3.7.94 \(\int \frac {1}{\sqrt [3]{x} (a+b x)^3} \, dx\) [694]

Optimal. Leaf size=140 \[ \frac {x^{2/3}}{2 a (a+b x)^2}+\frac {2 x^{2/3}}{3 a^2 (a+b x)}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{7/3} b^{2/3}}+\frac {\log (a+b x)}{9 a^{7/3} b^{2/3}} \]

[Out]

1/2*x^(2/3)/a/(b*x+a)^2+2/3*x^(2/3)/a^2/(b*x+a)-1/3*ln(a^(1/3)+b^(1/3)*x^(1/3))/a^(7/3)/b^(2/3)+1/9*ln(b*x+a)/
a^(7/3)/b^(2/3)-2/9*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^(1/3))/a^(1/3)*3^(1/2))/a^(7/3)/b^(2/3)*3^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {2 \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{7/3} b^{2/3}}+\frac {\log (a+b x)}{9 a^{7/3} b^{2/3}}+\frac {2 x^{2/3}}{3 a^2 (a+b x)}+\frac {x^{2/3}}{2 a (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(1/3)*(a + b*x)^3),x]

[Out]

x^(2/3)/(2*a*(a + b*x)^2) + (2*x^(2/3))/(3*a^2*(a + b*x)) - (2*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a
^(1/3))])/(3*Sqrt[3]*a^(7/3)*b^(2/3)) - Log[a^(1/3) + b^(1/3)*x^(1/3)]/(3*a^(7/3)*b^(2/3)) + Log[a + b*x]/(9*a
^(7/3)*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)^3} \, dx &=\frac {x^{2/3}}{2 a (a+b x)^2}+\frac {2 \int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx}{3 a}\\ &=\frac {x^{2/3}}{2 a (a+b x)^2}+\frac {2 x^{2/3}}{3 a^2 (a+b x)}+\frac {2 \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{9 a^2}\\ &=\frac {x^{2/3}}{2 a (a+b x)^2}+\frac {2 x^{2/3}}{3 a^2 (a+b x)}+\frac {\log (a+b x)}{9 a^{7/3} 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 )}{3 a^2 b}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{3 a^{7/3} b^{2/3}}\\ &=\frac {x^{2/3}}{2 a (a+b x)^2}+\frac {2 x^{2/3}}{3 a^2 (a+b x)}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{7/3} b^{2/3}}+\frac {\log (a+b x)}{9 a^{7/3} b^{2/3}}+\frac {2 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{3 a^{7/3} b^{2/3}}\\ &=\frac {x^{2/3}}{2 a (a+b x)^2}+\frac {2 x^{2/3}}{3 a^2 (a+b x)}-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{7/3} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{7/3} b^{2/3}}+\frac {\log (a+b x)}{9 a^{7/3} b^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 142, normalized size = 1.01 \begin {gather*} \frac {\frac {3 \sqrt [3]{a} x^{2/3} (7 a+4 b x)}{(a+b x)^2}-\frac {4 \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 {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{2/3}}+\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{b^{2/3}}}{18 a^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(1/3)*(a + b*x)^3),x]

[Out]

((3*a^(1/3)*x^(2/3)*(7*a + 4*b*x))/(a + b*x)^2 - (4*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]])
/b^(2/3) - (4*Log[a^(1/3) + b^(1/3)*x^(1/3)])/b^(2/3) + (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(
2/3)])/b^(2/3))/(18*a^(7/3))

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Maple [A]
time = 0.11, size = 139, normalized size = 0.99

method	result	size
derivativedivides	\(\frac {x^{\frac {2}{3}}}{2 a \left (b x +a \right )^{2}}+\frac {\frac {2 x^{\frac {2}{3}}}{3 a \left (b x +a \right )}+\frac {2 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 a}}{a}\)	\(139\)
default	\(\frac {x^{\frac {2}{3}}}{2 a \left (b x +a \right )^{2}}+\frac {\frac {2 x^{\frac {2}{3}}}{3 a \left (b x +a \right )}+\frac {2 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 a}}{a}\)	\(139\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 151, normalized size = 1.08 \begin {gather*} \frac {4 \, b x^{\frac {5}{3}} + 7 \, a x^{\frac {2}{3}}}{6 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} + \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 )}{9 \, a^{2} 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 )}{9 \, a^{2} b \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 )}{9 \, a^{2} 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)^3,x, algorithm="maxima")

