3.687 \(\int \frac{1}{x^{2/3} (a+b x)^2} \, dx\)

Optimal. Leaf size=113 \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{5/3} \sqrt [3]{b}}-\frac{\log (a+b x)}{3 a^{5/3} \sqrt [3]{b}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{\sqrt [3]{x}}{a (a+b x)} \]

[Out]

x^(1/3)/(a*(a + b*x)) - (2*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)*b^(1/3))
+ Log[a^(1/3) + b^(1/3)*x^(1/3)]/(a^(5/3)*b^(1/3)) - Log[a + b*x]/(3*a^(5/3)*b^(1/3))

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Rubi [A]  time = 0.0404265, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {51, 58, 617, 204, 31} \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{5/3} \sqrt [3]{b}}-\frac{\log (a+b x)}{3 a^{5/3} \sqrt [3]{b}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{\sqrt [3]{x}}{a (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(2/3)*(a + b*x)^2),x]

[Out]

x^(1/3)/(a*(a + b*x)) - (2*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)*b^(1/3))
+ Log[a^(1/3) + b^(1/3)*x^(1/3)]/(a^(5/3)*b^(1/3)) - Log[a + b*x]/(3*a^(5/3)*b^(1/3))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, -Sim
p[Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d
*x)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x
] && NegQ[(b*c - a*d)/b]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [C]  time = 0.0046421, size = 25, normalized size = 0.22 \[ \frac{3 \sqrt [3]{x} \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};-\frac{b x}{a}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(2/3)*(a + b*x)^2),x]

[Out]

(3*x^(1/3)*Hypergeometric2F1[1/3, 2, 4/3, -((b*x)/a)])/a^2

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Maple [A]  time = 0.007, size = 120, normalized size = 1.1 \begin{align*}{\frac{1}{a \left ( bx+a \right ) }\sqrt [3]{x}}+{\frac{2}{3\,ab}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{3\,ab}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{{\frac{a}{b}}}\sqrt [3]{x}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}}{3\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.33606, size = 991, normalized size = 8.77 \begin{align*} \left [\frac{3 \, a^{2} b x^{\frac{1}{3}} + 3 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x + a^{2} b\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x - a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{\frac{2}{3}} - \left (a^{2} b\right )^{\frac{1}{3}} a + \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac{1}{3}} a x^{\frac{1}{3}}}{b x + a}\right ) - \left (a^{2} b\right )^{\frac{2}{3}}{\left (b x + a\right )} \log \left (a b x^{\frac{2}{3}} + \left (a^{2} b\right )^{\frac{1}{3}} a - \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right ) + 2 \, \left (a^{2} b\right )^{\frac{2}{3}}{\left (b x + a\right )} \log \left (a b x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{3 \,{\left (a^{3} b^{2} x + a^{4} b\right )}}, \frac{3 \, a^{2} b x^{\frac{1}{3}} + 6 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x + a^{2} b\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (\left (a^{2} b\right )^{\frac{1}{3}} a - 2 \, \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) - \left (a^{2} b\right )^{\frac{2}{3}}{\left (b x + a\right )} \log \left (a b x^{\frac{2}{3}} + \left (a^{2} b\right )^{\frac{1}{3}} a - \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right ) + 2 \, \left (a^{2} b\right )^{\frac{2}{3}}{\left (b x + a\right )} \log \left (a b x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{3 \,{\left (a^{3} b^{2} x + a^{4} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/3*(3*a^2*b*x^(1/3) + 3*sqrt(1/3)*(a*b^2*x + a^2*b)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x - a^2 + 3*sqrt(1/3)*
(2*a*b*x^(2/3) - (a^2*b)^(1/3)*a + (a^2*b)^(2/3)*x^(1/3))*sqrt(-(a^2*b)^(1/3)/b) - 3*(a^2*b)^(1/3)*a*x^(1/3))/
(b*x + a)) - (a^2*b)^(2/3)*(b*x + a)*log(a*b*x^(2/3) + (a^2*b)^(1/3)*a - (a^2*b)^(2/3)*x^(1/3)) + 2*(a^2*b)^(2
/3)*(b*x + a)*log(a*b*x^(1/3) + (a^2*b)^(2/3)))/(a^3*b^2*x + a^4*b), 1/3*(3*a^2*b*x^(1/3) + 6*sqrt(1/3)*(a*b^2
*x + a^2*b)*sqrt((a^2*b)^(1/3)/b)*arctan(-sqrt(1/3)*((a^2*b)^(1/3)*a - 2*(a^2*b)^(2/3)*x^(1/3))*sqrt((a^2*b)^(
1/3)/b)/a^2) - (a^2*b)^(2/3)*(b*x + a)*log(a*b*x^(2/3) + (a^2*b)^(1/3)*a - (a^2*b)^(2/3)*x^(1/3)) + 2*(a^2*b)^
(2/3)*(b*x + a)*log(a*b*x^(1/3) + (a^2*b)^(2/3)))/(a^3*b^2*x + a^4*b)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.08328, size = 178, normalized size = 1.58 \begin{align*} -\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} + \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2} b} + \frac{x^{\frac{1}{3}}}{{\left (b x + a\right )} a} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \log \left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{3 \, a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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