Optimal. Leaf size=42 \[ -\frac {1}{2 a x^2 \sqrt {a+b x^4}}-\frac {b x^2}{a^2 \sqrt {a+b x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270}
\begin {gather*} -\frac {b x^2}{a^2 \sqrt {a+b x^4}}-\frac {1}{2 a x^2 \sqrt {a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^4\right )^{3/2}} \, dx &=-\frac {1}{2 a x^2 \sqrt {a+b x^4}}-\frac {(2 b) \int \frac {x}{\left (a+b x^4\right )^{3/2}} \, dx}{a}\\ &=-\frac {1}{2 a x^2 \sqrt {a+b x^4}}-\frac {b x^2}{a^2 \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 31, normalized size = 0.74 \begin {gather*} \frac {-a-2 b x^4}{2 a^2 x^2 \sqrt {a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 26, normalized size = 0.62
method | result | size |
gosper | \(-\frac {2 b \,x^{4}+a}{2 x^{2} \sqrt {b \,x^{4}+a}\, a^{2}}\) | \(26\) |
default | \(-\frac {2 b \,x^{4}+a}{2 x^{2} \sqrt {b \,x^{4}+a}\, a^{2}}\) | \(26\) |
trager | \(-\frac {2 b \,x^{4}+a}{2 x^{2} \sqrt {b \,x^{4}+a}\, a^{2}}\) | \(26\) |
elliptic | \(-\frac {2 b \,x^{4}+a}{2 x^{2} \sqrt {b \,x^{4}+a}\, a^{2}}\) | \(26\) |
risch | \(-\frac {\sqrt {b \,x^{4}+a}}{2 a^{2} x^{2}}-\frac {b \,x^{2}}{2 a^{2} \sqrt {b \,x^{4}+a}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 36, normalized size = 0.86 \begin {gather*} -\frac {b x^{2}}{2 \, \sqrt {b x^{4} + a} a^{2}} - \frac {\sqrt {b x^{4} + a}}{2 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 37, normalized size = 0.88 \begin {gather*} -\frac {{\left (2 \, b x^{4} + a\right )} \sqrt {b x^{4} + a}}{2 \, {\left (a^{2} b x^{6} + a^{3} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.43, size = 46, normalized size = 1.10 \begin {gather*} - \frac {1}{2 a \sqrt {b} x^{4} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {\sqrt {b}}{a^{2} \sqrt {\frac {a}{b x^{4}} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 53, normalized size = 1.26 \begin {gather*} -\frac {b x^{2}}{2 \, \sqrt {b x^{4} + a} a^{2}} + \frac {\sqrt {b}}{{\left ({\left (\sqrt {b} x^{2} - \sqrt {b x^{4} + a}\right )}^{2} - a\right )} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 25, normalized size = 0.60 \begin {gather*} -\frac {2\,b\,x^4+a}{2\,a^2\,x^2\,\sqrt {b\,x^4+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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