Optimal. Leaf size=18 \[ \frac {\log \left (\sqrt {a+b x^2}+1\right )}{b} \]
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Rubi [A] time = 0.07, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2155, 31} \[ \frac {\log \left (\sqrt {a+b x^2}+1\right )}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2155
Rubi steps
\begin {align*} \int \frac {x}{a+b x^2+\sqrt {a+b x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{a+b x+\sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt {a+b x^2}\right )}{b}\\ &=\frac {\log \left (1+\sqrt {a+b x^2}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 18, normalized size = 1.00 \[ \frac {\log \left (\sqrt {a+b x^2}+1\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 67, normalized size = 3.72 \[ \frac {2 \, \log \left (b x^{2} + a - 1\right ) + \log \left (\frac {b x^{2} + a + 2 \, \sqrt {b x^{2} + a} + 1}{x^{2}}\right ) - \log \left (\frac {b x^{2} + a - 2 \, \sqrt {b x^{2} + a} + 1}{x^{2}}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 16, normalized size = 0.89 \[ \frac {\log \left (\sqrt {b x^{2} + a} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1059, normalized size = 58.83 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 16, normalized size = 0.89 \[ \frac {\log \left (\sqrt {b x^{2} + a} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.38, size = 26, normalized size = 1.44 \[ \frac {\mathrm {atanh}\left (\sqrt {b\,x^2+a}\right )+\frac {\ln \left (b\,x^2+a-1\right )}{2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.39, size = 14, normalized size = 0.78 \[ \frac {\log {\left (\sqrt {a + b x^{2}} + 1 \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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