Optimal. Leaf size=147 \[ \frac {1}{2 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) \log (x)}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A]
time = 0.05, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1126, 272, 46}
\begin {gather*} \frac {1}{4 a \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{2 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\log (x) \left (a+b x^2\right )}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 1126
Rubi steps
\begin {align*} \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x \left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \left (a b+b^2 x\right )^3} \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \left (\frac {1}{a^3 b^3 x}-\frac {1}{a b^2 (a+b x)^3}-\frac {1}{a^2 b^2 (a+b x)^2}-\frac {1}{a^3 b^2 (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{2 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) \log (x)}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 74, normalized size = 0.50 \begin {gather*} \frac {a \left (3 a+2 b x^2\right )+4 \left (a+b x^2\right )^2 \log (x)-2 \left (a+b x^2\right )^2 \log \left (a+b x^2\right )}{4 a^3 \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 = 0.03, size = 107, normalized size = 0.73
method | result | size |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (\frac {b \,x^{2}}{2 a^{2}}+\frac {3}{4 a}\right )}{\left (b \,x^{2}+a \right )^{3}}+\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \ln \left (x \right )}{\left (b \,x^{2}+a \right ) a^{3}}-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \ln \left (b \,x^{2}+a \right )}{2 \left (b \,x^{2}+a \right ) a^{3}}\) | \(97\) |
default | \(-\frac {\left (2 \ln \left (b \,x^{2}+a \right ) b^{2} x^{4}-4 \ln \left (x \right ) b^{2} x^{4}+4 \ln \left (b \,x^{2}+a \right ) a b \,x^{2}-8 \ln \left (x \right ) a b \,x^{2}-2 a b \,x^{2}+2 a^{2} \ln \left (b \,x^{2}+a \right )-4 a^{2} \ln \left (x \right )-3 a^{2}\right ) \left (b \,x^{2}+a \right )}{4 a^{3} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 57, normalized size = 0.39 \begin {gather*} \frac {2 \, b x^{2} + 3 \, a}{4 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} - \frac {\log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac {\log \left (x\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 90, normalized size = 0.61 \begin {gather*} \frac {2 \, a b x^{2} + 3 \, a^{2} - 2 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\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 {1}{x \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 = 89, normalized size = 0.61 \begin {gather*} \frac {\log \left (x^{2}\right )}{2 \, a^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3 \, b^{2} x^{4} + 8 \, a b x^{2} + 6 \, a^{2}}{4 \, {\left (b x^{2} + a\right )}^{2} a^{3} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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