Optimal. Leaf size=50 \[ \frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {\sqrt {a+b x^2}}{2 a x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {5, 266, 51, 63, 208} \[ \frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {\sqrt {a+b x^2}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 5
Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac {1}{x^3 \sqrt {a+b x^2}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {a+b x^2}}{2 a x^2}-\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {\sqrt {a+b x^2}}{2 a x^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 a}\\ &=-\frac {\sqrt {a+b x^2}}{2 a x^2}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 1.22 \[ \frac {b \sqrt {a+b x^2} \left (\frac {\tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )}{2 \sqrt {\frac {b x^2}{a}+1}}-\frac {a}{2 b x^2}\right )}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 105, normalized size = 2.10 \[ \left [\frac {\sqrt {a} b x^{2} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, \sqrt {b x^{2} + a} a}{4 \, a^{2} x^{2}}, -\frac {\sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + \sqrt {b x^{2} + a} a}{2 \, a^{2} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 51, normalized size = 1.02 \[ -\frac {\frac {b^{2} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {\sqrt {b x^{2} + a} b}{a x^{2}}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 0.96 \[ \frac {b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {3}{2}}}-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 36, normalized size = 0.72 \[ \frac {b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {\sqrt {b x^{2} + a}}{2 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.54, size = 38, normalized size = 0.76 \[ \frac {b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {\sqrt {b\,x^2+a}}{2\,a\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.47, size = 42, normalized size = 0.84 \[ - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 a x} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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