3.7.86 \(\int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx\) [686]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, 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 = {44, 58, 631, 210, 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 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[L
og[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] && NegQ
[(b*c - a*d)/b]

Rule 210

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

Rule 631

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 {\text {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 {\text {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 {\text {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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.14, size = 133, normalized size = 1.15 \begin {gather*} \frac {\frac {6 \sqrt [3]{a} x^{2/3}}{a+b x}-\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{b^{2/3}}}{6 a^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(1/3)*(a + b*x)^2),x]

[Out]

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

________________________________________________________________________________________

Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 60.09, size = 515, normalized size = 4.44 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\frac {1}{x^{\frac {4}{3}}}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {-3}{4 b^2 x^{\frac {4}{3}}},a\text {==}0\right \},\left \{\frac {3 x^{\frac {2}{3}}}{2 a^2},b\text {==}0\right \}\right \},-\frac {a \text {Log}\left [4 x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}}+4 x^{\frac {2}{3}}+4 \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ]}{6 a^2 b \left (-\frac {a}{b}\right )^{\frac {1}{3}}+6 a b^2 x \left (-\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 \sqrt {3} a \text {ArcTan}\left [\frac {\sqrt {3}}{3}+\frac {2 \sqrt {3} x^{\frac {1}{3}}}{3 \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right ]}{6 a^2 b \left (-\frac {a}{b}\right )^{\frac {1}{3}}+6 a b^2 x \left (-\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 a \text {Log}\left [2\right ]}{6 a^2 b \left (-\frac {a}{b}\right )^{\frac {1}{3}}+6 a b^2 x \left (-\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 a \text {Log}\left [x^{\frac {1}{3}}-\left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ]}{6 a^2 b \left (-\frac {a}{b}\right )^{\frac {1}{3}}+6 a b^2 x \left (-\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {6 b x^{\frac {2}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}}}{6 a^2 b \left (-\frac {a}{b}\right )^{\frac {1}{3}}+6 a b^2 x \left (-\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {b x \text {Log}\left [4 x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}}+4 x^{\frac {2}{3}}+4 \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ]}{6 a^2 b \left (-\frac {a}{b}\right )^{\frac {1}{3}}+6 a b^2 x \left (-\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 \sqrt {3} b x \text {ArcTan}\left [\frac {\sqrt {3}}{3}+\frac {2 \sqrt {3} x^{\frac {1}{3}}}{3 \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right ]}{6 a^2 b \left (-\frac {a}{b}\right )^{\frac {1}{3}}+6 a b^2 x \left (-\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 b x \text {Log}\left [2\right ]}{6 a^2 b \left (-\frac {a}{b}\right )^{\frac {1}{3}}+6 a b^2 x \left (-\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 b x \text {Log}\left [x^{\frac {1}{3}}-\left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ]}{6 a^2 b \left (-\frac {a}{b}\right )^{\frac {1}{3}}+6 a b^2 x \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(x^(1/3)*(a + b*x)^2),x]')

[Out]

