {10}{9} a^{3} x^{4} b^{2}-\frac {5}{11} x^{2} a^{4} b -\frac {1}{13} a^{5}\right )}{\left (b \,x^{2}+a \right ) x^{13}}\) | \(79\) |
gosper | \(-\frac {\left (3003 x^{10} b^{5}+9009 a \,x^{8} b^{4}+12870 a^{2} x^{6} b^{3}+10010 a^{3} x^{4} b^{2}+4095 x^{2} a^{4} b +693 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{9009 x^{13} \left (b \,x^{2}+a \right )^{5}}\) | \(80\) |
default | \(-\frac {\left (3003 x^{10} b^{5}+9009 a \,x^{8} b^{4}+12870 a^{2} x^{6} b^{3}+10010 a^{3} x^{4} b^{2}+4095 x^{2} a^{4} b +693 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{9009 x^{13} \left (b \,x^{2}+a \right )^{5}}\) | \(80\) |
orering | \(-\frac {\left (3003 x^{10} b^{5}+9009 a \,x^{8} b^{4}+12870 a^{2} x^{6} b^{3}+10010 a^{3} x^{4} b^{2}+4095 x^{2} a^{4} b +693 a^{5}\right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {5}{2}}}{9009 x^{13} \left (b \,x^{2}+a \right )^{5}}\) | \(89\) |
Input:
int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^14,x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)^2)^(1/2)/(b*x^2+a)*(-1/3*x^10*b^5-a*x^8*b^4-10/7*a^2*x^6*b^3-10 /9*a^3*x^4*b^2-5/11*x^2*a^4*b-1/13*a^5)/x^13
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{14}} \, dx=-\frac {3003 \, b^{5} x^{10} + 9009 \, a b^{4} x^{8} + 12870 \, a^{2} b^{3} x^{6} + 10010 \, a^{3} b^{2} x^{4} + 4095 \, a^{4} b x^{2} + 693 \, a^{5}}{9009 \, x^{13}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^14,x, algorithm="fricas")
Output:
-1/9009*(3003*b^5*x^10 + 9009*a*b^4*x^8 + 12870*a^2*b^3*x^6 + 10010*a^3*b^ 2*x^4 + 4095*a^4*b*x^2 + 693*a^5)/x^13
\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{14}} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{x^{14}}\, dx \] Input:
integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**14,x)
Output:
Integral(((a + b*x**2)**2)**(5/2)/x**14, x)
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{14}} \, dx=-\frac {b^{5}}{3 \, x^{3}} - \frac {a b^{4}}{x^{5}} - \frac {10 \, a^{2} b^{3}}{7 \, x^{7}} - \frac {10 \, a^{3} b^{2}}{9 \, x^{9}} - \frac {5 \, a^{4} b}{11 \, x^{11}} - \frac {a^{5}}{13 \, x^{13}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^14,x, algorithm="maxima")
Output:
-1/3*b^5/x^3 - a*b^4/x^5 - 10/7*a^2*b^3/x^7 - 10/9*a^3*b^2/x^9 - 5/11*a^4* b/x^11 - 1/13*a^5/x^13
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{14}} \, dx=-\frac {3003 \, b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 9009 \, a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 12870 \, a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 10010 \, a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 4095 \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 693 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{9009 \, x^{13}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^14,x, algorithm="giac")
Output:
-1/9009*(3003*b^5*x^10*sgn(b*x^2 + a) + 9009*a*b^4*x^8*sgn(b*x^2 + a) + 12 870*a^2*b^3*x^6*sgn(b*x^2 + a) + 10010*a^3*b^2*x^4*sgn(b*x^2 + a) + 4095*a ^4*b*x^2*sgn(b*x^2 + a) + 693*a^5*sgn(b*x^2 + a))/x^13
Time = 18.96 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{14}} \, dx=-\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{13\,x^{13}\,\left (b\,x^2+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{3\,x^3\,\left (b\,x^2+a\right )}-\frac {a\,b^4\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{x^5\,\left (b\,x^2+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{11\,x^{11}\,\left (b\,x^2+a\right )}-\frac {10\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{7\,x^7\,\left (b\,x^2+a\right )}-\frac {10\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{9\,x^9\,\left (b\,x^2+a\right )} \] Input:
int((a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)/x^14,x)
Output:
- (a^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(13*x^13*(a + b*x^2)) - (b^5*(a^ 2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(3*x^3*(a + b*x^2)) - (a*b^4*(a^2 + b^2*x^ 4 + 2*a*b*x^2)^(1/2))/(x^5*(a + b*x^2)) - (5*a^4*b*(a^2 + b^2*x^4 + 2*a*b* x^2)^(1/2))/(11*x^11*(a + b*x^2)) - (10*a^2*b^3*(a^2 + b^2*x^4 + 2*a*b*x^2 )^(1/2))/(7*x^7*(a + b*x^2)) - (10*a^3*b^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/ 2))/(9*x^9*(a + b*x^2))
Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{14}} \, dx=\frac {-3003 b^{5} x^{10}-9009 a \,b^{4} x^{8}-12870 a^{2} b^{3} x^{6}-10010 a^{3} b^{2} x^{4}-4095 a^{4} b \,x^{2}-693 a^{5}}{9009 x^{13}} \] Input:
int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^14,x)
Output:
( - 693*a**5 - 4095*a**4*b*x**2 - 10010*a**3*b**2*x**4 - 12870*a**2*b**3*x **6 - 9009*a*b**4*x**8 - 3003*b**5*x**10)/(9009*x**13)