Integrand size = 15, antiderivative size = 179 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=-\frac {3}{11 b x^{11/3}}+\frac {3 a}{10 b^2 x^{10/3}}-\frac {a^2}{3 b^3 x^3}+\frac {3 a^3}{8 b^4 x^{8/3}}-\frac {3 a^4}{7 b^5 x^{7/3}}+\frac {a^5}{2 b^6 x^2}-\frac {3 a^6}{5 b^7 x^{5/3}}+\frac {3 a^7}{4 b^8 x^{4/3}}-\frac {a^8}{b^9 x}+\frac {3 a^9}{2 b^{10} x^{2/3}}-\frac {3 a^{10}}{b^{11} \sqrt [3]{x}}+\frac {3 a^{11} \log \left (b+a \sqrt [3]{x}\right )}{b^{12}}-\frac {a^{11} \log (x)}{b^{12}} \]
-3/11/b/x^(11/3)+3/10*a/b^2/x^(10/3)-1/3*a^2/b^3/x^3+3/8*a^3/b^4/x^(8/3)-3 /7*a^4/b^5/x^(7/3)+1/2*a^5/b^6/x^2-3/5*a^6/b^7/x^(5/3)+3/4*a^7/b^8/x^(4/3) -a^8/b^9/x+3/2*a^9/b^10/x^(2/3)-3*a^10/b^11/x^(1/3)+3*a^11*ln(b+a*x^(1/3)) /b^12-a^11*ln(x)/b^12
Time = 0.19 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=\frac {\frac {b \left (-2520 b^{10}+2772 a b^9 \sqrt [3]{x}-3080 a^2 b^8 x^{2/3}+3465 a^3 b^7 x-3960 a^4 b^6 x^{4/3}+4620 a^5 b^5 x^{5/3}-5544 a^6 b^4 x^2+6930 a^7 b^3 x^{7/3}-9240 a^8 b^2 x^{8/3}+13860 a^9 b x^3-27720 a^{10} x^{10/3}\right )}{x^{11/3}}+27720 a^{11} \log \left (b+a \sqrt [3]{x}\right )-9240 a^{11} \log (x)}{9240 b^{12}} \]
((b*(-2520*b^10 + 2772*a*b^9*x^(1/3) - 3080*a^2*b^8*x^(2/3) + 3465*a^3*b^7 *x - 3960*a^4*b^6*x^(4/3) + 4620*a^5*b^5*x^(5/3) - 5544*a^6*b^4*x^2 + 6930 *a^7*b^3*x^(7/3) - 9240*a^8*b^2*x^(8/3) + 13860*a^9*b*x^3 - 27720*a^10*x^( 10/3)))/x^(11/3) + 27720*a^11*Log[b + a*x^(1/3)] - 9240*a^11*Log[x])/(9240 *b^12)
Time = 0.30 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {795, 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.
