Optimal. Leaf size=124 \[ -\frac {b^2 \log (x)}{9 a^{2/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{2/3}}-\frac {2 b^2 \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3}}-\frac {(a+b x)^{4/3}}{2 x^2}-\frac {2 b \sqrt [3]{a+b x}}{3 x} \]
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Rubi [A] time = 0.04, antiderivative size = 124, 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 = {47, 57, 617, 204, 31} \begin {gather*} -\frac {b^2 \log (x)}{9 a^{2/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{2/3}}-\frac {2 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^{2/3}}-\frac {(a+b x)^{4/3}}{2 x^2}-\frac {2 b \sqrt [3]{a+b x}}{3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 57
Rule 204
Rule 617
Rubi steps
\begin {align*} \int \frac {(a+b x)^{4/3}}{x^3} \, dx &=-\frac {(a+b x)^{4/3}}{2 x^2}+\frac {1}{3} (2 b) \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx\\ &=-\frac {2 b \sqrt [3]{a+b x}}{3 x}-\frac {(a+b x)^{4/3}}{2 x^2}+\frac {1}{9} \left (2 b^2\right ) \int \frac {1}{x (a+b x)^{2/3}} \, dx\\ &=-\frac {2 b \sqrt [3]{a+b x}}{3 x}-\frac {(a+b x)^{4/3}}{2 x^2}-\frac {b^2 \log (x)}{9 a^{2/3}}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{3 a^{2/3}}-\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 )}{3 \sqrt [3]{a}}\\ &=-\frac {2 b \sqrt [3]{a+b x}}{3 x}-\frac {(a+b x)^{4/3}}{2 x^2}-\frac {b^2 \log (x)}{9 a^{2/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{2/3}}+\frac {\left (2 b^2\right ) \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^{2/3}}\\ &=-\frac {2 b \sqrt [3]{a+b x}}{3 x}-\frac {(a+b x)^{4/3}}{2 x^2}-\frac {2 b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{2/3}}-\frac {b^2 \log (x)}{9 a^{2/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{2/3}}\\ \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)^{7/3} \, _2F_1\left (\frac {7}{3},3;\frac {10}{3};\frac {b x}{a}+1\right )}{7 a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 146, normalized size = 1.18 \begin {gather*} \frac {2 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{9 a^{2/3}}-\frac {b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{9 a^{2/3}}-\frac {2 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^{2/3}}-\frac {\sqrt [3]{a+b x} (7 (a+b x)-4 a)}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 162, normalized size = 1.31 \begin {gather*} -\frac {4 \, \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 ) + 2 \, {\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 ) - 4 \, {\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 (7 \, a^{2} b x + 3 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{18 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.94, size = 127, normalized size = 1.02 \begin {gather*} -\frac {\frac {4 \, \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 {2}{3}}} + \frac {2 \, 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 {2}{3}}} - \frac {4 \, b^{3} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {2}{3}}} + \frac {3 \, {\left (7 \, {\left (b x + a\right )}^{\frac {4}{3}} b^{3} - 4 \, {\left (b x + a\right )}^{\frac {1}{3}} a b^{3}\right )}}{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 = 111, normalized size = 0.90 \begin {gather*} -\frac {2 \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 {2}{3}}}+\frac {2 b^{2} \ln \left (-a^{\frac {1}{3}}+\left (b x +a \right )^{\frac {1}{3}}\right )}{9 a^{\frac {2}{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 )}{9 a^{\frac {2}{3}}}+\frac {2 \left (b x +a \right )^{\frac {1}{3}} a}{3 x^{2}}-\frac {7 \left (b x +a \right )^{\frac {4}{3}}}{6 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.06, size = 136, normalized size = 1.10 \begin {gather*} -\frac {2 \, \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 {2}{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 )}{9 \, a^{\frac {2}{3}}} + \frac {2 \, b^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {2}{3}}} - \frac {7 \, {\left (b x + a\right )}^{\frac {4}{3}} b^{2} - 4 \, {\left (b x + a\right )}^{\frac {1}{3}} a b^{2}}{6 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 174, normalized size = 1.40 \begin {gather*} \frac {2\,b^2\,\ln \left (2\,b^2\,{\left (a+b\,x\right )}^{1/3}-2\,a^{1/3}\,b^2\right )}{9\,a^{2/3}}-\frac {\frac {7\,b^2\,{\left (a+b\,x\right )}^{4/3}}{6}-\frac {2\,a\,b^2\,{\left (a+b\,x\right )}^{1/3}}{3}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2}-\frac {\ln \left (2\,b^2\,{\left (a+b\,x\right )}^{1/3}+a^{1/3}\,\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )\right )\,\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )}{9\,a^{2/3}}+\frac {b^2\,\ln \left (2\,b^2\,{\left (a+b\,x\right )}^{1/3}-9\,a^{1/3}\,b^2\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}{a^{2/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.74, size = 2266, normalized size = 18.27
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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