Integrand size = 15, antiderivative size = 125 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=\frac {3 b^6}{a^7 \left (a+b \sqrt [3]{x}\right )}-\frac {1}{2 a^2 x^2}+\frac {6 b}{5 a^3 x^{5/3}}-\frac {9 b^2}{4 a^4 x^{4/3}}+\frac {4 b^3}{a^5 x}-\frac {15 b^4}{2 a^6 x^{2/3}}+\frac {18 b^5}{a^7 \sqrt [3]{x}}-\frac {21 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^8}+\frac {7 b^6 \log (x)}{a^8} \]
3*b^6/a^7/(a+b*x^(1/3))-1/2/a^2/x^2+6/5*b/a^3/x^(5/3)-9/4*b^2/a^4/x^(4/3)+ 4*b^3/a^5/x-15/2*b^4/a^6/x^(2/3)+18*b^5/a^7/x^(1/3)-21*b^6*ln(a+b*x^(1/3)) /a^8+7*b^6*ln(x)/a^8
Time = 0.16 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=\frac {\frac {a \left (-10 a^6+14 a^5 b \sqrt [3]{x}-21 a^4 b^2 x^{2/3}+35 a^3 b^3 x-70 a^2 b^4 x^{4/3}+210 a b^5 x^{5/3}+420 b^6 x^2\right )}{\left (a+b \sqrt [3]{x}\right ) x^2}-420 b^6 \log \left (a+b \sqrt [3]{x}\right )+140 b^6 \log (x)}{20 a^8} \]
((a*(-10*a^6 + 14*a^5*b*x^(1/3) - 21*a^4*b^2*x^(2/3) + 35*a^3*b^3*x - 70*a ^2*b^4*x^(4/3) + 210*a*b^5*x^(5/3) + 420*b^6*x^2))/((a + b*x^(1/3))*x^2) - 420*b^6*Log[a + b*x^(1/3)] + 140*b^6*Log[x])/(20*a^8)
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a+b \sqrt [3]{x}\right )^2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^{7/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 3 \int \left (-\frac {7 b^7}{a^8 \left (a+b \sqrt [3]{x}\right )}-\frac {b^7}{a^7 \left (a+b \sqrt [3]{x}\right )^2}+\frac {7 b^6}{a^8 \sqrt [3]{x}}-\frac {6 b^5}{a^7 x^{2/3}}+\frac {5 b^4}{a^6 x}-\frac {4 b^3}{a^5 x^{4/3}}+\frac {3 b^2}{a^4 x^{5/3}}-\frac {2 b}{a^3 x^2}+\frac {1}{a^2 x^{7/3}}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {7 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^8}+\frac {7 b^6 \log \left (\sqrt [3]{x}\right )}{a^8}+\frac {b^6}{a^7 \left (a+b \sqrt [3]{x}\right )}+\frac {6 b^5}{a^7 \sqrt [3]{x}}-\frac {5 b^4}{2 a^6 x^{2/3}}+\frac {4 b^3}{3 a^5 x}-\frac {3 b^2}{4 a^4 x^{4/3}}+\frac {2 b}{5 a^3 x^{5/3}}-\frac {1}{6 a^2 x^2}\right )\) |
3*(b^6/(a^7*(a + b*x^(1/3))) - 1/(6*a^2*x^2) + (2*b)/(5*a^3*x^(5/3)) - (3* b^2)/(4*a^4*x^(4/3)) + (4*b^3)/(3*a^5*x) - (5*b^4)/(2*a^6*x^(2/3)) + (6*b^ 5)/(a^7*x^(1/3)) - (7*b^6*Log[a + b*x^(1/3)])/a^8 + (7*b^6*Log[x^(1/3)])/a ^8)
