Optimal. Leaf size=72 \[ -\frac {\left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 a x^6}+\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{12 a^2 x^6} \]
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Rubi [A]
time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1124}
\begin {gather*} \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{12 a^2 x^6}-\frac {\left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 a x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 1124
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^7} \, dx &=-\frac {\left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 a x^6}+\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{12 a^2 x^6}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.54 \begin {gather*} -\frac {\sqrt {\left (a+b x^2\right )^2} \left (2 a+3 b x^2\right )}{12 x^6 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 36, normalized size = 0.50
method | result | size |
risch | \(\frac {\left (-\frac {b \,x^{2}}{4}-\frac {a}{6}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{x^{6} \left (b \,x^{2}+a \right )}\) | \(35\) |
gosper | \(-\frac {\left (3 b \,x^{2}+2 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{12 x^{6} \left (b \,x^{2}+a \right )}\) | \(36\) |
default | \(-\frac {\left (3 b \,x^{2}+2 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{12 x^{6} \left (b \,x^{2}+a \right )}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 15, normalized size = 0.21 \begin {gather*} -\frac {3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 15, normalized size = 0.21 \begin {gather*} -\frac {3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 15, normalized size = 0.21 \begin {gather*} \frac {- 2 a - 3 b x^{2}}{12 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.32, size = 31, normalized size = 0.43 \begin {gather*} -\frac {3 \, b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, a \mathrm {sgn}\left (b x^{2} + a\right )}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.24, size = 35, normalized size = 0.49 \begin {gather*} -\frac {\left (3\,b\,x^2+2\,a\right )\,\sqrt {{\left (b\,x^2+a\right )}^2}}{12\,x^6\,\left (b\,x^2+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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