Integrand size = 15, antiderivative size = 83 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3} \, dx=-\frac {3}{5 b x^{5/3}}+\frac {3 a}{4 b^2 x^{4/3}}-\frac {a^2}{b^3 x}+\frac {3 a^3}{2 b^4 x^{2/3}}-\frac {3 a^4}{b^5 \sqrt [3]{x}}+\frac {3 a^5 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{b^6} \] Output:
-3/5/b/x^(5/3)+3/4*a/b^2/x^(4/3)-a^2/b^3/x+3/2*a^3/b^4/x^(2/3)-3*a^4/b^5/x ^(1/3)+3*a^5*ln(a+b/x^(1/3))/b^6
Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3} \, dx=\frac {\frac {b \left (-12 b^4+15 a b^3 \sqrt [3]{x}-20 a^2 b^2 x^{2/3}+30 a^3 b x-60 a^4 x^{4/3}\right )}{x^{5/3}}+60 a^5 \log \left (b+a \sqrt [3]{x}\right )-20 a^5 \log (x)}{20 b^6} \] Input:
Integrate[1/((a + b/x^(1/3))*x^3),x]
Output:
((b*(-12*b^4 + 15*a*b^3*x^(1/3) - 20*a^2*b^2*x^(2/3) + 30*a^3*b*x - 60*a^4 *x^(4/3)))/x^(5/3) + 60*a^5*Log[b + a*x^(1/3)] - 20*a^5*Log[x])/(20*b^6)
Time = 0.37 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.20, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a+\frac {b}{\sqrt [3]{x}}\right )} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {1}{x^{8/3} \left (a \sqrt [3]{x}+b\right )}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {1}{\left (\sqrt [3]{x} a+b\right ) x^2}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 3 \int \left (\frac {a^6}{b^6 \left (\sqrt [3]{x} a+b\right )}-\frac {a^5}{b^6 \sqrt [3]{x}}+\frac {a^4}{b^5 x^{2/3}}-\frac {a^3}{b^4 x}+\frac {a^2}{b^3 x^{4/3}}-\frac {a}{b^2 x^{5/3}}+\frac {1}{b x^2}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {a^5 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac {a^5 \log \left (\sqrt [3]{x}\right )}{b^6}-\frac {a^4}{b^5 \sqrt [3]{x}}+\frac {a^3}{2 b^4 x^{2/3}}-\frac {a^2}{3 b^3 x}+\frac {a}{4 b^2 x^{4/3}}-\frac {1}{5 b x^{5/3}}\right )\) |
Input:
Int[1/((a + b/x^(1/3))*x^3),x]
Output:
3*(-1/5*1/(b*x^(5/3)) + a/(4*b^2*x^(4/3)) - a^2/(3*b^3*x) + a^3/(2*b^4*x^( 2/3)) - a^4/(b^5*x^(1/3)) + (a^5*Log[b + a*x^(1/3)])/b^6 - (a^5*Log[x^(1/3 )])/b^6)
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 = 0.44 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {3 a^{5} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{6}}-\frac {3}{5 b \,x^{\frac {5}{3}}}-\frac {a^{2}}{b^{3} x}-\frac {3 a^{4}}{b^{5} x^{\frac {1}{3}}}+\frac {3 a}{4 b^{2} x^{\frac {4}{3}}}+\frac {3 a^{3}}{2 b^{4} x^{\frac {2}{3}}}-\frac {a^{5} \ln \left (x \right )}{b^{6}}\) | \(78\) |
default | \(\frac {3 a^{5} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{6}}-\frac {3}{5 b \,x^{\frac {5}{3}}}-\frac {a^{2}}{b^{3} x}-\frac {3 a^{4}}{b^{5} x^{\frac {1}{3}}}+\frac {3 a}{4 b^{2} x^{\frac {4}{3}}}+\frac {3 a^{3}}{2 b^{4} x^{\frac {2}{3}}}-\frac {a^{5} \ln \left (x \right )}{b^{6}}\) | \(78\) |
Input:
int(1/(a+b/x^(1/3))/x^3,x,method=_RETURNVERBOSE)
Output:
3*a^5/b^6*ln(b+a*x^(1/3))-3/5/b/x^(5/3)-a^2/b^3/x-3*a^4/b^5/x^(1/3)+3/4*a/ b^2/x^(4/3)+3/2*a^3/b^4/x^(2/3)-a^5/b^6*ln(x)
Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3} \, dx=\frac {60 \, a^{5} x^{2} \log \left (a x^{\frac {1}{3}} + b\right ) - 60 \, a^{5} x^{2} \log \left (x^{\frac {1}{3}}\right ) - 20 \, a^{2} b^{3} x - 15 \, {\left (4 \, a^{4} b x - a b^{4}\right )} x^{\frac {2}{3}} + 6 \, {\left (5 \, a^{3} b^{2} x - 2 \, b^{5}\right )} x^{\frac {1}{3}}}{20 \, b^{6} x^{2}} \] Input:
integrate(1/(a+b/x^(1/3))/x^3,x, algorithm="fricas")
Output:
