Optimal. Leaf size=140 \[ \frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}-\frac {\log (a+b x)}{9 a^{2/3} b^{7/3}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3} b^{7/3}}-\frac {2 \sqrt [3]{x}}{3 b^2 (a+b x)}-\frac {x^{4/3}}{2 b (a+b x)^2} \]
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Rubi [A] time = 0.05, 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 = {47, 58, 617, 204, 31} \[ \frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}-\frac {\log (a+b x)}{9 a^{2/3} b^{7/3}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3} b^{7/3}}-\frac {2 \sqrt [3]{x}}{3 b^2 (a+b x)}-\frac {x^{4/3}}{2 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 58
Rule 204
Rule 617
Rubi steps
\begin {align*} \int \frac {x^{4/3}}{(a+b x)^3} \, dx &=-\frac {x^{4/3}}{2 b (a+b x)^2}+\frac {2 \int \frac {\sqrt [3]{x}}{(a+b x)^2} \, dx}{3 b}\\ &=-\frac {x^{4/3}}{2 b (a+b x)^2}-\frac {2 \sqrt [3]{x}}{3 b^2 (a+b x)}+\frac {2 \int \frac {1}{x^{2/3} (a+b x)} \, dx}{9 b^2}\\ &=-\frac {x^{4/3}}{2 b (a+b x)^2}-\frac {2 \sqrt [3]{x}}{3 b^2 (a+b x)}-\frac {\log (a+b x)}{9 a^{2/3} b^{7/3}}+\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 )}{3 \sqrt [3]{a} b^{8/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}\\ &=-\frac {x^{4/3}}{2 b (a+b x)^2}-\frac {2 \sqrt [3]{x}}{3 b^2 (a+b x)}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}-\frac {\log (a+b x)}{9 a^{2/3} b^{7/3}}+\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 )}{3 a^{2/3} b^{7/3}}\\ &=-\frac {x^{4/3}}{2 b (a+b x)^2}-\frac {2 \sqrt [3]{x}}{3 b^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^{2/3} b^{7/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}-\frac {\log (a+b x)}{9 a^{2/3} b^{7/3}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 27, normalized size = 0.19 \[ \frac {3 x^{7/3} \, _2F_1\left (\frac {7}{3},3;\frac {10}{3};-\frac {b x}{a}\right )}{7 a^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 503, normalized size = 3.59 \[ \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^{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 ) - 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 {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) + 4 \, {\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 (7 \, a^{2} b^{2} x + 4 \, a^{3} b\right )} x^{\frac {1}{3}}}{18 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\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^{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 ) - 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 {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) + 4 \, {\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 (7 \, a^{2} b^{2} x + 4 \, a^{3} b\right )} x^{\frac {1}{3}}}{18 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 146, normalized size = 1.04 \[ -\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{2}} + \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{3}} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a b^{3}} - \frac {7 \, b x^{\frac {4}{3}} + 4 \, a x^{\frac {1}{3}}}{6 \, {\left (b x + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 124, normalized size = 0.89 \[ \frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {-\frac {7 x^{\frac {4}{3}}}{6 b}-\frac {2 a \,x^{\frac {1}{3}}}{3 b^{2}}}{\left (b x +a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.88, size = 143, normalized size = 1.02 \[ -\frac {7 \, b x^{\frac {4}{3}} + 4 \, a x^{\frac {1}{3}}}{6 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 139, normalized size = 0.99 \[ \frac {2\,\ln \left (2\,x^{1/3}+\frac {2\,a^{1/3}}{b^{1/3}}\right )}{9\,a^{2/3}\,b^{7/3}}-\frac {\frac {7\,x^{4/3}}{6\,b}+\frac {2\,a\,x^{1/3}}{3\,b^2}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {\ln \left (2\,x^{1/3}+\frac {a^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{b^{1/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{2/3}\,b^{7/3}}-\frac {\ln \left (2\,x^{1/3}-\frac {a^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{b^{1/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{2/3}\,b^{7/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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