Optimal. Leaf size=130 \[ -\frac {\sqrt [3]{a+b x}}{2 a x^2}+\frac {5 b \sqrt [3]{a+b x}}{6 a^2 x}-\frac {5 b^2 \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3}}-\frac {5 b^2 \log (x)}{18 a^{8/3}}+\frac {5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{8/3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 130, 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 = {44, 59, 631,
210, 31} \begin {gather*} -\frac {5 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^{8/3}}-\frac {5 b^2 \log (x)}{18 a^{8/3}}+\frac {5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{8/3}}+\frac {5 b \sqrt [3]{a+b x}}{6 a^2 x}-\frac {\sqrt [3]{a+b x}}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 59
Rule 210
Rule 631
Rubi steps
\begin {align*} \int \frac {1}{x^3 (a+b x)^{2/3}} \, dx &=-\frac {\sqrt [3]{a+b x}}{2 a x^2}-\frac {(5 b) \int \frac {1}{x^2 (a+b x)^{2/3}} \, dx}{6 a}\\ &=-\frac {\sqrt [3]{a+b x}}{2 a x^2}+\frac {5 b \sqrt [3]{a+b x}}{6 a^2 x}+\frac {\left (5 b^2\right ) \int \frac {1}{x (a+b x)^{2/3}} \, dx}{9 a^2}\\ &=-\frac {\sqrt [3]{a+b x}}{2 a x^2}+\frac {5 b \sqrt [3]{a+b x}}{6 a^2 x}-\frac {5 b^2 \log (x)}{18 a^{8/3}}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{6 a^{8/3}}-\frac {\left (5 b^2\right ) \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^{7/3}}\\ &=-\frac {\sqrt [3]{a+b x}}{2 a x^2}+\frac {5 b \sqrt [3]{a+b x}}{6 a^2 x}-\frac {5 b^2 \log (x)}{18 a^{8/3}}+\frac {5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{8/3}}+\frac {\left (5 b^2\right ) \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^{8/3}}\\ &=-\frac {\sqrt [3]{a+b x}}{2 a x^2}+\frac {5 b \sqrt [3]{a+b x}}{6 a^2 x}-\frac {5 b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{8/3}}-\frac {5 b^2 \log (x)}{18 a^{8/3}}+\frac {5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{8/3}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 149, normalized size = 1.15 \begin {gather*} -\frac {\sqrt [3]{a+b x} (8 a-5 (a+b x))}{6 a^2 x^2}-\frac {5 b^2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3}}+\frac {5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{9 a^{8/3}}-\frac {5 b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{18 a^{8/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 130, normalized size = 1.00
method | result | size |
derivativedivides | \(3 b^{2} \left (-\frac {\left (b x +a \right )^{\frac {1}{3}}}{6 a \,b^{2} x^{2}}-\frac {5 \left (-\frac {\left (b x +a \right )^{\frac {1}{3}}}{3 a b x}+\frac {-\frac {2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}}{a}\right )}{6 a}\right )\) | \(130\) |
default | \(3 b^{2} \left (-\frac {\left (b x +a \right )^{\frac {1}{3}}}{6 a \,b^{2} x^{2}}-\frac {5 \left (-\frac {\left (b x +a \right )^{\frac {1}{3}}}{3 a b x}+\frac {-\frac {2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}}{a}\right )}{6 a}\right )\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 142, normalized size = 1.09 \begin {gather*} -\frac {5 \, \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 {8}{3}}} - \frac {5 \, 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 {8}{3}}} + \frac {5 \, b^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {8}{3}}} + \frac {5 \, {\left (b x + a\right )}^{\frac {4}{3}} b^{2} - 8 \, {\left (b x + a\right )}^{\frac {1}{3}} a b^{2}}{6 \, {\left ({\left (b x + a\right )}^{2} a^{2} - 2 \, {\left (b x + a\right )} a^{3} + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.50, size = 162, normalized size = 1.25 \begin {gather*} -\frac {10 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{6}} a b^{2} x^{2} \arctan \left (\frac {{\left (a^{2}\right )}^{\frac {1}{6}} {\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 )}}{3 \, a^{2}}\right ) + 5 \, {\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 ) - 10 \, {\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 (5 \, a^{2} b x - 3 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{18 \, a^{4} 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 = 2.03, size = 2728, normalized size = 20.98 \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 = 0.99, size = 130, normalized size = 1.00 \begin {gather*} -\frac {\frac {10 \, \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 {8}{3}}} + \frac {5 \, 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 {8}{3}}} - \frac {10 \, b^{3} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {8}{3}}} - \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {4}{3}} b^{3} - 8 \, {\left (b x + a\right )}^{\frac {1}{3}} a b^{3}\right )}}{a^{2} 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 = 0.13, size = 175, normalized size = 1.35 \begin {gather*} \frac {5\,b^2\,\ln \left ({\left (a+b\,x\right )}^{1/3}-a^{1/3}\right )}{9\,a^{8/3}}-\frac {\frac {4\,b^2\,{\left (a+b\,x\right )}^{1/3}}{3\,a}-\frac {5\,b^2\,{\left (a+b\,x\right )}^{4/3}}{6\,a^2}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2}+\frac {5\,b^2\,\ln \left (\frac {5\,b^2\,{\left (a+b\,x\right )}^{1/3}}{a^2}-\frac {5\,b^2\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{5/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{8/3}}-\frac {5\,b^2\,\ln \left (\frac {5\,b^2\,{\left (a+b\,x\right )}^{1/3}}{a^2}+\frac {5\,b^2\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{5/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{8/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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