Optimal. Leaf size=71 \[ \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}-\frac {b \sqrt {a+b x^2}}{8 a x^2}-\frac {\sqrt {a+b x^2}}{4 x^4} \]
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Rubi [A] time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 208} \begin {gather*} \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}-\frac {b \sqrt {a+b x^2}}{8 a x^2}-\frac {\sqrt {a+b x^2}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2}}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {a+b x^2}}{4 x^4}+\frac {1}{8} b \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {a+b x^2}}{4 x^4}-\frac {b \sqrt {a+b x^2}}{8 a x^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{16 a}\\ &=-\frac {\sqrt {a+b x^2}}{4 x^4}-\frac {b \sqrt {a+b x^2}}{8 a x^2}-\frac {b \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{8 a}\\ &=-\frac {\sqrt {a+b x^2}}{4 x^4}-\frac {b \sqrt {a+b x^2}}{8 a x^2}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 39, normalized size = 0.55 \begin {gather*} -\frac {b^2 \left (a+b x^2\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {b x^2}{a}+1\right )}{3 a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 62, normalized size = 0.87 \begin {gather*} \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {\left (-2 a-b x^2\right ) \sqrt {a+b x^2}}{8 a x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 131, normalized size = 1.85 \begin {gather*} \left [\frac {\sqrt {a} b^{2} x^{4} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (a b x^{2} + 2 \, a^{2}\right )} \sqrt {b x^{2} + a}}{16 \, a^{2} x^{4}}, -\frac {\sqrt {-a} b^{2} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (a b x^{2} + 2 \, a^{2}\right )} \sqrt {b x^{2} + a}}{8 \, a^{2} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 72, normalized size = 1.01 \begin {gather*} -\frac {\frac {b^{3} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3} + \sqrt {b x^{2} + a} a b^{3}}{a b^{2} x^{4}}}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 85, normalized size = 1.20 \begin {gather*} \frac {b^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {3}{2}}}-\frac {\sqrt {b \,x^{2}+a}\, b^{2}}{8 a^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} b}{8 a^{2} x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 73, normalized size = 1.03 \begin {gather*} \frac {b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {3}{2}}} - \frac {\sqrt {b x^{2} + a} b^{2}}{8 \, a^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{4 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.91, size = 54, normalized size = 0.76 \begin {gather*} \frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{3/2}}-\frac {\sqrt {b\,x^2+a}}{8\,x^4}-\frac {{\left (b\,x^2+a\right )}^{3/2}}{8\,a\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.73, size = 92, normalized size = 1.30 \begin {gather*} - \frac {a}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {b^{\frac {3}{2}}}{8 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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