Optimal. Leaf size=43 \[ -\frac {1}{3 a x^3}+\frac {b}{a^2 x}+\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1598, 331, 211}
\begin {gather*} \frac {b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {b}{a^2 x}-\frac {1}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 331
Rule 1598
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a x+b x^3\right )} \, dx &=\int \frac {1}{x^4 \left (a+b x^2\right )} \, dx\\ &=-\frac {1}{3 a x^3}-\frac {b \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{a}\\ &=-\frac {1}{3 a x^3}+\frac {b}{a^2 x}+\frac {b^2 \int \frac {1}{a+b x^2} \, dx}{a^2}\\ &=-\frac {1}{3 a x^3}+\frac {b}{a^2 x}+\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 1.00 \begin {gather*} -\frac {1}{3 a x^3}+\frac {b}{a^2 x}+\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 39, normalized size = 0.91
method | result | size |
default | \(\frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}-\frac {1}{3 a \,x^{3}}+\frac {b}{a^{2} x}\) | \(39\) |
risch | \(\frac {\frac {b \,x^{2}}{a^{2}}-\frac {1}{3 a}}{x^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{5} \textit {\_Z}^{2}+b^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (3 a^{5} \textit {\_R}^{2}+2 b^{3}\right ) x -a^{3} b \textit {\_R} \right )\right )}{2}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 40, normalized size = 0.93 \begin {gather*} \frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, b x^{2} - a}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.29, size = 106, normalized size = 2.47 \begin {gather*} \left [\frac {3 \, b x^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 6 \, b x^{2} - 2 \, a}{6 \, a^{2} x^{3}}, \frac {3 \, b x^{3} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 3 \, b x^{2} - a}{3 \, a^{2} x^{3}}\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. 87 vs.
\(2 (37) = 74\).
time = 0.10, size = 87, normalized size = 2.02 \begin {gather*} - \frac {\sqrt {- \frac {b^{3}}{a^{5}}} \log {\left (- \frac {a^{3} \sqrt {- \frac {b^{3}}{a^{5}}}}{b^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{3}}{a^{5}}} \log {\left (\frac {a^{3} \sqrt {- \frac {b^{3}}{a^{5}}}}{b^{2}} + x \right )}}{2} + \frac {- a + 3 b x^{2}}{3 a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.92, size = 40, normalized size = 0.93 \begin {gather*} \frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, b x^{2} - a}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.94, size = 37, normalized size = 0.86 \begin {gather*} \frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {1}{3\,a}-\frac {b\,x^2}{a^2}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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