\(\displaystyle \int \frac {1}{x^5 \left (a+\frac {b}{\sqrt [3]{x}}\right )} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {1}{x^{14/3} \left (a \sqrt [3]{x}+b\right )}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {1}{\left (\sqrt [3]{x} a+b\right ) x^4}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 3 \int \left (\frac {a^{12}}{b^{12} \left (\sqrt [3]{x} a+b\right )}-\frac {a^{11}}{b^{12} \sqrt [3]{x}}+\frac {a^{10}}{b^{11} x^{2/3}}-\frac {a^9}{b^{10} x}+\frac {a^8}{b^9 x^{4/3}}-\frac {a^7}{b^8 x^{5/3}}+\frac {a^6}{b^7 x^2}-\frac {a^5}{b^6 x^{7/3}}+\frac {a^4}{b^5 x^{8/3}}-\frac {a^3}{b^4 x^3}+\frac {a^2}{b^3 x^{10/3}}-\frac {a}{b^2 x^{11/3}}+\frac {1}{b x^4}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {a^{11} \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}-\frac {a^{11} \log \left (\sqrt [3]{x}\right )}{b^{12}}-\frac {a^{10}}{b^{11} \sqrt [3]{x}}+\frac {a^9}{2 b^{10} x^{2/3}}-\frac {a^8}{3 b^9 x}+\frac {a^7}{4 b^8 x^{4/3}}-\frac {a^6}{5 b^7 x^{5/3}}+\frac {a^5}{6 b^6 x^2}-\frac {a^4}{7 b^5 x^{7/3}}+\frac {a^3}{8 b^4 x^{8/3}}-\frac {a^2}{9 b^3 x^3}+\frac {a}{10 b^2 x^{10/3}}-\frac {1}{11 b x^{11/3}}\right )\) |
3*(-1/11*1/(b*x^(11/3)) + a/(10*b^2*x^(10/3)) - a^2/(9*b^3*x^3) + a^3/(8*b ^4*x^(8/3)) - a^4/(7*b^5*x^(7/3)) + a^5/(6*b^6*x^2) - a^6/(5*b^7*x^(5/3)) + a^7/(4*b^8*x^(4/3)) - a^8/(3*b^9*x) + a^9/(2*b^10*x^(2/3)) - a^10/(b^11* x^(1/3)) + (a^11*Log[b + a*x^(1/3)])/b^12 - (a^11*Log[x^(1/3)])/b^12)
3.25.28.3.1 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] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
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 = 3.75 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-\frac {3}{11 b \,x^{\frac {11}{3}}}+\frac {3 a}{10 b^{2} x^{\frac {10}{3}}}-\frac {a^{2}}{3 b^{3} x^{3}}+\frac {3 a^{3}}{8 b^{4} x^{\frac {8}{3}}}-\frac {3 a^{4}}{7 b^{5} x^{\frac {7}{3}}}+\frac {a^{5}}{2 b^{6} x^{2}}-\frac {3 a^{6}}{5 b^{7} x^{\frac {5}{3}}}+\frac {3 a^{7}}{4 b^{8} x^{\frac {4}{3}}}-\frac {a^{8}}{b^{9} x}+\frac {3 a^{9}}{2 b^{10} x^{\frac {2}{3}}}-\frac {3 a^{10}}{b^{11} x^{\frac {1}{3}}}+\frac {3 a^{11} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{12}}-\frac {a^{11} \ln \left (x \right )}{b^{12}}\) | \(144\) |
default | \(-\frac {3}{11 b \,x^{\frac {11}{3}}}+\frac {3 a}{10 b^{2} x^{\frac {10}{3}}}-\frac {a^{2}}{3 b^{3} x^{3}}+\frac {3 a^{3}}{8 b^{4} x^{\frac {8}{3}}}-\frac {3 a^{4}}{7 b^{5} x^{\frac {7}{3}}}+\frac {a^{5}}{2 b^{6} x^{2}}-\frac {3 a^{6}}{5 b^{7} x^{\frac {5}{3}}}+\frac {3 a^{7}}{4 b^{8} x^{\frac {4}{3}}}-\frac {a^{8}}{b^{9} x}+\frac {3 a^{9}}{2 b^{10} x^{\frac {2}{3}}}-\frac {3 a^{10}}{b^{11} x^{\frac {1}{3}}}+\frac {3 a^{11} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{12}}-\frac {a^{11} \ln \left (x \right )}{b^{12}}\) | \(144\) |