1/20*(60*a^5*x^2*log(a*x^(1/3) + b) - 60*a^5*x^2*log(x^(1/3)) - 20*a^2*b^3 *x - 15*(4*a^4*b*x - a*b^4)*x^(2/3) + 6*(5*a^3*b^2*x - 2*b^5)*x^(1/3))/(b^ 6*x^2)
Time = 0.99 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{5 b x^{\frac {5}{3}}} & \text {for}\: a = 0 \\- \frac {1}{2 a x^{2}} & \text {for}\: b = 0 \\- \frac {a^{5} \log {\left (x \right )}}{b^{6}} + \frac {3 a^{5} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{b^{6}} - \frac {3 a^{4}}{b^{5} \sqrt [3]{x}} + \frac {3 a^{3}}{2 b^{4} x^{\frac {2}{3}}} - \frac {a^{2}}{b^{3} x} + \frac {3 a}{4 b^{2} x^{\frac {4}{3}}} - \frac {3}{5 b x^{\frac {5}{3}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x**(1/3))/x**3,x)
Output:
Piecewise((zoo/x**(5/3), Eq(a, 0) & Eq(b, 0)), (-3/(5*b*x**(5/3)), Eq(a, 0 )), (-1/(2*a*x**2), Eq(b, 0)), (-a**5*log(x)/b**6 + 3*a**5*log(x**(1/3) + b/a)/b**6 - 3*a**4/(b**5*x**(1/3)) + 3*a**3/(2*b**4*x**(2/3)) - a**2/(b**3 *x) + 3*a/(4*b**2*x**(4/3)) - 3/(5*b*x**(5/3)), True))
Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3} \, dx=\frac {3 \, a^{5} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{6}} - \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5}}{5 \, b^{6}} + \frac {15 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a}{4 \, b^{6}} - \frac {10 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{2}}{b^{6}} + \frac {15 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{3}}{b^{6}} - \frac {15 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{4}}{b^{6}} \] Input:
integrate(1/(a+b/x^(1/3))/x^3,x, algorithm="maxima")
Output:
3*a^5*log(a + b/x^(1/3))/b^6 - 3/5*(a + b/x^(1/3))^5/b^6 + 15/4*(a + b/x^( 1/3))^4*a/b^6 - 10*(a + b/x^(1/3))^3*a^2/b^6 + 15*(a + b/x^(1/3))^2*a^3/b^ 6 - 15*(a + b/x^(1/3))*a^4/b^6
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3} \, dx=\frac {3 \, a^{5} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{6}} - \frac {a^{5} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac {60 \, a^{4} b x^{\frac {4}{3}} - 30 \, a^{3} b^{2} x + 20 \, a^{2} b^{3} x^{\frac {2}{3}} - 15 \, a b^{4} x^{\frac {1}{3}} + 12 \, b^{5}}{20 \, b^{6} x^{\frac {5}{3}}} \] Input:
integrate(1/(a+b/x^(1/3))/x^3,x, algorithm="giac")
Output:
3*a^5*log(abs(a*x^(1/3) + b))/b^6 - a^5*log(abs(x))/b^6 - 1/20*(60*a^4*b*x ^(4/3) - 30*a^3*b^2*x + 20*a^2*b^3*x^(2/3) - 15*a*b^4*x^(1/3) + 12*b^5)/(b ^6*x^(5/3))
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3} \, dx=\frac {6\,a^5\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^6}-\frac {\frac {3}{5\,b}-\frac {3\,a\,x^{1/3}}{4\,b^2}-\frac {3\,a^3\,x}{2\,b^4}+\frac {a^2\,x^{2/3}}{b^3}+\frac {3\,a^4\,x^{4/3}}{b^5}}{x^{5/3}} \] Input:
int(1/(x^3*(a + b/x^(1/3))),x)
Output:
(6*a^5*atanh((2*a*x^(1/3))/b + 1))/b^6 - (3/(5*b) - (3*a*x^(1/3))/(4*b^2) - (3*a^3*x)/(2*b^4) + (a^2*x^(2/3))/b^3 + (3*a^4*x^(4/3))/b^5)/x^(5/3)
Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3} \, dx=\frac {-60 x^{\frac {5}{3}} \mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{5}+60 x^{\frac {5}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{5}-20 x^{\frac {2}{3}} a^{2} b^{3}-60 x^{\frac {4}{3}} a^{4} b +15 x^{\frac {1}{3}} a \,b^{4}+30 a^{3} b^{2} x -12 b^{5}}{20 x^{\frac {5}{3}} b^{6}} \] Input:
int(1/(a+b/x^(1/3))/x^3,x)
Output:
( - 60*x**(2/3)*log(x**(1/3))*a**5*x + 60*x**(2/3)*log(x**(1/3)*a + b)*a** 5*x - 20*x**(2/3)*a**2*b**3 - 60*x**(1/3)*a**4*b*x + 15*x**(1/3)*a*b**4 + 30*a**3*b**2*x - 12*b**5)/(20*x**(2/3)*b**6*x)