Integrand size = 15, antiderivative size = 126 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {9 b}{14 a^2 \left (a+b x^2\right )^{7/2}}-\frac {1}{2 a x^2 \left (a+b x^2\right )^{7/2}}-\frac {9 b}{10 a^3 \left (a+b x^2\right )^{5/2}}-\frac {3 b}{2 a^4 \left (a+b x^2\right )^{3/2}}-\frac {9 b}{2 a^5 \sqrt {a+b x^2}}+\frac {9 b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{11/2}} \] Output:
-9/14*b/a^2/(b*x^2+a)^(7/2)-1/2/a/x^2/(b*x^2+a)^(7/2)-9/10*b/a^3/(b*x^2+a) ^(5/2)-3/2*b/a^4/(b*x^2+a)^(3/2)-9/2*b/a^5/(b*x^2+a)^(1/2)+9/2*b*arctanh(( b*x^2+a)^(1/2)/a^(1/2))/a^(11/2)
Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=\frac {-35 a^4-528 a^3 b x^2-1218 a^2 b^2 x^4-1050 a b^3 x^6-315 b^4 x^8}{70 a^5 x^2 \left (a+b x^2\right )^{7/2}}+\frac {9 b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{11/2}} \] Input:
Integrate[1/(x^3*(a + b*x^2)^(9/2)),x]
Output:
(-35*a^4 - 528*a^3*b*x^2 - 1218*a^2*b^2*x^4 - 1050*a*b^3*x^6 - 315*b^4*x^8 )/(70*a^5*x^2*(a + b*x^2)^(7/2)) + (9*b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/ (2*a^(11/2))
Time = 0.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {243, 52, 61, 61, 61, 61, 73, 221}
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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^2+a\right )^{9/2}}dx^2\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{2} \left (-\frac {9 b \int \frac {1}{x^2 \left (b x^2+a\right )^{9/2}}dx^2}{2 a}-\frac {1}{a x^2 \left (a+b x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{2} \left (-\frac {9 b \left (\frac {\int \frac {1}{x^2 \left (b x^2+a\right )^{7/2}}dx^2}{a}+\frac {2}{7 a \left (a+b x^2\right )^{7/2}}\right )}{2 a}-\frac {1}{a x^2 \left (a+b x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{2} \left (-\frac {9 b \left (\frac {\frac {\int \frac {1}{x^2 \left (b x^2+a\right )^{5/2}}dx^2}{a}+\frac {2}{5 a \left (a+b x^2\right )^{5/2}}}{a}+\frac {2}{7 a \left (a+b x^2\right )^{7/2}}\right )}{2 a}-\frac {1}{a x^2 \left (a+b x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{2} \left (-\frac {9 b \left (\frac {\frac {\frac {\int \frac {1}{x^2 \left (b x^2+a\right )^{3/2}}dx^2}{a}+\frac {2}{3 a \left (a+b x^2\right )^{3/2}}}{a}+\frac {2}{5 a \left (a+b x^2\right )^{5/2}}}{a}+\frac {2}{7 a \left (a+b x^2\right )^{7/2}}\right )}{2 a}-\frac {1}{a x^2 \left (a+b x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{2} \left (-\frac {9 b \left (\frac {\frac {\frac {\frac {\int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2}{a}+\frac {2}{a \sqrt {a+b x^2}}}{a}+\frac {2}{3 a \left (a+b x^2\right )^{3/2}}}{a}+\frac {2}{5 a \left (a+b x^2\right )^{5/2}}}{a}+\frac {2}{7 a \left (a+b x^2\right )^{7/2}}\right )}{2 a}-\frac {1}{a x^2 \left (a+b x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (-\frac {9 b \left (\frac {\frac {\frac {\frac {2 \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{a b}+\frac {2}{a \sqrt {a+b x^2}}}{a}+\frac {2}{3 a \left (a+b x^2\right )^{3/2}}}{a}+\frac {2}{5 a \left (a+b x^2\right )^{5/2}}}{a}+\frac {2}{7 a \left (a+b x^2\right )^{7/2}}\right )}{2 a}-\frac {1}{a x^2 \left (a+b x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (-\frac {9 b \left (\frac {\frac {\frac {\frac {2}{a \sqrt {a+b x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a \left (a+b x^2\right )^{3/2}}}{a}+\frac {2}{5 a \left (a+b x^2\right )^{5/2}}}{a}+\frac {2}{7 a \left (a+b x^2\right )^{7/2}}\right )}{2 a}-\frac {1}{a x^2 \left (a+b x^2\right )^{7/2}}\right )\) |
Input:
Int[1/(x^3*(a + b*x^2)^(9/2)),x]
Output:
(-(1/(a*x^2*(a + b*x^2)^(7/2))) - (9*b*(2/(7*a*(a + b*x^2)^(7/2)) + (2/(5* a*(a + b*x^2)^(5/2)) + (2/(3*a*(a + b*x^2)^(3/2)) + (2/(a*Sqrt[a + b*x^2]) - (2*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(3/2))/a)/a)/a))/(2*a))/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Time = 0.47 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.75
