Optimal. Leaf size=44 \[ -\frac {\sqrt {a+b x^4}}{6 a x^6}+\frac {b \sqrt {a+b x^4}}{3 a^2 x^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 44, 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 \sqrt {a+b x^4}}{3 a^2 x^2}-\frac {\sqrt {a+b x^4}}{6 a x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^7 \sqrt {a+b x^4}} \, dx &=-\frac {\sqrt {a+b x^4}}{6 a x^6}-\frac {(2 b) \int \frac {1}{x^3 \sqrt {a+b x^4}} \, dx}{3 a}\\ &=-\frac {\sqrt {a+b x^4}}{6 a x^6}+\frac {b \sqrt {a+b x^4}}{3 a^2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 31, normalized size = 0.70 \begin {gather*} \frac {\sqrt {a+b x^4} \left (-a+2 b x^4\right )}{6 a^2 x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 26, normalized size = 0.59
method | result | size |
gosper | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-2 b \,x^{4}+a \right )}{6 a^{2} x^{6}}\) | \(26\) |
default | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-2 b \,x^{4}+a \right )}{6 a^{2} x^{6}}\) | \(26\) |
trager | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-2 b \,x^{4}+a \right )}{6 a^{2} x^{6}}\) | \(26\) |
risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-2 b \,x^{4}+a \right )}{6 a^{2} x^{6}}\) | \(26\) |
elliptic | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-2 b \,x^{4}+a \right )}{6 a^{2} x^{6}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 35, normalized size = 0.80 \begin {gather*} \frac {\frac {3 \, \sqrt {b x^{4} + a} b}{x^{2}} - \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}}}{x^{6}}}{6 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 27, normalized size = 0.61 \begin {gather*} \frac {{\left (2 \, b x^{4} - a\right )} \sqrt {b x^{4} + a}}{6 \, a^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.48, size = 44, normalized size = 1.00 \begin {gather*} - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{4}} + 1}}{6 a x^{4}} + \frac {b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{4}} + 1}}{3 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.78, size = 59, normalized size = 1.34 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {b} x^{2} - \sqrt {b x^{4} + a}\right )}^{2} - a\right )} b^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {b} x^{2} - \sqrt {b x^{4} + a}\right )}^{2} - a\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 25, normalized size = 0.57 \begin {gather*} -\frac {\sqrt {b\,x^4+a}\,\left (a-2\,b\,x^4\right )}{6\,a^2\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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