Optimal. Leaf size=58 \[ \frac {(A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {A \sqrt {a+b x^2}}{2 a x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {446, 78, 63, 208} \[ \frac {(A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {A \sqrt {a+b x^2}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^3 \sqrt {a+b x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=-\frac {A \sqrt {a+b x^2}}{2 a x^2}+\frac {\left (-\frac {A b}{2}+a B\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {A \sqrt {a+b x^2}}{2 a x^2}+\frac {\left (-\frac {A b}{2}+a B\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{a b}\\ &=-\frac {A \sqrt {a+b x^2}}{2 a x^2}+\frac {(A b-2 a 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 = 0.03, size = 60, normalized size = 1.03 \[ \frac {1}{2} \left (-\frac {2 \left (a B-\frac {A b}{2}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {A \sqrt {a+b x^2}}{a x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 124, normalized size = 2.14 \[ \left [-\frac {{\left (2 \, B a - A b\right )} \sqrt {a} x^{2} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, \sqrt {b x^{2} + a} A a}{4 \, a^{2} x^{2}}, \frac {{\left (2 \, B a - A b\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - \sqrt {b x^{2} + a} A a}{2 \, a^{2} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 62, normalized size = 1.07 \[ \frac {\frac {{\left (2 \, B a b - A b^{2}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {\sqrt {b x^{2} + a} A b}{a x^{2}}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 79, normalized size = 1.36 \[ \frac {A b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {3}{2}}}-\frac {B \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{\sqrt {a}}-\frac {\sqrt {b \,x^{2}+a}\, A}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 56, normalized size = 0.97 \[ -\frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{\sqrt {a}} + \frac {A b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {\sqrt {b x^{2} + a} A}{2 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 60, normalized size = 1.03 \[ \frac {A\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {A\,\sqrt {b\,x^2+a}}{2\,a\,x^2}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 39.42, size = 66, normalized size = 1.14 \[ - \frac {A \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 a x} + \frac {A b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {3}{2}}} - \frac {B \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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