-3/11/b/x^(11/3)+3/10*a/b^2/x^(10/3)-1/3*a^2/b^3/x^3+3/8*a^3/b^4/x^(8/3)-3 /7*a^4/b^5/x^(7/3)+1/2*a^5/b^6/x^2-3/5*a^6/b^7/x^(5/3)+3/4*a^7/b^8/x^(4/3) -a^8/b^9/x+3/2*a^9/b^10/x^(2/3)-3*a^10/b^11/x^(1/3)+3*a^11*ln(b+a*x^(1/3)) /b^12-a^11*ln(x)/b^12
Time = 0.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=\frac {27720 \, a^{11} x^{4} \log \left (a x^{\frac {1}{3}} + b\right ) - 27720 \, a^{11} x^{4} \log \left (x^{\frac {1}{3}}\right ) - 9240 \, a^{8} b^{3} x^{3} + 4620 \, a^{5} b^{6} x^{2} - 3080 \, a^{2} b^{9} x - 198 \, {\left (140 \, a^{10} b x^{3} - 35 \, a^{7} b^{4} x^{2} + 20 \, a^{4} b^{7} x - 14 \, a b^{10}\right )} x^{\frac {2}{3}} + 63 \, {\left (220 \, a^{9} b^{2} x^{3} - 88 \, a^{6} b^{5} x^{2} + 55 \, a^{3} b^{8} x - 40 \, b^{11}\right )} x^{\frac {1}{3}}}{9240 \, b^{12} x^{4}} \]
1/9240*(27720*a^11*x^4*log(a*x^(1/3) + b) - 27720*a^11*x^4*log(x^(1/3)) - 9240*a^8*b^3*x^3 + 4620*a^5*b^6*x^2 - 3080*a^2*b^9*x - 198*(140*a^10*b*x^3 - 35*a^7*b^4*x^2 + 20*a^4*b^7*x - 14*a*b^10)*x^(2/3) + 63*(220*a^9*b^2*x^ 3 - 88*a^6*b^5*x^2 + 55*a^3*b^8*x - 40*b^11)*x^(1/3))/(b^12*x^4)
Time = 3.09 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {11}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{11 b x^{\frac {11}{3}}} & \text {for}\: a = 0 \\- \frac {1}{4 a x^{4}} & \text {for}\: b = 0 \\- \frac {a^{11} \log {\left (x \right )}}{b^{12}} + \frac {3 a^{11} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{b^{12}} - \frac {3 a^{10}}{b^{11} \sqrt [3]{x}} + \frac {3 a^{9}}{2 b^{10} x^{\frac {2}{3}}} - \frac {a^{8}}{b^{9} x} + \frac {3 a^{7}}{4 b^{8} x^{\frac {4}{3}}} - \frac {3 a^{6}}{5 b^{7} x^{\frac {5}{3}}} + \frac {a^{5}}{2 b^{6} x^{2}} - \frac {3 a^{4}}{7 b^{5} x^{\frac {7}{3}}} + \frac {3 a^{3}}{8 b^{4} x^{\frac {8}{3}}} - \frac {a^{2}}{3 b^{3} x^{3}} + \frac {3 a}{10 b^{2} x^{\frac {10}{3}}} - \frac {3}{11 b x^{\frac {11}{3}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo/x**(11/3), Eq(a, 0) & Eq(b, 0)), (-3/(11*b*x**(11/3)), Eq(a , 0)), (-1/(4*a*x**4), Eq(b, 0)), (-a**11*log(x)/b**12 + 3*a**11*log(x**(1 /3) + b/a)/b**12 - 3*a**10/(b**11*x**(1/3)) + 3*a**9/(2*b**10*x**(2/3)) - a**8/(b**9*x) + 3*a**7/(4*b**8*x**(4/3)) - 3*a**6/(5*b**7*x**(5/3)) + a**5 /(2*b**6*x**2) - 3*a**4/(7*b**5*x**(7/3)) + 3*a**3/(8*b**4*x**(8/3)) - a** 2/(3*b**3*x**3) + 3*a/(10*b**2*x**(10/3)) - 3/(11*b*x**(11/3)), True))
