Optimal. Leaf size=55 \[ \frac {3 x}{2 b^2}-\frac {x^3}{2 b \left (a+b x^2\right )}-\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {294, 327, 211}
\begin {gather*} -\frac {3 \sqrt {a} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2}}-\frac {x^3}{2 b \left (a+b x^2\right )}+\frac {3 x}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 294
Rule 327
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+b x^2\right )^2} \, dx &=-\frac {x^3}{2 b \left (a+b x^2\right )}+\frac {3 \int \frac {x^2}{a+b x^2} \, dx}{2 b}\\ &=\frac {3 x}{2 b^2}-\frac {x^3}{2 b \left (a+b x^2\right )}-\frac {(3 a) \int \frac {1}{a+b x^2} \, dx}{2 b^2}\\ &=\frac {3 x}{2 b^2}-\frac {x^3}{2 b \left (a+b x^2\right )}-\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 51, normalized size = 0.93 \begin {gather*} \frac {x}{b^2}+\frac {a x}{2 b^2 \left (a+b x^2\right )}-\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 42, normalized size = 0.76
method | result | size |
default | \(\frac {x}{b^{2}}-\frac {a \left (-\frac {x}{2 \left (b \,x^{2}+a \right )}+\frac {3 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{2}}\) | \(42\) |
risch | \(\frac {x}{b^{2}}+\frac {a x}{2 b^{2} \left (b \,x^{2}+a \right )}+\frac {3 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x -a \right )}{4 b^{3}}-\frac {3 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x -a \right )}{4 b^{3}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 45, normalized size = 0.82 \begin {gather*} \frac {a x}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} - \frac {3 \, a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} + \frac {x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.00, size = 136, normalized size = 2.47 \begin {gather*} \left [\frac {4 \, b x^{3} + 3 \, {\left (b x^{2} + a\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 6 \, a x}{4 \, {\left (b^{3} x^{2} + a b^{2}\right )}}, \frac {2 \, b x^{3} - 3 \, {\left (b x^{2} + a\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 3 \, a x}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 83, normalized size = 1.51 \begin {gather*} \frac {a x}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {3 \sqrt {- \frac {a}{b^{5}}} \log {\left (- b^{2} \sqrt {- \frac {a}{b^{5}}} + x \right )}}{4} - \frac {3 \sqrt {- \frac {a}{b^{5}}} \log {\left (b^{2} \sqrt {- \frac {a}{b^{5}}} + x \right )}}{4} + \frac {x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.30, size = 42, normalized size = 0.76 \begin {gather*} -\frac {3 \, a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} + \frac {a x}{2 \, {\left (b x^{2} + a\right )} b^{2}} + \frac {x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.59, size = 43, normalized size = 0.78 \begin {gather*} \frac {x}{b^2}+\frac {a\,x}{2\,\left (b^3\,x^2+a\,b^2\right )}-\frac {3\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,b^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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