3.24.71.3.1 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 = 6.01 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {3 b^{6}}{a^{7} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {1}{2 a^{2} x^{2}}+\frac {6 b}{5 a^{3} x^{\frac {5}{3}}}-\frac {9 b^{2}}{4 a^{4} x^{\frac {4}{3}}}+\frac {4 b^{3}}{a^{5} x}-\frac {15 b^{4}}{2 a^{6} x^{\frac {2}{3}}}+\frac {18 b^{5}}{a^{7} x^{\frac {1}{3}}}-\frac {21 b^{6} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{8}}+\frac {7 b^{6} \ln \left (x \right )}{a^{8}}\) | \(106\) |
default | \(\frac {3 b^{6}}{a^{7} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {1}{2 a^{2} x^{2}}+\frac {6 b}{5 a^{3} x^{\frac {5}{3}}}-\frac {9 b^{2}}{4 a^{4} x^{\frac {4}{3}}}+\frac {4 b^{3}}{a^{5} x}-\frac {15 b^{4}}{2 a^{6} x^{\frac {2}{3}}}+\frac {18 b^{5}}{a^{7} x^{\frac {1}{3}}}-\frac {21 b^{6} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{8}}+\frac {7 b^{6} \ln \left (x \right )}{a^{8}}\) | \(106\) |
3*b^6/a^7/(a+b*x^(1/3))-1/2/a^2/x^2+6/5*b/a^3/x^(5/3)-9/4*b^2/a^4/x^(4/3)+ 4*b^3/a^5/x-15/2*b^4/a^6/x^(2/3)+18*b^5/a^7/x^(1/3)-21*b^6*ln(a+b*x^(1/3)) /a^8+7*b^6*ln(x)/a^8
Time = 0.30 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=\frac {140 \, a^{3} b^{6} x^{2} + 70 \, a^{6} b^{3} x - 10 \, a^{9} - 420 \, {\left (b^{9} x^{3} + a^{3} b^{6} x^{2}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + 420 \, {\left (b^{9} x^{3} + a^{3} b^{6} x^{2}\right )} \log \left (x^{\frac {1}{3}}\right ) + 15 \, {\left (28 \, a b^{8} x^{2} + 21 \, a^{4} b^{5} x - 3 \, a^{7} b^{2}\right )} x^{\frac {2}{3}} - 6 \, {\left (35 \, a^{2} b^{7} x^{2} + 21 \, a^{5} b^{4} x - 4 \, a^{8} b\right )} x^{\frac {1}{3}}}{20 \, {\left (a^{8} b^{3} x^{3} + a^{11} x^{2}\right )}} \]
1/20*(140*a^3*b^6*x^2 + 70*a^6*b^3*x - 10*a^9 - 420*(b^9*x^3 + a^3*b^6*x^2 )*log(b*x^(1/3) + a) + 420*(b^9*x^3 + a^3*b^6*x^2)*log(x^(1/3)) + 15*(28*a *b^8*x^2 + 21*a^4*b^5*x - 3*a^7*b^2)*x^(2/3) - 6*(35*a^2*b^7*x^2 + 21*a^5* b^4*x - 4*a^8*b)*x^(1/3))/(a^8*b^3*x^3 + a^11*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (124) = 248\).
