Integrand size = 13, antiderivative size = 85 \[ \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx=-\frac {3 a x}{b^4}+\frac {x^3}{3 b^3}+\frac {a^3 x}{4 b^4 \left (a+b x^2\right )^2}-\frac {13 a^2 x}{8 b^4 \left (a+b x^2\right )}+\frac {35 a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{9/2}} \] Output:
-3*a*x/b^4+1/3*x^3/b^3+1/4*a^3*x/b^4/(b*x^2+a)^2-13/8*a^2*x/b^4/(b*x^2+a)+ 35/8*a^(3/2)*arctan(b^(1/2)*x/a^(1/2))/b^(9/2)
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx=-\frac {105 a^3 x+175 a^2 b x^3+56 a b^2 x^5-8 b^3 x^7}{24 b^4 \left (a+b x^2\right )^2}+\frac {35 a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{9/2}} \] Input:
Integrate[x^8/(a + b*x^2)^3,x]
Output:
-1/24*(105*a^3*x + 175*a^2*b*x^3 + 56*a*b^2*x^5 - 8*b^3*x^7)/(b^4*(a + b*x ^2)^2) + (35*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(9/2))
Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.13, 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.
\(\displaystyle \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {7 \int \frac {x^6}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {x^7}{4 b \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {7 \left (\frac {5 \int \frac {x^4}{b x^2+a}dx}{2 b}-\frac {x^5}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^7}{4 b \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {7 \left (\frac {5 \int \left (\frac {a^2}{b^2 \left (b x^2+a\right )}-\frac {a}{b^2}+\frac {x^2}{b}\right )dx}{2 b}-\frac {x^5}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^7}{4 b \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}-\frac {a x}{b^2}+\frac {x^3}{3 b}\right )}{2 b}-\frac {x^5}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^7}{4 b \left (a+b x^2\right )^2}\) |
Input:
Int[x^8/(a + b*x^2)^3,x]
Output:
-1/4*x^7/(b*(a + b*x^2)^2) + (7*(-1/2*x^5/(b*(a + b*x^2)) + (5*(-((a*x)/b^ 2) + x^3/(3*b) + (a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2)))/(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 = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.74
method | result | size |
default | \(-\frac {-\frac {1}{3} b \,x^{3}+3 a x}{b^{4}}+\frac {a^{2} \left (\frac {-\frac {13}{8} b \,x^{3}-\frac {11}{8} a x}{\left (b \,x^{2}+a \right )^{2}}+\frac {35 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4}}\) | \(63\) |
risch | \(\frac {x^{3}}{3 b^{3}}-\frac {3 a x}{b^{4}}+\frac {-\frac {13}{8} a^{2} b \,x^{3}-\frac {11}{8} a^{3} x}{b^{4} \left (b \,x^{2}+a \right )^{2}}+\frac {35 \sqrt {-a b}\, a \ln \left (-\sqrt {-a b}\, x +a \right )}{16 b^{5}}-\frac {35 \sqrt {-a b}\, a \ln \left (\sqrt {-a b}\, x +a \right )}{16 b^{5}}\) | \(93\) |
Input:
int(x^8/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
-1/b^4*(-1/3*b*x^3+3*a*x)+a^2/b^4*((-13/8*b*x^3-11/8*a*x)/(b*x^2+a)^2+35/8 /(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.07 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.71 \[ \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx=\left [\frac {16 \, b^{3} x^{7} - 112 \, a b^{2} x^{5} - 350 \, a^{2} b x^{3} - 210 \, a^{3} x + 105 \, {\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{48 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}, \frac {8 \, b^{3} x^{7} - 56 \, a b^{2} x^{5} - 175 \, a^{2} b x^{3} - 105 \, a^{3} x + 105 \, {\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{24 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}\right ] \] Input:
integrate(x^8/(b*x^2+a)^3,x, algorithm="fricas")
Output:
