Optimal. Leaf size=112 \[ \frac {4 \sqrt {a^2 x^2+b^2} \left (a^2 x^3-b^2 x\right )}{3 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}}+\frac {2 \left (10 a^4 x^4-5 a^2 b^2 x^2-7 b^4\right )}{15 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}} \]
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Rubi [A] time = 0.13, 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 = {2122, 270} \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 270
Rule 2122
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 {\operatorname {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 {\operatorname {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 = 0.06, size = 93, normalized size = 0.83 \begin {gather*} \frac {-\frac {4 b^2}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {2}{3} \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}-\frac {2 b^4}{5 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}}}{4 a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 112, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt {a^2 x^2+b^2} \left (a^2 x^3-b^2 x\right )}{3 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}}+\frac {2 \left (10 a^4 x^4-5 a^2 b^2 x^2-7 b^4\right )}{15 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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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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giac [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}}}}\,{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 {\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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maxima [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}}}}\,{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 {\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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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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