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

Optimal. Leaf size=116 \[ -\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3} b^{2/3}}+\frac {\log (a+b x)}{6 a^{4/3} b^{2/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} b^{2/3}}+\frac {x^{2/3}}{a (a+b x)} \]

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Rubi [A]  time = 0.04, antiderivative size = 116, normalized size of antiderivative = 1.00, 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, 56, 617, 204, 31} \begin {gather*} -\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3} b^{2/3}}+\frac {\log (a+b x)}{6 a^{4/3} b^{2/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} b^{2/3}}+\frac {x^{2/3}}{a (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(1/3)*(a + b*x)^2),x]

[Out]

x^(2/3)/(a*(a + b*x)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(4/3)*b^(2/3)) - Lo
g[a^(1/3) + b^(1/3)*x^(1/3)]/(2*a^(4/3)*b^(2/3)) + Log[a + b*x]/(6*a^(4/3)*b^(2/3))

Rule 31

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

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 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 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 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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx &=\frac {x^{2/3}}{a (a+b x)}+\frac {\int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{3 a}\\ &=\frac {x^{2/3}}{a (a+b x)}+\frac {\log (a+b x)}{6 a^{4/3} b^{2/3}}+\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 )}{2 a b}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 a^{4/3} b^{2/3}}\\ &=\frac {x^{2/3}}{a (a+b x)}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3} b^{2/3}}+\frac {\log (a+b x)}{6 a^{4/3} b^{2/3}}+\frac {\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^{4/3} b^{2/3}}\\ &=\frac {x^{2/3}}{a (a+b x)}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{4/3} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3} b^{2/3}}+\frac {\log (a+b x)}{6 a^{4/3} b^{2/3}}\\ \end {align*}

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Mathematica [C]  time = 0.00, size = 27, normalized size = 0.23 \begin {gather*} \frac {3 x^{2/3} \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {b x}{a}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(1/3)*(a + b*x)^2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.16, size = 144, normalized size = 1.24 \begin {gather*} \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{6 a^{4/3} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{4/3} b^{2/3}}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} b^{2/3}}+\frac {x^{2/3}}{a (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^(2/3)/(a*(a + b*x)) - ArcTan[1/Sqrt[3] - (2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(4/3)*b^(2/3)) -
Log[a^(1/3) + b^(1/3)*x^(1/3)]/(3*a^(4/3)*b^(2/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2/3)]/
(6*a^(4/3)*b^(2/3))

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fricas [B]  time = 1.44, size = 396, normalized size = 3.41 \begin {gather*} \left [\frac {6 \, a b^{2} x^{\frac {2}{3}} + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x + a^{2} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x - a b + 3 \, \sqrt {\frac {1}{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}} {\left (b x + a\right )} \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}} {\left (b x + a\right )} \log \left (b x^{\frac {1}{3}} - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{6 \, {\left (a^{2} b^{3} x + a^{3} b^{2}\right )}}, \frac {6 \, a b^{2} x^{\frac {2}{3}} + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x + a^{2} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{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}}}{b}\right ) + \left (-a b^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )} \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}} {\left (b x + a\right )} \log \left (b x^{\frac {1}{3}} - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{6 \, {\left (a^{2} b^{3} x + a^{3} b^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.11, size = 132, normalized size = 1.14 \begin {gather*} -\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 )}{3 \, a^{2}} + \frac {x^{\frac {2}{3}}}{{\left (b x + a\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 )}{3 \, a^{2} 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 )}{6 \, a^{2} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 120, normalized size = 1.03 \begin {gather*} \frac {x^{\frac {2}{3}}}{\left (b x +a \right ) a}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.96, size = 127, normalized size = 1.09 \begin {gather*} \frac {x^{\frac {2}{3}}}{a b x + a^{2}} + \frac {\sqrt {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 b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\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 )}{6 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.36, size = 144, normalized size = 1.24 \begin {gather*} \frac {x^{2/3}}{a\,\left (a+b\,x\right )}+\frac {{\left (-1\right )}^{1/3}\,\ln \left (\frac {{\left (-1\right )}^{2/3}\,b^{2/3}}{a^{5/3}}+\frac {b\,x^{1/3}}{a^2}\right )}{3\,a^{4/3}\,b^{2/3}}-\frac {{\left (-1\right )}^{1/3}\,\ln \left (\frac {b\,x^{1/3}}{a^2}+\frac {{\left (-1\right )}^{2/3}\,b^{2/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{a^{5/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}\,b^{2/3}}+\frac {{\left (-1\right )}^{1/3}\,\ln \left (\frac {b\,x^{1/3}}{a^2}+\frac {9\,{\left (-1\right )}^{2/3}\,b^{2/3}\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2}{a^{5/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{4/3}\,b^{2/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/3)*(a + b*x)^2),x)

[Out]

