Integrand size = 17, antiderivative size = 101 \[ \int \frac {x^{-1-2 n}}{\left (a+b x^n\right )^3} \, dx=-\frac {x^{-2 n}}{2 a^3 n}+\frac {3 b x^{-n}}{a^4 n}+\frac {b^2}{2 a^3 n \left (a+b x^n\right )^2}+\frac {3 b^2}{a^4 n \left (a+b x^n\right )}+\frac {6 b^2 \log (x)}{a^5}-\frac {6 b^2 \log \left (a+b x^n\right )}{a^5 n} \]
-1/2/a^3/n/(x^(2*n))+3*b/a^4/n/(x^n)+1/2*b^2/a^3/n/(a+b*x^n)^2+3*b^2/a^4/n /(a+b*x^n)+6*b^2*ln(x)/a^5-6*b^2*ln(a+b*x^n)/a^5/n
Time = 0.18 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.84 \[ \int \frac {x^{-1-2 n}}{\left (a+b x^n\right )^3} \, dx=\frac {\frac {a x^{-2 n} \left (-a^3+4 a^2 b x^n+18 a b^2 x^{2 n}+12 b^3 x^{3 n}\right )}{\left (a+b x^n\right )^2}+12 b^2 \log \left (x^n\right )-12 b^2 \log \left (a+b x^n\right )}{2 a^5 n} \]
((a*(-a^3 + 4*a^2*b*x^n + 18*a*b^2*x^(2*n) + 12*b^3*x^(3*n)))/(x^(2*n)*(a + b*x^n)^2) + 12*b^2*Log[x^n] - 12*b^2*Log[a + b*x^n])/(2*a^5*n)
Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {798, 54, 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.
\(\displaystyle \int \frac {x^{-2 n-1}}{\left (a+b x^n\right )^3} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \frac {x^{-3 n}}{\left (b x^n+a\right )^3}dx^n}{n}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {\int \left (\frac {x^{-3 n}}{a^3}-\frac {3 b x^{-2 n}}{a^4}+\frac {6 b^2 x^{-n}}{a^5}-\frac {6 b^3}{a^5 \left (b x^n+a\right )}-\frac {3 b^3}{a^4 \left (b x^n+a\right )^2}-\frac {b^3}{a^3 \left (b x^n+a\right )^3}\right )dx^n}{n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {6 b^2 \log \left (x^n\right )}{a^5}-\frac {6 b^2 \log \left (a+b x^n\right )}{a^5}+\frac {3 b^2}{a^4 \left (a+b x^n\right )}+\frac {3 b x^{-n}}{a^4}+\frac {b^2}{2 a^3 \left (a+b x^n\right )^2}-\frac {x^{-2 n}}{2 a^3}}{n}\) |
(-1/2*1/(a^3*x^(2*n)) + (3*b)/(a^4*x^n) + b^2/(2*a^3*(a + b*x^n)^2) + (3*b ^2)/(a^4*(a + b*x^n)) + (6*b^2*Log[x^n])/a^5 - (6*b^2*Log[a + b*x^n])/a^5) /n
3.27.38.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 3.78 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {3 b \,x^{-n}}{a^{4} n}-\frac {x^{-2 n}}{2 a^{3} n}+\frac {6 b^{2} \ln \left (x \right )}{a^{5}}+\frac {b^{2} \left (6 b \,x^{n}+7 a \right )}{2 a^{4} n \left (a +b \,x^{n}\right )^{2}}-\frac {6 b^{2} \ln \left (x^{n}+\frac {a}{b}\right )}{a^{5} n}\) | \(90\) |
