Optimal. Leaf size=72 \[ \frac {A b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {A \sqrt {a+b x^2}}{2 a x^2}-\frac {B \sqrt {a+b x^2}}{a x} \]
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Rubi [A] time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {835, 807, 266, 63, 208} \begin {gather*} \frac {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}-\frac {B \sqrt {a+b x^2}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rubi steps
\begin {align*} \int \frac {A+B x}{x^3 \sqrt {a+b x^2}} \, dx &=-\frac {A \sqrt {a+b x^2}}{2 a x^2}-\frac {\int \frac {-2 a B+A b x}{x^2 \sqrt {a+b x^2}} \, dx}{2 a}\\ &=-\frac {A \sqrt {a+b x^2}}{2 a x^2}-\frac {B \sqrt {a+b x^2}}{a x}-\frac {(A b) \int \frac {1}{x \sqrt {a+b x^2}} \, dx}{2 a}\\ &=-\frac {A \sqrt {a+b x^2}}{2 a x^2}-\frac {B \sqrt {a+b x^2}}{a x}-\frac {(A b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {A \sqrt {a+b x^2}}{2 a x^2}-\frac {B \sqrt {a+b x^2}}{a x}-\frac {A \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 a}\\ &=-\frac {A \sqrt {a+b x^2}}{2 a x^2}-\frac {B \sqrt {a+b x^2}}{a x}+\frac {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 = 0.14, size = 63, normalized size = 0.88 \begin {gather*} \frac {\sqrt {a+b x^2} \left (\frac {A b \tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )}{\sqrt {\frac {b x^2}{a}+1}}-\frac {a (A+2 B x)}{x^2}\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 71, normalized size = 0.99 \begin {gather*} \frac {\sqrt {a+b x^2} (-A-2 B x)}{2 a x^2}-\frac {A b \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 123, normalized size = 1.71 \begin {gather*} \left [\frac {A \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 \, {\left (2 \, B a x + A a\right )} \sqrt {b x^{2} + a}}{4 \, a^{2} x^{2}}, -\frac {A \sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (2 \, B a x + A a\right )} \sqrt {b x^{2} + a}}{2 \, a^{2} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 146, normalized size = 2.03 \begin {gather*} -\frac {A b \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b - 2 \, B a^{2} \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 0.94 \begin {gather*} \frac {A 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}\, B}{a x}-\frac {\sqrt {b \,x^{2}+a}\, A}{2 a \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 56, normalized size = 0.78 \begin {gather*} \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} B}{a x} - \frac {\sqrt {b x^{2} + a} A}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 58, normalized size = 0.81 \begin {gather*} \frac {A\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {B\,\sqrt {b\,x^2+a}}{a\,x}-\frac {A\,\sqrt {b\,x^2+a}}{2\,a\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.06, size = 66, normalized size = 0.92 \begin {gather*} - \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 \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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