Optimal. Leaf size=75 \[ -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1126, 14}
\begin {gather*} \frac {b \log (x) \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1126
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^3} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {a b+b^2 x^2}{x^3} \, dx}{a b+b^2 x^2}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (\frac {a b}{x^3}+\frac {b^2}{x}\right ) \, dx}{a b+b^2 x^2}\\ &=-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.52 \begin {gather*} -\frac {\sqrt {\left (a+b x^2\right )^2} \left (a-2 b x^2 \log (x)\right )}{2 x^2 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 38, normalized size = 0.51
method | result | size |
default | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (2 b \ln \left (x \right ) x^{2}-a \right )}{2 x^{2} \left (b \,x^{2}+a \right )}\) | \(38\) |
risch | \(-\frac {a \sqrt {\left (b \,x^{2}+a \right )^{2}}}{2 x^{2} \left (b \,x^{2}+a \right )}+\frac {b \ln \left (x \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b \,x^{2}+a}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 14, normalized size = 0.19 \begin {gather*} \frac {1}{2} \, b \log \left (x^{2}\right ) - \frac {a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 17, normalized size = 0.23 \begin {gather*} \frac {2 \, b x^{2} \log \left (x\right ) - a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 10, normalized size = 0.13 \begin {gather*} - \frac {a}{2 x^{2}} + b \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.60, size = 45, normalized size = 0.60 \begin {gather*} \frac {1}{2} \, b \log \left (x^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a \mathrm {sgn}\left (b x^{2} + a\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.45, size = 112, normalized size = 1.49 \begin {gather*} \frac {\ln \left (a\,b+\sqrt {{\left (b\,x^2+a\right )}^2}\,\sqrt {b^2}+b^2\,x^2\right )\,\sqrt {b^2}}{2}-\frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2\,x^2}-\frac {a\,b\,\ln \left (a\,b+\frac {a^2}{x^2}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{x^2}\right )}{2\,\sqrt {a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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