Integrand size = 15, antiderivative size = 106 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\frac {1}{7 a x \left (a+b x^2\right )^{7/2}}+\frac {8}{35 a^2 x \left (a+b x^2\right )^{5/2}}+\frac {16}{35 a^3 x \left (a+b x^2\right )^{3/2}}+\frac {64}{35 a^4 x \sqrt {a+b x^2}}-\frac {128 \sqrt {a+b x^2}}{35 a^5 x} \] Output:
1/7/a/x/(b*x^2+a)^(7/2)+8/35/a^2/x/(b*x^2+a)^(5/2)+16/35/a^3/x/(b*x^2+a)^( 3/2)+64/35/a^4/x/(b*x^2+a)^(1/2)-128/35*(b*x^2+a)^(1/2)/a^5/x
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.60 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\frac {-35 a^4-280 a^3 b x^2-560 a^2 b^2 x^4-448 a b^3 x^6-128 b^4 x^8}{35 a^5 x \left (a+b x^2\right )^{7/2}} \] Input:
Integrate[1/(x^2*(a + b*x^2)^(9/2)),x]
Output:
(-35*a^4 - 280*a^3*b*x^2 - 560*a^2*b^2*x^4 - 448*a*b^3*x^6 - 128*b^4*x^8)/ (35*a^5*x*(a + b*x^2)^(7/2))
Time = 0.18 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {245, 209, 209, 209, 208}
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.
\(\displaystyle \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {8 b \int \frac {1}{\left (b x^2+a\right )^{9/2}}dx}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -\frac {8 b \left (\frac {6 \int \frac {1}{\left (b x^2+a\right )^{7/2}}dx}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -\frac {8 b \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -\frac {8 b \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle -\frac {8 b \left (\frac {6 \left (\frac {4 \left (\frac {2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\) |
Input:
Int[1/(x^2*(a + b*x^2)^(9/2)),x]
Output:
-(1/(a*x*(a + b*x^2)^(7/2))) - (8*b*(x/(7*a*(a + b*x^2)^(7/2)) + (6*(x/(5* a*(a + b*x^2)^(5/2)) + (4*(x/(3*a*(a + b*x^2)^(3/2)) + (2*x)/(3*a^2*Sqrt[a + b*x^2])))/(5*a)))/(7*a)))/a
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Time = 0.42 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.56
method | result | size |
pseudoelliptic | \(-\frac {\frac {128}{35} b^{4} x^{8}+\frac {64}{5} a \,b^{3} x^{6}+16 a^{2} b^{2} x^{4}+8 a^{3} b \,x^{2}+a^{4}}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} x \,a^{5}}\) | \(59\) |
gosper | \(-\frac {128 b^{4} x^{8}+448 a \,b^{3} x^{6}+560 a^{2} b^{2} x^{4}+280 a^{3} b \,x^{2}+35 a^{4}}{35 x \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{5}}\) | \(61\) |
trager | \(-\frac {128 b^{4} x^{8}+448 a \,b^{3} x^{6}+560 a^{2} b^{2} x^{4}+280 a^{3} b \,x^{2}+35 a^{4}}{35 x \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{5}}\) | \(61\) |
orering | \(-\frac {128 b^{4} x^{8}+448 a \,b^{3} x^{6}+560 a^{2} b^{2} x^{4}+280 a^{3} b \,x^{2}+35 a^{4}}{35 x \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{5}}\) | \(61\) |
default | \(-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\) | \(98\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}}{a^{5} x}-\frac {\sqrt {b \,x^{2}+a}\, x \left (93 b^{3} x^{6}+308 a \,b^{2} x^{4}+350 a^{2} b \,x^{2}+140 a^{3}\right ) b}{35 \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right ) a^{5}}\) | \(109\) |
Input:
int(1/x^2/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
Output:
-(128/35*b^4*x^8+64/5*a*b^3*x^6+16*a^2*b^2*x^4+8*a^3*b*x^2+a^4)/(b*x^2+a)^ (7/2)/x/a^5
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {{\left (128 \, b^{4} x^{8} + 448 \, a b^{3} x^{6} + 560 \, a^{2} b^{2} x^{4} + 280 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt {b x^{2} + a}}{35 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}} \] Input:
integrate(1/x^2/(b*x^2+a)^(9/2),x, algorithm="fricas")
Output:
-1/35*(128*b^4*x^8 + 448*a*b^3*x^6 + 560*a^2*b^2*x^4 + 280*a^3*b*x^2 + 35* a^4)*sqrt(b*x^2 + a)/(a^5*b^4*x^9 + 4*a^6*b^3*x^7 + 6*a^7*b^2*x^5 + 4*a^8* b*x^3 + a^9*x)
Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (90) = 180\).
