Optimal. Leaf size=41 \[ \frac {x^4}{4 a \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.68, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1125, 654, 621}
\begin {gather*} \frac {a}{4 b^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 621
Rule 654
Rule 1125
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {1}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a \text {Subst}\left (\int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,x^2\right )}{2 b}\\ &=-\frac {1}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a}{4 b^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.95 \begin {gather*} \frac {-a-2 b x^2}{4 b^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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Maple [A]
time = 0.02, size = 32, normalized size = 0.78
method | result | size |
gosper | \(-\frac {\left (b \,x^{2}+a \right ) \left (2 b \,x^{2}+a \right )}{4 b^{2} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}\) | \(32\) |
default | \(-\frac {\left (b \,x^{2}+a \right ) \left (2 b \,x^{2}+a \right )}{4 b^{2} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}\) | \(32\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {x^{2}}{2 b}-\frac {a}{4 b^{2}}\right )}{\left (b \,x^{2}+a \right )^{3}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 36, normalized size = 0.88 \begin {gather*} -\frac {2 \, b x^{2} + a}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 36, normalized size = 0.88 \begin {gather*} -\frac {2 \, b x^{2} + a}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\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 {x^{3}}{\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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Giac [A]
time = 4.63, size = 32, normalized size = 0.78 \begin {gather*} -\frac {2 \, b x^{2} + a}{4 \, {\left (b x^{2} + a\right )}^{2} b^{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 [B]
time = 4.24, size = 42, normalized size = 1.02 \begin {gather*} -\frac {\left (2\,b\,x^2+a\right )\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,b^2\,{\left (b\,x^2+a\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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