Optimal. Leaf size=135 \[ \frac {2 \sqrt {a^2 x^2+b^2} \left (4 a^4 x^5+19 a^2 b^2 x^3+7 b^4 x\right )}{5 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}}+\frac {2 \left (28 a^6 x^6+147 a^4 b^2 x^4+112 a^2 b^4 x^2+9 b^6\right )}{35 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 130, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2122, 270} \begin {gather*} \frac {3 b^2 \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{4 a}+\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}}{20 a}-\frac {b^6}{28 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}}-\frac {b^4}{4 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 270
Rule 2122
Rubi steps
\begin {align*} \int \frac {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 )^3}{x^{9/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{8 a}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b^6}{x^{9/2}}+\frac {3 b^4}{x^{5/2}}+\frac {3 b^2}{\sqrt {x}}+x^{3/2}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{8 a}\\ &=-\frac {b^6}{28 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}-\frac {b^4}{4 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {3 b^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{4 a}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}{20 a}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 111, normalized size = 0.82 \begin {gather*} \frac {105 b^2 \left (\sqrt {a^2 x^2+b^2}+a x\right )^4+7 \left (\sqrt {a^2 x^2+b^2}+a x\right )^6-35 b^4 \left (\sqrt {a^2 x^2+b^2}+a x\right )^2-5 b^6}{140 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 135, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a^2 x^2+b^2} \left (4 a^4 x^5+19 a^2 b^2 x^3+7 b^4 x\right )}{5 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}}+\frac {2 \left (28 a^6 x^6+147 a^4 b^2 x^4+112 a^2 b^4 x^2+9 b^6\right )}{35 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 83, normalized size = 0.61 \begin {gather*} -\frac {2 \, {\left (5 \, a^{4} x^{4} + 12 \, a^{2} b^{2} x^{2} - 9 \, b^{4} - {\left (5 \, a^{3} x^{3} + 13 \, a b^{2} x\right )} \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{35 \, 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 {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 {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 {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 {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 {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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