\(\int \frac {1}{(a+\frac {b}{\sqrt [3]{x}})^3 x^3} \, dx\) [2442]

Optimal result

Integrand size = 15, antiderivative size = 103 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=-\frac {3 a^3}{2 b^4 \left (b+a \sqrt [3]{x}\right )^2}-\frac {12 a^3}{b^5 \left (b+a \sqrt [3]{x}\right )}-\frac {1}{b^3 x}+\frac {9 a}{2 b^4 x^{2/3}}-\frac {18 a^2}{b^5 \sqrt [3]{x}}+\frac {30 a^3 \log \left (b+a \sqrt [3]{x}\right )}{b^6}-\frac {10 a^3 \log (x)}{b^6} \]

[Out]

-3/2*a^3/b^4/(b+a*x^(1/3))^2-12*a^3/b^5/(b+a*x^(1/3))-1/b^3/x+9/2*a/b^4/x^(2/3)-18*a^2/b^5/x^(1/3)+30*a^3*ln(b
+a*x^(1/3))/b^6-10*a^3*ln(x)/b^6

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {269, 272, 46} \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=\frac {30 a^3 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac {10 a^3 \log (x)}{b^6}-\frac {12 a^3}{b^5 \left (a \sqrt [3]{x}+b\right )}-\frac {3 a^3}{2 b^4 \left (a \sqrt [3]{x}+b\right )^2}-\frac {18 a^2}{b^5 \sqrt [3]{x}}+\frac {9 a}{2 b^4 x^{2/3}}-\frac {1}{b^3 x} \]

[In]

Int[1/((a + b/x^(1/3))^3*x^3),x]

[Out]

(-3*a^3)/(2*b^4*(b + a*x^(1/3))^2) - (12*a^3)/(b^5*(b + a*x^(1/3))) - 1/(b^3*x) + (9*a)/(2*b^4*x^(2/3)) - (18*
a^2)/(b^5*x^(1/3)) + (30*a^3*Log[b + a*x^(1/3)])/b^6 - (10*a^3*Log[x])/b^6

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 269

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]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (b+a \sqrt [3]{x}\right )^3 x^2} \, dx \\ & = 3 \text {Subst}\left (\int \frac {1}{x^4 (b+a x)^3} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {1}{b^3 x^4}-\frac {3 a}{b^4 x^3}+\frac {6 a^2}{b^5 x^2}-\frac {10 a^3}{b^6 x}+\frac {a^4}{b^4 (b+a x)^3}+\frac {4 a^4}{b^5 (b+a x)^2}+\frac {10 a^4}{b^6 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {3 a^3}{2 b^4 \left (b+a \sqrt [3]{x}\right )^2}-\frac {12 a^3}{b^5 \left (b+a \sqrt [3]{x}\right )}-\frac {1}{b^3 x}+\frac {9 a}{2 b^4 x^{2/3}}-\frac {18 a^2}{b^5 \sqrt [3]{x}}+\frac {30 a^3 \log \left (b+a \sqrt [3]{x}\right )}{b^6}-\frac {10 a^3 \log (x)}{b^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=-\frac {\frac {b \left (2 b^4-5 a b^3 \sqrt [3]{x}+20 a^2 b^2 x^{2/3}+90 a^3 b x+60 a^4 x^{4/3}\right )}{\left (b+a \sqrt [3]{x}\right )^2 x}-60 a^3 \log \left (b+a \sqrt [3]{x}\right )+20 a^3 \log (x)}{2 b^6} \]

[In]

Integrate[1/((a + b/x^(1/3))^3*x^3),x]

[Out]

-1/2*((b*(2*b^4 - 5*a*b^3*x^(1/3) + 20*a^2*b^2*x^(2/3) + 90*a^3*b*x + 60*a^4*x^(4/3)))/((b + a*x^(1/3))^2*x) -
60*a^3*Log[b + a*x^(1/3)] + 20*a^3*Log[x])/b^6

Maple [A] (verified)

