Integrand size = 20, antiderivative size = 43 \[ \int \frac {x \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {(A b-a B) \sqrt {a+b x^2}}{b^2}+\frac {B \left (a+b x^2\right )^{3/2}}{3 b^2} \]
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Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {455, 45} \[ \int \frac {x \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} (A b-a B)}{b^2}+\frac {B \left (a+b x^2\right )^{3/2}}{3 b^2} \]
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Rule 45
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{\sqrt {a+b x}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {A b-a B}{b \sqrt {a+b x}}+\frac {B \sqrt {a+b x}}{b}\right ) \, dx,x,x^2\right ) \\ & = \frac {(A b-a B) \sqrt {a+b x^2}}{b^2}+\frac {B \left (a+b x^2\right )^{3/2}}{3 b^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77 \[ \int \frac {x \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (3 A b-2 a B+b B x^2\right )}{3 b^2} \]
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Time = 2.80 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.70
method | result | size |
gosper | \(\frac {\sqrt {b \,x^{2}+a}\, \left (b B \,x^{2}+3 A b -2 B a \right )}{3 b^{2}}\) | \(30\) |
trager | \(\frac {\sqrt {b \,x^{2}+a}\, \left (b B \,x^{2}+3 A b -2 B a \right )}{3 b^{2}}\) | \(30\) |
risch | \(\frac {\sqrt {b \,x^{2}+a}\, \left (b B \,x^{2}+3 A b -2 B a \right )}{3 b^{2}}\) | \(30\) |
pseudoelliptic | \(\frac {\left (\left (x^{2} B +3 A \right ) b -2 B a \right ) \sqrt {b \,x^{2}+a}}{3 b^{2}}\) | \(31\) |
default | \(B \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )+\frac {A \sqrt {b \,x^{2}+a}}{b}\) | \(51\) |
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none
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.67 \[ \int \frac {x \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {{\left (B b x^{2} - 2 \, B a + 3 \, A b\right )} \sqrt {b x^{2} + a}}{3 \, b^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.63 \[ \int \frac {x \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\begin {cases} \frac {A \sqrt {a + b x^{2}}}{b} - \frac {2 B a \sqrt {a + b x^{2}}}{3 b^{2}} + \frac {B x^{2} \sqrt {a + b x^{2}}}{3 b} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{2}}{2} + \frac {B x^{4}}{4}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {x \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} B x^{2}}{3 \, b} - \frac {2 \, \sqrt {b x^{2} + a} B a}{3 \, b^{2}} + \frac {\sqrt {b x^{2} + a} A}{b} \]
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none
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int \frac {x \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{3 \, b^{2}} - \frac {\sqrt {b x^{2} + a} {\left (B a - A b\right )}}{b^{2}} \]
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Time = 5.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \frac {x \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\left (\frac {3\,A\,b-2\,B\,a}{3\,b^2}+\frac {B\,x^2}{3\,b}\right )\,\sqrt {b\,x^2+a} \]
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