Optimal. Leaf size=128 \[ -\frac {5}{2 a^2 x^{2/3}}+\frac {1}{a x^{2/3} (a+b x)}+\frac {5 b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{8/3}}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac {5 b^{2/3} \log (a+b x)}{6 a^{8/3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 128, 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 = {44, 53, 60, 631,
210, 31} \begin {gather*} \frac {5 b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{8/3}}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac {5 b^{2/3} \log (a+b x)}{6 a^{8/3}}-\frac {5}{2 a^2 x^{2/3}}+\frac {1}{a x^{2/3} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 53
Rule 60
Rule 210
Rule 631
Rubi steps
\begin {align*} \int \frac {1}{x^{5/3} (a+b x)^2} \, dx &=\frac {1}{a x^{2/3} (a+b x)}+\frac {5 \int \frac {1}{x^{5/3} (a+b x)} \, dx}{3 a}\\ &=-\frac {5}{2 a^2 x^{2/3}}+\frac {1}{a x^{2/3} (a+b x)}-\frac {(5 b) \int \frac {1}{x^{2/3} (a+b x)} \, dx}{3 a^2}\\ &=-\frac {5}{2 a^2 x^{2/3}}+\frac {1}{a x^{2/3} (a+b x)}+\frac {5 b^{2/3} \log (a+b x)}{6 a^{8/3}}-\frac {\left (5 \sqrt [3]{b}\right ) \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 a^{7/3}}-\frac {\left (5 b^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 a^{8/3}}\\ &=-\frac {5}{2 a^2 x^{2/3}}+\frac {1}{a x^{2/3} (a+b x)}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac {5 b^{2/3} \log (a+b x)}{6 a^{8/3}}-\frac {\left (5 b^{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 )}{a^{8/3}}\\ &=-\frac {5}{2 a^2 x^{2/3}}+\frac {1}{a x^{2/3} (a+b x)}+\frac {5 b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{8/3}}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac {5 b^{2/3} \log (a+b x)}{6 a^{8/3}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 142, normalized size = 1.11 \begin {gather*} \frac {-\frac {3 a^{2/3} (3 a+5 b x)}{x^{2/3} (a+b x)}+10 \sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-10 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{6 a^{8/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 124, normalized size = 0.97
method | result | size |
derivativedivides | \(-\frac {3}{2 a^{2} x^{\frac {2}{3}}}-\frac {3 b \left (\frac {x^{\frac {1}{3}}}{3 b x +3 a}+\frac {5 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {5 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{a^{2}}\) | \(124\) |
default | \(-\frac {3}{2 a^{2} x^{\frac {2}{3}}}-\frac {3 b \left (\frac {x^{\frac {1}{3}}}{3 b x +3 a}+\frac {5 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {5 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{a^{2}}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 132, normalized size = 1.03 \begin {gather*} -\frac {5 \, b x + 3 \, a}{2 \, {\left (a^{2} b x^{\frac {5}{3}} + a^{3} x^{\frac {2}{3}}\right )}} - \frac {5 \, \sqrt {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 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {5 \, \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 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs.
\(2 (91) = 182\).
time = 1.64, size = 189, normalized size = 1.48 \begin {gather*} \frac {10 \, \sqrt {3} {\left (b x^{2} + a x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x^{\frac {1}{3}} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 5 \, {\left (b x^{2} + a x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{\frac {2}{3}} + a b x^{\frac {1}{3}} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 10 \, {\left (b x^{2} + a x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x^{\frac {1}{3}} - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 3 \, {\left (5 \, b x + 3 \, a\right )} x^{\frac {1}{3}}}{6 \, {\left (a^{2} b x^{2} + a^{3} x\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. 590 vs.
\(2 (126) = 252\).
time = 130.18, size = 590, normalized size = 4.61 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {8}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{2 a^{2} x^{\frac {2}{3}}} & \text {for}\: b = 0 \\- \frac {3}{8 b^{2} x^{\frac {8}{3}}} & \text {for}\: a = 0 \\- \frac {9 a^{2}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} + \frac {10 a b x^{\frac {2}{3}} \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} - \frac {5 a b x^{\frac {2}{3}} \sqrt [3]{- \frac {a}{b}} \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^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} - \frac {10 \sqrt {3} a b x^{\frac {2}{3}} \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} + \frac {10 a b x^{\frac {2}{3}} \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} - \frac {15 a b x}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} + \frac {10 b^{2} x^{\frac {5}{3}} \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} - \frac {5 b^{2} x^{\frac {5}{3}} \sqrt [3]{- \frac {a}{b}} \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^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} - \frac {10 \sqrt {3} b^{2} x^{\frac {5}{3}} \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} + \frac {10 b^{2} x^{\frac {5}{3}} \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.32, size = 137, normalized size = 1.07 \begin {gather*} \frac {5 \, b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{3}} - \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{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 \, a^{3}} - \frac {b x^{\frac {1}{3}}}{{\left (b x + a\right )} a^{2}} - \frac {5 \, \left (-a b^{2}\right )^{\frac {1}{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 \, a^{3}} - \frac {3}{2 \, a^{2} x^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 166, normalized size = 1.30 \begin {gather*} \frac {5\,{\left (-1\right )}^{1/3}\,b^{2/3}\,\ln \left (15\,{\left (-1\right )}^{1/3}\,a^{13/3}\,b^{8/3}-15\,a^4\,b^3\,x^{1/3}\right )}{3\,a^{8/3}}-\frac {\frac {3}{2\,a}+\frac {5\,b\,x}{2\,a^2}}{a\,x^{2/3}+b\,x^{5/3}}+\frac {5\,{\left (-1\right )}^{1/3}\,b^{2/3}\,\ln \left (15\,a^4\,b^3\,x^{1/3}-15\,{\left (-1\right )}^{1/3}\,a^{13/3}\,b^{8/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{8/3}}-\frac {5\,{\left (-1\right )}^{1/3}\,b^{2/3}\,\ln \left (15\,a^4\,b^3\,x^{1/3}+15\,{\left (-1\right )}^{1/3}\,a^{13/3}\,b^{8/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{8/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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