[1/48*(16*b^3*x^7 - 112*a*b^2*x^5 - 350*a^2*b*x^3 - 210*a^3*x + 105*(a*b^2 *x^4 + 2*a^2*b*x^2 + a^3)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b *x^2 + a)))/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4), 1/24*(8*b^3*x^7 - 56*a*b^2* x^5 - 175*a^2*b*x^3 - 105*a^3*x + 105*(a*b^2*x^4 + 2*a^2*b*x^2 + a^3)*sqrt (a/b)*arctan(b*x*sqrt(a/b)/a))/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)]
Time = 0.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.56 \[ \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx=- \frac {3 a x}{b^{4}} - \frac {35 \sqrt {- \frac {a^{3}}{b^{9}}} \log {\left (x - \frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}}}{a} \right )}}{16} + \frac {35 \sqrt {- \frac {a^{3}}{b^{9}}} \log {\left (x + \frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}}}{a} \right )}}{16} + \frac {- 11 a^{3} x - 13 a^{2} b x^{3}}{8 a^{2} b^{4} + 16 a b^{5} x^{2} + 8 b^{6} x^{4}} + \frac {x^{3}}{3 b^{3}} \] Input:
integrate(x**8/(b*x**2+a)**3,x)
Output:
-3*a*x/b**4 - 35*sqrt(-a**3/b**9)*log(x - b**4*sqrt(-a**3/b**9)/a)/16 + 35 *sqrt(-a**3/b**9)*log(x + b**4*sqrt(-a**3/b**9)/a)/16 + (-11*a**3*x - 13*a **2*b*x**3)/(8*a**2*b**4 + 16*a*b**5*x**2 + 8*b**6*x**4) + x**3/(3*b**3)
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96 \[ \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx=-\frac {13 \, a^{2} b x^{3} + 11 \, a^{3} x}{8 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} + \frac {35 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{4}} + \frac {b x^{3} - 9 \, a x}{3 \, b^{4}} \] Input:
integrate(x^8/(b*x^2+a)^3,x, algorithm="maxima")
Output:
-1/8*(13*a^2*b*x^3 + 11*a^3*x)/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4) + 35/8*a^ 2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/3*(b*x^3 - 9*a*x)/b^4
Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx=\frac {35 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{4}} - \frac {13 \, a^{2} b x^{3} + 11 \, a^{3} x}{8 \, {\left (b x^{2} + a\right )}^{2} b^{4}} + \frac {b^{6} x^{3} - 9 \, a b^{5} x}{3 \, b^{9}} \] Input:
integrate(x^8/(b*x^2+a)^3,x, algorithm="giac")
Output:
35/8*a^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) - 1/8*(13*a^2*b*x^3 + 11*a^ 3*x)/((b*x^2 + a)^2*b^4) + 1/3*(b^6*x^3 - 9*a*b^5*x)/b^9
Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx=\frac {x^3}{3\,b^3}-\frac {\frac {11\,a^3\,x}{8}+\frac {13\,b\,a^2\,x^3}{8}}{a^2\,b^4+2\,a\,b^5\,x^2+b^6\,x^4}+\frac {35\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,b^{9/2}}-\frac {3\,a\,x}{b^4} \] Input:
int(x^8/(a + b*x^2)^3,x)
Output:
x^3/(3*b^3) - ((11*a^3*x)/8 + (13*a^2*b*x^3)/8)/(a^2*b^4 + b^6*x^4 + 2*a*b ^5*x^2) + (35*a^(3/2)*atan((b^(1/2)*x)/a^(1/2)))/(8*b^(9/2)) - (3*a*x)/b^4
Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.55 \[ \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx=\frac {105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3}+210 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b \,x^{2}+105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} x^{4}-105 a^{3} b x -175 a^{2} b^{2} x^{3}-56 a \,b^{3} x^{5}+8 b^{4} x^{7}}{24 b^{5} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int(x^8/(b*x^2+a)^3,x)
Output:
(105*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3 + 210*sqrt(b)*sqrt (a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b*x**2 + 105*sqrt(b)*sqrt(a)*atan(( b*x)/(sqrt(b)*sqrt(a)))*a*b**2*x**4 - 105*a**3*b*x - 175*a**2*b**2*x**3 - 56*a*b**3*x**5 + 8*b**4*x**7)/(24*b**5*(a**2 + 2*a*b*x**2 + b**2*x**4))