Optimal. Leaf size=154 \[ -\frac {x}{6 b \left (a+b x^3\right )^2}+\frac {x}{18 a b \left (a+b x^3\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {294, 205, 206,
31, 648, 631, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{4/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3}}+\frac {x}{18 a b \left (a+b x^3\right )}-\frac {x}{6 b \left (a+b x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 205
Rule 206
Rule 210
Rule 294
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^3\right )^3} \, dx &=-\frac {x}{6 b \left (a+b x^3\right )^2}+\frac {\int \frac {1}{\left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac {x}{6 b \left (a+b x^3\right )^2}+\frac {x}{18 a b \left (a+b x^3\right )}+\frac {\int \frac {1}{a+b x^3} \, dx}{9 a b}\\ &=-\frac {x}{6 b \left (a+b x^3\right )^2}+\frac {x}{18 a b \left (a+b x^3\right )}+\frac {\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{5/3} b}+\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{5/3} b}\\ &=-\frac {x}{6 b \left (a+b x^3\right )^2}+\frac {x}{18 a b \left (a+b x^3\right )}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3}}-\frac {\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{5/3} b^{4/3}}+\frac {\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{4/3} b}\\ &=-\frac {x}{6 b \left (a+b x^3\right )^2}+\frac {x}{18 a b \left (a+b x^3\right )}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{5/3} b^{4/3}}\\ &=-\frac {x}{6 b \left (a+b x^3\right )^2}+\frac {x}{18 a b \left (a+b x^3\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 138, normalized size = 0.90 \begin {gather*} \frac {-\frac {9 \sqrt [3]{b} x}{\left (a+b x^3\right )^2}+\frac {3 \sqrt [3]{b} x}{a^2+a b x^3}-\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}}{54 b^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 125, normalized size = 0.81
method | result | size |
risch | \(\frac {\frac {x^{4}}{18 a}-\frac {x}{9 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 b^{2} a}\) | \(56\) |
default | \(\frac {\frac {x^{4}}{18 a}-\frac {x}{9 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{9 b a}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 146, normalized size = 0.95 \begin {gather*} \frac {b x^{4} - 2 \, a x}{18 \, {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 231 vs.
\(2 (113) = 226\).
time = 0.37, size = 503, normalized size = 3.27 \begin {gather*} \left [\frac {3 \, a^{2} b^{2} x^{4} - 6 \, a^{3} b x + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{3} b^{4} x^{6} + 2 \, a^{4} b^{3} x^{3} + a^{5} b^{2}\right )}}, \frac {3 \, a^{2} b^{2} x^{4} - 6 \, a^{3} b x + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{3} b^{4} x^{6} + 2 \, a^{4} b^{3} x^{3} + a^{5} b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 65, normalized size = 0.42 \begin {gather*} \frac {- 2 a x + b x^{4}}{18 a^{3} b + 36 a^{2} b^{2} x^{3} + 18 a b^{3} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a^{5} b^{4} - 1, \left ( t \mapsto t \log {\left (27 t a^{2} b + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.89, size = 142, normalized size = 0.92 \begin {gather*} -\frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{2} b} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b^{2}} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b^{2}} + \frac {b x^{4} - 2 \, a x}{18 \, {\left (b x^{3} + a\right )}^{2} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.13, size = 145, normalized size = 0.94 \begin {gather*} \frac {\ln \left (\frac {b^{2/3}}{3\,a^{2/3}}+\frac {b\,x}{3\,a}\right )}{27\,a^{5/3}\,b^{4/3}}-\frac {\frac {x}{9\,b}-\frac {x^4}{18\,a}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\frac {\ln \left (\frac {b\,x}{3\,a}+\frac {b^{2/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{54\,a^{5/3}\,b^{4/3}}-\frac {\ln \left (\frac {b\,x}{3\,a}-\frac {b^{2/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{54\,a^{5/3}\,b^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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