Integrand size = 13, antiderivative size = 109 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^3} \, dx=-\frac {10 a^3 x}{b^6}+\frac {2 a^2 x^3}{b^5}-\frac {3 a x^5}{5 b^4}+\frac {x^7}{7 b^3}+\frac {a^5 x}{4 b^6 \left (a+b x^2\right )^2}-\frac {21 a^4 x}{8 b^6 \left (a+b x^2\right )}+\frac {99 a^{7/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}} \] Output:
-10*a^3*x/b^6+2*a^2*x^3/b^5-3/5*a*x^5/b^4+1/7*x^7/b^3+1/4*a^5*x/b^6/(b*x^2 +a)^2-21/8*a^4*x/b^6/(b*x^2+a)+99/8*a^(7/2)*arctan(b^(1/2)*x/a^(1/2))/b^(1 3/2)
Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^3} \, dx=-\frac {3465 a^5 x+5775 a^4 b x^3+1848 a^3 b^2 x^5-264 a^2 b^3 x^7+88 a b^4 x^9-40 b^5 x^{11}}{280 b^6 \left (a+b x^2\right )^2}+\frac {99 a^{7/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}} \] Input:
Integrate[x^12/(a + b*x^2)^3,x]
Output:
-1/280*(3465*a^5*x + 5775*a^4*b*x^3 + 1848*a^3*b^2*x^5 - 264*a^2*b^3*x^7 + 88*a*b^4*x^9 - 40*b^5*x^11)/(b^6*(a + b*x^2)^2) + (99*a^(7/2)*ArcTan[(Sqr t[b]*x)/Sqrt[a]])/(8*b^(13/2))
Time = 0.22 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {252, 252, 254, 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.
\(\displaystyle \int \frac {x^{12}}{\left (a+b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {11 \int \frac {x^{10}}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {x^{11}}{4 b \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {11 \left (\frac {9 \int \frac {x^8}{b x^2+a}dx}{2 b}-\frac {x^9}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^{11}}{4 b \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {11 \left (\frac {9 \int \left (\frac {x^6}{b}-\frac {a x^4}{b^2}+\frac {a^2 x^2}{b^3}+\frac {a^4}{b^4 \left (b x^2+a\right )}-\frac {a^3}{b^4}\right )dx}{2 b}-\frac {x^9}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^{11}}{4 b \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {a^{7/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}}-\frac {a^3 x}{b^4}+\frac {a^2 x^3}{3 b^3}-\frac {a x^5}{5 b^2}+\frac {x^7}{7 b}\right )}{2 b}-\frac {x^9}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^{11}}{4 b \left (a+b x^2\right )^2}\) |
Input:
Int[x^12/(a + b*x^2)^3,x]
Output:
-1/4*x^11/(b*(a + b*x^2)^2) + (11*(-1/2*x^9/(b*(a + b*x^2)) + (9*(-((a^3*x )/b^4) + (a^2*x^3)/(3*b^3) - (a*x^5)/(5*b^2) + x^7/(7*b) + (a^(7/2)*ArcTan [(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)))/(2*b)))/(4*b)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Time = 0.32 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.78
method | result | size |
default | \(-\frac {-\frac {1}{7} b^{3} x^{7}+\frac {3}{5} a \,b^{2} x^{5}-2 a^{2} b \,x^{3}+10 a^{3} x}{b^{6}}+\frac {a^{4} \left (\frac {-\frac {21}{8} b \,x^{3}-\frac {19}{8} a x}{\left (b \,x^{2}+a \right )^{2}}+\frac {99 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{6}}\) | \(85\) |
risch | \(\frac {x^{7}}{7 b^{3}}-\frac {3 a \,x^{5}}{5 b^{4}}+\frac {2 a^{2} x^{3}}{b^{5}}-\frac {10 a^{3} x}{b^{6}}+\frac {-\frac {21}{8} a^{4} b \,x^{3}-\frac {19}{8} a^{5} x}{b^{6} \left (b \,x^{2}+a \right )^{2}}+\frac {99 \sqrt {-a b}\, a^{3} \ln \left (-\sqrt {-a b}\, x +a \right )}{16 b^{7}}-\frac {99 \sqrt {-a b}\, a^{3} \ln \left (\sqrt {-a b}\, x +a \right )}{16 b^{7}}\) | \(119\) |
Input:
int(x^12/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
-1/b^6*(-1/7*b^3*x^7+3/5*a*b^2*x^5-2*a^2*b*x^3+10*a^3*x)+a^4/b^6*((-21/8*b *x^3-19/8*a*x)/(b*x^2+a)^2+99/8/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.07 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.55 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^3} \, dx=\left [\frac {80 \, b^{5} x^{11} - 176 \, a b^{4} x^{9} + 528 \, a^{2} b^{3} x^{7} - 3696 \, a^{3} b^{2} x^{5} - 11550 \, a^{4} b x^{3} - 6930 \, a^{5} x + 3465 \, {\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{560 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}, \frac {40 \, b^{5} x^{11} - 88 \, a b^{4} x^{9} + 264 \, a^{2} b^{3} x^{7} - 1848 \, a^{3} b^{2} x^{5} - 5775 \, a^{4} b x^{3} - 3465 \, a^{5} x + 3465 \, {\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{280 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}\right ] \] Input:
integrate(x^12/(b*x^2+a)^3,x, algorithm="fricas")
