Optimal. Leaf size=129 \[ \frac {5 x^{2/3}}{2 b^2}-\frac {x^{5/3}}{b (a+b x)}+\frac {5 a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{8/3}}+\frac {5 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}-\frac {5 a^{2/3} \log (a+b x)}{6 b^{8/3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 129, 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, 52, 58, 631,
210, 31} \begin {gather*} \frac {5 a^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{8/3}}+\frac {5 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}-\frac {5 a^{2/3} \log (a+b x)}{6 b^{8/3}}-\frac {x^{5/3}}{b (a+b x)}+\frac {5 x^{2/3}}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 52
Rule 58
Rule 210
Rule 631
Rubi steps
\begin {align*} \int \frac {x^{5/3}}{(a+b x)^2} \, dx &=-\frac {x^{5/3}}{b (a+b x)}+\frac {5 \int \frac {x^{2/3}}{a+b x} \, dx}{3 b}\\ &=\frac {5 x^{2/3}}{2 b^2}-\frac {x^{5/3}}{b (a+b x)}-\frac {(5 a) \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{3 b^2}\\ &=\frac {5 x^{2/3}}{2 b^2}-\frac {x^{5/3}}{b (a+b x)}-\frac {5 a^{2/3} \log (a+b x)}{6 b^{8/3}}-\frac {(5 a) \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 b^3}+\frac {\left (5 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 b^{8/3}}\\ &=\frac {5 x^{2/3}}{2 b^2}-\frac {x^{5/3}}{b (a+b x)}+\frac {5 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}-\frac {5 a^{2/3} \log (a+b x)}{6 b^{8/3}}-\frac {\left (5 a^{2/3}\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 )}{b^{8/3}}\\ &=\frac {5 x^{2/3}}{2 b^2}-\frac {x^{5/3}}{b (a+b x)}+\frac {5 a^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} b^{8/3}}+\frac {5 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}-\frac {5 a^{2/3} \log (a+b x)}{6 b^{8/3}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 142, normalized size = 1.10 \begin {gather*} \frac {\frac {3 b^{2/3} x^{2/3} (5 a+3 b x)}{a+b x}+10 \sqrt {3} a^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+10 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-5 a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{6 b^{8/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 124, normalized size = 0.96
method | result | size |
risch | \(\frac {3 x^{\frac {2}{3}}}{2 b^{2}}+\frac {a \,x^{\frac {2}{3}}}{b^{2} \left (b x +a \right )}+\frac {5 a \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {5 a \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {5 a \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}}\) | \(123\) |
derivativedivides | \(\frac {3 x^{\frac {2}{3}}}{2 b^{2}}-\frac {3 a \left (-\frac {x^{\frac {2}{3}}}{3 \left (b x +a \right )}-\frac {5 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {5 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2}}\) | \(124\) |
default | \(\frac {3 x^{\frac {2}{3}}}{2 b^{2}}-\frac {3 a \left (-\frac {x^{\frac {2}{3}}}{3 \left (b x +a \right )}-\frac {5 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {5 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2}}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 133, normalized size = 1.03 \begin {gather*} \frac {a x^{\frac {2}{3}}}{b^{3} x + a b^{2}} - \frac {5 \, \sqrt {3} a \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 )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {3 \, x^{\frac {2}{3}}}{2 \, b^{2}} - \frac {5 \, a \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 )}{6 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {5 \, a \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.60, size = 162, normalized size = 1.26 \begin {gather*} -\frac {10 \, \sqrt {3} {\left (b x + a\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{\frac {1}{3}} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) + 5 \, {\left (b x + a\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (-b x^{\frac {1}{3}} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a x^{\frac {2}{3}} + a \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 10 \, {\left (b x + a\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (b \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a x^{\frac {1}{3}}\right ) - 3 \, {\left (3 \, b x + 5 \, a\right )} x^{\frac {2}{3}}}{6 \, {\left (b^{3} x + a b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 595 vs.
\(2 (124) = 248\).
time = 89.40, size = 595, normalized size = 4.61 \begin {gather*} \begin {cases} \tilde {\infty } x^{\frac {2}{3}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {8}{3}}}{8 a^{2}} & \text {for}\: b = 0 \\\frac {3 x^{\frac {2}{3}}}{2 b^{2}} & \text {for}\: a = 0 \\- \frac {10 a^{2} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{6 a b^{3} \sqrt [3]{- \frac {a}{b}} + 6 b^{4} x \sqrt [3]{- \frac {a}{b}}} + \frac {5 a^{2} \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 )}}{6 a b^{3} \sqrt [3]{- \frac {a}{b}} + 6 b^{4} x \sqrt [3]{- \frac {a}{b}}} - \frac {10 \sqrt {3} a^{2} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{6 a b^{3} \sqrt [3]{- \frac {a}{b}} + 6 b^{4} x \sqrt [3]{- \frac {a}{b}}} - \frac {10 a^{2} \log {\left (2 \right )}}{6 a b^{3} \sqrt [3]{- \frac {a}{b}} + 6 b^{4} x \sqrt [3]{- \frac {a}{b}}} + \frac {15 a b x^{\frac {2}{3}} \sqrt [3]{- \frac {a}{b}}}{6 a b^{3} \sqrt [3]{- \frac {a}{b}} + 6 b^{4} x \sqrt [3]{- \frac {a}{b}}} - \frac {10 a b x \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{6 a b^{3} \sqrt [3]{- \frac {a}{b}} + 6 b^{4} x \sqrt [3]{- \frac {a}{b}}} + \frac {5 a b x \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 )}}{6 a b^{3} \sqrt [3]{- \frac {a}{b}} + 6 b^{4} x \sqrt [3]{- \frac {a}{b}}} - \frac {10 \sqrt {3} a b x \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{6 a b^{3} \sqrt [3]{- \frac {a}{b}} + 6 b^{4} x \sqrt [3]{- \frac {a}{b}}} - \frac {10 a b x \log {\left (2 \right )}}{6 a b^{3} \sqrt [3]{- \frac {a}{b}} + 6 b^{4} x \sqrt [3]{- \frac {a}{b}}} + \frac {9 b^{2} x^{\frac {5}{3}} \sqrt [3]{- \frac {a}{b}}}{6 a b^{3} \sqrt [3]{- \frac {a}{b}} + 6 b^{4} x \sqrt [3]{- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 135, normalized size = 1.05 \begin {gather*} \frac {5 \, \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 )}{3 \, b^{2}} + \frac {a x^{\frac {2}{3}}}{{\left (b x + a\right )} b^{2}} + \frac {3 \, x^{\frac {2}{3}}}{2 \, b^{2}} + \frac {5 \, \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 )}{3 \, b^{4}} - \frac {5 \, \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 )}{6 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 150, normalized size = 1.16 \begin {gather*} \frac {3\,x^{2/3}}{2\,b^2}+\frac {5\,a^{2/3}\,\ln \left (\frac {25\,a^{7/3}}{b^{10/3}}+\frac {25\,a^2\,x^{1/3}}{b^3}\right )}{3\,b^{8/3}}+\frac {a\,x^{2/3}}{x\,b^3+a\,b^2}+\frac {5\,a^{2/3}\,\ln \left (\frac {25\,a^{7/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{10/3}}+\frac {25\,a^2\,x^{1/3}}{b^3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^{8/3}}-\frac {5\,a^{2/3}\,\ln \left (\frac {25\,a^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{10/3}}+\frac {25\,a^2\,x^{1/3}}{b^3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^{8/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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