Time = 2.52 (sec) , antiderivative size = 405, normalized size of antiderivative = 3.24 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {8}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{2 a^{2} x^{2}} & \text {for}\: b = 0 \\- \frac {3}{8 b^{2} x^{\frac {8}{3}}} & \text {for}\: a = 0 \\- \frac {10 a^{7} x^{\frac {2}{3}}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} + \frac {14 a^{6} b x}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} - \frac {21 a^{5} b^{2} x^{\frac {4}{3}}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} + \frac {35 a^{4} b^{3} x^{\frac {5}{3}}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} - \frac {70 a^{3} b^{4} x^{2}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} + \frac {210 a^{2} b^{5} x^{\frac {7}{3}}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} + \frac {140 a b^{6} x^{\frac {8}{3}} \log {\left (x \right )}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} - \frac {420 a b^{6} x^{\frac {8}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} + \frac {420 a b^{6} x^{\frac {8}{3}}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} + \frac {140 b^{7} x^{3} \log {\left (x \right )}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} - \frac {420 b^{7} x^{3} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} & \text {otherwise} \end {cases} \]
Piecewise((zoo/x**(8/3), Eq(a, 0) & Eq(b, 0)), (-1/(2*a**2*x**2), Eq(b, 0) ), (-3/(8*b**2*x**(8/3)), Eq(a, 0)), (-10*a**7*x**(2/3)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 14*a**6*b*x/(20*a**9*x**(8/3) + 20*a**8*b*x**3) - 21*a **5*b**2*x**(4/3)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 35*a**4*b**3*x**(5 /3)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) - 70*a**3*b**4*x**2/(20*a**9*x**(8 /3) + 20*a**8*b*x**3) + 210*a**2*b**5*x**(7/3)/(20*a**9*x**(8/3) + 20*a**8 *b*x**3) + 140*a*b**6*x**(8/3)*log(x)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) - 420*a*b**6*x**(8/3)*log(a/b + x**(1/3))/(20*a**9*x**(8/3) + 20*a**8*b*x* *3) + 420*a*b**6*x**(8/3)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 140*b**7*x **3*log(x)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) - 420*b**7*x**3*log(a/b + x **(1/3))/(20*a**9*x**(8/3) + 20*a**8*b*x**3), True))
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=\frac {420 \, b^{6} x^{2} + 210 \, a b^{5} x^{\frac {5}{3}} - 70 \, a^{2} b^{4} x^{\frac {4}{3}} + 35 \, a^{3} b^{3} x - 21 \, a^{4} b^{2} x^{\frac {2}{3}} + 14 \, a^{5} b x^{\frac {1}{3}} - 10 \, a^{6}}{20 \, {\left (a^{7} b x^{\frac {7}{3}} + a^{8} x^{2}\right )}} - \frac {21 \, b^{6} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{8}} + \frac {7 \, b^{6} \log \left (x\right )}{a^{8}} \]
1/20*(420*b^6*x^2 + 210*a*b^5*x^(5/3) - 70*a^2*b^4*x^(4/3) + 35*a^3*b^3*x - 21*a^4*b^2*x^(2/3) + 14*a^5*b*x^(1/3) - 10*a^6)/(a^7*b*x^(7/3) + a^8*x^2 ) - 21*b^6*log(b*x^(1/3) + a)/a^8 + 7*b^6*log(x)/a^8
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=-\frac {21 \, b^{6} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{8}} + \frac {7 \, b^{6} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac {420 \, a b^{6} x^{2} + 210 \, a^{2} b^{5} x^{\frac {5}{3}} - 70 \, a^{3} b^{4} x^{\frac {4}{3}} + 35 \, a^{4} b^{3} x - 21 \, a^{5} b^{2} x^{\frac {2}{3}} + 14 \, a^{6} b x^{\frac {1}{3}} - 10 \, a^{7}}{20 \, {\left (b x^{\frac {1}{3}} + a\right )} a^{8} x^{2}} \]
-21*b^6*log(abs(b*x^(1/3) + a))/a^8 + 7*b^6*log(abs(x))/a^8 + 1/20*(420*a* b^6*x^2 + 210*a^2*b^5*x^(5/3) - 70*a^3*b^4*x^(4/3) + 35*a^4*b^3*x - 21*a^5 *b^2*x^(2/3) + 14*a^6*b*x^(1/3) - 10*a^7)/((b*x^(1/3) + a)*a^8*x^2)
Time = 5.89 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=\frac {\frac {7\,b\,x^{1/3}}{10\,a^2}-\frac {1}{2\,a}+\frac {7\,b^3\,x}{4\,a^4}-\frac {21\,b^2\,x^{2/3}}{20\,a^3}+\frac {21\,b^6\,x^2}{a^7}-\frac {7\,b^4\,x^{4/3}}{2\,a^5}+\frac {21\,b^5\,x^{5/3}}{2\,a^6}}{a\,x^2+b\,x^{7/3}}-\frac {42\,b^6\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a^8} \]