Integrand size = 15, antiderivative size = 149 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right ) x^4} \, dx=-\frac {1}{3 a x^3}+\frac {3 b}{8 a^2 x^{8/3}}-\frac {3 b^2}{7 a^3 x^{7/3}}+\frac {b^3}{2 a^4 x^2}-\frac {3 b^4}{5 a^5 x^{5/3}}+\frac {3 b^5}{4 a^6 x^{4/3}}-\frac {b^6}{a^7 x}+\frac {3 b^7}{2 a^8 x^{2/3}}-\frac {3 b^8}{a^9 \sqrt [3]{x}}+\frac {3 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{10}}-\frac {b^9 \log (x)}{a^{10}} \]
-1/3/a/x^3+3/8*b/a^2/x^(8/3)-3/7*b^2/a^3/x^(7/3)+1/2*b^3/a^4/x^2-3/5*b^4/a ^5/x^(5/3)+3/4*b^5/a^6/x^(4/3)-b^6/a^7/x+3/2*b^7/a^8/x^(2/3)-3*b^8/a^9/x^( 1/3)+3*b^9*ln(a+b*x^(1/3))/a^10-b^9*ln(x)/a^10
Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right ) x^4} \, dx=\frac {\frac {a \left (-280 a^8+315 a^7 b \sqrt [3]{x}-360 a^6 b^2 x^{2/3}+420 a^5 b^3 x-504 a^4 b^4 x^{4/3}+630 a^3 b^5 x^{5/3}-840 a^2 b^6 x^2+1260 a b^7 x^{7/3}-2520 b^8 x^{8/3}\right )}{x^3}+2520 b^9 \log \left (a+b \sqrt [3]{x}\right )-840 b^9 \log (x)}{840 a^{10}} \]
((a*(-280*a^8 + 315*a^7*b*x^(1/3) - 360*a^6*b^2*x^(2/3) + 420*a^5*b^3*x - 504*a^4*b^4*x^(4/3) + 630*a^3*b^5*x^(5/3) - 840*a^2*b^6*x^2 + 1260*a*b^7*x ^(7/3) - 2520*b^8*x^(8/3)))/x^3 + 2520*b^9*Log[a + b*x^(1/3)] - 840*b^9*Lo g[x])/(840*a^10)
Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.
\(\displaystyle \int \frac {1}{x^4 \left (a+b \sqrt [3]{x}\right )} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {1}{\left (a+b \sqrt [3]{x}\right ) x^{10/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 3 \int \left (\frac {b^{10}}{a^{10} \left (a+b \sqrt [3]{x}\right )}-\frac {b^9}{a^{10} \sqrt [3]{x}}+\frac {b^8}{a^9 x^{2/3}}-\frac {b^7}{a^8 x}+\frac {b^6}{a^7 x^{4/3}}-\frac {b^5}{a^6 x^{5/3}}+\frac {b^4}{a^5 x^2}-\frac {b^3}{a^4 x^{7/3}}+\frac {b^2}{a^3 x^{8/3}}-\frac {b}{a^2 x^3}+\frac {1}{a x^{10/3}}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{10}}-\frac {b^9 \log \left (\sqrt [3]{x}\right )}{a^{10}}-\frac {b^8}{a^9 \sqrt [3]{x}}+\frac {b^7}{2 a^8 x^{2/3}}-\frac {b^6}{3 a^7 x}+\frac {b^5}{4 a^6 x^{4/3}}-\frac {b^4}{5 a^5 x^{5/3}}+\frac {b^3}{6 a^4 x^2}-\frac {b^2}{7 a^3 x^{7/3}}+\frac {b}{8 a^2 x^{8/3}}-\frac {1}{9 a x^3}\right )\) |
3*(-1/9*1/(a*x^3) + b/(8*a^2*x^(8/3)) - b^2/(7*a^3*x^(7/3)) + b^3/(6*a^4*x ^2) - b^4/(5*a^5*x^(5/3)) + b^5/(4*a^6*x^(4/3)) - b^6/(3*a^7*x) + b^7/(2*a ^8*x^(2/3)) - b^8/(a^9*x^(1/3)) + (b^9*Log[a + b*x^(1/3)])/a^10 - (b^9*Log [x^(1/3)])/a^10)