[Out]

1/6*(4*b*x^(5/3) + 7*a*x^(2/3))/(a^2*b^2*x^2 + 2*a^3*b*x + a^4) + 2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/3) -
(a/b)^(1/3))/(a/b)^(1/3))/(a^2*b*(a/b)^(1/3)) + 1/9*log(x^(2/3) - x^(1/3)*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b*(a
/b)^(1/3)) - 2/9*log(x^(1/3) + (a/b)^(1/3))/(a^2*b*(a/b)^(1/3))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (99) = 198\).
time = 0.88, size = 510, normalized size = 3.64 \begin {gather*} \left [\frac {6 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} 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 ) + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\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 ) - 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x^{\frac {1}{3}} - \left (-a b^{2}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b^{3} x + 7 \, a^{2} b^{2}\right )} x^{\frac {2}{3}}}{18 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}}, \frac {12 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} 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 ) + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\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 ) - 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x^{\frac {1}{3}} - \left (-a b^{2}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b^{3} x + 7 \, a^{2} b^{2}\right )} x^{\frac {2}{3}}}{18 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} 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)^3,x, algorithm="fricas")

[Out]

[1/18*(6*sqrt(1/3)*(a*b^3*x^2 + 2*a^2*b^2*x + a^3*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)) + 2*(b^2*x^2 + 2*a*b*x + a^2)*(-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)) - 4*(b^2*x^2 + 2*a*b*x + a^2)*(-a*b^2)^(2/3)*log(b*x^(1/3) - (-a*b^2)^(1/3)) + 3*(4*a*b^3*x + 7*a^2*b^2
)*x^(2/3))/(a^3*b^4*x^2 + 2*a^4*b^3*x + a^5*b^2), 1/18*(12*sqrt(1/3)*(a*b^3*x^2 + 2*a^2*b^2*x + a^3*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) + 2*(b^2*x^2 + 2*a
*b*x + a^2)*(-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)) - 4*(b^2*x^2 + 2*a*b*x
+ a^2)*(-a*b^2)^(2/3)*log(b*x^(1/3) - (-a*b^2)^(1/3)) + 3*(4*a*b^3*x + 7*a^2*b^2)*x^(2/3))/(a^3*b^4*x^2 + 2*a
^4*b^3*x + a^5*b^2)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (129) = 258\).
time = 140.71, size = 1175, normalized size = 8.39 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {2}{3}}}{2 a^{3}} & \text {for}\: b = 0 \\- \frac {3}{7 b^{3} x^{\frac {7}{3}}} & \text {for}\: a = 0 \\\frac {4 a^{2} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} - \frac {2 a^{2} \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 )}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {4 \sqrt {3} a^{2} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {4 a^{2} \log {\left (2 \right )}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {21 a b x^{\frac {2}{3}} \sqrt [3]{- \frac {a}{b}}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {8 a b x \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} - \frac {4 a 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 )}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {8 \sqrt {3} a b x \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {8 a b x \log {\left (2 \right )}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {12 b^{2} x^{\frac {5}{3}} \sqrt [3]{- \frac {a}{b}}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {4 b^{2} x^{2} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} - \frac {2 b^{2} x^{2} \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 )}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {4 \sqrt {3} b^{2} x^{2} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {4 b^{2} x^{2} \log {\left (2 \right )}}{18 a^{4} b \sqrt [3]{- \frac {a}{b}} + 36 a^{3} b^{2} x \sqrt [3]{- \frac {a}{b}} + 18 a^{2} b^{3} x^{2} \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)**3,x)

[Out]

