Optimal. Leaf size=198 \[ \frac {7 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{64 \sqrt {2} b^{9/2}}+\frac {\sqrt {\sqrt {a x^2+b^2}+b} \left (a^{3/2} \left (14 b^2 x^2-70 b x^2 \sqrt {a x^2+b^2}\right )+105 a^{5/2} x^4+\sqrt {a} \left (48 b^3 \sqrt {a x^2+b^2}-432 b^4\right )\right )}{1920 \sqrt {a} b^4 x^5} \]
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Rubi [F] time = 0.15, 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} \, 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} \, dx &=\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6} \, dx\\ \end {align*}
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Mathematica [C] time = 0.25, size = 95, normalized size = 0.48 \begin {gather*} \frac {\sqrt {\sqrt {a x^2+b^2}+b} \left (\left (2 b \sqrt {a x^2+b^2}+a x^2+2 b^2\right ) \, _2F_1\left (-\frac {5}{2},2;-\frac {3}{2};\frac {b-\sqrt {b^2+a x^2}}{2 b}\right )-20 b^2\right )}{80 b^2 x^5} \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 {b+\sqrt {b^2+a x^2}} \left (105 a^{5/2} x^4+\sqrt {a} \left (-432 b^4+48 b^3 \sqrt {b^2+a x^2}\right )+a^{3/2} \left (14 b^2 x^2-70 b x^2 \sqrt {b^2+a x^2}\right )\right )}{1920 \sqrt {a} b^4 x^5}+\frac {7 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{128 \sqrt {2} b^{9/2}} \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}}}}{x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 31, normalized size = 0.16
method | result | size |
meijerg | \(-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \hypergeom \left (\left [-\frac {5}{2}, -\frac {1}{4}, \frac {1}{4}\right ], \left [-\frac {3}{2}, \frac {1}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{5 x^{5}}\) | \(31\) |
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}}}}{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} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.58, size = 51, normalized size = 0.26 \begin {gather*} \frac {\sqrt {b} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {5}{2}, - \frac {1}{4}, \frac {1}{4} \\ - \frac {3}{2}, \frac {1}{2} \end {matrix}\middle | {\frac {a x^{2} e^{i \pi }}{b^{2}}} \right )}}{20 \pi x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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