Optimal. Leaf size=173 \[ \frac {2 x \sqrt {a x^2+b^2} \left (15 a x^2+46 b^2\right )}{105 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 \sqrt {2} b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 x \left (3 a b x^2+46 b^3\right )}{105 \sqrt {\sqrt {a x^2+b^2}+b}} \]
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Rubi [F] time = 0.24, 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 {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx &=\int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.64, size = 260, normalized size = 1.50 \begin {gather*} \frac {60 a^3 x^6+232 a^2 b^2 x^4+48 a^2 b x^4 \sqrt {a x^2+b^2}-210 a b^4 x^2+105 \sqrt {2} b^{7/2} \sqrt {\sqrt {a x^2+b^2}-b} \left (2 b \sqrt {a x^2+b^2}+a x^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a x^2+b^2}-b}}{\sqrt {2} \sqrt {b}}\right )-420 b^5 \sqrt {a x^2+b^2}+210 b^4 \left (2 b \sqrt {a x^2+b^2}+a x^2+2 b^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b-\sqrt {b^2+a x^2}}{2 b}\right )-420 b^6}{210 a x \left (\sqrt {a x^2+b^2}+b\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 140, normalized size = 0.81 \begin {gather*} \frac {2 x \sqrt {b^2+a x^2} \left (46 b^2+15 a x^2\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {2 x \left (46 b^3+3 a b x^2\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {\sqrt {2} b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {{\left (a x^{2} + b^{2}\right )}^{\frac {3}{2}}}{\sqrt {b + \sqrt {a 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 = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}+b^{2}\right )^{\frac {3}{2}}}{\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}\, dx\]
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*} \int \frac {{\left (a x^{2} + b^{2}\right )}^{\frac {3}{2}}}{\sqrt {b + \sqrt {a 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 {{\left (b^2+a\,x^2\right )}^{3/2}}{\sqrt {b+\sqrt {b^2+a\,x^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 {\left (a x^{2} + b^{2}\right )^{\frac {3}{2}}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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