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

Optimal. Leaf size=100 \[ -\frac{3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac{\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0284094, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {56, 617, 204, 31} \[ -\frac{3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac{\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(1/3)*(a + b*x)),x]

[Out]

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

Rule 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(
1/3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && Ne
gQ[(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}{\sqrt [3]{x} (a+b x)} \, dx &=\frac{\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}+\frac{3 \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 )}{2 b}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}\\ &=-\frac{3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac{\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{2/3}}\\ &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac{3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac{\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0054872, size = 27, normalized size = 0.27 \[ \frac{3 x^{2/3} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b x}{a}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(1/3)*(a + b*x)),x]

[Out]

(3*x^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, -((b*x)/a)])/(2*a)

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Maple [A]  time = 0.004, size = 96, normalized size = 1. \begin{align*} -{\frac{1}{b}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{2\,b}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{{\frac{a}{b}}}\sqrt [3]{x}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}}{b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.87932, size = 801, normalized size = 8.01 \begin{align*} \left [\frac{\sqrt{3} a b \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b^{2} x - a b + \sqrt{3}{\left (a b x^{\frac{1}{3}} + \left (-a b^{2}\right )^{\frac{1}{3}} a + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{\frac{2}{3}}\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{\frac{1}{3}}}{b x + a}\right ) + \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{\frac{2}{3}} + \left (-a b^{2}\right )^{\frac{1}{3}} b x^{\frac{1}{3}} + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x^{\frac{1}{3}} - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{2 \, a b^{2}}, \frac{2 \, \sqrt{3} a b \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{\sqrt{3}{\left (2 \, b x^{\frac{1}{3}} + \left (-a b^{2}\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}}}{3 \, b}\right ) + \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{\frac{2}{3}} + \left (-a b^{2}\right )^{\frac{1}{3}} b x^{\frac{1}{3}} + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x^{\frac{1}{3}} - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{2 \, a b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 9.68224, size = 218, normalized size = 2.18 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt [3]{x}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{3 x^{\frac{2}{3}}}{2 a} & \text{for}\: b = 0 \\- \frac{3}{b \sqrt [3]{x}} & \text{for}\: a = 0 \\- \frac{\left (-1\right )^{\frac{2}{3}} \log{\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} + \sqrt [3]{x} \right )}}{\sqrt [3]{a} b^{2} \left (\frac{1}{b}\right )^{\frac{4}{3}}} + \frac{\left (-1\right )^{\frac{2}{3}} \log{\left (4 \left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (\frac{1}{b}\right )^{\frac{2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{\frac{1}{b}} + 4 x^{\frac{2}{3}} \right )}}{2 \sqrt [3]{a} b^{2} \left (\frac{1}{b}\right )^{\frac{4}{3}}} - \frac{\left (-1\right )^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2 \left (-1\right )^{\frac{2}{3}} \sqrt{3} \sqrt [3]{x}}{3 \sqrt [3]{a} \sqrt [3]{\frac{1}{b}}} \right )}}{\sqrt [3]{a} b^{2} \left (\frac{1}{b}\right )^{\frac{4}{3}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(1/3), Eq(a, 0) & Eq(b, 0)), (3*x**(2/3)/(2*a), Eq(b, 0)), (-3/(b*x**(1/3)), Eq(a, 0)), (-(-
1)**(2/3)*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x**(1/3))/(a**(1/3)*b**2*(1/b)**(4/3)) + (-1)**(2/3)*log(4*
(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x**(1/3)*(1/b)**(1/3) + 4*x**(2/3))/(2*a**(1/3)*b**
2*(1/b)**(4/3)) - (-1)**(2/3)*sqrt(3)*atan(sqrt(3)/3 - 2*(-1)**(2/3)*sqrt(3)*x**(1/3)/(3*a**(1/3)*(1/b)**(1/3)
))/(a**(1/3)*b**2*(1/b)**(4/3)), True))

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Giac [A]  time = 1.09265, size = 159, normalized size = 1.59 \begin{align*} -\frac{\left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{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 )}{a b^{2}} + \frac{\left (-a b^{2}\right )^{\frac{2}{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 )}{2 \, a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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