Optimal. Leaf size=234 \[ \frac {x^5}{6 a \left (a+b x^6\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}}-\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}} \]
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Rubi [A]
time = 0.36, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {296, 301, 648,
632, 210, 642, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac {\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac {\log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}}-\frac {\log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}}+\frac {x^5}{6 a \left (a+b x^6\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 296
Rule 301
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+b x^6\right )^2} \, dx &=\frac {x^5}{6 a \left (a+b x^6\right )}+\frac {\int \frac {x^4}{a+b x^6} \, dx}{6 a}\\ &=\frac {x^5}{6 a \left (a+b x^6\right )}+\frac {\int \frac {-\frac {\sqrt [6]{a}}{2}+\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{18 a^{7/6} b^{2/3}}+\frac {\int \frac {-\frac {\sqrt [6]{a}}{2}-\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{18 a^{7/6} b^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{18 a b^{2/3}}\\ &=\frac {x^5}{6 a \left (a+b x^6\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}+\frac {\int \frac {-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{24 \sqrt {3} a^{7/6} b^{5/6}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{24 \sqrt {3} a^{7/6} b^{5/6}}+\frac {\int \frac {1}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a b^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a b^{2/3}}\\ &=\frac {x^5}{6 a \left (a+b x^6\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}+\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}}-\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{36 \sqrt {3} a^{7/6} b^{5/6}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{36 \sqrt {3} a^{7/6} b^{5/6}}\\ &=\frac {x^5}{6 a \left (a+b x^6\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac {\tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}}-\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 193, normalized size = 0.82 \begin {gather*} \frac {\frac {12 \sqrt [6]{a} x^5}{a+b x^6}+\frac {4 \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}-\frac {2 \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac {2 \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac {\sqrt {3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{b^{5/6}}-\frac {\sqrt {3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{b^{5/6}}}{72 a^{7/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(345\) vs.
\(2(158)=316\).
time = 0.56, size = 346, normalized size = 1.48
method | result | size |
risch | \(\frac {x^{5}}{6 a \left (b \,x^{6}+a \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{36 b a}\) | \(48\) |
default | \(\frac {x}{18 b a \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {3}}{36 b a \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}-\frac {\left (\frac {a}{b}\right )^{\frac {5}{6}} \sqrt {3}\, \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a^{2}}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{36 b a \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {x}{18 b a \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {3}}{36 b a \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}+\frac {\left (\frac {a}{b}\right )^{\frac {5}{6}} \sqrt {3}\, \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a^{2}}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{36 b a \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {x}{18 b a \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}+\frac {\arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 b a \left (\frac {a}{b}\right )^{\frac {1}{6}}}\) | \(346\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 206, normalized size = 0.88 \begin {gather*} \frac {x^{5}}{6 \, {\left (a b x^{6} + a^{2}\right )}} - \frac {\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}}{72 \, a} \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. 450 vs.
\(2 (160) = 320\).
time = 0.38, size = 450, normalized size = 1.92 \begin {gather*} \frac {12 \, x^{5} - 4 \, \sqrt {3} {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2}{3} \, \sqrt {3} a b x \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} + \frac {2}{3} \, \sqrt {3} \sqrt {a^{6} b^{4} x \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {5}{6}} - a^{5} b^{3} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {2}{3}} + x^{2}} a b \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - 4 \, \sqrt {3} {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2}{3} \, \sqrt {3} a b x \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} + \frac {2}{3} \, \sqrt {3} \sqrt {-a^{6} b^{4} x \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {5}{6}} - a^{5} b^{3} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {2}{3}} + x^{2}} a b \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} \log \left (a^{6} b^{4} x \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {5}{6}} - a^{5} b^{3} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {2}{3}} + x^{2}\right ) - {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} \log \left (-a^{6} b^{4} x \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {5}{6}} - a^{5} b^{3} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {2}{3}} + x^{2}\right ) + 2 \, {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} \log \left (a^{6} b^{4} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {5}{6}} + x\right ) - 2 \, {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} \log \left (-a^{6} b^{4} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {5}{6}} + x\right )}{72 \, {\left (a b x^{6} + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 46, normalized size = 0.20 \begin {gather*} \frac {x^{5}}{6 a^{2} + 6 a b x^{6}} + \operatorname {RootSum} {\left (2176782336 t^{6} a^{7} b^{5} + 1, \left ( t \mapsto t \log {\left (60466176 t^{5} a^{6} b^{4} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.37, size = 207, normalized size = 0.88 \begin {gather*} \frac {x^{5}}{6 \, {\left (b x^{6} + a\right )} a} - \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{2} b^{5}} + \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{2} b^{5}} + \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{2} b^{5}} + \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{2} b^{5}} + \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, a^{2} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 160, normalized size = 0.68 \begin {gather*} \frac {x^5}{6\,a\,\left (b\,x^6+a\right )}-\frac {\mathrm {atan}\left (\frac {b^{1/6}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/6}}\right )\,1{}\mathrm {i}}{18\,{\left (-a\right )}^{7/6}\,b^{5/6}}-\frac {\mathrm {atan}\left (\frac {b^{3/2}\,x\,1{}\mathrm {i}}{3888\,{\left (-a\right )}^{5/2}\,\left (\frac {b^{4/3}}{7776\,{\left (-a\right )}^{7/3}}-\frac {\sqrt {3}\,b^{4/3}\,1{}\mathrm {i}}{7776\,{\left (-a\right )}^{7/3}}\right )}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{36\,{\left (-a\right )}^{7/6}\,b^{5/6}}+\frac {\mathrm {atan}\left (\frac {b^{3/2}\,x\,1{}\mathrm {i}}{3888\,{\left (-a\right )}^{5/2}\,\left (\frac {b^{4/3}}{7776\,{\left (-a\right )}^{7/3}}+\frac {\sqrt {3}\,b^{4/3}\,1{}\mathrm {i}}{7776\,{\left (-a\right )}^{7/3}}\right )}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{36\,{\left (-a\right )}^{7/6}\,b^{5/6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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