Optimal. Leaf size=168 \[ \sqrt {a^2 x^2+b^2} \left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{3 a}-\frac {b}{2 a}\right )+\frac {\left (a x+3 b^2\right ) \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{3 a}+\frac {b \log \left (\sqrt {a^2 x^2+b^2}+a x\right )}{2 a}+\frac {\left (-b^3-b\right ) \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}+b\right )}{a}-\frac {b x}{2} \]
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Rubi [F] time = 13.17, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {a x+\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a x+\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\int \left (\frac {a x}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx\\ &=a \int \frac {x}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx+\int \frac {\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\\ &=a \int \left (\frac {-1-b^2}{4 a b}+\frac {x}{2 b}+\frac {-1+b^4}{4 a b \left (-1+b^2-2 a x\right )}-\frac {\sqrt {b^2+a^2 x^2}}{2 a b}+\frac {\left (1-b^2\right ) \sqrt {b^2+a^2 x^2}}{2 a b \left (1-b^2+2 a x\right )}+\frac {\left (1+\frac {1}{b^2}\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{4 a}-\frac {x \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 b^2}-\frac {\left (1-b^4\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{4 a b^2 \left (1-b^2+2 a x\right )}+\frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 a b^2}+\frac {\left (1-\frac {1}{b^2}\right ) \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 a \left (1-b^2+2 a x\right )}\right ) \, dx+\int \left (\frac {a x \sqrt {b^2+a^2 x^2}}{b (1+2 a x)}+\frac {b (1+a x) \sqrt {b^2+a^2 x^2}}{\left (-1+b^2-2 a x\right ) (1+2 a x)}-\frac {b^2+a^2 x^2}{b (1+2 a x)}+\frac {b \left (b^2+a^2 x^2\right )}{\left (-1+b^2-2 a x\right ) (1+2 a x)}-\frac {a x \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 (1+2 a x)}-\frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\left (-1+b^2-2 a x\right ) (1+2 a x)}+\frac {a x \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{(1+2 a x) \left (1-b^2+2 a x\right )}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 (1+2 a x)}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{(1+2 a x) \left (1-b^2+2 a x\right )}\right ) \, dx\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}
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Mathematica [B] time = 3.79, size = 423, normalized size = 2.52 \begin {gather*} \frac {b \left (-2 \sqrt {a^2 x^2+b^2}+b^2 \log \left (b^2 \left (\sqrt {a^2 x^2+b^2}+a x+2\right )+\sqrt {a^2 x^2+b^2}-a x\right )-\left (b^2-1\right ) \log \left (\sqrt {a^2 x^2+b^2}+a x\right )+\log \left (b^2 \left (\sqrt {a^2 x^2+b^2}+a x+2\right )+\sqrt {a^2 x^2+b^2}-a x\right )-2 \left (b^2+1\right ) \log \left (2 a x-b^2+1\right )-2 a x\right )}{4 a}+\frac {a b^2 x \left (a x \left (6 \sqrt {a^2 x^2+b^2}+5\right )+3 \sqrt {a^2 x^2+b^2}+6 a^2 x^2\right )-3 \left (b^2+1\right ) b \sqrt {\sqrt {a^2 x^2+b^2}+a x} \left (a x \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{b}\right )+b^4 \left (3 \sqrt {a^2 x^2+b^2}+6 a x+1\right )+4 a^3 x^3 \left (\sqrt {a^2 x^2+b^2}+a x\right )}{3 a \sqrt {\sqrt {a^2 x^2+b^2}+a x} \left (a x \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 168, normalized size = 1.00 \begin {gather*} \sqrt {a^2 x^2+b^2} \left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{3 a}-\frac {b}{2 a}\right )+\frac {\left (a x+3 b^2\right ) \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{3 a}+\frac {b \log \left (\sqrt {a^2 x^2+b^2}+a x\right )}{2 a}+\frac {\left (-b^3-b\right ) \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}+b\right )}{a}-\frac {b x}{2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 122, normalized size = 0.73 \begin {gather*} -\frac {3 \, a b x + 6 \, {\left (b^{3} + b\right )} \log \left (b + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) - 6 \, b \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) - 2 \, {\left (3 \, b^{2} + a x + \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} + 3 \, \sqrt {a^{2} x^{2} + b^{2}} b}{6 \, a} \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 {a x + \sqrt {a^{2} x^{2} + b^{2}}}{b + \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] time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x +\sqrt {a^{2} x^{2}+b^{2}}}{b +\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*} \frac {a x^{2}}{4 \, b} + \frac {\frac {b^{2} \operatorname {arsinh}\left (\frac {a x}{b}\right )}{2 \, a} + \frac {1}{2} \, \sqrt {a^{2} x^{2} + b^{2}} x}{2 \, b} - \int -\frac {a b^{2} x - 2 \, a^{2} x^{2} - b^{2} + \sqrt {a^{2} x^{2} + b^{2}} {\left (b^{2} - 2 \, a x\right )}}{2 \, {\left (b^{3} + a b x + 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} b^{2} + \sqrt {a^{2} x^{2} + b^{2}} b\right )}}\,{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 = -0.01 \begin {gather*} \int \frac {a\,x+\sqrt {a^2\,x^2+b^2}}{b+\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 {a x + \sqrt {a^{2} x^{2} + b^{2}}}{b + \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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