| method | result | size |
| pseudoelliptic | \(\frac {\frac {9 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right ) \left (b \,x^{2}+a \right )^{\frac {7}{2}} b \,x^{2}}{2}-\frac {9 \sqrt {a}\, b^{4} x^{8}}{2}-15 a^{\frac {3}{2}} b^{3} x^{6}-\frac {87 a^{\frac {5}{2}} b^{2} x^{4}}{5}-\frac {264 a^{\frac {7}{2}} b \,x^{2}}{35}-\frac {a^{\frac {9}{2}}}{2}}{x^{2} a^{\frac {11}{2}} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\) | \(94\) |
| default | \(-\frac {1}{2 a \,x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 b \left (\frac {1}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {1}{5 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}}{a}}{a}\right )}{2 a}\) | \(124\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}}{2 a^{5} x^{2}}+\frac {9 b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {11}{2}}}-\frac {2629 b \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{1120 a^{5} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {2629 b \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{1120 a^{5} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {19 \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{280 a^{4} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{3}}+\frac {389 \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{1120 a^{5} \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {19 \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{280 a^{4} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{3}}+\frac {389 \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{1120 a^{5} \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{112 a^{4} b \left (x -\frac {\sqrt {-a b}}{b}\right )^{4}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{112 a^{4} b \left (x +\frac {\sqrt {-a b}}{b}\right )^{4}}\) | \(556\) |
Input:
int(1/x^3/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
Output:
9/2*(arctanh((b*x^2+a)^(1/2)/a^(1/2))*(b*x^2+a)^(7/2)*b*x^2-a^(1/2)*b^4*x^ 8-10/3*a^(3/2)*b^3*x^6-58/15*a^(5/2)*b^2*x^4-176/105*a^(7/2)*b*x^2-1/9*a^( 9/2))/(b*x^2+a)^(7/2)/a^(11/2)/x^2
Time = 0.10 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.98 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=\left [\frac {315 \, {\left (b^{5} x^{10} + 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} + 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (315 \, a b^{4} x^{8} + 1050 \, a^{2} b^{3} x^{6} + 1218 \, a^{3} b^{2} x^{4} + 528 \, a^{4} b x^{2} + 35 \, a^{5}\right )} \sqrt {b x^{2} + a}}{140 \, {\left (a^{6} b^{4} x^{10} + 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} + 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}}, -\frac {315 \, {\left (b^{5} x^{10} + 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} + 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (315 \, a b^{4} x^{8} + 1050 \, a^{2} b^{3} x^{6} + 1218 \, a^{3} b^{2} x^{4} + 528 \, a^{4} b x^{2} + 35 \, a^{5}\right )} \sqrt {b x^{2} + a}}{70 \, {\left (a^{6} b^{4} x^{10} + 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} + 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}}\right ] \] Input:
integrate(1/x^3/(b*x^2+a)^(9/2),x, algorithm="fricas")
Output:
[1/140*(315*(b^5*x^10 + 4*a*b^4*x^8 + 6*a^2*b^3*x^6 + 4*a^3*b^2*x^4 + a^4* b*x^2)*sqrt(a)*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(31 5*a*b^4*x^8 + 1050*a^2*b^3*x^6 + 1218*a^3*b^2*x^4 + 528*a^4*b*x^2 + 35*a^5 )*sqrt(b*x^2 + a))/(a^6*b^4*x^10 + 4*a^7*b^3*x^8 + 6*a^8*b^2*x^6 + 4*a^9*b *x^4 + a^10*x^2), -1/70*(315*(b^5*x^10 + 4*a*b^4*x^8 + 6*a^2*b^3*x^6 + 4*a ^3*b^2*x^4 + a^4*b*x^2)*sqrt(-a)*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (315 *a*b^4*x^8 + 1050*a^2*b^3*x^6 + 1218*a^3*b^2*x^4 + 528*a^4*b*x^2 + 35*a^5) *sqrt(b*x^2 + a))/(a^6*b^4*x^10 + 4*a^7*b^3*x^8 + 6*a^8*b^2*x^6 + 4*a^9*b* x^4 + a^10*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 5540 vs. \(2 (119) = 238\).
