Optimal. Leaf size=84 \[ \frac {\sqrt {a+b x^2} (2 a B+A b)}{2 a}-\frac {(2 a B+A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {A \left (a+b x^2\right )^{3/2}}{2 a x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 50, 63, 208} \begin {gather*} \frac {\sqrt {a+b x^2} (2 a B+A b)}{2 a}-\frac {(2 a B+A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {A \left (a+b x^2\right )^{3/2}}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x} (A+B x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^{3/2}}{2 a x^2}+\frac {(A b+2 a B) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )}{4 a}\\ &=\frac {(A b+2 a B) \sqrt {a+b x^2}}{2 a}-\frac {A \left (a+b x^2\right )^{3/2}}{2 a x^2}+\frac {1}{4} (A b+2 a B) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {(A b+2 a B) \sqrt {a+b x^2}}{2 a}-\frac {A \left (a+b x^2\right )^{3/2}}{2 a x^2}+\frac {(A b+2 a B) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 b}\\ &=\frac {(A b+2 a B) \sqrt {a+b x^2}}{2 a}-\frac {A \left (a+b x^2\right )^{3/2}}{2 a x^2}-\frac {(A b+2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 0.75 \begin {gather*} \frac {1}{2} \left (\frac {\sqrt {a+b x^2} \left (2 B x^2-A\right )}{x^2}-\frac {(2 a B+A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 65, normalized size = 0.77 \begin {gather*} \frac {\sqrt {a+b x^2} \left (2 B x^2-A\right )}{2 x^2}+\frac {(-2 a B-A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 141, normalized size = 1.68 \begin {gather*} \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 \, {\left (2 \, B a x^{2} - A a\right )} \sqrt {b x^{2} + a}}{4 \, a 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 ) + {\left (2 \, B a x^{2} - A a\right )} \sqrt {b x^{2} + a}}{2 \, a x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 68, normalized size = 0.81 \begin {gather*} \frac {2 \, \sqrt {b x^{2} + a} B b + \frac {{\left (2 \, B a b + A b^{2}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {b x^{2} + a} A b}{x^{2}}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 106, normalized size = 1.26 \begin {gather*} -\frac {A b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 \sqrt {a}}-B \sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )+\frac {\sqrt {b \,x^{2}+a}\, A b}{2 a}+\sqrt {b \,x^{2}+a}\, B -\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A}{2 a \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 83, normalized size = 0.99 \begin {gather*} -B \sqrt {a} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) - \frac {A b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, \sqrt {a}} + \sqrt {b x^{2} + a} B + \frac {\sqrt {b x^{2} + a} A b}{2 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} 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 = 68, normalized size = 0.81 \begin {gather*} B\,\sqrt {b\,x^2+a}-\frac {A\,\sqrt {b\,x^2+a}}{2\,x^2}-B\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )-\frac {A\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 43.70, size = 107, normalized size = 1.27 \begin {gather*} - \frac {A \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {A b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 \sqrt {a}} - B \sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {B a}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B \sqrt {b} x}{\sqrt {\frac {a}{b x^{2}} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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