Optimal. Leaf size=129 \[ \frac {x}{8 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {x}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.05, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1112, 288, 199, 205} \begin {gather*} \frac {x}{8 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {x}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 288
Rule 1112
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {x^2}{\left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {x}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a b+b^2 x^2\right ) \int \frac {1}{\left (a b+b^2 x^2\right )^2} \, dx}{4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {x}{8 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {x}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a b+b^2 x^2\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{8 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {x}{8 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {x}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 81, normalized size = 0.63 \begin {gather*} \frac {\sqrt {a} \sqrt {b} x \left (b x^2-a\right )+\left (a+b x^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{3/2} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.91, size = 77, normalized size = 0.60 \begin {gather*} \frac {\left (a+b x^2\right ) \left (\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{3/2}}-\frac {x \left (a-b x^2\right )}{8 a b \left (a+b x^2\right )^2}\right )}{\sqrt {\left (a+b x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 190, normalized size = 1.47 \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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 97, normalized size = 0.75 \begin {gather*} \frac {\left (b^{2} x^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+2 a b \,x^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+\sqrt {a b}\, b \,x^{3}+a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )-\sqrt {a b}\, a x \right ) \left (b \,x^{2}+a \right )}{8 \sqrt {a b}\, \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 62, normalized size = 0.48 \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 {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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