Time = 7.61 (sec) , antiderivative size = 5540, normalized size of antiderivative = 43.97 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x**3/(b*x**2+a)**(9/2),x)
Output:
-70*a**49*sqrt(1 + b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2 )*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 1 6800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b** 9*x**20 + 140*a**(87/2)*b**10*x**22) - 1476*a**48*b*x**2*sqrt(1 + b*x**2/a )/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x** 6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97 /2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b** 10*x**22) - 315*a**48*b*x**2*log(b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a** (105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)* b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400 *a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) + 630*a**48*b*x**2*log( sqrt(1 + b*x**2/a) + 1)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 63 00*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b** 4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800* a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x** 20 + 140*a**(87/2)*b**10*x**22) - 9822*a**47*b**2*x**4*sqrt(1 + b*x**2/a)/ (140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x*...
Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=\frac {9 \, b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {11}{2}}} - \frac {9 \, b}{2 \, \sqrt {b x^{2} + a} a^{5}} - \frac {3 \, b}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {9 \, b}{10 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {9 \, b}{14 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} - \frac {1}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{2}} \] Input:
integrate(1/x^3/(b*x^2+a)^(9/2),x, algorithm="maxima")
Output:
9/2*b*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(11/2) - 9/2*b/(sqrt(b*x^2 + a)*a^5) - 3/2*b/((b*x^2 + a)^(3/2)*a^4) - 9/10*b/((b*x^2 + a)^(5/2)*a^3) - 9/14*b /((b*x^2 + a)^(7/2)*a^2) - 1/2/((b*x^2 + a)^(7/2)*a*x^2)
Time = 0.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {9 \, b \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a^{5}} - \frac {\sqrt {b x^{2} + a}}{2 \, a^{5} x^{2}} - \frac {140 \, {\left (b x^{2} + a\right )}^{3} b + 35 \, {\left (b x^{2} + a\right )}^{2} a b + 14 \, {\left (b x^{2} + a\right )} a^{2} b + 5 \, a^{3} b}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{5}} \] Input:
integrate(1/x^3/(b*x^2+a)^(9/2),x, algorithm="giac")
Output:
-9/2*b*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^5) - 1/2*sqrt(b*x^2 + a)/(a^5*x^2) - 1/35*(140*(b*x^2 + a)^3*b + 35*(b*x^2 + a)^2*a*b + 14*(b*x^ 2 + a)*a^2*b + 5*a^3*b)/((b*x^2 + a)^(7/2)*a^5)
Time = 0.62 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=\frac {9\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{11/2}}-\frac {\frac {b}{7\,a}+\frac {3\,b\,{\left (b\,x^2+a\right )}^2}{5\,a^3}+\frac {3\,b\,{\left (b\,x^2+a\right )}^3}{a^4}-\frac {9\,b\,{\left (b\,x^2+a\right )}^4}{2\,a^5}+\frac {9\,b\,\left (b\,x^2+a\right )}{35\,a^2}}{a\,{\left (b\,x^2+a\right )}^{7/2}-{\left (b\,x^2+a\right )}^{9/2}} \] Input:
int(1/(x^3*(a + b*x^2)^(9/2)),x)
Output:
(9*b*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(2*a^(11/2)) - (b/(7*a) + (3*b*(a + b*x^2)^2)/(5*a^3) + (3*b*(a + b*x^2)^3)/a^4 - (9*b*(a + b*x^2)^4)/(2*a^5) + (9*b*(a + b*x^2))/(35*a^2))/(a*(a + b*x^2)^(7/2) - (a + b*x^2)^(9/2))
Time = 0.46 (sec) , antiderivative size = 472, normalized size of antiderivative = 3.75 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=\frac {-35 \sqrt {b \,x^{2}+a}\, a^{5}-528 \sqrt {b \,x^{2}+a}\, a^{4} b \,x^{2}-1218 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{4}-1050 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{6}-315 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{8}-315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b \,x^{2}-1260 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{2} x^{4}-1890 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{3} x^{6}-1260 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} x^{8}-315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{10}+315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b \,x^{2}+1260 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{2} x^{4}+1890 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{3} x^{6}+1260 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} x^{8}+315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{10}}{70 a^{6} x^{2} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int(1/x^3/(b*x^2+a)^(9/2),x)
Output:
( - 35*sqrt(a + b*x**2)*a**5 - 528*sqrt(a + b*x**2)*a**4*b*x**2 - 1218*sqr t(a + b*x**2)*a**3*b**2*x**4 - 1050*sqrt(a + b*x**2)*a**2*b**3*x**6 - 315* sqrt(a + b*x**2)*a*b**4*x**8 - 315*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a**4*b*x**2 - 1260*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a**3*b**2*x**4 - 1890*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a**2*b**3*x**6 - 1260*sqrt(a)*log( (sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**4*x**8 - 315*sqrt(a )*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**5*x**10 + 315*s qrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a**4*b*x**2 + 1260*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a**3*b **2*x**4 + 1890*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt( a))*a**2*b**3*x**6 + 1260*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b )*x)/sqrt(a))*a*b**4*x**8 + 315*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**5*x**10)/(70*a**6*x**2*(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))