3.25.41 \(\int \frac {1}{(a+\frac {b}{\sqrt [3]{x}})^3 x^2} \, dx\) [2441]

Optimal. Leaf size=56 \[ \frac {3}{2 b \left (b+a \sqrt [3]{x}\right )^2}+\frac {3}{b^2 \left (b+a \sqrt [3]{x}\right )}-\frac {3 \log \left (b+a \sqrt [3]{x}\right )}{b^3}+\frac {\log (x)}{b^3} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {269, 272, 46} \begin {gather*} -\frac {3 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac {3}{b^2 \left (a \sqrt [3]{x}+b\right )}+\frac {3}{2 b \left (a \sqrt [3]{x}+b\right )^2}+\frac {\log (x)}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))^3*x^2),x]

[Out]

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

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

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Mathematica [A]
time = 0.07, size = 51, normalized size = 0.91 \begin {gather*} \frac {3 \left (\frac {b \left (3 b+2 a \sqrt [3]{x}\right )}{\left (b+a \sqrt [3]{x}\right )^2}-2 \log \left (b+a \sqrt [3]{x}\right )+\frac {2 \log (x)}{3}\right )}{2 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^(1/3))^3*x^2),x]

[Out]

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

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Maple [A]
time = 0.21, size = 49, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {3}{2 b \left (b +a \,x^{\frac {1}{3}}\right )^{2}}+\frac {3}{b^{2} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {3 \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{3}}+\frac {\ln \left (x \right )}{b^{3}}\)	\(49\)
default	\(\frac {3}{2 b \left (b +a \,x^{\frac {1}{3}}\right )^{2}}+\frac {3}{b^{2} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {3 \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{3}}+\frac {\ln \left (x \right )}{b^{3}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 46, normalized size = 0.82 \begin {gather*} -\frac {3 \, \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{3}} - \frac {6 \, a}{{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} b^{3}} + \frac {3 \, a^{2}}{2 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (48) = 96\).
time = 0.39, size = 129, normalized size = 2.30 \begin {gather*} \frac {3 \, {\left (3 \, b^{6} - 2 \, {\left (a^{6} x^{2} + 2 \, a^{3} b^{3} x + b^{6}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) + 2 \, {\left (a^{6} x^{2} + 2 \, a^{3} b^{3} x + b^{6}\right )} \log \left (x^{\frac {1}{3}}\right ) + {\left (2 \, a^{5} b x + 5 \, a^{2} b^{4}\right )} x^{\frac {2}{3}} - {\left (a^{4} b^{2} x + 4 \, a b^{5}\right )} x^{\frac {1}{3}}\right )}}{2 \, {\left (a^{6} b^{3} x^{2} + 2 \, a^{3} b^{6} x + b^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (49) = 98\).
time = 1.34, size = 406, normalized size = 7.25 \begin {gather*} \begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\left (x \right )}}{b^{3}} & \text {for}\: a = 0 \\- \frac {1}{a^{3} x} & \text {for}\: b = 0 \\\frac {2 a^{2} x^{\frac {7}{3}} \log {\left (x \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} - \frac {6 a^{2} x^{\frac {7}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {4 a b x^{2} \log {\left (x \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} - \frac {12 a b x^{2} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {6 a b x^{2}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {2 b^{2} x^{\frac {5}{3}} \log {\left (x \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} - \frac {6 b^{2} x^{\frac {5}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {9 b^{2} x^{\frac {5}{3}}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0)), (log(x)/b**3, Eq(a, 0)), (-1/(a**3*x), Eq(b, 0)), (2*a**2*x**(7/3
)*log(x)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) - 6*a**2*x**(7/3)*log(x**(1/3) + b/a)/(2*a**
2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) + 4*a*b*x**2*log(x)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 +
2*b**5*x**(5/3)) - 12*a*b*x**2*log(x**(1/3) + b/a)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) +
6*a*b*x**2/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) + 2*b**2*x**(5/3)*log(x)/(2*a**2*b**3*x**
(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) - 6*b**2*x**(5/3)*log(x**(1/3) + b/a)/(2*a**2*b**3*x**(7/3) + 4*a*b**
4*x**2 + 2*b**5*x**(5/3)) + 9*b**2*x**(5/3)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)), True))

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Giac [A]
time = 1.37, size = 49, normalized size = 0.88 \begin {gather*} -\frac {3 \, \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{3}} + \frac {\log \left ({\left | x \right |}\right )}{b^{3}} + \frac {3 \, {\left (2 \, a b x^{\frac {1}{3}} + 3 \, b^{2}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.14, size = 54, normalized size = 0.96 \begin {gather*} \frac {\frac {9}{2\,b}+\frac {3\,a\,x^{1/3}}{b^2}}{b^2+a^2\,x^{2/3}+2\,a\,b\,x^{1/3}}-\frac {6\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b/x^(1/3))^3),x)

[Out]

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

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