Optimal. Leaf size=198 \[ \frac {\sqrt {\sqrt {a x^2+b^2}+b} \left (a^{3/2} \left (210 b x^2 \sqrt {a x^2+b^2}-42 b^2 x^2\right )-315 a^{5/2} x^4+\sqrt {a} \left (16 b^4-144 b^3 \sqrt {a x^2+b^2}\right )\right )}{640 \sqrt {a} b^5 x^5}-\frac {63 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{64 \sqrt {2} b^{11/2}} \]
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Rubi [F] time = 0.68, 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 {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \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 {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx &=\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.25, size = 81, normalized size = 0.41 \begin {gather*} -\frac {a^3 x \, _2F_1\left (-\frac {5}{2},3;-\frac {3}{2};\frac {b-\sqrt {b^2+a x^2}}{2 b}\right )}{20 b^3 \left (\sqrt {a x^2+b^2}-b\right )^3 \sqrt {\sqrt {a x^2+b^2}+b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 166, normalized size = 0.84 \begin {gather*} \frac {\sqrt {\sqrt {a x^2+b^2}+b} \left (a^{3/2} \left (210 b x^2 \sqrt {a x^2+b^2}-42 b^2 x^2\right )-315 a^{5/2} x^4+\sqrt {a} \left (16 b^4-144 b^3 \sqrt {a x^2+b^2}\right )\right )}{640 \sqrt {a} b^5 x^5}-\frac {63 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{128 \sqrt {2} b^{11/2}} \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 {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{\sqrt {a x^{2} + b^{2}} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{x^{6} \sqrt {a \,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 {b + \sqrt {a x^{2} + b^{2}}}}{\sqrt {a x^{2} + b^{2}} x^{6}}\,{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 {b+\sqrt {b^2+a\,x^2}}}{x^6\,\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 [C] time = 1.99, size = 49, normalized size = 0.25 \begin {gather*} - \frac {\Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {5}{2}, \frac {1}{4}, \frac {3}{4} \\ - \frac {3}{2}, \frac {1}{2} \end {matrix}\middle | {\frac {a x^{2} e^{i \pi }}{b^{2}}} \right )}}{5 \pi \sqrt {b} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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