Optimal. Leaf size=61 \[ \frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a}-\frac {b^2}{3 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2117, 14} \begin {gather*} \frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a}-\frac {b^2}{3 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2117
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^2+x^2}{x^{5/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b^2}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 57, normalized size = 0.93 \begin {gather*} \frac {2 \left (3 a x \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^2\right )}{3 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 61, normalized size = 1.00 \begin {gather*} \frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a}-\frac {b^2}{3 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 57, normalized size = 0.93 \begin {gather*} -\frac {2 \, {\left (a^{2} x^{2} - \sqrt {a^{2} x^{2} + b^{2}} a x - b^{2}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{3 \, a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\,{d x} \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 = -0.02 \begin {gather*} \int \frac {1}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \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}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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