3.2445 \(\int \frac{1}{1+\frac{b}{\sqrt [3]{x}}} \, dx\)

Optimal. Leaf size=44 \[ 3 b^2 \sqrt [3]{x}-3 b^3 \log \left (\frac{b}{\sqrt [3]{x}}+1\right )-b^3 \log (x)-\frac{3}{2} b x^{2/3}+x \]

[Out]

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

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Rubi [A]  time = 0.0238145, antiderivative size = 44, normalized size of antiderivative = 1., 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 = {190, 44} \[ 3 b^2 \sqrt [3]{x}-3 b^3 \log \left (\frac{b}{\sqrt [3]{x}}+1\right )-b^3 \log (x)-\frac{3}{2} b x^{2/3}+x \]

Antiderivative was successfully verified.

[In]

Int[(1 + b/x^(1/3))^(-1),x]

[Out]

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

Rule 190

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]

Rule 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0175819, size = 35, normalized size = 0.8 \[ 3 b^2 \sqrt [3]{x}-3 b^3 \log \left (b+\sqrt [3]{x}\right )-\frac{3}{2} b x^{2/3}+x \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + b/x^(1/3))^(-1),x]

[Out]

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

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Maple [A]  time = 0.002, size = 28, normalized size = 0.6 \begin{align*} x-{\frac{3\,b}{2}{x}^{{\frac{2}{3}}}}+3\,{b}^{2}\sqrt [3]{x}-3\,{b}^{3}\ln \left ( \sqrt [3]{x}+b \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+b/x^(1/3)),x)

[Out]

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

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Maxima [A]  time = 1.06248, size = 54, normalized size = 1.23 \begin{align*} -b^{3} \log \left (x\right ) - 3 \, b^{3} \log \left (\frac{b}{x^{\frac{1}{3}}} + 1\right ) + \frac{1}{2} \,{\left (\frac{6 \, b^{2}}{x^{\frac{2}{3}}} - \frac{3 \, b}{x^{\frac{1}{3}}} + 2\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.52849, size = 82, normalized size = 1.86 \begin{align*} -3 \, b^{3} \log \left (b + x^{\frac{1}{3}}\right ) + 3 \, b^{2} x^{\frac{1}{3}} - \frac{3}{2} \, b x^{\frac{2}{3}} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.172669, size = 34, normalized size = 0.77 \begin{align*} - 3 b^{3} \log{\left (b + \sqrt [3]{x} \right )} + 3 b^{2} \sqrt [3]{x} - \frac{3 b x^{\frac{2}{3}}}{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19857, size = 38, normalized size = 0.86 \begin{align*} -3 \, b^{3} \log \left ({\left | b + x^{\frac{1}{3}} \right |}\right ) + 3 \, b^{2} x^{\frac{1}{3}} - \frac{3}{2} \, b x^{\frac{2}{3}} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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