norman | \(\frac {\left (\frac {9 b^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{a^{3} n}-\frac {1}{2 a n}+\frac {6 b^{2} \ln \left (x \right ) {\mathrm e}^{2 n \ln \left (x \right )}}{a^{3}}+\frac {2 b \,{\mathrm e}^{n \ln \left (x \right )}}{a^{2} n}+\frac {12 b^{3} \ln \left (x \right ) {\mathrm e}^{3 n \ln \left (x \right )}}{a^{4}}+\frac {6 b^{4} \ln \left (x \right ) {\mathrm e}^{4 n \ln \left (x \right )}}{a^{5}}+\frac {6 b^{3} {\mathrm e}^{3 n \ln \left (x \right )}}{a^{4} n}\right ) {\mathrm e}^{-2 n \ln \left (x \right )}}{\left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}}-\frac {6 b^{2} \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{a^{5} n}\) | \(152\) |
3*b/a^4/n/(x^n)-1/2/a^3/n/(x^n)^2+6*b^2*ln(x)/a^5+1/2*b^2*(6*b*x^n+7*a)/a^ 4/n/(a+b*x^n)^2-6*b^2/a^5/n*ln(x^n+a/b)
Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.58 \[ \int \frac {x^{-1-2 n}}{\left (a+b x^n\right )^3} \, dx=\frac {12 \, b^{4} n x^{4 \, n} \log \left (x\right ) + 4 \, a^{3} b x^{n} - a^{4} + 12 \, {\left (2 \, a b^{3} n \log \left (x\right ) + a b^{3}\right )} x^{3 \, n} + 6 \, {\left (2 \, a^{2} b^{2} n \log \left (x\right ) + 3 \, a^{2} b^{2}\right )} x^{2 \, n} - 12 \, {\left (b^{4} x^{4 \, n} + 2 \, a b^{3} x^{3 \, n} + a^{2} b^{2} x^{2 \, n}\right )} \log \left (b x^{n} + a\right )}{2 \, {\left (a^{5} b^{2} n x^{4 \, n} + 2 \, a^{6} b n x^{3 \, n} + a^{7} n x^{2 \, n}\right )}} \]
1/2*(12*b^4*n*x^(4*n)*log(x) + 4*a^3*b*x^n - a^4 + 12*(2*a*b^3*n*log(x) + a*b^3)*x^(3*n) + 6*(2*a^2*b^2*n*log(x) + 3*a^2*b^2)*x^(2*n) - 12*(b^4*x^(4 *n) + 2*a*b^3*x^(3*n) + a^2*b^2*x^(2*n))*log(b*x^n + a))/(a^5*b^2*n*x^(4*n ) + 2*a^6*b*n*x^(3*n) + a^7*n*x^(2*n))
Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (90) = 180\).
Time = 41.65 (sec) , antiderivative size = 629, normalized size of antiderivative = 6.23 \[ \int \frac {x^{-1-2 n}}{\left (a+b x^n\right )^3} \, dx=\begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac {x x^{- 2 n - 1}}{2 a^{3} n} & \text {for}\: b = 0 \\- \frac {x x^{- 3 n} x^{- 2 n - 1}}{5 b^{3} n} & \text {for}\: a = 0 \\\frac {\tilde {\infty } x x^{- 2 n - 1}}{n} & \text {for}\: b = - a x^{- n} \\\frac {\log {\left (x \right )}}{\left (a + b\right )^{3}} & \text {for}\: n = 0 \\- \frac {a^{4}}{2 a^{7} n x^{2 n} + 4 a^{6} b n x^{3 n} + 2 a^{5} b^{2} n x^{4 n}} + \frac {4 a^{3} b x^{n}}{2 a^{7} n x^{2 n} + 4 a^{6} b n x^{3 n} + 2 a^{5} b^{2} n x^{4 n}} + \frac {12 a^{2} b^{2} n x^{2 n} \log {\left (x \right )}}{2 a^{7} n x^{2 n} + 4 a^{6} b n x^{3 n} + 2 a^{5} b^{2} n x^{4 n}} - \frac {12 a^{2} b^{2} x^{2 n} \log {\left (\frac {a}{b} + x^{n} \right )}}{2 a^{7} n x^{2 n} + 4 a^{6} b n x^{3 n} + 2 a^{5} b^{2} n x^{4 n}} + \frac {18 a^{2} b^{2} x^{2 n}}{2 a^{7} n x^{2 n} + 4 a^{6} b n x^{3 n} + 2 a^{5} b^{2} n x^{4 n}} + \frac {24 a b^{3} n x^{3 n} \log {\left (x \right )}}{2 a^{7} n