Time = 0.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=\frac {3 \, a^{11} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{12}} - \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{11}}{11 \, b^{12}} + \frac {33 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{10} a}{10 \, b^{12}} - \frac {55 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{9} a^{2}}{3 \, b^{12}} + \frac {495 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{8} a^{3}}{8 \, b^{12}} - \frac {990 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{7} a^{4}}{7 \, b^{12}} + \frac {231 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{6} a^{5}}{b^{12}} - \frac {1386 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5} a^{6}}{5 \, b^{12}} + \frac {495 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a^{7}}{2 \, b^{12}} - \frac {165 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{8}}{b^{12}} + \frac {165 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{9}}{2 \, b^{12}} - \frac {33 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{10}}{b^{12}} \]
3*a^11*log(a + b/x^(1/3))/b^12 - 3/11*(a + b/x^(1/3))^11/b^12 + 33/10*(a + b/x^(1/3))^10*a/b^12 - 55/3*(a + b/x^(1/3))^9*a^2/b^12 + 495/8*(a + b/x^( 1/3))^8*a^3/b^12 - 990/7*(a + b/x^(1/3))^7*a^4/b^12 + 231*(a + b/x^(1/3))^ 6*a^5/b^12 - 1386/5*(a + b/x^(1/3))^5*a^6/b^12 + 495/2*(a + b/x^(1/3))^4*a ^7/b^12 - 165*(a + b/x^(1/3))^3*a^8/b^12 + 165/2*(a + b/x^(1/3))^2*a^9/b^1 2 - 33*(a + b/x^(1/3))*a^10/b^12
Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=\frac {3 \, a^{11} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{12}} - \frac {a^{11} \log \left ({\left | x \right |}\right )}{b^{12}} - \frac {27720 \, a^{10} b x^{\frac {10}{3}} - 13860 \, a^{9} b^{2} x^{3} + 9240 \, a^{8} b^{3} x^{\frac {8}{3}} - 6930 \, a^{7} b^{4} x^{\frac {7}{3}} + 5544 \, a^{6} b^{5} x^{2} - 4620 \, a^{5} b^{6} x^{\frac {5}{3}} + 3960 \, a^{4} b^{7} x^{\frac {4}{3}} - 3465 \, a^{3} b^{8} x + 3080 \, a^{2} b^{9} x^{\frac {2}{3}} - 2772 \, a b^{10} x^{\frac {1}{3}} + 2520 \, b^{11}}{9240 \, b^{12} x^{\frac {11}{3}}} \]
3*a^11*log(abs(a*x^(1/3) + b))/b^12 - a^11*log(abs(x))/b^12 - 1/9240*(2772 0*a^10*b*x^(10/3) - 13860*a^9*b^2*x^3 + 9240*a^8*b^3*x^(8/3) - 6930*a^7*b^ 4*x^(7/3) + 5544*a^6*b^5*x^2 - 4620*a^5*b^6*x^(5/3) + 3960*a^4*b^7*x^(4/3) - 3465*a^3*b^8*x + 3080*a^2*b^9*x^(2/3) - 2772*a*b^10*x^(1/3) + 2520*b^11 )/(b^12*x^(11/3))
Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=-\frac {2520\,b^{11}-55440\,a^{11}\,x^{11/3}\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )-3465\,a^3\,b^8\,x-2772\,a\,b^{10}\,x^{1/3}+27720\,a^{10}\,b\,x^{10/3}+5544\,a^6\,b^5\,x^2-13860\,a^9\,b^2\,x^3+3080\,a^2\,b^9\,x^{2/3}+3960\,a^4\,b^7\,x^{4/3}-4620\,a^5\,b^6\,x^{5/3}-6930\,a^7\,b^4\,x^{7/3}+9240\,a^8\,b^3\,x^{8/3}}{9240\,b^{12}\,x^{11/3}} \]
-(2520*b^11 - 55440*a^11*x^(11/3)*atanh((2*a*x^(1/3))/b + 1) - 3465*a^3*b^ 8*x - 2772*a*b^10*x^(1/3) + 27720*a^10*b*x^(10/3) + 5544*a^6*b^5*x^2 - 138 60*a^9*b^2*x^3 + 3080*a^2*b^9*x^(2/3) + 3960*a^4*b^7*x^(4/3) - 4620*a^5*b^ 6*x^(5/3) - 6930*a^7*b^4*x^(7/3) + 9240*a^8*b^3*x^(8/3))/(9240*b^12*x^(11/ 3))