Optimal. Leaf size=55 \[ \frac {b^2 x}{c^3}-\frac {b x^3}{3 c^2}+\frac {x^5}{5 c}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1598, 308, 211}
\begin {gather*} -\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{7/2}}+\frac {b^2 x}{c^3}-\frac {b x^3}{3 c^2}+\frac {x^5}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 308
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^8}{b x^2+c x^4} \, dx &=\int \frac {x^6}{b+c x^2} \, dx\\ &=\int \left (\frac {b^2}{c^3}-\frac {b x^2}{c^2}+\frac {x^4}{c}-\frac {b^3}{c^3 \left (b+c x^2\right )}\right ) \, dx\\ &=\frac {b^2 x}{c^3}-\frac {b x^3}{3 c^2}+\frac {x^5}{5 c}-\frac {b^3 \int \frac {1}{b+c x^2} \, dx}{c^3}\\ &=\frac {b^2 x}{c^3}-\frac {b x^3}{3 c^2}+\frac {x^5}{5 c}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 55, normalized size = 1.00 \begin {gather*} \frac {b^2 x}{c^3}-\frac {b x^3}{3 c^2}+\frac {x^5}{5 c}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 49, normalized size = 0.89
method | result | size |
default | \(\frac {\frac {1}{5} c^{2} x^{5}-\frac {1}{3} b c \,x^{3}+b^{2} x}{c^{3}}-\frac {b^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{c^{3} \sqrt {b c}}\) | \(49\) |
risch | \(\frac {x^{5}}{5 c}-\frac {b \,x^{3}}{3 c^{2}}+\frac {b^{2} x}{c^{3}}+\frac {\sqrt {-b c}\, b^{2} \ln \left (-\sqrt {-b c}\, x -b \right )}{2 c^{4}}-\frac {\sqrt {-b c}\, b^{2} \ln \left (\sqrt {-b c}\, x -b \right )}{2 c^{4}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 50, normalized size = 0.91 \begin {gather*} -\frac {b^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{3}} + \frac {3 \, c^{2} x^{5} - 5 \, b c x^{3} + 15 \, b^{2} x}{15 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 126, normalized size = 2.29 \begin {gather*} \left [\frac {6 \, c^{2} x^{5} - 10 \, b c x^{3} + 15 \, b^{2} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right ) + 30 \, b^{2} x}{30 \, c^{3}}, \frac {3 \, c^{2} x^{5} - 5 \, b c x^{3} - 15 \, b^{2} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) + 15 \, b^{2} x}{15 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 95, normalized size = 1.73 \begin {gather*} \frac {b^{2} x}{c^{3}} - \frac {b x^{3}}{3 c^{2}} + \frac {\sqrt {- \frac {b^{5}}{c^{7}}} \log {\left (x - \frac {c^{3} \sqrt {- \frac {b^{5}}{c^{7}}}}{b^{2}} \right )}}{2} - \frac {\sqrt {- \frac {b^{5}}{c^{7}}} \log {\left (x + \frac {c^{3} \sqrt {- \frac {b^{5}}{c^{7}}}}{b^{2}} \right )}}{2} + \frac {x^{5}}{5 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.64, size = 55, normalized size = 1.00 \begin {gather*} -\frac {b^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{3}} + \frac {3 \, c^{4} x^{5} - 5 \, b c^{3} x^{3} + 15 \, b^{2} c^{2} x}{15 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 43, normalized size = 0.78 \begin {gather*} \frac {x^5}{5\,c}-\frac {b\,x^3}{3\,c^2}+\frac {b^2\,x}{c^3}-\frac {b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{c^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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