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

Optimal. Leaf size=115 \[ -\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{a} b^{5/3}}+\frac {\log (a+b x)}{3 \sqrt [3]{a} b^{5/3}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{5/3}}-\frac {x^{2/3}}{b (a+b x)} \]

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Rubi [A]  time = 0.04, antiderivative size = 115, 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 = {47, 56, 617, 204, 31} \begin {gather*} -\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{a} b^{5/3}}+\frac {\log (a+b x)}{3 \sqrt [3]{a} b^{5/3}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{5/3}}-\frac {x^{2/3}}{b (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(2/3)/(a + b*x)^2,x]

[Out]

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

Rule 31

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && 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 {x^{2/3}}{(a+b x)^2} \, dx &=-\frac {x^{2/3}}{b (a+b x)}+\frac {2 \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{3 b}\\ &=-\frac {x^{2/3}}{b (a+b x)}+\frac {\log (a+b x)}{3 \sqrt [3]{a} b^{5/3}}+\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 )}{b^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{a} b^{5/3}}\\ &=-\frac {x^{2/3}}{b (a+b x)}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{a} b^{5/3}}+\frac {\log (a+b x)}{3 \sqrt [3]{a} b^{5/3}}+\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 )}{\sqrt [3]{a} b^{5/3}}\\ &=-\frac {x^{2/3}}{b (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} \sqrt [3]{a} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{a} b^{5/3}}+\frac {\log (a+b x)}{3 \sqrt [3]{a} b^{5/3}}\\ \end {align*}

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Mathematica [C]  time = 0.00, size = 27, normalized size = 0.23 \begin {gather*} \frac {3 x^{5/3} \, _2F_1\left (\frac {5}{3},2;\frac {8}{3};-\frac {b x}{a}\right )}{5 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(2/3)/(a + b*x)^2,x]

[Out]

(3*x^(5/3)*Hypergeometric2F1[5/3, 2, 8/3, -((b*x)/a)])/(5*a^2)

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IntegrateAlgebraic [A]  time = 0.16, size = 145, normalized size = 1.26 \begin {gather*} \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{3 \sqrt [3]{a} b^{5/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 \sqrt [3]{a} b^{5/3}}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{5/3}}-\frac {x^{2/3}}{b (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.32, size = 394, normalized size = 3.43 \begin {gather*} \left [-\frac {3 \, 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 )}{3 \, {\left (a b^{4} x + a^{2} b^{3}\right )}}, -\frac {3 \, 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 )}{3 \, {\left (a b^{4} x + a^{2} b^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.20, size = 136, normalized size = 1.18 \begin {gather*} -\frac {2 \, \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 b} - \frac {x^{\frac {2}{3}}}{{\left (b x + a\right )} b} - \frac {2 \, \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 b^{3}} + \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 )}{3 \, a b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 112, normalized size = 0.97 \begin {gather*} -\frac {x^{\frac {2}{3}}}{\left (b x +a \right ) b}+\frac {2 \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}} b^{2}}-\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}+\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 )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^(2/3)/b/(b*x+a)-2/3/b^2/(a/b)^(1/3)*ln(x^(1/3)+(a/b)^(1/3))+1/3/b^2/(a/b)^(1/3)*ln(x^(2/3)-(a/b)^(1/3)*x^(1
/3)+(a/b)^(2/3))+2/3/b^2*3^(1/2)/(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 = 2.99, size = 120, normalized size = 1.04 \begin {gather*} -\frac {x^{\frac {2}{3}}}{b^{2} x + a b} + \frac {2 \, \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 \, b^{2} \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 )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.24, size = 142, normalized size = 1.23 \begin {gather*} \frac {2\,\ln \left (\frac {4\,x^{1/3}}{b}-\frac {4\,{\left (-a\right )}^{1/3}}{b^{4/3}}\right )}{3\,{\left (-a\right )}^{1/3}\,b^{5/3}}-\frac {x^{2/3}}{b\,\left (a+b\,x\right )}+\frac {\ln \left (\frac {4\,x^{1/3}}{b}-\frac {{\left (-a\right )}^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{b^{4/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{3\,{\left (-a\right )}^{1/3}\,b^{5/3}}-\frac {\ln \left (\frac {4\,x^{1/3}}{b}-\frac {{\left (-a\right )}^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{b^{4/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{3\,{\left (-a\right )}^{1/3}\,b^{5/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2/3)/(a + b*x)^2,x)

[Out]