3.24.63.3.1 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 = 5.97 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(-\frac {1}{3 a \,x^{3}}+\frac {3 b}{8 a^{2} x^{\frac {8}{3}}}-\frac {3 b^{2}}{7 a^{3} x^{\frac {7}{3}}}+\frac {b^{3}}{2 a^{4} x^{2}}-\frac {3 b^{4}}{5 a^{5} x^{\frac {5}{3}}}+\frac {3 b^{5}}{4 a^{6} x^{\frac {4}{3}}}-\frac {b^{6}}{a^{7} x}+\frac {3 b^{7}}{2 a^{8} x^{\frac {2}{3}}}-\frac {3 b^{8}}{a^{9} x^{\frac {1}{3}}}+\frac {3 b^{9} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{10}}-\frac {b^{9} \ln \left (x \right )}{a^{10}}\) | \(122\) |
default | \(-\frac {1}{3 a \,x^{3}}+\frac {3 b}{8 a^{2} x^{\frac {8}{3}}}-\frac {3 b^{2}}{7 a^{3} x^{\frac {7}{3}}}+\frac {b^{3}}{2 a^{4} x^{2}}-\frac {3 b^{4}}{5 a^{5} x^{\frac {5}{3}}}+\frac {3 b^{5}}{4 a^{6} x^{\frac {4}{3}}}-\frac {b^{6}}{a^{7} x}+\frac {3 b^{7}}{2 a^{8} x^{\frac {2}{3}}}-\frac {3 b^{8}}{a^{9} x^{\frac {1}{3}}}+\frac {3 b^{9} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{10}}-\frac {b^{9} \ln \left (x \right )}{a^{10}}\) | \(122\) |
-1/3/a/x^3+3/8*b/a^2/x^(8/3)-3/7*b^2/a^3/x^(7/3)+1/2*b^3/a^4/x^2-3/5*b^4/a ^5/x^(5/3)+3/4*b^5/a^6/x^(4/3)-b^6/a^7/x+3/2*b^7/a^8/x^(2/3)-3*b^8/a^9/x^( 1/3)+3*b^9*ln(a+b*x^(1/3))/a^10-b^9*ln(x)/a^10
Time = 0.29 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right ) x^4} \, dx=\frac {2520 \, b^{9} x^{3} \log \left (b x^{\frac {1}{3}} + a\right ) - 2520 \, b^{9} x^{3} \log \left (x^{\frac {1}{3}}\right ) - 840 \, a^{3} b^{6} x^{2} + 420 \, a^{6} b^{3} x - 280 \, a^{9} - 90 \, {\left (28 \, a b^{8} x^{2} - 7 \, a^{4} b^{5} x + 4 \, a^{7} b^{2}\right )} x^{\frac {2}{3}} + 63 \, {\left (20 \, a^{2} b^{7} x^{2} - 8 \, a^{5} b^{4} x + 5 \, a^{8} b\right )} x^{\frac {1}{3}}}{840 \, a^{10} x^{3}} \]
1/840*(2520*b^9*x^3*log(b*x^(1/3) + a) - 2520*b^9*x^3*log(x^(1/3)) - 840*a ^3*b^6*x^2 + 420*a^6*b^3*x - 280*a^9 - 90*(28*a*b^8*x^2 - 7*a^4*b^5*x + 4* a^7*b^2)*x^(2/3) + 63*(20*a^2*b^7*x^2 - 8*a^5*b^4*x + 5*a^8*b)*x^(1/3))/(a ^10*x^3)
Time = 2.20 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right ) x^4} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {10}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{10 b x^{\frac {10}{3}}} & \text {for}\: a = 0 \\- \frac {1}{3 a x^{3}} & \text {for}\: b = 0 \\- \frac {1}{3 a x^{3}} + \frac {3 b}{8 a^{2} x^{\frac {8}{3}}} - \frac {3 b^{2}}{7 a^{3} x^{\frac {7}{3}}} + \frac {b^{3}}{2 a^{4} x^{2}} - \frac {3 b^{4}}{5 a^{5} x^{\frac {5}{3}}} + \frac {3 b^{5}}{4 a^{6} x^{\frac {4}{3}}} - \frac {b^{6}}{a^{7} x} + \frac {3 b^{7}}{2 a^{8} x^{\frac {2}{3}}} - \frac {3 b^{8}}{a^{9} \sqrt [3]{x}} - \frac {b^{9} \log {\left (x \right )}}{a^{10}} + \frac {3 b^{9} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{a^{10}} & \text {otherwise} \end {cases} \]
