Optimal. Leaf size=111 \[ -\frac {3}{2 a x^{2/3}}+\frac {\sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3}}-\frac {3 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{5/3}}+\frac {b^{2/3} \log (a+b x)}{2 a^{5/3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {53, 60, 631,
210, 31} \begin {gather*} -\frac {3 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{5/3}}+\frac {b^{2/3} \log (a+b x)}{2 a^{5/3}}+\frac {\sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3}}-\frac {3}{2 a x^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 53
Rule 60
Rule 210
Rule 631
Rubi steps
\begin {align*} \int \frac {1}{x^{5/3} (a+b x)} \, dx &=-\frac {3}{2 a x^{2/3}}-\frac {b \int \frac {1}{x^{2/3} (a+b x)} \, dx}{a}\\ &=-\frac {3}{2 a x^{2/3}}+\frac {b^{2/3} \log (a+b x)}{2 a^{5/3}}-\frac {\left (3 \sqrt [3]{b}\right ) \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 )}{2 a^{4/3}}-\frac {\left (3 b^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 a^{5/3}}\\ &=-\frac {3}{2 a x^{2/3}}-\frac {3 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{5/3}}+\frac {b^{2/3} \log (a+b x)}{2 a^{5/3}}-\frac {\left (3 b^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{a^{5/3}}\\ &=-\frac {3}{2 a x^{2/3}}+\frac {\sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}-\frac {3 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{5/3}}+\frac {b^{2/3} \log (a+b x)}{2 a^{5/3}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 126, normalized size = 1.14 \begin {gather*} \frac {-\frac {3 a^{2/3}}{x^{2/3}}+2 \sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{2 a^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 41.43, size = 156, normalized size = 1.41 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\frac {1}{x^{\frac {5}{3}}}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {-3}{5 b x^{\frac {5}{3}}},a\text {==}0\right \},\left \{\frac {-3}{2 a x^{\frac {2}{3}}},b\text {==}0\right \}\right \},-\frac {\text {Log}\left [x^{\frac {1}{3}}-\left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ]}{a \left (-\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\text {Log}\left [4 x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}}+4 x^{\frac {2}{3}}+4 \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ]}{2 a \left (-\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3} \text {ArcTan}\left [\frac {\sqrt {3}}{3}+\frac {2 \sqrt {3} x^{\frac {1}{3}}}{3 \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right ]}{a \left (-\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {3}{2 a x^{\frac {2}{3}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.17, size = 112, normalized size = 1.01
method | result | size |
derivativedivides | \(-\frac {3 \left (\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b}{a}-\frac {3}{2 a \,x^{\frac {2}{3}}}\) | \(112\) |
default | \(-\frac {3 \left (\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b}{a}-\frac {3}{2 a \,x^{\frac {2}{3}}}\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 112, normalized size = 1.01 \begin {gather*} -\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 )}{a \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 )}{2 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {3}{2 \, a x^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 147, normalized size = 1.32 \begin {gather*} \frac {2 \, \sqrt {3} x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x^{\frac {1}{3}} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{\frac {2}{3}} + a b x^{\frac {1}{3}} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x^{\frac {1}{3}} - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 3 \, x^{\frac {1}{3}}}{2 \, a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 41.42, size = 155, normalized size = 1.40 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{5 b x^{\frac {5}{3}}} & \text {for}\: a = 0 \\- \frac {3}{2 a x^{\frac {2}{3}}} & \text {for}\: b = 0 \\- \frac {\log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{a \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\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 )}}{2 a \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{a \left (- \frac {a}{b}\right )^{\frac {2}{3}}} - \frac {3}{2 a x^{\frac {2}{3}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 173, normalized size = 1.56 \begin {gather*} 3 \left (-\frac {\left (-a b^{2}\right )^{\frac {1}{3}} \ln \left (\left (x^{\frac {1}{3}}\right )^{2}+\left (-\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (-\frac {a}{b}\right )^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 a^{2}}-\frac {\left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (x^{\frac {1}{3}}+\frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}}}{2}\right )}{\sqrt {3} \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a^{2}}+\frac {b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \ln \left |x^{\frac {1}{3}}-\left (-\frac {a}{b}\right )^{\frac {1}{3}}\right |}{3 a^{2}}-\frac {\frac {1}{2}}{a \left (x^{\frac {1}{3}}\right )^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 138, normalized size = 1.24 \begin {gather*} \frac {b^{2/3}\,\ln \left (9\,{\left (-a\right )}^{7/3}\,b^{8/3}-9\,a^2\,b^3\,x^{1/3}\right )}{{\left (-a\right )}^{5/3}}-\frac {3}{2\,a\,x^{2/3}}+\frac {b^{2/3}\,\ln \left (9\,{\left (-a\right )}^{7/3}\,b^{8/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-9\,a^2\,b^3\,x^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{{\left (-a\right )}^{5/3}}-\frac {b^{2/3}\,\ln \left (9\,{\left (-a\right )}^{7/3}\,b^{8/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+9\,a^2\,b^3\,x^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{{\left (-a\right )}^{5/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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