Time = 3.66 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(-\frac {3 a^{3}}{2 b^{4} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}-\frac {12 a^{3}}{b^{5} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {1}{b^{3} x}+\frac {9 a}{2 b^{4} x^{\frac {2}{3}}}-\frac {18 a^{2}}{b^{5} x^{\frac {1}{3}}}+\frac {30 a^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{6}}-\frac {10 a^{3} \ln \left (x \right )}{b^{6}}\)	\(90\)
default	\(-\frac {3 a^{3}}{2 b^{4} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}-\frac {12 a^{3}}{b^{5} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {1}{b^{3} x}+\frac {9 a}{2 b^{4} x^{\frac {2}{3}}}-\frac {18 a^{2}}{b^{5} x^{\frac {1}{3}}}+\frac {30 a^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{6}}-\frac {10 a^{3} \ln \left (x \right )}{b^{6}}\)	\(90\)

[In]

int(1/(a+b/x^(1/3))^3/x^3,x,method=_RETURNVERBOSE)

[Out]

-3/2*a^3/b^4/(b+a*x^(1/3))^2-12*a^3/b^5/(b+a*x^(1/3))-1/b^3/x+9/2*a/b^4/x^(2/3)-18*a^2/b^5/x^(1/3)+30*a^3*ln(b
+a*x^(1/3))/b^6-10*a^3*ln(x)/b^6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (89) = 178\).

Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=-\frac {20 \, a^{6} b^{3} x^{2} + 31 \, a^{3} b^{6} x + 2 \, b^{9} - 60 \, {\left (a^{9} x^{3} + 2 \, a^{6} b^{3} x^{2} + a^{3} b^{6} x\right )} \log \left (a x^{\frac {1}{3}} + b\right ) + 60 \, {\left (a^{9} x^{3} + 2 \, a^{6} b^{3} x^{2} + a^{3} b^{6} x\right )} \log \left (x^{\frac {1}{3}}\right ) + 3 \, {\left (20 \, a^{8} b x^{2} + 35 \, a^{5} b^{4} x + 12 \, a^{2} b^{7}\right )} x^{\frac {2}{3}} - 3 \, {\left (10 \, a^{7} b^{2} x^{2} + 16 \, a^{4} b^{5} x + 3 \, a b^{8}\right )} x^{\frac {1}{3}}}{2 \, {\left (a^{6} b^{6} x^{3} + 2 \, a^{3} b^{9} x^{2} + b^{12} x\right )}} \]

[In]

integrate(1/(a+b/x^(1/3))^3/x^3,x, algorithm="fricas")

[Out]

-1/2*(20*a^6*b^3*x^2 + 31*a^3*b^6*x + 2*b^9 - 60*(a^9*x^3 + 2*a^6*b^3*x^2 + a^3*b^6*x)*log(a*x^(1/3) + b) + 60
*(a^9*x^3 + 2*a^6*b^3*x^2 + a^3*b^6*x)*log(x^(1/3)) + 3*(20*a^8*b*x^2 + 35*a^5*b^4*x + 12*a^2*b^7)*x^(2/3) - 3
*(10*a^7*b^2*x^2 + 16*a^4*b^5*x + 3*a*b^8)*x^(1/3))/(a^6*b^6*x^3 + 2*a^3*b^9*x^2 + b^12*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (100) = 200\).

Time = 3.56 (sec) , antiderivative size = 561, normalized size of antiderivative = 5.45 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{b^{3} x} & \text {for}\: a = 0 \\- \frac {1}{2 a^{3} x^{2}} & \text {for}\: b = 0 \\- \frac {20 a^{5} x^{\frac {10}{3}} \log {\left (x \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} + \frac {60 a^{5} x^{\frac {10}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {40 a^{4} b x^{3} \log {\left (x \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} + \frac {120 a^{4} b x^{3} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {60 a^{4} b x^{3}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {20 a^{3} b^{2} x^{\frac {8}{3}} \log {\left (x \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} + \frac {60 a^{3} b^{2} x^{\frac {8}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {90 a^{3} b^{2} x^{\frac {8}{3}}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {20 a^{2} b^{3} x^{\frac {7}{3}}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} + \frac {5 a b^{4} x^{2}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {2 b^{5} x^{\frac {5}{3}}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(a+b/x**(1/3))**3/x**3,x)

