Optimal. Leaf size=80 \[ -\frac {\sqrt {a+b x^2} (A+2 B x)}{2 x^2}-\frac {A b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}}+\sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {811, 844, 217, 206, 266, 63, 208} \begin {gather*} -\frac {\sqrt {a+b x^2} (A+2 B x)}{2 x^2}-\frac {A b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}}+\sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 811
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+b x^2}}{x^3} \, dx &=-\frac {(A+2 B x) \sqrt {a+b x^2}}{2 x^2}-\frac {\int \frac {-2 a A b-4 a b B x}{x \sqrt {a+b x^2}} \, dx}{4 a}\\ &=-\frac {(A+2 B x) \sqrt {a+b x^2}}{2 x^2}+\frac {1}{2} (A b) \int \frac {1}{x \sqrt {a+b x^2}} \, dx+(b B) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=-\frac {(A+2 B x) \sqrt {a+b x^2}}{2 x^2}+\frac {1}{4} (A b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )+(b B) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=-\frac {(A+2 B x) \sqrt {a+b x^2}}{2 x^2}+\sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {1}{2} A \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )\\ &=-\frac {(A+2 B x) \sqrt {a+b x^2}}{2 x^2}+\sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {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 = 0.09, size = 108, normalized size = 1.35 \begin {gather*} -\frac {\sqrt {a+b x^2} \left (a \sqrt {\frac {b x^2}{a}+1} (A+2 B x)+A b x^2 \tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )-2 \sqrt {a} \sqrt {b} B x^2 \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right )}{2 a x^2 \sqrt {\frac {b x^2}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 96, normalized size = 1.20 \begin {gather*} \frac {\sqrt {a+b x^2} (-A-2 B x)}{2 x^2}+\frac {A b \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\sqrt {b} B \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 377, normalized size = 4.71 \begin {gather*} \left [\frac {2 \, B a \sqrt {b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 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 x^{2}}, -\frac {4 \, B a \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 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 x^{2}}, \frac {A \sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + B a \sqrt {b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - {\left (2 \, B a x + A a\right )} \sqrt {b x^{2} + a}}{2 \, a x^{2}}, -\frac {2 \, B a \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 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 x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 163, normalized size = 2.04 \begin {gather*} \frac {A b \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - B \sqrt {b} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 121, normalized size = 1.51 \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 {b}\, \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+\frac {\sqrt {b \,x^{2}+a}\, B b x}{a}+\frac {\sqrt {b \,x^{2}+a}\, A b}{2 a}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B}{a x}-\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 = 1.36, size = 83, normalized size = 1.04 \begin {gather*} B \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {A b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, \sqrt {a}} + \frac {\sqrt {b x^{2} + a} A b}{2 \, a} - \frac {\sqrt {b x^{2} + a} B}{x} - \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 = 1.79, size = 94, normalized size = 1.18 \begin {gather*} -\frac {A\,\sqrt {b\,x^2+a}}{2\,x^2}-\frac {B\,\sqrt {b\,x^2+a}}{x}-\frac {A\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,\sqrt {a}}-\frac {B\,\sqrt {b}\,\mathrm {asin}\left (\frac {\sqrt {b}\,x\,1{}\mathrm {i}}{\sqrt {a}}\right )\,\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}\,\sqrt {\frac {b\,x^2}{a}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.50, size = 107, normalized size = 1.34 \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}} - \frac {B \sqrt {a}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + B \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {B b x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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