Optimal. Leaf size=67 \[ \frac {x}{4 b \left (a-b x^2\right )^2}-\frac {x}{8 a b \left (a-b x^2\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {294, 205, 214}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{3/2}}-\frac {x}{8 a b \left (a-b x^2\right )}+\frac {x}{4 b \left (a-b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 214
Rule 294
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a-b x^2\right )^3} \, dx &=\frac {x}{4 b \left (a-b x^2\right )^2}-\frac {\int \frac {1}{\left (a-b x^2\right )^2} \, dx}{4 b}\\ &=\frac {x}{4 b \left (a-b x^2\right )^2}-\frac {x}{8 a b \left (a-b x^2\right )}-\frac {\int \frac {1}{a-b x^2} \, dx}{8 a b}\\ &=\frac {x}{4 b \left (a-b x^2\right )^2}-\frac {x}{8 a b \left (a-b x^2\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 56, normalized size = 0.84 \begin {gather*} \frac {x \left (a+b x^2\right )}{8 a b \left (a-b x^2\right )^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 50, normalized size = 0.75
method | result | size |
default | \(\frac {\frac {x^{3}}{8 a}+\frac {x}{8 b}}{\left (-b \,x^{2}+a \right )^{2}}-\frac {\arctanh \left (\frac {b x}{\sqrt {a b}}\right )}{8 b a \sqrt {a b}}\) | \(50\) |
risch | \(\frac {\frac {x^{3}}{8 a}+\frac {x}{8 b}}{\left (-b \,x^{2}+a \right )^{2}}+\frac {\ln \left (b x -\sqrt {a b}\right )}{16 \sqrt {a b}\, b a}-\frac {\ln \left (-b x -\sqrt {a b}\right )}{16 \sqrt {a b}\, b a}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 76, normalized size = 1.13 \begin {gather*} \frac {b x^{3} + a x}{8 \, {\left (a b^{3} x^{4} - 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} + \frac {\log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{16 \, \sqrt {a b} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 188, normalized size = 2.81 \begin {gather*} \left [\frac {2 \, a b^{2} x^{3} + 2 \, a^{2} b x + {\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \sqrt {a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {a b} x + a}{b x^{2} - a}\right )}{16 \, {\left (a^{2} b^{4} x^{4} - 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}}, \frac {a b^{2} x^{3} + a^{2} b x + {\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \sqrt {-a b} \arctan \left (\frac {\sqrt {-a b} x}{a}\right )}{8 \, {\left (a^{2} b^{4} x^{4} - 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}}\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. 105 vs.
\(2 (51) = 102\).
time = 0.13, size = 105, normalized size = 1.57 \begin {gather*} \frac {\sqrt {\frac {1}{a^{3} b^{3}}} \log {\left (- a^{2} b \sqrt {\frac {1}{a^{3} b^{3}}} + x \right )}}{16} - \frac {\sqrt {\frac {1}{a^{3} b^{3}}} \log {\left (a^{2} b \sqrt {\frac {1}{a^{3} b^{3}}} + x \right )}}{16} - \frac {- a x - b x^{3}}{8 a^{3} b - 16 a^{2} b^{2} x^{2} + 8 a b^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 53, normalized size = 0.79 \begin {gather*} \frac {\arctan \left (\frac {b x}{\sqrt {-a b}}\right )}{8 \, \sqrt {-a b} a b} + \frac {b x^{3} + a x}{8 \, {\left (b x^{2} - a\right )}^{2} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.62, size = 54, normalized size = 0.81 \begin {gather*} \frac {\frac {x}{8\,b}+\frac {x^3}{8\,a}}{a^2-2\,a\,b\,x^2+b^2\,x^4}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,a^{3/2}\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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