Optimal. Leaf size=112 \[ \frac {4 \sqrt {b^2+a^2 x^2} \left (-b^2 x+a^2 x^3\right )}{3 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}+\frac {2 \left (-7 b^4-5 a^2 b^2 x^2+10 a^4 x^4\right )}{15 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}} \]
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Rubi [A]
time = 0.19, antiderivative size = 95, normalized size of antiderivative = 0.85, number of steps
used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2147, 276}
\begin {gather*} -\frac {b^2}{a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{6 a}-\frac {b^4}{10 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2147
Rubi steps
\begin {align*} \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (b^2+x^2\right )^2}{x^{7/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{4 a}\\ &=\frac {\text {Subst}\left (\int \left (\frac {b^4}{x^{7/2}}+\frac {2 b^2}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{4 a}\\ &=-\frac {b^4}{10 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{6 a}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 91, normalized size = 0.81 \begin {gather*} -\frac {2 \left (7 b^4-10 a^3 x^3 \left (a x+\sqrt {b^2+a^2 x^2}\right )+5 a b^2 x \left (a x+2 \sqrt {b^2+a^2 x^2}\right )\right )}{15 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}+b^{2}}}{\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 72, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left (3 \, a^{3} x^{3} + 11 \, a b^{2} x - {\left (3 \, a^{2} x^{2} + 7 \, b^{2}\right )} \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{15 \, a b^{2}} \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 {\sqrt {a^{2} x^{2} + b^{2}}}{\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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\sqrt {a^2\,x^2+b^2}}{\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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