Output:
[1/560*(80*b^5*x^11 - 176*a*b^4*x^9 + 528*a^2*b^3*x^7 - 3696*a^3*b^2*x^5 - 11550*a^4*b*x^3 - 6930*a^5*x + 3465*(a^3*b^2*x^4 + 2*a^4*b*x^2 + a^5)*sqr t(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^8*x^4 + 2*a*b^ 7*x^2 + a^2*b^6), 1/280*(40*b^5*x^11 - 88*a*b^4*x^9 + 264*a^2*b^3*x^7 - 18 48*a^3*b^2*x^5 - 5775*a^4*b*x^3 - 3465*a^5*x + 3465*(a^3*b^2*x^4 + 2*a^4*b *x^2 + a^5)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^8*x^4 + 2*a*b^7*x^2 + a^ 2*b^6)]
Time = 0.26 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.49 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^3} \, dx=- \frac {10 a^{3} x}{b^{6}} + \frac {2 a^{2} x^{3}}{b^{5}} - \frac {3 a x^{5}}{5 b^{4}} - \frac {99 \sqrt {- \frac {a^{7}}{b^{13}}} \log {\left (x - \frac {b^{6} \sqrt {- \frac {a^{7}}{b^{13}}}}{a^{3}} \right )}}{16} + \frac {99 \sqrt {- \frac {a^{7}}{b^{13}}} \log {\left (x + \frac {b^{6} \sqrt {- \frac {a^{7}}{b^{13}}}}{a^{3}} \right )}}{16} + \frac {- 19 a^{5} x - 21 a^{4} b x^{3}}{8 a^{2} b^{6} + 16 a b^{7} x^{2} + 8 b^{8} x^{4}} + \frac {x^{7}}{7 b^{3}} \] Input:
integrate(x**12/(b*x**2+a)**3,x)
Output:
-10*a**3*x/b**6 + 2*a**2*x**3/b**5 - 3*a*x**5/(5*b**4) - 99*sqrt(-a**7/b** 13)*log(x - b**6*sqrt(-a**7/b**13)/a**3)/16 + 99*sqrt(-a**7/b**13)*log(x + b**6*sqrt(-a**7/b**13)/a**3)/16 + (-19*a**5*x - 21*a**4*b*x**3)/(8*a**2*b **6 + 16*a*b**7*x**2 + 8*b**8*x**4) + x**7/(7*b**3)
Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.96 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^3} \, dx=-\frac {21 \, a^{4} b x^{3} + 19 \, a^{5} x}{8 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} + \frac {99 \, a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} + \frac {5 \, b^{3} x^{7} - 21 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} - 350 \, a^{3} x}{35 \, b^{6}} \] Input:
integrate(x^12/(b*x^2+a)^3,x, algorithm="maxima")
Output:
-1/8*(21*a^4*b*x^3 + 19*a^5*x)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6) + 99/8*a^ 4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/35*(5*b^3*x^7 - 21*a*b^2*x^5 + 70*a^2*b*x^3 - 350*a^3*x)/b^6
Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^3} \, dx=\frac {99 \, a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} - \frac {21 \, a^{4} b x^{3} + 19 \, a^{5} x}{8 \, {\left (b x^{2} + a\right )}^{2} b^{6}} + \frac {5 \, b^{18} x^{7} - 21 \, a b^{17} x^{5} + 70 \, a^{2} b^{16} x^{3} - 350 \, a^{3} b^{15} x}{35 \, b^{21}} \] Input:
integrate(x^12/(b*x^2+a)^3,x, algorithm="giac")
Output:
99/8*a^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) - 1/8*(21*a^4*b*x^3 + 19*a^ 5*x)/((b*x^2 + a)^2*b^6) + 1/35*(5*b^18*x^7 - 21*a*b^17*x^5 + 70*a^2*b^16* x^3 - 350*a^3*b^15*x)/b^21
Time = 0.16 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^3} \, dx=\frac {x^7}{7\,b^3}-\frac {\frac {19\,a^5\,x}{8}+\frac {21\,b\,a^4\,x^3}{8}}{a^2\,b^6+2\,a\,b^7\,x^2+b^8\,x^4}-\frac {3\,a\,x^5}{5\,b^4}-\frac {10\,a^3\,x}{b^6}+\frac {99\,a^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,b^{13/2}}+\frac {2\,a^2\,x^3}{b^5} \] Input:
int(x^12/(a + b*x^2)^3,x)
Output:
x^7/(7*b^3) - ((19*a^5*x)/8 + (21*a^4*b*x^3)/8)/(a^2*b^6 + b^8*x^4 + 2*a*b ^7*x^2) - (3*a*x^5)/(5*b^4) - (10*a^3*x)/b^6 + (99*a^(7/2)*atan((b^(1/2)*x )/a^(1/2)))/(8*b^(13/2)) + (2*a^2*x^3)/b^5
Time = 0.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.43 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^3} \, dx=\frac {3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5}+6930 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b \,x^{2}+3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} x^{4}-3465 a^{5} b x -5775 a^{4} b^{2} x^{3}-1848 a^{3} b^{3} x^{5}+264 a^{2} b^{4} x^{7}-88 a \,b^{5} x^{9}+40 b^{6} x^{11}}{280 b^{7} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int(x^12/(b*x^2+a)^3,x)
Output:
(3465*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5 + 6930*sqrt(b)*sq rt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*x**2 + 3465*sqrt(b)*sqrt(a)*ata n((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*x**4 - 3465*a**5*b*x - 5775*a**4*b**2 *x**3 - 1848*a**3*b**3*x**5 + 264*a**2*b**4*x**7 - 88*a*b**5*x**9 + 40*b** 6*x**11)/(280*b**7*(a**2 + 2*a*b*x**2 + b**2*x**4))