3.2368 \(\int \frac{1}{(a+b \sqrt [3]{x})^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{3 a^2}{b^3 \left (a+b \sqrt [3]{x}\right )}-\frac{6 a \log \left (a+b \sqrt [3]{x}\right )}{b^3}+\frac{3 \sqrt [3]{x}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0259325, antiderivative size = 46, 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, 43} \[ -\frac{3 a^2}{b^3 \left (a+b \sqrt [3]{x}\right )}-\frac{6 a \log \left (a+b \sqrt [3]{x}\right )}{b^3}+\frac{3 \sqrt [3]{x}}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^(-2),x]

[Out]

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

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 43

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])

Rubi steps

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

Mathematica [A]  time = 0.0306636, size = 42, normalized size = 0.91 \[ \frac{3 \left (-\frac{a^2}{a+b \sqrt [3]{x}}-2 a \log \left (a+b \sqrt [3]{x}\right )+b \sqrt [3]{x}\right )}{b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^(-2),x]

[Out]

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

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Maple [B]  time = 0.042, size = 160, normalized size = 3.5 \begin{align*} -3\,{\frac{{a}^{4}}{ \left ({b}^{3}x+{a}^{3} \right ){b}^{3}}}+3\,{\frac{\sqrt [3]{x}}{{b}^{2}}}-{\frac{{a}^{2}}{{b}^{2}}\sqrt [3]{x} \left ({b}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{a}^{2} \right ) ^{-1}}+2\,{\frac{{a}^{3}}{{b}^{3} \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) }}+2\,{\frac{a\ln \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) }{{b}^{3}}}-4\,{\frac{a\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{3}}}-2\,{\frac{{a}^{2}}{{b}^{3} \left ( a+b\sqrt [3]{x} \right ) }}-2\,{\frac{a\ln \left ({b}^{3}x+{a}^{3} \right ) }{{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.981196, size = 59, normalized size = 1.28 \begin{align*} -\frac{6 \, a \log \left (b x^{\frac{1}{3}} + a\right )}{b^{3}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}}{b^{3}} - \frac{3 \, a^{2}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.48487, size = 153, normalized size = 3.33 \begin{align*} -\frac{3 \,{\left (a^{2} b^{2} x^{\frac{2}{3}} + a^{4} + 2 \,{\left (a b^{3} x + a^{4}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) -{\left (b^{4} x + 2 \, a^{3} b\right )} x^{\frac{1}{3}}\right )}}{b^{6} x + a^{3} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.414846, size = 109, normalized size = 2.37 \begin{align*} \begin{cases} - \frac{6 a^{2} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a b^{3} + b^{4} \sqrt [3]{x}} - \frac{6 a^{2}}{a b^{3} + b^{4} \sqrt [3]{x}} - \frac{6 a b \sqrt [3]{x} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a b^{3} + b^{4} \sqrt [3]{x}} + \frac{3 b^{2} x^{\frac{2}{3}}}{a b^{3} + b^{4} \sqrt [3]{x}} & \text{for}\: b \neq 0 \\\frac{x}{a^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20783, size = 55, normalized size = 1.2 \begin{align*} -\frac{6 \, a \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{3}} + \frac{3 \, x^{\frac{1}{3}}}{b^{2}} - \frac{3 \, a^{2}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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