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

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sympy [A]  time = 79.66, size = 774, normalized size = 6.67 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {4}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {2}{3}}}{2 a^{2}} & \text {for}\: b = 0 \\- \frac {3}{4 b^{2} x^{\frac {4}{3}}} & \text {for}\: a = 0 \\\frac {6 \sqrt [3]{-1} \sqrt [3]{a} b x^{\frac {2}{3}} \sqrt [3]{\frac {1}{b}}}{6 \sqrt [3]{-1} a^{\frac {7}{3}} b \sqrt [3]{\frac {1}{b}} + 6 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} x \sqrt [3]{\frac {1}{b}}} + \frac {2 a \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + \sqrt [3]{x} \right )}}{6 \sqrt [3]{-1} a^{\frac {7}{3}} b \sqrt [3]{\frac {1}{b}} + 6 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} x \sqrt [3]{\frac {1}{b}}} - \frac {a \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 )}}{6 \sqrt [3]{-1} a^{\frac {7}{3}} b \sqrt [3]{\frac {1}{b}} + 6 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} x \sqrt [3]{\frac {1}{b}}} + \frac {2 \sqrt {3} a \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 )}}{6 \sqrt [3]{-1} a^{\frac {7}{3}} b \sqrt [3]{\frac {1}{b}} + 6 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} x \sqrt [3]{\frac {1}{b}}} + \frac {2 a \log {\relax (2 )}}{6 \sqrt [3]{-1} a^{\frac {7}{3}} b \sqrt [3]{\frac {1}{b}} + 6 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} x \sqrt [3]{\frac {1}{b}}} + \frac {2 b x \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + \sqrt [3]{x} \right )}}{6 \sqrt [3]{-1} a^{\frac {7}{3}} b \sqrt [3]{\frac {1}{b}} + 6 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} x \sqrt [3]{\frac {1}{b}}} - \frac {b x \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 )}}{6 \sqrt [3]{-1} a^{\frac {7}{3}} b \sqrt [3]{\frac {1}{b}} + 6 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} x \sqrt [3]{\frac {1}{b}}} + \frac {2 \sqrt {3} b x \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 )}}{6 \sqrt [3]{-1} a^{\frac {7}{3}} b \sqrt [3]{\frac {1}{b}} + 6 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} x \sqrt [3]{\frac {1}{b}}} + \frac {2 b x \log {\relax (2 )}}{6 \sqrt [3]{-1} a^{\frac {7}{3}} b \sqrt [3]{\frac {1}{b}} + 6 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} x \sqrt [3]{\frac {1}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(4/3), Eq(a, 0) & Eq(b, 0)), (3*x**(2/3)/(2*a**2), Eq(b, 0)), (-3/(4*b**2*x**(4/3)), Eq(a, 0
)), (6*(-1)**(1/3)*a**(1/3)*b*x**(2/3)*(1/b)**(1/3)/(6*(-1)**(1/3)*a**(7/3)*b*(1/b)**(1/3) + 6*(-1)**(1/3)*a**
(4/3)*b**2*x*(1/b)**(1/3)) + 2*a*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x**(1/3))/(6*(-1)**(1/3)*a**(7/3)*b*
(1/b)**(1/3) + 6*(-1)**(1/3)*a**(4/3)*b**2*x*(1/b)**(1/3)) - a*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))/(6*(-1)**(1/3)*a**(7/3)*b*(1/b)**(1/3) + 6*(-1)**(1/3)*a
**(4/3)*b**2*x*(1/b)**(1/3)) + 2*sqrt(3)*a*atan(sqrt(3)/3 - 2*(-1)**(2/3)*sqrt(3)*x**(1/3)/(3*a**(1/3)*(1/b)**
(1/3)))/(6*(-1)**(1/3)*a**(7/3)*b*(1/b)**(1/3) + 6*(-1)**(1/3)*a**(4/3)*b**2*x*(1/b)**(1/3)) + 2*a*log(2)/(6*(
-1)**(1/3)*a**(7/3)*b*(1/b)**(1/3) + 6*(-1)**(1/3)*a**(4/3)*b**2*x*(1/b)**(1/3)) + 2*b*x*log(-(-1)**(1/3)*a**(
1/3)*(1/b)**(1/3) + x**(1/3))/(6*(-1)**(1/3)*a**(7/3)*b*(1/b)**(1/3) + 6*(-1)**(1/3)*a**(4/3)*b**2*x*(1/b)**(1
/3)) - b*x*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)
)/(6*(-1)**(1/3)*a**(7/3)*b*(1/b)**(1/3) + 6*(-1)**(1/3)*a**(4/3)*b**2*x*(1/b)**(1/3)) + 2*sqrt(3)*b*x*atan(sq
rt(3)/3 - 2*(-1)**(2/3)*sqrt(3)*x**(1/3)/(3*a**(1/3)*(1/b)**(1/3)))/(6*(-1)**(1/3)*a**(7/3)*b*(1/b)**(1/3) + 6
*(-1)**(1/3)*a**(4/3)*b**2*x*(1/b)**(1/3)) + 2*b*x*log(2)/(6*(-1)**(1/3)*a**(7/3)*b*(1/b)**(1/3) + 6*(-1)**(1/
3)*a**(4/3)*b**2*x*(1/b)**(1/3)), True))

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