Time = 1.59 (sec) , antiderivative size = 400, normalized size of antiderivative = 3.77 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=- \frac {35 a^{4} b^{\frac {33}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac {280 a^{3} b^{\frac {35}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac {560 a^{2} b^{\frac {37}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac {448 a b^{\frac {39}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} - \frac {128 b^{\frac {41}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} + 140 a^{8} b^{17} x^{2} + 210 a^{7} b^{18} x^{4} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{8}} \] Input:
integrate(1/x**2/(b*x**2+a)**(9/2),x)
Output:
-35*a**4*b**(33/2)*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17*x* *2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 280 *a**3*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17* x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 5 60*a**2*b**(37/2)*x**4*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**1 7*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 448*a*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17 *x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 128*b**(41/2)*x**8*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17*x* *2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8)
Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {128 \, b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} - \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} \] Input:
integrate(1/x^2/(b*x^2+a)^(9/2),x, algorithm="maxima")
Output:
-128/35*b*x/(sqrt(b*x^2 + a)*a^5) - 64/35*b*x/((b*x^2 + a)^(3/2)*a^4) - 48 /35*b*x/((b*x^2 + a)^(5/2)*a^3) - 8/7*b*x/((b*x^2 + a)^(7/2)*a^2) - 1/((b* x^2 + a)^(7/2)*a*x)
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {{\left ({\left (x^{2} {\left (\frac {93 \, b^{4} x^{2}}{a^{5}} + \frac {308 \, b^{3}}{a^{4}}\right )} + \frac {350 \, b^{2}}{a^{3}}\right )} x^{2} + \frac {140 \, b}{a^{2}}\right )} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \] Input:
integrate(1/x^2/(b*x^2+a)^(9/2),x, algorithm="giac")
Output:
-1/35*((x^2*(93*b^4*x^2/a^5 + 308*b^3/a^4) + 350*b^2/a^3)*x^2 + 140*b/a^2) *x/(b*x^2 + a)^(7/2) + 2*sqrt(b)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*a^ 4)
Time = 0.48 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {\frac {1}{a^4}+\frac {128\,b\,x^2}{35\,a^5}}{x\,\sqrt {b\,x^2+a}}-\frac {29\,b\,x}{35\,a^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {13\,b\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {b\,x}{7\,a^2\,{\left (b\,x^2+a\right )}^{7/2}} \] Input:
int(1/(x^2*(a + b*x^2)^(9/2)),x)
Output:
- (1/a^4 + (128*b*x^2)/(35*a^5))/(x*(a + b*x^2)^(1/2)) - (29*b*x)/(35*a^4* (a + b*x^2)^(3/2)) - (13*b*x)/(35*a^3*(a + b*x^2)^(5/2)) - (b*x)/(7*a^2*(a + b*x^2)^(7/2))
Time = 0.20 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.75 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\frac {-35 \sqrt {b \,x^{2}+a}\, a^{4}-280 \sqrt {b \,x^{2}+a}\, a^{3} b \,x^{2}-560 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} x^{4}-448 \sqrt {b \,x^{2}+a}\, a \,b^{3} x^{6}-128 \sqrt {b \,x^{2}+a}\, b^{4} x^{8}+128 \sqrt {b}\, a^{4} x +512 \sqrt {b}\, a^{3} b \,x^{3}+768 \sqrt {b}\, a^{2} b^{2} x^{5}+512 \sqrt {b}\, a \,b^{3} x^{7}+128 \sqrt {b}\, b^{4} x^{9}}{35 a^{5} x \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int(1/x^2/(b*x^2+a)^(9/2),x)
Output:
( - 35*sqrt(a + b*x**2)*a**4 - 280*sqrt(a + b*x**2)*a**3*b*x**2 - 560*sqrt (a + b*x**2)*a**2*b**2*x**4 - 448*sqrt(a + b*x**2)*a*b**3*x**6 - 128*sqrt( a + b*x**2)*b**4*x**8 + 128*sqrt(b)*a**4*x + 512*sqrt(b)*a**3*b*x**3 + 768 *sqrt(b)*a**2*b**2*x**5 + 512*sqrt(b)*a*b**3*x**7 + 128*sqrt(b)*b**4*x**9) /(35*a**5*x*(a**4 + 4*a**3*b*x**2 + 6*a**2*b**2*x**4 + 4*a*b**3*x**6 + b** 4*x**8))