x^{2 n} + 4 a^{6} b n x^{3 n} + 2 a^{5} b^{2} n x^{4 n}} - \frac {24 a b^{3} x^{3 n} \log {\left (\frac {a}{b} + x^{n} \right )}}{2 a^{7} n x^{2 n} + 4 a^{6} b n x^{3 n} + 2 a^{5} b^{2} n x^{4 n}} + \frac {12 a b^{3} x^{3 n}}{2 a^{7} n x^{2 n} + 4 a^{6} b n x^{3 n} + 2 a^{5} b^{2} n x^{4 n}} + \frac {12 b^{4} n x^{4 n} \log {\left (x \right )}}{2 a^{7} n x^{2 n} + 4 a^{6} b n x^{3 n} + 2 a^{5} b^{2} n x^{4 n}} - \frac {12 b^{4} x^{4 n} \log {\left (\frac {a}{b} + x^{n} \right )}}{2 a^{7} n x^{2 n} + 4 a^{6} b n x^{3 n} + 2 a^{5} b^{2} n x^{4 n}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (-x*x**(-2*n - 1)/ (2*a**3*n), Eq(b, 0)), (-x*x**(-2*n - 1)/(5*b**3*n*x**(3*n)), Eq(a, 0)), ( zoo*x*x**(-2*n - 1)/n, Eq(b, -a/x**n)), (log(x)/(a + b)**3, Eq(n, 0)), (-a **4/(2*a**7*n*x**(2*n) + 4*a**6*b*n*x**(3*n) + 2*a**5*b**2*n*x**(4*n)) + 4 *a**3*b*x**n/(2*a**7*n*x**(2*n) + 4*a**6*b*n*x**(3*n) + 2*a**5*b**2*n*x**( 4*n)) + 12*a**2*b**2*n*x**(2*n)*log(x)/(2*a**7*n*x**(2*n) + 4*a**6*b*n*x** (3*n) + 2*a**5*b**2*n*x**(4*n)) - 12*a**2*b**2*x**(2*n)*log(a/b + x**n)/(2 *a**7*n*x**(2*n) + 4*a**6*b*n*x**(3*n) + 2*a**5*b**2*n*x**(4*n)) + 18*a**2 *b**2*x**(2*n)/(2*a**7*n*x**(2*n) + 4*a**6*b*n*x**(3*n) + 2*a**5*b**2*n*x* *(4*n)) + 24*a*b**3*n*x**(3*n)*log(x)/(2*a**7*n*x**(2*n) + 4*a**6*b*n*x**( 3*n) + 2*a**5*b**2*n*x**(4*n)) - 24*a*b**3*x**(3*n)*log(a/b + x**n)/(2*a** 7*n*x**(2*n) + 4*a**6*b*n*x**(3*n) + 2*a**5*b**2*n*x**(4*n)) + 12*a*b**3*x **(3*n)/(2*a**7*n*x**(2*n) + 4*a**6*b*n*x**(3*n) + 2*a**5*b**2*n*x**(4*n)) + 12*b**4*n*x**(4*n)*log(x)/(2*a**7*n*x**(2*n) + 4*a**6*b*n*x**(3*n) + 2* a**5*b**2*n*x**(4*n)) - 12*b**4*x**(4*n)*log(a/b + x**n)/(2*a**7*n*x**(2*n ) + 4*a**6*b*n*x**(3*n) + 2*a**5*b**2*n*x**(4*n)), True))
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.09 \[ \int \frac {x^{-1-2 n}}{\left (a+b x^n\right )^3} \, dx=\frac {12 \, b^{3} x^{3 \, n} + 18 \, a b^{2} x^{2 \, n} + 4 \, a^{2} b x^{n} - a^{3}}{2 \, {\left (a^{4} b^{2} n x^{4 \, n} + 2 \, a^{5} b n x^{3 \, n} + a^{6} n x^{2 \, n}\right )}} + \frac {6 \, b^{2} \log \left (x\right )}{a^{5}} - \frac {6 \, b^{2} \log \left (\frac {b x^{n} + a}{b}\right )}{a^{5} n} \]
1/2*(12*b^3*x^(3*n) + 18*a*b^2*x^(2*n) + 4*a^2*b*x^n - a^3)/(a^4*b^2*n*x^( 4*n) + 2*a^5*b*n*x^(3*n) + a^6*n*x^(2*n)) + 6*b^2*log(x)/a^5 - 6*b^2*log(( b*x^n + a)/b)/(a^5*n)
\[ \int \frac {x^{-1-2 n}}{\left (a+b x^n\right )^3} \, dx=\int { \frac {x^{-2 \, n - 1}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^{-1-2 n}}{\left (a+b x^n\right )^3} \, dx=\int \frac {1}{x^{2\,n+1}\,{\left (a+b\,x^n\right )}^3} \,d x \]