Optimal. Leaf size=135 \[ \frac {x \left (a+b x^2\right )}{4 a \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac {3 x \left (a+b x^2\right )^2}{8 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac {3 \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1102, 205, 211}
\begin {gather*} \frac {3 x \left (a+b x^2\right )^2}{8 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac {x \left (a+b x^2\right )}{4 a \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac {3 \left (a+b x^2\right )^3 \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 211
Rule 1102
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {\left (2 a b+2 b^2 x^2\right )^3 \int \frac {1}{\left (2 a b+2 b^2 x^2\right )^3} \, dx}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}\\ &=\frac {x \left (a+b x^2\right )}{4 a \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac {\left (3 \left (2 a b+2 b^2 x^2\right )^3\right ) \int \frac {1}{\left (2 a b+2 b^2 x^2\right )^2} \, dx}{8 a b \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}\\ &=\frac {x \left (a+b x^2\right )}{4 a \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac {3 x \left (a+b x^2\right )^2}{8 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac {\left (3 \left (2 a b+2 b^2 x^2\right )^3\right ) \int \frac {1}{2 a b+2 b^2 x^2} \, dx}{32 a^2 b^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}\\ &=\frac {x \left (a+b x^2\right )}{4 a \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac {3 x \left (a+b x^2\right )^2}{8 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac {3 \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 83, normalized size = 0.61 \begin {gather*} \frac {\sqrt {a} \sqrt {b} x \left (5 a+3 b x^2\right )+3 \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 97, normalized size = 0.72
method | result | size |
default | \(\frac {\left (3 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) b^{2} x^{4}+3 \sqrt {a b}\, b \,x^{3}+6 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a b \,x^{2}+5 \sqrt {a b}\, a x +3 a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )\right ) \left (b \,x^{2}+a \right )}{8 \sqrt {a b}\, a^{2} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}\) | \(97\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (\frac {3 b \,x^{3}}{8 a^{2}}+\frac {5 x}{8 a}\right )}{\left (b \,x^{2}+a \right )^{3}}-\frac {3 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \ln \left (b x +\sqrt {-a b}\right )}{16 \left (b \,x^{2}+a \right ) \sqrt {-a b}\, a^{2}}+\frac {3 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \ln \left (-b x +\sqrt {-a b}\right )}{16 \left (b \,x^{2}+a \right ) \sqrt {-a b}\, a^{2}}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 58, normalized size = 0.43 \begin {gather*} \frac {3 \, b x^{3} + 5 \, a x}{8 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} + \frac {3 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 188, normalized size = 1.39 \begin {gather*} \left [\frac {6 \, a b^{2} x^{3} + 10 \, a^{2} b x - 3 \, {\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^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}}, \frac {3 \, a b^{2} x^{3} + 5 \, a^{2} b x + 3 \, {\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^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}}\right ] \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 {1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.18, size = 65, normalized size = 0.48 \begin {gather*} \frac {3 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3 \, b x^{3} + 5 \, a x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \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 {1}{{\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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