Optimal. Leaf size=143 \[ -\frac {\sqrt [3]{x}}{2 b (a+b x)^2}+\frac {\sqrt [3]{x}}{6 a b (a+b x)}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{5/3} b^{4/3}}-\frac {\log (a+b x)}{18 a^{5/3} b^{4/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {43, 44, 60, 631,
210, 31} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{5/3} b^{4/3}}-\frac {\log (a+b x)}{18 a^{5/3} b^{4/3}}+\frac {\sqrt [3]{x}}{6 a b (a+b x)}-\frac {\sqrt [3]{x}}{2 b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 43
Rule 44
Rule 60
Rule 210
Rule 631
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{x}}{(a+b x)^3} \, dx &=-\frac {\sqrt [3]{x}}{2 b (a+b x)^2}+\frac {\int \frac {1}{x^{2/3} (a+b x)^2} \, dx}{6 b}\\ &=-\frac {\sqrt [3]{x}}{2 b (a+b x)^2}+\frac {\sqrt [3]{x}}{6 a b (a+b x)}+\frac {\int \frac {1}{x^{2/3} (a+b x)} \, dx}{9 a b}\\ &=-\frac {\sqrt [3]{x}}{2 b (a+b x)^2}+\frac {\sqrt [3]{x}}{6 a b (a+b x)}-\frac {\log (a+b x)}{18 a^{5/3} b^{4/3}}+\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 )}{6 a^{4/3} b^{5/3}}+\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{6 a^{5/3} b^{4/3}}\\ &=-\frac {\sqrt [3]{x}}{2 b (a+b x)^2}+\frac {\sqrt [3]{x}}{6 a b (a+b x)}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{5/3} b^{4/3}}-\frac {\log (a+b x)}{18 a^{5/3} b^{4/3}}+\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 )}{3 a^{5/3} b^{4/3}}\\ &=-\frac {\sqrt [3]{x}}{2 b (a+b x)^2}+\frac {\sqrt [3]{x}}{6 a b (a+b x)}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{5/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{5/3} b^{4/3}}-\frac {\log (a+b x)}{18 a^{5/3} b^{4/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.22, size = 136, normalized size = 0.95 \begin {gather*} \frac {\frac {3 a^{2/3} \sqrt [3]{b} \sqrt [3]{x} (-2 a+b x)}{(a+b x)^2}-2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{18 a^{5/3} b^{4/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 132, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {\frac {x^{\frac {4}{3}}}{6 a}-\frac {x^{\frac {1}{3}}}{3 b}}{\left (b x +a \right )^{2}}+\frac {\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{3 a b}\) | \(132\) |
default | \(\frac {\frac {x^{\frac {4}{3}}}{6 a}-\frac {x^{\frac {1}{3}}}{3 b}}{\left (b x +a \right )^{2}}+\frac {\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{3 a b}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 152, normalized size = 1.06 \begin {gather*} \frac {b x^{\frac {4}{3}} - 2 \, a x^{\frac {1}{3}}}{6 \, {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )}} + \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 )}{9 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{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 )}{18 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 231 vs.
\(2 (102) = 204\).
time = 1.05, size = 501, normalized size = 3.50 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{\frac {2}{3}} - \left (a^{2} b\right )^{\frac {1}{3}} a + \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x^{\frac {1}{3}}}{b x + a}\right ) - {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{\frac {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{\frac {1}{3}} + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 3 \, {\left (a^{2} b^{2} x - 2 \, a^{3} b\right )} x^{\frac {1}{3}}}{18 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}}, \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^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (a^{2} b\right )^{\frac {1}{3}} a - 2 \, \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{\frac {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{\frac {1}{3}} + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 3 \, {\left (a^{2} b^{2} x - 2 \, a^{3} b\right )} x^{\frac {1}{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]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 899 vs.
\(2 (126) = 252\).
time = 129.92, size = 899, normalized size = 6.29 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {4}{3}}}{4 a^{3}} & \text {for}\: b = 0 \\- \frac {3}{5 b^{3} x^{\frac {5}{3}}} & \text {for}\: a = 0 \\- \frac {6 a^{2} \sqrt [3]{x}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} - \frac {2 a^{2} \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {a^{2} \sqrt [3]{- \frac {a}{b}} \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 + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {2 \sqrt {3} a^{2} \sqrt [3]{- \frac {a}{b}} \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 + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} - \frac {2 a^{2} \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {3 a b x^{\frac {4}{3}}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} - \frac {4 a b x \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {2 a b x \sqrt [3]{- \frac {a}{b}} \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 + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {4 \sqrt {3} a b x \sqrt [3]{- \frac {a}{b}} \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 + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} - \frac {4 a b x \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} - \frac {2 b^{2} x^{2} \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {b^{2} x^{2} \sqrt [3]{- \frac {a}{b}} \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 + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {2 \sqrt {3} b^{2} x^{2} \sqrt [3]{- \frac {a}{b}} \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 + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} - \frac {2 b^{2} x^{2} \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.16, size = 148, normalized size = 1.03 \begin {gather*} -\frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2} b} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{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^{2}} + \frac {\left (-a b^{2}\right )^{\frac {1}{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 )}{18 \, a^{2} b^{2}} + \frac {b x^{\frac {4}{3}} - 2 \, a x^{\frac {1}{3}}}{6 \, {\left (b x + a\right )}^{2} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.24, size = 146, normalized size = 1.02 \begin {gather*} \frac {\frac {x^{4/3}}{6\,a}-\frac {x^{1/3}}{3\,b}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {\ln \left (\frac {b^{2/3}}{a^{2/3}}+\frac {b\,x^{1/3}}{a}\right )}{9\,a^{5/3}\,b^{4/3}}+\frac {\ln \left (\frac {b\,x^{1/3}}{a}+\frac {b^{2/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{5/3}\,b^{4/3}}-\frac {\ln \left (\frac {b\,x^{1/3}}{a}-\frac {b^{2/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{5/3}\,b^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________