Piecewise((zoo/x**(10/3), Eq(a, 0) & Eq(b, 0)), (-3/(10*b*x**(10/3)), Eq(a , 0)), (-1/(3*a*x**3), Eq(b, 0)), (-1/(3*a*x**3) + 3*b/(8*a**2*x**(8/3)) - 3*b**2/(7*a**3*x**(7/3)) + b**3/(2*a**4*x**2) - 3*b**4/(5*a**5*x**(5/3)) + 3*b**5/(4*a**6*x**(4/3)) - b**6/(a**7*x) + 3*b**7/(2*a**8*x**(2/3)) - 3* b**8/(a**9*x**(1/3)) - b**9*log(x)/a**10 + 3*b**9*log(a/b + x**(1/3))/a**1 0, True))
Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right ) x^4} \, dx=\frac {3 \, b^{9} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{10}} - \frac {b^{9} \log \left (x\right )}{a^{10}} - \frac {2520 \, b^{8} x^{\frac {8}{3}} - 1260 \, a b^{7} x^{\frac {7}{3}} + 840 \, a^{2} b^{6} x^{2} - 630 \, a^{3} b^{5} x^{\frac {5}{3}} + 504 \, a^{4} b^{4} x^{\frac {4}{3}} - 420 \, a^{5} b^{3} x + 360 \, a^{6} b^{2} x^{\frac {2}{3}} - 315 \, a^{7} b x^{\frac {1}{3}} + 280 \, a^{8}}{840 \, a^{9} x^{3}} \]
3*b^9*log(b*x^(1/3) + a)/a^10 - b^9*log(x)/a^10 - 1/840*(2520*b^8*x^(8/3) - 1260*a*b^7*x^(7/3) + 840*a^2*b^6*x^2 - 630*a^3*b^5*x^(5/3) + 504*a^4*b^4 *x^(4/3) - 420*a^5*b^3*x + 360*a^6*b^2*x^(2/3) - 315*a^7*b*x^(1/3) + 280*a ^8)/(a^9*x^3)
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right ) x^4} \, dx=\frac {3 \, b^{9} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{10}} - \frac {b^{9} \log \left ({\left | x \right |}\right )}{a^{10}} - \frac {2520 \, a b^{8} x^{\frac {8}{3}} - 1260 \, a^{2} b^{7} x^{\frac {7}{3}} + 840 \, a^{3} b^{6} x^{2} - 630 \, a^{4} b^{5} x^{\frac {5}{3}} + 504 \, a^{5} b^{4} x^{\frac {4}{3}} - 420 \, a^{6} b^{3} x + 360 \, a^{7} b^{2} x^{\frac {2}{3}} - 315 \, a^{8} b x^{\frac {1}{3}} + 280 \, a^{9}}{840 \, a^{10} x^{3}} \]
3*b^9*log(abs(b*x^(1/3) + a))/a^10 - b^9*log(abs(x))/a^10 - 1/840*(2520*a* b^8*x^(8/3) - 1260*a^2*b^7*x^(7/3) + 840*a^3*b^6*x^2 - 630*a^4*b^5*x^(5/3) + 504*a^5*b^4*x^(4/3) - 420*a^6*b^3*x + 360*a^7*b^2*x^(2/3) - 315*a^8*b*x ^(1/3) + 280*a^9)/(a^10*x^3)
Time = 5.81 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right ) x^4} \, dx=-\frac {280\,a^9-5040\,b^9\,x^3\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )-420\,a^6\,b^3\,x-315\,a^8\,b\,x^{1/3}+2520\,a\,b^8\,x^{8/3}+840\,a^3\,b^6\,x^2+360\,a^7\,b^2\,x^{2/3}+504\,a^5\,b^4\,x^{4/3}-630\,a^4\,b^5\,x^{5/3}-1260\,a^2\,b^7\,x^{7/3}}{840\,a^{10}\,x^3} \]