Optimal. Leaf size=133 \[ \frac {x^2}{2 b}+\frac {\sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} b^{4/3}}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 b^{4/3}} \]
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Rubi [A]
time = 0.09, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {281, 327, 206,
31, 648, 631, 210, 642} \begin {gather*} \frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 b^{4/3}}+\frac {\sqrt [3]{a} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} b^{4/3}}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac {x^2}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 281
Rule 327
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x^7}{a+b x^6} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{a+b x^3} \, dx,x,x^2\right )\\ &=\frac {x^2}{2 b}-\frac {a \text {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,x^2\right )}{2 b}\\ &=\frac {x^2}{2 b}-\frac {\sqrt [3]{a} \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{6 b}-\frac {\sqrt [3]{a} \text {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{6 b}\\ &=\frac {x^2}{2 b}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac {\sqrt [3]{a} \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{12 b^{4/3}}-\frac {a^{2/3} \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{4 b}\\ &=\frac {x^2}{2 b}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 b^{4/3}}-\frac {\sqrt [3]{a} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{2 b^{4/3}}\\ &=\frac {x^2}{2 b}+\frac {\sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} b^{4/3}}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 b^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 186, normalized size = 1.40 \begin {gather*} \frac {6 \sqrt [3]{b} x^2+2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )+\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 b^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 112, normalized size = 0.84
method | result | size |
risch | \(\frac {x^{2}}{2 b}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\textit {\_R} \ln \left (x^{2}-\textit {\_R} \right )}{6 b}\) | \(36\) |
default | \(\frac {x^{2}}{2 b}-\frac {\left (\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a}{2 b}\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 115, normalized size = 0.86 \begin {gather*} \frac {x^{2}}{2 \, b} - \frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {a \log \left (x^{4} - x^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {a \log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b^{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 [A]
time = 0.36, size = 114, normalized size = 0.86 \begin {gather*} \frac {6 \, x^{2} + 2 \, \sqrt {3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{2} \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 27, normalized size = 0.20 \begin {gather*} \operatorname {RootSum} {\left (216 t^{3} b^{4} + a, \left ( t \mapsto t \log {\left (- 6 t b + x^{2} \right )} \right )\right )} + \frac {x^{2}}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.28, size = 120, normalized size = 0.90 \begin {gather*} \frac {x^{2}}{2 \, b} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{6 \, b} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, b^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 130, normalized size = 0.98 \begin {gather*} \frac {x^2}{2\,b}+\frac {{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{10/3}+a^3\,b^{1/3}\,x^2\right )}{6\,b^{4/3}}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (6\,a^3\,b\,x^2+6\,{\left (-a\right )}^{10/3}\,b^{2/3}\,\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 )}{6\,b^{4/3}}-\frac {{\left (-a\right )}^{1/3}\,\ln \left (6\,a^3\,b\,x^2-6\,{\left (-a\right )}^{10/3}\,b^{2/3}\,\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 )}{6\,b^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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