Optimal. Leaf size=127 \[ -\frac {\sqrt [3]{a+b x}}{2 x^2}-\frac {b \sqrt [3]{a+b x}}{6 a x}+\frac {b^2 \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3}}+\frac {b^2 \log (x)}{18 a^{5/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{5/3}} \]
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Rubi [A]
time = 0.04, antiderivative size = 127, 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 = {43, 44, 59, 631,
210, 31} \begin {gather*} \frac {b^2 \text {ArcTan}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3}}+\frac {b^2 \log (x)}{18 a^{5/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{5/3}}-\frac {\sqrt [3]{a+b x}}{2 x^2}-\frac {b \sqrt [3]{a+b x}}{6 a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 44
Rule 59
Rule 210
Rule 631
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x}}{x^3} \, dx &=-\frac {\sqrt [3]{a+b x}}{2 x^2}+\frac {1}{6} b \int \frac {1}{x^2 (a+b x)^{2/3}} \, dx\\ &=-\frac {\sqrt [3]{a+b x}}{2 x^2}-\frac {b \sqrt [3]{a+b x}}{6 a x}-\frac {b^2 \int \frac {1}{x (a+b x)^{2/3}} \, dx}{9 a}\\ &=-\frac {\sqrt [3]{a+b x}}{2 x^2}-\frac {b \sqrt [3]{a+b x}}{6 a x}+\frac {b^2 \log (x)}{18 a^{5/3}}+\frac {b^2 \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{6 a^{5/3}}+\frac {b^2 \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{6 a^{4/3}}\\ &=-\frac {\sqrt [3]{a+b x}}{2 x^2}-\frac {b \sqrt [3]{a+b x}}{6 a x}+\frac {b^2 \log (x)}{18 a^{5/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{5/3}}-\frac {b^2 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{3 a^{5/3}}\\ &=-\frac {\sqrt [3]{a+b x}}{2 x^2}-\frac {b \sqrt [3]{a+b x}}{6 a x}+\frac {b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{5/3}}+\frac {b^2 \log (x)}{18 a^{5/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{5/3}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 129, normalized size = 1.02 \begin {gather*} \frac {-\frac {3 a^{2/3} \sqrt [3]{a+b x} (3 a+b x)}{x^2}+2 \sqrt {3} b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{18 a^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 118, normalized size = 0.93
method | result | size |
derivativedivides | \(3 b^{2} \left (-\frac {\frac {\left (b x +a \right )^{\frac {4}{3}}}{18 a}+\frac {\left (b x +a \right )^{\frac {1}{3}}}{9}}{b^{2} x^{2}}-\frac {\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}}{9 a}\right )\) | \(118\) |
default | \(3 b^{2} \left (-\frac {\frac {\left (b x +a \right )^{\frac {4}{3}}}{18 a}+\frac {\left (b x +a \right )^{\frac {1}{3}}}{9}}{b^{2} x^{2}}-\frac {\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}}{9 a}\right )\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 139, normalized size = 1.09 \begin {gather*} \frac {\sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {5}{3}}} + \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{18 \, a^{\frac {5}{3}}} - \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {5}{3}}} - \frac {{\left (b x + a\right )}^{\frac {4}{3}} b^{2} + 2 \, {\left (b x + a\right )}^{\frac {1}{3}} a b^{2}}{6 \, {\left ({\left (b x + a\right )}^{2} a - 2 \, {\left (b x + a\right )} a^{2} + a^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 187, normalized size = 1.47 \begin {gather*} \frac {2 \, \sqrt {3} a b^{2} x^{2} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {{\left (\sqrt {3} \left (-a^{2}\right )^{\frac {1}{3}} a - 2 \, \sqrt {3} \left (-a^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}}}{3 \, a^{2}}\right ) + \left (-a^{2}\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} a - \left (-a^{2}\right )^{\frac {1}{3}} a + \left (-a^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right ) - 2 \, \left (-a^{2}\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} a - \left (-a^{2}\right )^{\frac {2}{3}}\right ) - 3 \, {\left (a^{2} b x + 3 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{18 \, a^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.66, size = 2266, normalized size = 17.84 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.73, size = 128, normalized size = 1.01 \begin {gather*} \frac {\frac {2 \, \sqrt {3} b^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {5}{3}}} + \frac {b^{3} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {5}{3}}} - \frac {2 \, b^{3} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {5}{3}}} - \frac {3 \, {\left ({\left (b x + a\right )}^{\frac {4}{3}} b^{3} + 2 \, {\left (b x + a\right )}^{\frac {1}{3}} a b^{3}\right )}}{a b^{2} x^{2}}}{18 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 196, normalized size = 1.54 \begin {gather*} \frac {b^2\,\ln \left (\frac {b^2}{{\left (-a\right )}^{2/3}}-\frac {b^2\,{\left (a+b\,x\right )}^{1/3}}{a}\right )}{9\,{\left (-a\right )}^{5/3}}-\frac {\ln \left (\frac {b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{2/3}}+\frac {b^2\,{\left (a+b\,x\right )}^{1/3}}{a}\right )\,\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )}{18\,{\left (-a\right )}^{5/3}}-\frac {\frac {b^2\,{\left (a+b\,x\right )}^{1/3}}{3}+\frac {b^2\,{\left (a+b\,x\right )}^{4/3}}{6\,a}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2}+\frac {b^2\,\ln \left (\frac {b^2\,{\left (a+b\,x\right )}^{1/3}}{a}-\frac {b^2\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{{\left (-a\right )}^{2/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,{\left (-a\right )}^{5/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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