[Out]

Piecewise((zoo/x, Eq(a, 0) & Eq(b, 0)), (-1/(b**3*x), Eq(a, 0)), (-1/(2*a**3*x**2), Eq(b, 0)), (-20*a**5*x**(1
0/3)*log(x)/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)) + 60*a**5*x**(10/3)*log(x**(1/3) + b/a)/
(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)) - 40*a**4*b*x**3*log(x)/(2*a**2*b**6*x**(10/3) + 4*a
*b**7*x**3 + 2*b**8*x**(8/3)) + 120*a**4*b*x**3*log(x**(1/3) + b/a)/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2
*b**8*x**(8/3)) - 60*a**4*b*x**3/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)) - 20*a**3*b**2*x**(
8/3)*log(x)/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)) + 60*a**3*b**2*x**(8/3)*log(x**(1/3) + b
/a)/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)) - 90*a**3*b**2*x**(8/3)/(2*a**2*b**6*x**(10/3) +
4*a*b**7*x**3 + 2*b**8*x**(8/3)) - 20*a**2*b**3*x**(7/3)/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(
8/3)) + 5*a*b**4*x**2/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)) - 2*b**5*x**(5/3)/(2*a**2*b**6
*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=\frac {30 \, a^{3} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{6}} - \frac {{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3}}{b^{6}} + \frac {15 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a}{2 \, b^{6}} - \frac {30 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{2}}{b^{6}} + \frac {15 \, a^{4}}{{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} b^{6}} - \frac {3 \, a^{5}}{2 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} b^{6}} \]

[In]

integrate(1/(a+b/x^(1/3))^3/x^3,x, algorithm="maxima")

[Out]

30*a^3*log(a + b/x^(1/3))/b^6 - (a + b/x^(1/3))^3/b^6 + 15/2*(a + b/x^(1/3))^2*a/b^6 - 30*(a + b/x^(1/3))*a^2/
b^6 + 15*a^4/((a + b/x^(1/3))*b^6) - 3/2*a^5/((a + b/x^(1/3))^2*b^6)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=\frac {30 \, a^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{6}} - \frac {10 \, a^{3} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac {60 \, a^{4} b x^{\frac {4}{3}} + 90 \, a^{3} b^{2} x + 20 \, a^{2} b^{3} x^{\frac {2}{3}} - 5 \, a b^{4} x^{\frac {1}{3}} + 2 \, b^{5}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} b^{6} x} \]

[In]

integrate(1/(a+b/x^(1/3))^3/x^3,x, algorithm="giac")

[Out]

30*a^3*log(abs(a*x^(1/3) + b))/b^6 - 10*a^3*log(abs(x))/b^6 - 1/2*(60*a^4*b*x^(4/3) + 90*a^3*b^2*x + 20*a^2*b^
3*x^(2/3) - 5*a*b^4*x^(1/3) + 2*b^5)/((a*x^(1/3) + b)^2*b^6*x)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=\frac {60\,a^3\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^6}-\frac {\frac {1}{b}-\frac {5\,a\,x^{1/3}}{2\,b^2}+\frac {45\,a^3\,x}{b^4}+\frac {10\,a^2\,x^{2/3}}{b^3}+\frac {30\,a^4\,x^{4/3}}{b^5}}{b^2\,x+a^2\,x^{5/3}+2\,a\,b\,x^{4/3}} \]

[In]

int(1/(x^3*(a + b/x^(1/3))^3),x)

[Out]

(60*a^3*atanh((2*a*x^(1/3))/b + 1))/b^6 - (1/b - (5*a*x^(1/3))/(2*b^2) + (45*a^3*x)/b^4 + (10*a^2*x^(2/3))/b^3
+ (30*a^4*x^(4/3))/b^5)/(b^2*x + a^2*x^(5/3) + 2*a*b*x^(4/3))