Optimal. Leaf size=127 \[ \frac {b^2 \log (x)}{18 a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{4/3}}-\frac {b^2 \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3}}-\frac {(a+b x)^{2/3}}{2 x^2}-\frac {b (a+b x)^{2/3}}{3 a x} \]
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Rubi [A] time = 0.05, 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 = {47, 51, 55, 617, 204, 31} \begin {gather*} \frac {b^2 \log (x)}{18 a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{4/3}}-\frac {b^2 \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3}}-\frac {(a+b x)^{2/3}}{2 x^2}-\frac {b (a+b x)^{2/3}}{3 a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 51
Rule 55
Rule 204
Rule 617
Rubi steps
\begin {align*} \int \frac {(a+b x)^{2/3}}{x^3} \, dx &=-\frac {(a+b x)^{2/3}}{2 x^2}+\frac {1}{3} b \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx\\ &=-\frac {(a+b x)^{2/3}}{2 x^2}-\frac {b (a+b x)^{2/3}}{3 a x}-\frac {b^2 \int \frac {1}{x \sqrt [3]{a+b x}} \, dx}{9 a}\\ &=-\frac {(a+b x)^{2/3}}{2 x^2}-\frac {b (a+b x)^{2/3}}{3 a x}+\frac {b^2 \log (x)}{18 a^{4/3}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{6 a^{4/3}}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{6 a}\\ &=-\frac {(a+b x)^{2/3}}{2 x^2}-\frac {b (a+b x)^{2/3}}{3 a x}+\frac {b^2 \log (x)}{18 a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{4/3}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{3 a^{4/3}}\\ &=-\frac {(a+b x)^{2/3}}{2 x^2}-\frac {b (a+b x)^{2/3}}{3 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^{4/3}}+\frac {b^2 \log (x)}{18 a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{4/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 35, normalized size = 0.28 \begin {gather*} -\frac {3 b^2 (a+b x)^{5/3} \, _2F_1\left (\frac {5}{3},3;\frac {8}{3};\frac {b x}{a}+1\right )}{5 a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 147, normalized size = 1.16 \begin {gather*} -\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{9 a^{4/3}}+\frac {b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{18 a^{4/3}}-\frac {b^2 \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3}}-\frac {(a+b x)^{2/3} (2 (a+b x)+a)}{6 a x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 350, normalized size = 2.76 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} a b^{2} x^{2} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + 3 \, a}{x}\right ) + \left (-a\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 3 \, {\left (2 \, a b x + 3 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{18 \, a^{2} x^{2}}, -\frac {6 \, \sqrt {\frac {1}{3}} a b^{2} x^{2} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) - \left (-a\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 2 \, \left (-a\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) + 3 \, {\left (2 \, a b x + 3 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{18 \, a^{2} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.47, size = 129, normalized size = 1.02 \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 {4}{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 {4}{3}}} + \frac {2 \, b^{3} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {4}{3}}} + \frac {3 \, {\left (2 \, {\left (b x + a\right )}^{\frac {5}{3}} b^{3} + {\left (b x + a\right )}^{\frac {2}{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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maple [A] time = 0.01, size = 113, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {3}\, b^{2} \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 {4}{3}}}-\frac {b^{2} \ln \left (-a^{\frac {1}{3}}+\left (b x +a \right )^{\frac {1}{3}}\right )}{9 a^{\frac {4}{3}}}+\frac {b^{2} \ln \left (a^{\frac {2}{3}}+\left (b x +a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+\left (b x +a \right )^{\frac {2}{3}}\right )}{18 a^{\frac {4}{3}}}-\frac {\left (b x +a \right )^{\frac {2}{3}}}{6 x^{2}}-\frac {\left (b x +a \right )^{\frac {5}{3}}}{3 a \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, 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 {4}{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 {4}{3}}} - \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {4}{3}}} - \frac {2 \, {\left (b x + a\right )}^{\frac {5}{3}} b^{2} + {\left (b x + a\right )}^{\frac {2}{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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mupad [B] time = 0.33, size = 194, normalized size = 1.53 \begin {gather*} \frac {{\left (-1\right )}^{1/3}\,b^2\,\ln \left ({\left (a+b\,x\right )}^{1/3}-{\left (-1\right )}^{2/3}\,a^{1/3}\right )}{9\,a^{4/3}}-\frac {\frac {b^2\,{\left (a+b\,x\right )}^{2/3}}{6}+\frac {b^2\,{\left (a+b\,x\right )}^{5/3}}{3\,a}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2}+\frac {{\left (-1\right )}^{1/3}\,b^2\,\ln \left (\frac {b^4\,{\left (a+b\,x\right )}^{1/3}}{9\,a^2}-\frac {{\left (-1\right )}^{2/3}\,b^4\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{9\,a^{5/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{4/3}}-\frac {{\left (-1\right )}^{1/3}\,b^2\,\ln \left (\frac {b^4\,{\left (a+b\,x\right )}^{1/3}}{9\,a^2}-\frac {{\left (-1\right )}^{2/3}\,b^4\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{9\,a^{5/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.65, size = 2266, normalized size = 17.84
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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