Optimal. Leaf size=125 \[ -\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}+\frac {4 \sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{7/3}}-\frac {x^{4/3}}{b (a+b x)}+\frac {4 \sqrt [3]{x}}{b^2} \]
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Rubi [A] time = 0.05, antiderivative size = 125, 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 = {47, 50, 58, 617, 204, 31} \[ -\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}+\frac {4 \sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{7/3}}-\frac {x^{4/3}}{b (a+b x)}+\frac {4 \sqrt [3]{x}}{b^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 50
Rule 58
Rule 204
Rule 617
Rubi steps
\begin {align*} \int \frac {x^{4/3}}{(a+b x)^2} \, dx &=-\frac {x^{4/3}}{b (a+b x)}+\frac {4 \int \frac {\sqrt [3]{x}}{a+b x} \, dx}{3 b}\\ &=\frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}-\frac {(4 a) \int \frac {1}{x^{2/3} (a+b x)} \, dx}{3 b^2}\\ &=\frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}-\frac {\left (2 a^{2/3}\right ) \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 )}{b^{8/3}}-\frac {\left (2 \sqrt [3]{a}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{b^{7/3}}\\ &=\frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}-\frac {\left (4 \sqrt [3]{a}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{b^{7/3}}\\ &=\frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}+\frac {4 \sqrt [3]{a} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} b^{7/3}}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 27, normalized size = 0.22 \[ \frac {3 x^{7/3} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {b x}{a}\right )}{7 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 147, normalized size = 1.18 \[ \frac {4 \, \sqrt {3} {\left (b x + a\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - 2 \, {\left (b x + a\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 4 \, {\left (b x + a\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (3 \, b x + 4 \, a\right )} x^{\frac {1}{3}}}{3 \, {\left (b^{3} x + a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 135, normalized size = 1.08 \[ \frac {4 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b^{2}} - \frac {4 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{3}} + \frac {a x^{\frac {1}{3}}}{{\left (b x + a\right )} b^{2}} + \frac {3 \, x^{\frac {1}{3}}}{b^{2}} - \frac {2 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 123, normalized size = 0.98 \[ \frac {a \,x^{\frac {1}{3}}}{\left (b x +a \right ) b^{2}}-\frac {4 \sqrt {3}\, a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {4 a \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {2 a \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {3 x^{\frac {1}{3}}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 133, normalized size = 1.06 \[ \frac {a x^{\frac {1}{3}}}{b^{3} x + a b^{2}} - \frac {4 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {3 \, x^{\frac {1}{3}}}{b^{2}} + \frac {2 \, a \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {4 \, a \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 142, normalized size = 1.14 \[ \frac {3\,x^{1/3}}{b^2}+\frac {a\,x^{1/3}}{x\,b^3+a\,b^2}+\frac {4\,{\left (-a\right )}^{1/3}\,\ln \left (\frac {12\,{\left (-a\right )}^{4/3}}{b^{1/3}}+12\,a\,x^{1/3}\right )}{3\,b^{7/3}}-\frac {4\,{\left (-a\right )}^{1/3}\,\ln \left (12\,a\,x^{1/3}-\frac {12\,{\left (-a\right )}^{4/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{1/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^{7/3}}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (12\,a\,x^{1/3}+\frac {9\,{\left (-a\right )}^{4/3}\,\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}{b^{1/3}}\right )\,\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}{b^{7/3}} \]
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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