Piecewise((zoo/x**(7/3), Eq(a, 0) & Eq(b, 0)), (3*x**(2/3)/(2*a**3), Eq(b, 0)), (-3/(7*b**3*x**(7/3)), Eq(a, 0
)), (4*a**2*log(x**(1/3) - (-a/b)**(1/3))/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b*
*3*x**2*(-a/b)**(1/3)) - 2*a**2*log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(18*a**4*b*(-a/b)
**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) + 4*sqrt(3)*a**2*atan(2*sqrt(3)*x**(
1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**
2*(-a/b)**(1/3)) + 4*a**2*log(2)/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(
-a/b)**(1/3)) + 21*a*b*x**(2/3)*(-a/b)**(1/3)/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**
2*b**3*x**2*(-a/b)**(1/3)) + 8*a*b*x*log(x**(1/3) - (-a/b)**(1/3))/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(
-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) - 4*a*b*x*log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)
**(2/3))/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) + 8*sqrt(3
)*a*b*x*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b
)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) + 8*a*b*x*log(2)/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)*
*(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) + 12*b**2*x**(5/3)*(-a/b)**(1/3)/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*
b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) + 4*b**2*x**2*log(x**(1/3) - (-a/b)**(1/3))/(18*a**4*b
*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) - 2*b**2*x**2*log(4*x**(2/3)
+ 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**
2*b**3*x**2*(-a/b)**(1/3)) + 4*sqrt(3)*b**2*x**2*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(18*a*
*4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) + 4*b**2*x**2*log(2)/(18*
a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)), True))

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Giac [A]
time = 2.09, size = 143, normalized size = 1.02 \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 )}{9 \, a^{3}} - \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 )}{9 \, a^{3} b^{2}} + \frac {4 \, b x^{\frac {5}{3}} + 7 \, a x^{\frac {2}{3}}}{6 \, {\left (b x + a\right )}^{2} a^{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 )}{9 \, a^{3} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*(-a/b)^(2/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/a^3 - 2/9*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^3*b^2) + 1/6*(4*b*x^(5/3) + 7*a*x^(2/3))/((b*x + a)^2*a^2) + 1/9*(-a*b^2)
^(2/3)*log(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^2)

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Mupad [B]
time = 0.19, size = 167, normalized size = 1.19 \begin {gather*} \frac {\frac {7\,x^{2/3}}{6\,a}+\frac {2\,b\,x^{5/3}}{3\,a^2}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {2\,\ln \left (\frac {4\,b\,x^{1/3}}{9\,a^4}-\frac {4\,b^{2/3}}{9\,{\left (-a\right )}^{11/3}}\right )}{9\,{\left (-a\right )}^{7/3}\,b^{2/3}}+\frac {\ln \left (\frac {4\,b\,x^{1/3}}{9\,a^4}-\frac {b^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{9\,{\left (-a\right )}^{11/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,{\left (-a\right )}^{7/3}\,b^{2/3}}-\frac {\ln \left (\frac {4\,b\,x^{1/3}}{9\,a^4}-\frac {b^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{9\,{\left (-a\right )}^{11/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,{\left (-a\right )}^{7/3}\,b^{2/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/3)*(a + b*x)^3),x)

[Out]

((7*x^(2/3))/(6*a) + (2*b*x^(5/3))/(3*a^2))/(a^2 + b^2*x^2 + 2*a*b*x) + (2*log((4*b*x^(1/3))/(9*a^4) - (4*b^(2
/3))/(9*(-a)^(11/3))))/(9*(-a)^(7/3)*b^(2/3)) + (log((4*b*x^(1/3))/(9*a^4) - (b^(2/3)*(3^(1/2)*1i - 1)^2)/(9*(
-a)^(11/3)))*(3^(1/2)*1i - 1))/(9*(-a)^(7/3)*b^(2/3)) - (log((4*b*x^(1/3))/(9*a^4) - (b^(2/3)*(3^(1/2)*1i + 1)
^2)/(9*(-a)^(11/3)))*(3^(1/2)*1i + 1))/(9*(-a)^(7/3)*b^(2/3))

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