\(\int \frac {1}{(a+b \sqrt [3]{x})^3} \, dx\) [2376]

Optimal result

Integrand size = 11, antiderivative size = 54 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=-\frac {3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac {6 a}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{b^3} \]

[Out]

-3/2*a^2/b^3/(a+b*x^(1/3))^2+6*a/b^3/(a+b*x^(1/3))+3*ln(a+b*x^(1/3))/b^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {196, 45} \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=-\frac {3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac {6 a}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{b^3} \]

[In]

Int[(a + b*x^(1/3))^(-3),x]

[Out]

(-3*a^2)/(2*b^3*(a + b*x^(1/3))^2) + (6*a)/(b^3*(a + b*x^(1/3))) + (3*Log[a + b*x^(1/3)])/b^3

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 196

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=\frac {3 a \left (3 a+4 b \sqrt [3]{x}\right )}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{b^3} \]

[In]

Integrate[(a + b*x^(1/3))^(-3),x]

[Out]

(3*a*(3*a + 4*b*x^(1/3)))/(2*b^3*(a + b*x^(1/3))^2) + (3*Log[a + b*x^(1/3)])/b^3

Maple [A] (verified)

Time = 3.78 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(-\frac {3 a^{2}}{2 b^{3} \left (a +b \,x^{\frac {1}{3}}\right )^{2}}+\frac {6 a}{b^{3} \left (a +b \,x^{\frac {1}{3}}\right )}+\frac {3 \ln \left (a +b \,x^{\frac {1}{3}}\right )}{b^{3}}\)	\(47\)
default	\(-\frac {a^{6}}{\left (b^{3} x +a^{3}\right )^{2} b^{3}}+\frac {2 a^{3}}{b^{3} \left (b^{3} x +a^{3}\right )}+\frac {\ln \left (b^{3} x +a^{3}\right )}{b^{3}}-7 a^{3} b^{3} \left (-\frac {1}{\left (b^{3} x +a^{3}\right ) b^{6}}+\frac {a^{3}}{2 b^{6} \left (b^{3} x +a^{3}\right )^{2}}\right )-3 a^{5} b \left (\frac {\ln \left (a +b \,x^{\frac {1}{3}}\right )}{9 a^{5} b^{4}}+\frac {1}{18 a^{3} b^{4} \left (a +b \,x^{\frac {1}{3}}\right )^{2}}-\frac {\frac {-a^{2} x^{\frac {2}{3}}+\frac {a^{3} x^{\frac {1}{3}}}{b}+\frac {a^{4}}{2 b^{2}}}{\left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2}}+\frac {\frac {\ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )}{2 b}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 b^{2} x^{\frac {1}{3}}-a b \right ) \sqrt {3}}{3 a b}\right )}{b}}{b}}{9 a^{5} b^{2}}\right )+6 a^{4} b^{2} \left (\frac {1}{9 a^{3} b^{5} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {\ln \left (a +b \,x^{\frac {1}{3}}\right )}{9 a^{4} b^{5}}-\frac {1}{18 b^{5} a^{2} \left (a +b \,x^{\frac {1}{3}}\right )^{2}}+\frac {\frac {2 a \,b^{2} x -\frac {5 a^{2} b \,x^{\frac {2}{3}}}{2}+a^{3} x^{\frac {1}{3}}-\frac {a^{4}}{2 b}}{\left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2}}+\frac {\ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )}{2 b}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 b^{2} x^{\frac {1}{3}}-a b \right ) \sqrt {3}}{3 a b}\right )}{b}}{9 b^{4} a^{4}}\right )+6 a^{2} b^{4} \left (-\frac {1}{18 b^{7} \left (a +b \,x^{\frac {1}{3}}\right )^{2}}+\frac {1}{3 a \,b^{7} \left (a +b \,x^{\frac {1}{3}}\right )}+\frac {2 \ln \left (a +b \,x^{\frac {1}{3}}\right )}{9 a^{2} b^{7}}-\frac {\frac {3 a b x +a^{2} x^{\frac {2}{3}}-\frac {a^{3} x^{\frac {1}{3}}}{b}+\frac {5 a^{4}}{2 b^{2}}}{\left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2}}+\frac {\frac {\ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )}{b}-\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 b^{2} x^{\frac {1}{3}}-a b \right ) \sqrt {3}}{3 a b}\right )}{b}}{b}}{9 b^{5} a^{2}}\right )-3 a \,b^{5} \left (-\frac {4}{9 b^{8} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {5 \ln \left (a +b \,x^{\frac {1}{3}}\right )}{9 a \,b^{8}}+\frac {a}{18 b^{8} \left (a +b \,x^{\frac {1}{3}}\right )^{2}}+\frac {\frac {-8 a \,b^{2} x +\frac {23 a^{2} b \,x^{\frac {2}{3}}}{2}-10 a^{3} x^{\frac {1}{3}}+\frac {7 a^{4}}{2 b}}{\left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2}}+\frac {5 \ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )}{2 b}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\left (2 b^{2} x^{\frac {1}{3}}-a b \right ) \sqrt {3}}{3 a b}\right )}{b}}{9 a \,b^{7}}\right )\)	\(787\)