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

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sympy [A]  time = 106.98, size = 787, normalized size = 6.84 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{\sqrt [3]{x}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {5}{3}}}{5 a^{2}} & \text {for}\: b = 0 \\- \frac {3}{b^{2} \sqrt [3]{x}} & \text {for}\: a = 0 \\- \frac {3 \sqrt [3]{-1} \sqrt [3]{a} b x^{\frac {2}{3}} \sqrt [3]{\frac {1}{b}}}{3 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} \sqrt [3]{\frac {1}{b}} + 3 \sqrt [3]{-1} \sqrt [3]{a} b^{3} 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 )}}{3 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} \sqrt [3]{\frac {1}{b}} + 3 \sqrt [3]{-1} \sqrt [3]{a} b^{3} 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 )}}{3 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} \sqrt [3]{\frac {1}{b}} + 3 \sqrt [3]{-1} \sqrt [3]{a} b^{3} 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 )}}{3 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} \sqrt [3]{\frac {1}{b}} + 3 \sqrt [3]{-1} \sqrt [3]{a} b^{3} x \sqrt [3]{\frac {1}{b}}} + \frac {2 a \log {\relax (2 )}}{3 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} \sqrt [3]{\frac {1}{b}} + 3 \sqrt [3]{-1} \sqrt [3]{a} b^{3} 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 )}}{3 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} \sqrt [3]{\frac {1}{b}} + 3 \sqrt [3]{-1} \sqrt [3]{a} b^{3} 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 )}}{3 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} \sqrt [3]{\frac {1}{b}} + 3 \sqrt [3]{-1} \sqrt [3]{a} b^{3} 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 )}}{3 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} \sqrt [3]{\frac {1}{b}} + 3 \sqrt [3]{-1} \sqrt [3]{a} b^{3} x \sqrt [3]{\frac {1}{b}}} + \frac {2 b x \log {\relax (2 )}}{3 \sqrt [3]{-1} a^{\frac {4}{3}} b^{2} \sqrt [3]{\frac {1}{b}} + 3 \sqrt [3]{-1} \sqrt [3]{a} b^{3} x \sqrt [3]{\frac {1}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(1/3), Eq(a, 0) & Eq(b, 0)), (3*x**(5/3)/(5*a**2), Eq(b, 0)), (-3/(b**2*x**(1/3)), Eq(a, 0))
, (-3*(-1)**(1/3)*a**(1/3)*b*x**(2/3)*(1/b)**(1/3)/(3*(-1)**(1/3)*a**(4/3)*b**2*(1/b)**(1/3) + 3*(-1)**(1/3)*a
**(1/3)*b**3*x*(1/b)**(1/3)) + 2*a*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x**(1/3))/(3*(-1)**(1/3)*a**(4/3)*
b**2*(1/b)**(1/3) + 3*(-1)**(1/3)*a**(1/3)*b**3*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))/(3*(-1)**(1/3)*a**(4/3)*b**2*(1/b)**(1/3) + 3*(-1)*
*(1/3)*a**(1/3)*b**3*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)))/(3*(-1)**(1/3)*a**(4/3)*b**2*(1/b)**(1/3) + 3*(-1)**(1/3)*a**(1/3)*b**3*x*(1/b)**(1/3)) + 2*a*
log(2)/(3*(-1)**(1/3)*a**(4/3)*b**2*(1/b)**(1/3) + 3*(-1)**(1/3)*a**(1/3)*b**3*x*(1/b)**(1/3)) + 2*b*x*log(-(-
1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x**(1/3))/(3*(-1)**(1/3)*a**(4/3)*b**2*(1/b)**(1/3) + 3*(-1)**(1/3)*a**(1/3)
*b**3*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))/(3*(-1)**(1/3)*a**(4/3)*b**2*(1/b)**(1/3) + 3*(-1)**(1/3)*a**(1/3)*b**3*x*(1/b)**(1/3)) + 2
*sqrt(3)*b*x*atan(sqrt(3)/3 - 2*(-1)**(2/3)*sqrt(3)*x**(1/3)/(3*a**(1/3)*(1/b)**(1/3)))/(3*(-1)**(1/3)*a**(4/3
)*b**2*(1/b)**(1/3) + 3*(-1)**(1/3)*a**(1/3)*b**3*x*(1/b)**(1/3)) + 2*b*x*log(2)/(3*(-1)**(1/3)*a**(4/3)*b**2*
(1/b)**(1/3) + 3*(-1)**(1/3)*a**(1/3)*b**3*x*(1/b)**(1/3)), True))

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