Integrand size = 20, antiderivative size = 80 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^3} \, dx=-\frac {(A+2 B x) \sqrt {a+b x^2}}{2 x^2}+\sqrt {b} B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {A b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]
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Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {825, 858, 223, 212, 272, 65, 214} \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^3} \, dx=-\frac {A b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {\sqrt {a+b x^2} (A+2 B x)}{2 x^2}+\sqrt {b} B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
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Rule 65
Rule 212
Rule 214
Rule 223
Rule 272
Rule 825
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\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) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )+(b B) \text {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 \text {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*}
Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.12 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^3} \, dx=-\frac {(A+2 B x) \sqrt {a+b x^2}}{2 x^2}+\frac {A b \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\sqrt {b} B \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \]
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Time = 3.43 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91
method | result | size |
risch | \(-\frac {\left (2 B x +A \right ) \sqrt {b \,x^{2}+a}}{2 x^{2}}+\sqrt {b}\, B \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )-\frac {b A \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 \sqrt {a}}\) | \(73\) |
default | \(B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{a x}+\frac {2 b \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{a}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )\) | \(127\) |
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Time = 0.28 (sec) , antiderivative size = 377, normalized size of antiderivative = 4.71 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^3} \, dx=\left [\frac {2 \, B a \sqrt {b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + A \sqrt {a} b x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (2 \, B a x + A a\right )} \sqrt {b x^{2} + a}}{4 \, a x^{2}}, -\frac {4 \, B a \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - A \sqrt {a} b x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, B a x + A a\right )} \sqrt {b x^{2} + a}}{4 \, a x^{2}}, \frac {A \sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + B a \sqrt {b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - {\left (2 \, B a x + A a\right )} \sqrt {b x^{2} + a}}{2 \, a x^{2}}, -\frac {2 \, B a \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - A \sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (2 \, B a x + A a\right )} \sqrt {b x^{2} + a}}{2 \, a x^{2}}\right ] \]
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Time = 1.99 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^3} \, dx=- \frac {A \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {A b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 \sqrt {a}} - \frac {B \sqrt {a}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + B \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {B b x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
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Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^3} \, dx=B \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {A b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, \sqrt {a}} + \frac {\sqrt {b x^{2} + a} A b}{2 \, a} - \frac {\sqrt {b x^{2} + a} B}{x} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{2 \, a x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (62) = 124\).
Time = 0.31 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.04 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^3} \, dx=\frac {A b \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - B \sqrt {b} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b - 2 \, B a^{2} \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2}} \]
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Time = 6.45 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.18 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^3} \, dx=-\frac {A\,\sqrt {b\,x^2+a}}{2\,x^2}-\frac {B\,\sqrt {b\,x^2+a}}{x}-\frac {A\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,\sqrt {a}}-\frac {B\,\sqrt {b}\,\mathrm {asin}\left (\frac {\sqrt {b}\,x\,1{}\mathrm {i}}{\sqrt {a}}\right )\,\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}\,\sqrt {\frac {b\,x^2}{a}+1}} \]
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