[In]

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

[Out]

-3/2*a^2/b^3/(a+b*x^(1/3))^2+6*a/b^3/(a+b*x^(1/3))+3*ln(a+b*x^(1/3))/b^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (46) = 92\).

Time = 0.30 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.09 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=\frac {3 \, {\left (6 \, a^{3} b^{3} x + 3 \, a^{6} + 2 \, {\left (b^{6} x^{2} + 2 \, a^{3} b^{3} x + a^{6}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + {\left (4 \, a b^{5} x + a^{4} b^{2}\right )} x^{\frac {2}{3}} - {\left (5 \, a^{2} b^{4} x + 2 \, a^{5} b\right )} x^{\frac {1}{3}}\right )}}{2 \, {\left (b^{9} x^{2} + 2 \, a^{3} b^{6} x + a^{6} b^{3}\right )}} \]

[In]

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

[Out]

3/2*(6*a^3*b^3*x + 3*a^6 + 2*(b^6*x^2 + 2*a^3*b^3*x + a^6)*log(b*x^(1/3) + a) + (4*a*b^5*x + a^4*b^2)*x^(2/3)
- (5*a^2*b^4*x + 2*a^5*b)*x^(1/3))/(b^9*x^2 + 2*a^3*b^6*x + a^6*b^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (49) = 98\).

Time = 0.30 (sec) , antiderivative size = 228, normalized size of antiderivative = 4.22 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=\begin {cases} \frac {6 a^{2} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} + \frac {9 a^{2}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} + \frac {12 a b \sqrt [3]{x} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} + \frac {12 a b \sqrt [3]{x}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} + \frac {6 b^{2} x^{\frac {2}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} & \text {for}\: b \neq 0 \\\frac {x}{a^{3}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((6*a**2*log(a/b + x**(1/3))/(2*a**2*b**3 + 4*a*b**4*x**(1/3) + 2*b**5*x**(2/3)) + 9*a**2/(2*a**2*b**
3 + 4*a*b**4*x**(1/3) + 2*b**5*x**(2/3)) + 12*a*b*x**(1/3)*log(a/b + x**(1/3))/(2*a**2*b**3 + 4*a*b**4*x**(1/3
) + 2*b**5*x**(2/3)) + 12*a*b*x**(1/3)/(2*a**2*b**3 + 4*a*b**4*x**(1/3) + 2*b**5*x**(2/3)) + 6*b**2*x**(2/3)*l
og(a/b + x**(1/3))/(2*a**2*b**3 + 4*a*b**4*x**(1/3) + 2*b**5*x**(2/3)), Ne(b, 0)), (x/a**3, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=\frac {3 \, \log \left (b x^{\frac {1}{3}} + a\right )}{b^{3}} + \frac {6 \, a}{{\left (b x^{\frac {1}{3}} + a\right )} b^{3}} - \frac {3 \, a^{2}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{3}} \]

[In]

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

[Out]

3*log(b*x^(1/3) + a)/b^3 + 6*a/((b*x^(1/3) + a)*b^3) - 3/2*a^2/((b*x^(1/3) + a)^2*b^3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=\frac {3 \, \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{3}} + \frac {3 \, {\left (4 \, a x^{\frac {1}{3}} + \frac {3 \, a^{2}}{b}\right )}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{2}} \]

[In]

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

[Out]

3*log(abs(b*x^(1/3) + a))/b^3 + 3/2*(4*a*x^(1/3) + 3*a^2/b)/((b*x^(1/3) + a)^2*b^2)

Mupad [B] (verification not implemented)

Time = 5.90 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^3} \, dx=\frac {\frac {9\,a^2}{2\,b^3}+\frac {6\,a\,x^{1/3}}{b^2}}{a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}+\frac {3\,\ln \left (a+b\,x^{1/3}\right )}{b^3} \]

[In]

int(1/(a + b*x^(1/3))^3,x)

[Out]

((9*a^2)/(2*b^3) + (6*a*x^(1/3))/b^2)/(a^2 + b^2*x^(2/3) + 2*a*b*x^(1/3)) + (3*log(a + b*x^(1/3)))/b^3