Optimal. Leaf size=193 \[ -\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}}-\frac {\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}} \]
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Rubi [A]
time = 0.11, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {281, 303,
1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {a}+\sqrt {b} x^4\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}}-\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {a}+\sqrt {b} x^4\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 281
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^5}{a+b x^8} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,x^2\right )\\ &=-\frac {\text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,x^2\right )}{4 \sqrt {b}}+\frac {\text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,x^2\right )}{4 \sqrt {b}}\\ &=\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^2\right )}{8 b}+\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^2\right )}{8 b}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^2\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^2\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}}\\ &=\frac {\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}}-\frac {\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{3/4}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{3/4}}\\ &=-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}}-\frac {\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 279, normalized size = 1.45 \begin {gather*} -\frac {2 \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right )-2 \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right )+2 \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 110, normalized size = 0.57
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4} b^{3} a +1\right )}{\sum }\textit {\_R} \ln \left (-b \,x^{2} \textit {\_R} +1\right )\right )}{8}\) | \(29\) |
default | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{2} \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{4}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{2} \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(110\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 177, normalized size = 0.92 \begin {gather*} \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{8 \, \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{8 \, \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} x^{4} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x^{2} + \sqrt {a}\right )}{16 \, a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} x^{4} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x^{2} + \sqrt {a}\right )}{16 \, a^{\frac {1}{4}} b^{\frac {3}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 128, normalized size = 0.66 \begin {gather*} -\frac {1}{2} \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \arctan \left (-b x^{2} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} + \sqrt {x^{4} - a b \sqrt {-\frac {1}{a b^{3}}}} b \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}}\right ) + \frac {1}{8} \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (a b^{2} \left (-\frac {1}{a b^{3}}\right )^{\frac {3}{4}} + x^{2}\right ) - \frac {1}{8} \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (-a b^{2} \left (-\frac {1}{a b^{3}}\right )^{\frac {3}{4}} + x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 27, normalized size = 0.14 \begin {gather*} \operatorname {RootSum} {\left (4096 t^{4} a b^{3} + 1, \left ( t \mapsto t \log {\left (512 t^{3} a b^{2} + x^{2} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.11, size = 199, normalized size = 1.03 \begin {gather*} \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} x^{4} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{2} + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a b} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} x^{4} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{2} - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a b} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} x^{4} \log \left (x^{4} + \sqrt {2} x^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{16 \, a b} - \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} x^{4} \log \left (x^{4} - \sqrt {2} x^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{16 \, a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.11, size = 39, normalized size = 0.20 \begin {gather*} \frac {\mathrm {atan}\left (\frac {b^{1/4}\,x^2}{{\left (-a\right )}^{1/4}}\right )-\mathrm {atanh}\left (\frac {b^{1/4}\,x^2}{{\left (-a\right )}^{1/4}}\right )}{4\,{\left (-a\right )}^{1/4}\,b^{3/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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