3.25.31 \(\int \frac {1}{(a+\frac {b}{\sqrt [3]{x}})^2} \, dx\) [2431]

Optimal. Leaf size=77 \[ \frac {3 b^3}{a^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {9 b^2 \sqrt [3]{x}}{a^4}-\frac {3 b x^{2/3}}{a^3}+\frac {x}{a^2}-\frac {12 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^5}-\frac {4 b^3 \log (x)}{a^5} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {196, 46} \begin {gather*} -\frac {12 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^5}-\frac {4 b^3 \log (x)}{a^5}+\frac {3 b^3}{a^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {9 b^2 \sqrt [3]{x}}{a^4}-\frac {3 b x^{2/3}}{a^3}+\frac {x}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^(-2),x]

[Out]

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

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

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Mathematica [A]
time = 0.05, size = 80, normalized size = 1.04 \begin {gather*} \frac {-3 b^4+9 a b^3 \sqrt [3]{x}+6 a^2 b^2 x^{2/3}-2 a^3 b x+a^4 x^{4/3}}{a^5 \left (b+a \sqrt [3]{x}\right )}-\frac {12 b^3 \log \left (b+a \sqrt [3]{x}\right )}{a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^(-2),x]

[Out]

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

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Maple [A]
time = 0.20, size = 62, normalized size = 0.81

method	result	size
derivativedivides	\(\frac {a^{2} x -3 a b \,x^{\frac {2}{3}}+9 b^{2} x^{\frac {1}{3}}}{a^{4}}-\frac {3 b^{4}}{a^{5} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {12 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{5}}\)	\(62\)
default	\(\frac {a^{2} x -3 a b \,x^{\frac {2}{3}}+9 b^{2} x^{\frac {1}{3}}}{a^{4}}-\frac {3 b^{4}}{a^{5} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {12 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{5}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^(1/3))^2,x,method=_RETURNVERBOSE)

[Out]

3/a^4*(1/3*a^2*x-a*b*x^(2/3)+3*b^2*x^(1/3))-3*b^4/a^5/(b+a*x^(1/3))-12/a^5*b^3*ln(b+a*x^(1/3))

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Maxima [A]
time = 0.30, size = 76, normalized size = 0.99 \begin {gather*} \frac {a^{3} - \frac {2 \, a^{2} b}{x^{\frac {1}{3}}} + \frac {6 \, a b^{2}}{x^{\frac {2}{3}}} + \frac {12 \, b^{3}}{x}}{\frac {a^{5}}{x} + \frac {a^{4} b}{x^{\frac {4}{3}}}} - \frac {12 \, b^{3} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{5}} - \frac {4 \, b^{3} \log \left (x\right )}{a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))^2,x, algorithm="maxima")

[Out]

(a^3 - 2*a^2*b/x^(1/3) + 6*a*b^2/x^(2/3) + 12*b^3/x)/(a^5/x + a^4*b/x^(4/3)) - 12*b^3*log(a + b/x^(1/3))/a^5 -
4*b^3*log(x)/a^5

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Fricas [A]
time = 0.38, size = 100, normalized size = 1.30 \begin {gather*} \frac {a^{6} x^{2} + a^{3} b^{3} x - 3 \, b^{6} - 12 \, {\left (a^{3} b^{3} x + b^{6}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 3 \, {\left (a^{5} b x + 2 \, a^{2} b^{4}\right )} x^{\frac {2}{3}} + 3 \, {\left (3 \, a^{4} b^{2} x + 4 \, a b^{5}\right )} x^{\frac {1}{3}}}{a^{8} x + a^{5} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))^2,x, algorithm="fricas")

[Out]

(a^6*x^2 + a^3*b^3*x - 3*b^6 - 12*(a^3*b^3*x + b^6)*log(a*x^(1/3) + b) - 3*(a^5*b*x + 2*a^2*b^4)*x^(2/3) + 3*(
3*a^4*b^2*x + 4*a*b^5)*x^(1/3))/(a^8*x + a^5*b^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (76) = 152\).
time = 0.23, size = 165, normalized size = 2.14 \begin {gather*} \begin {cases} \frac {a^{4} x^{\frac {4}{3}}}{a^{6} \sqrt [3]{x} + a^{5} b} - \frac {2 a^{3} b x}{a^{6} \sqrt [3]{x} + a^{5} b} + \frac {6 a^{2} b^{2} x^{\frac {2}{3}}}{a^{6} \sqrt [3]{x} + a^{5} b} - \frac {12 a b^{3} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a^{6} \sqrt [3]{x} + a^{5} b} - \frac {12 b^{4} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a^{6} \sqrt [3]{x} + a^{5} b} - \frac {12 b^{4}}{a^{6} \sqrt [3]{x} + a^{5} b} & \text {for}\: a \neq 0 \\\frac {3 x^{\frac {5}{3}}}{5 b^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**(1/3))**2,x)

[Out]

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

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Giac [A]
time = 0.53, size = 65, normalized size = 0.84 \begin {gather*} -\frac {12 \, b^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{5}} - \frac {3 \, b^{4}}{{\left (a x^{\frac {1}{3}} + b\right )} a^{5}} + \frac {a^{4} x - 3 \, a^{3} b x^{\frac {2}{3}} + 9 \, a^{2} b^{2} x^{\frac {1}{3}}}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))^2,x, algorithm="giac")

[Out]

-12*b^3*log(abs(a*x^(1/3) + b))/a^5 - 3*b^4/((a*x^(1/3) + b)*a^5) + (a^4*x - 3*a^3*b*x^(2/3) + 9*a^2*b^2*x^(1/
3))/a^6

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Mupad [B]
time = 1.13, size = 65, normalized size = 0.84 \begin {gather*} \frac {x}{a^2}-\frac {3\,b^4}{a\,\left (a^4\,b+a^5\,x^{1/3}\right )}-\frac {3\,b\,x^{2/3}}{a^3}-\frac {12\,b^3\,\ln \left (b+a\,x^{1/3}\right )}{a^5}+\frac {9\,b^2\,x^{1/3}}{a^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/x^(1/3))^2,x)

[Out]

x/a^2 - (3*b^4)/(a*(a^4*b + a^5*x^(1/3))) - (3*b*x^(2/3))/a^3 - (12*b^3*log(b + a*x^(1/3)))/a^5 + (9*b^2*x^(1/
3))/a^4

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