Piecewise[{{DirectedInfinity[1 / x ^ (4 / 3)], a == 0 && b == 0}, {-3 / (4 b ^ 2 x ^ (4 / 3)), a == 0}, {3 x ^
(2 / 3) / (2 a ^ 2), b == 0}}, -a Log[4 x ^ (1 / 3) (-a / b) ^ (1 / 3) + 4 x ^ (2 / 3) + 4 (-a / b) ^ (2 / 3)
] / (6 a ^ 2 b (-a / b) ^ (1 / 3) + 6 a b ^ 2 x (-a / b) ^ (1 / 3)) + 2 Sqrt[3] a ArcTan[Sqrt[3] / 3 + 2 Sqrt[
3] x ^ (1 / 3) / (3 (-a / b) ^ (1 / 3))] / (6 a ^ 2 b (-a / b) ^ (1 / 3) + 6 a b ^ 2 x (-a / b) ^ (1 / 3)) + 2
a Log[2] / (6 a ^ 2 b (-a / b) ^ (1 / 3) + 6 a b ^ 2 x (-a / b) ^ (1 / 3)) + 2 a Log[x ^ (1 / 3) - (-a / b) ^
(1 / 3)] / (6 a ^ 2 b (-a / b) ^ (1 / 3) + 6 a b ^ 2 x (-a / b) ^ (1 / 3)) + 6 b x ^ (2 / 3) (-a / b) ^ (1 /
3) / (6 a ^ 2 b (-a / b) ^ (1 / 3) + 6 a b ^ 2 x (-a / b) ^ (1 / 3)) - b x Log[4 x ^ (1 / 3) (-a / b) ^ (1 / 3
) + 4 x ^ (2 / 3) + 4 (-a / b) ^ (2 / 3)] / (6 a ^ 2 b (-a / b) ^ (1 / 3) + 6 a b ^ 2 x (-a / b) ^ (1 / 3)) +
2 Sqrt[3] b x ArcTan[Sqrt[3] / 3 + 2 Sqrt[3] x ^ (1 / 3) / (3 (-a / b) ^ (1 / 3))] / (6 a ^ 2 b (-a / b) ^ (1
/ 3) + 6 a b ^ 2 x (-a / b) ^ (1 / 3)) + 2 b x Log[2] / (6 a ^ 2 b (-a / b) ^ (1 / 3) + 6 a b ^ 2 x (-a / b) ^
(1 / 3)) + 2 b x Log[x ^ (1 / 3) - (-a / b) ^ (1 / 3)] / (6 a ^ 2 b (-a / b) ^ (1 / 3) + 6 a b ^ 2 x (-a / b)
^ (1 / 3))]

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 116, normalized size = 1.00

method	result	size
derivativedivides	\(\frac {x^{\frac {2}{3}}}{a \left (b x +a \right )}+\frac {-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{a}\)	\(116\)
default	\(\frac {x^{\frac {2}{3}}}{a \left (b x +a \right )}+\frac {-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{a}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.36, 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))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (83) = 166\).
time = 0.33, 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)]

________________________________________________________________________________________

Sympy [A]
time = 61.77, size = 544, normalized size = 4.69 \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 {2 a \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} - \frac {a \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 \sqrt {3} a \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 a \log {\left (2 \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} + \frac {6 b x^{\frac {2}{3}} \sqrt [3]{- \frac {a}{b}}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 b x \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} - \frac {b x \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 \sqrt {3} b x \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 b x \log {\left (2 \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{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
)), (2*a*log(x**(1/3) - (-a/b)**(1/3))/(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*(-a/b)**(1/3)) - a*log(4*x**(2/3)
+ 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*(-a/b)**(1/3)) + 2*sqrt(3)*
a*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*(-a/b)**(1/3)) +
2*a*log(2)/(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*(-a/b)**(1/3)) + 6*b*x**(2/3)*(-a/b)**(1/3)/(6*a**2*b*(-a/b)*
*(1/3) + 6*a*b**2*x*(-a/b)**(1/3)) + 2*b*x*log(x**(1/3) - (-a/b)**(1/3))/(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*
(-a/b)**(1/3)) - b*x*log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(6*a**2*b*(-a/b)**(1/3) + 6*
a*b**2*x*(-a/b)**(1/3)) + 2*sqrt(3)*b*x*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(6*a**2*b*(-a/b
)**(1/3) + 6*a*b**2*x*(-a/b)**(1/3)) + 2*b*x*log(2)/(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*(-a/b)**(1/3)), True)
)

________________________________________________________________________________________

Giac [A]
time = 0.01, size = 200, normalized size = 1.72 \begin {gather*} 3 \left (\frac {\left (\left (-a b^{2}\right )^{\frac {1}{3}}\right )^{2} \ln \left (\left (x^{\frac {1}{3}}\right )^{2}+\left (-\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (-\frac {a}{b}\right )^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{18 a^{2} b^{2}}-\frac {\frac {1}{3} \left (\left (-a b^{2}\right )^{\frac {1}{3}}\right )^{2} \arctan \left (\frac {2 \left (x^{\frac {1}{3}}+\frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}}}{2}\right )}{\sqrt {3} \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a^{2} b^{2}}-\frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \ln \left |x^{\frac {1}{3}}-\left (-\frac {a}{b}\right )^{\frac {1}{3}}\right |}{3\cdot 3 a^{2}}+\frac {\frac {1}{3} \left (x^{\frac {1}{3}}\right )^{2}}{a \left (x b+a\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)

[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)

________________________________________________________________________________________

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))

________________________________________________________________________________________