Integrand size = 27, antiderivative size = 70 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^{11}} \, dx=-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{10 a^2 x^{10}}-\frac {b^2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{15 a^4 x^6} \] Output:
-1/10*(-b*x^2+a)^(3/2)*(b*x^2+a)^(3/2)/a^2/x^10-1/15*b^2*(-b*x^2+a)^(3/2)* (b*x^2+a)^(3/2)/a^4/x^6
Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^{11}} \, dx=-\frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2} \left (3 a^2+2 b^2 x^4\right )}{30 a^4 x^{10}} \] Input:
Integrate[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^11,x]
Output:
-1/30*((a - b*x^2)^(3/2)*(a + b*x^2)^(3/2)*(3*a^2 + 2*b^2*x^4))/(a^4*x^10)
Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.60, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {338, 108, 25, 27, 114, 27, 106}
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.
\(\displaystyle \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^{11}} \, dx\) |
\(\Big \downarrow \) 338 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {a-b x^2} \sqrt {b x^2+a}}{x^{12}}dx^2\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \int -\frac {b^2}{x^8 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{5 x^{10}}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{5} \int \frac {b^2}{x^8 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{5 x^{10}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{5} b^2 \int \frac {1}{x^8 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{5 x^{10}}\right )\) |
\(\Big \downarrow \) 114 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{5} b^2 \left (-\frac {\int -\frac {2 b^2}{x^4 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2}{3 a^2}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{3 a^2 x^6}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{5 x^{10}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{5} b^2 \left (\frac {2 b^2 \int \frac {1}{x^4 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2}{3 a^2}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{3 a^2 x^6}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{5 x^{10}}\right )\) |
\(\Big \downarrow \) 106 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{5} b^2 \left (-\frac {2 b^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}{3 a^4 x^2}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{3 a^2 x^6}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{5 x^{10}}\right )\) |
Input:
Int[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^11,x]
Output:
(-1/5*(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^10 - (b^2*(-1/3*(Sqrt[a - b*x^2] *Sqrt[a + b*x^2])/(a^2*x^6) - (2*b^2*Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/(3*a ^4*x^2)))/5)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b*c + a*d, 0] && IntegerQ[(m - 1)/2]
Time = 0.56 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(-\frac {\left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (2 b^{2} x^{4}+3 a^{2}\right )}{30 x^{10} a^{4}}\) | \(42\) |
default | \(-\frac {\left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (2 b^{2} x^{4}+3 a^{2}\right )}{30 x^{10} a^{4}}\) | \(42\) |
orering | \(-\frac {\left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (2 b^{2} x^{4}+3 a^{2}\right )}{30 x^{10} a^{4}}\) | \(42\) |
risch | \(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (-2 b^{4} x^{8}-a^{2} b^{2} x^{4}+3 a^{4}\right )}{30 x^{10} a^{4}}\) | \(53\) |
elliptic | \(-\frac {\left (-b^{2} x^{4}+a^{2}\right ) \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (2 b^{2} x^{4}+3 a^{2}\right )}{30 x^{10} a^{4}}\) | \(54\) |
Input:
int((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^11,x,method=_RETURNVERBOSE)
Output:
-1/30*(-b*x^2+a)^(3/2)*(b*x^2+a)^(3/2)*(2*b^2*x^4+3*a^2)/x^10/a^4
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^{11}} \, dx=\frac {{\left (2 \, b^{4} x^{8} + a^{2} b^{2} x^{4} - 3 \, a^{4}\right )} \sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}}{30 \, a^{4} x^{10}} \] Input:
integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^11,x, algorithm="fricas")
Output:
1/30*(2*b^4*x^8 + a^2*b^2*x^4 - 3*a^4)*sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)/(a ^4*x^10)
\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^{11}} \, dx=\int \frac {\sqrt {a - b x^{2}} \sqrt {a + b x^{2}}}{x^{11}}\, dx \] Input:
integrate((-b*x**2+a)**(1/2)*(b*x**2+a)**(1/2)/x**11,x)
Output:
Integral(sqrt(a - b*x**2)*sqrt(a + b*x**2)/x**11, x)
Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^{11}} \, dx=-\frac {{\left (-b^{2} x^{4} + a^{2}\right )}^{\frac {3}{2}} b^{2}}{15 \, a^{4} x^{6}} - \frac {{\left (-b^{2} x^{4} + a^{2}\right )}^{\frac {3}{2}}}{10 \, a^{2} x^{10}} \] Input:
integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^11,x, algorithm="maxima")
Output:
-1/15*(-b^2*x^4 + a^2)^(3/2)*b^2/(a^4*x^6) - 1/10*(-b^2*x^4 + a^2)^(3/2)/( a^2*x^10)
Exception generated. \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^{11}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^11,x, algorithm="giac")
Output:
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
Time = 1.69 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^{11}} \, dx=\frac {\sqrt {a-b\,x^2}\,\left (\frac {b^2\,x^4\,\sqrt {b\,x^2+a}}{30\,a^2}-\frac {\sqrt {b\,x^2+a}}{10}+\frac {b^4\,x^8\,\sqrt {b\,x^2+a}}{15\,a^4}\right )}{x^{10}} \] Input:
int(((a + b*x^2)^(1/2)*(a - b*x^2)^(1/2))/x^11,x)
Output:
((a - b*x^2)^(1/2)*((b^2*x^4*(a + b*x^2)^(1/2))/(30*a^2) - (a + b*x^2)^(1/ 2)/10 + (b^4*x^8*(a + b*x^2)^(1/2))/(15*a^4)))/x^10
Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^{11}} \, dx=\frac {\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, \left (2 b^{4} x^{8}+a^{2} b^{2} x^{4}-3 a^{4}\right )}{30 a^{4} x^{10}} \] Input:
int((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^11,x)
Output:
(sqrt(a + b*x**2)*sqrt(a - b*x**2)*( - 3*a**4 + a**2*b**2*x**4 + 2*b**4*x* *8))/(30*a**4*x**10)