Optimal. Leaf size=109 \[ -\frac {3}{a \sqrt [3]{x}}+\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{4/3}}+\frac {3 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 a^{4/3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 109, 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, 58, 631,
210, 31} \begin {gather*} \frac {\sqrt {3} \sqrt [3]{b} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{4/3}}+\frac {3 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 a^{4/3}}-\frac {3}{a \sqrt [3]{x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 53
Rule 58
Rule 210
Rule 631
Rubi steps
\begin {align*} \int \frac {1}{x^{4/3} (a+b x)} \, dx &=-\frac {3}{a \sqrt [3]{x}}-\frac {b \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{a}\\ &=-\frac {3}{a \sqrt [3]{x}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 a^{4/3}}-\frac {3 \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}+\frac {\left (3 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 a^{4/3}}\\ &=-\frac {3}{a \sqrt [3]{x}}+\frac {3 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 a^{4/3}}-\frac {\left (3 \sqrt [3]{b}\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^{4/3}}\\ &=-\frac {3}{a \sqrt [3]{x}}+\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3}}+\frac {3 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 a^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 127, normalized size = 1.17 \begin {gather*} \frac {-\frac {6 \sqrt [3]{a}}{\sqrt [3]{x}}+2 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{2 a^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 112, normalized size = 1.03
method | result | size |
risch | \(-\frac {3}{a \,x^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{a \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 a \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{a \left (\frac {a}{b}\right )^{\frac {1}{3}}}\) | \(104\) |
derivativedivides | \(-\frac {3}{a \,x^{\frac {1}{3}}}-\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b}{a}\) | \(112\) |
default | \(-\frac {3}{a \,x^{\frac {1}{3}}}-\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b}{a}\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 111, normalized size = 1.02 \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 {1}{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 {1}{3}}} + \frac {\log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {3}{a x^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.63, size = 113, normalized size = 1.04 \begin {gather*} -\frac {2 \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (-a x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {2}{3}} + b x^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, x \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (a \left (\frac {b}{a}\right )^{\frac {2}{3}} + b x^{\frac {1}{3}}\right ) + 6 \, x^{\frac {2}{3}}}{2 \, a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 8.90, size = 151, normalized size = 1.39 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {4}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{a \sqrt [3]{x}} & \text {for}\: b = 0 \\- \frac {3}{4 b x^{\frac {4}{3}}} & \text {for}\: a = 0 \\- \frac {\log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{a \sqrt [3]{- \frac {a}{b}}} + \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 \sqrt [3]{- \frac {a}{b}}} - \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 \sqrt [3]{- \frac {a}{b}}} - \frac {3}{a \sqrt [3]{x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.39, size = 125, normalized size = 1.15 \begin {gather*} \frac {b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a^{2}} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{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^{2} b} - \frac {3}{a x^{\frac {1}{3}}} - \frac {\left (-a b^{2}\right )^{\frac {2}{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 )}{2 \, a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 124, normalized size = 1.14 \begin {gather*} \frac {b^{1/3}\,\ln \left (9\,a^{4/3}\,b^{8/3}+9\,a\,b^3\,x^{1/3}\right )}{a^{4/3}}-\frac {3}{a\,x^{1/3}}+\frac {b^{1/3}\,\ln \left (9\,a\,b^3\,x^{1/3}+9\,a^{4/3}\,b^{8/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{4/3}}-\frac {b^{1/3}\,\ln \left (9\,a\,b^3\,x^{1/3}+9\,a^{4/3}\,b^{8/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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