Integrand size = 20, antiderivative size = 44 \[ \int \frac {x \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {-A b+a B}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {B}{b^2 \sqrt {a+b x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 44, 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} \[ \int \frac {x \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {A b-a B}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {B}{b^2 \sqrt {a+b x^2}} \]
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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}{(a+b x)^{5/2}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {A b-a B}{b (a+b x)^{5/2}}+\frac {B}{b (a+b x)^{3/2}}\right ) \, dx,x,x^2\right ) \\ & = -\frac {A b-a B}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {B}{b^2 \sqrt {a+b x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.77 \[ \int \frac {x \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {-A b-2 a B-3 b B x^2}{3 b^2 \left (a+b x^2\right )^{3/2}} \]
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Time = 2.91 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(-\frac {3 b B \,x^{2}+A b +2 B a}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2}}\) | \(30\) |
trager | \(-\frac {3 b B \,x^{2}+A b +2 B a}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2}}\) | \(30\) |
pseudoelliptic | \(-\frac {\left (3 x^{2} B +A \right ) b +2 B a}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2}}\) | \(30\) |
default | \(B \left (-\frac {x^{2}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\right )-\frac {A}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(52\) |
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none
Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.18 \[ \int \frac {x \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {{\left (3 \, B b x^{2} + 2 \, B a + A b\right )} \sqrt {b x^{2} + a}}{3 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (37) = 74\).
Time = 0.34 (sec) , antiderivative size = 143, normalized size of antiderivative = 3.25 \[ \int \frac {x \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\begin {cases} - \frac {A b}{3 a b^{2} \sqrt {a + b x^{2}} + 3 b^{3} x^{2} \sqrt {a + b x^{2}}} - \frac {2 B a}{3 a b^{2} \sqrt {a + b x^{2}} + 3 b^{3} x^{2} \sqrt {a + b x^{2}}} - \frac {3 B b x^{2}}{3 a b^{2} \sqrt {a + b x^{2}} + 3 b^{3} x^{2} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{2}}{2} + \frac {B x^{4}}{4}}{a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.14 \[ \int \frac {x \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {B x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {2 \, B a}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} - \frac {A}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} \]
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Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.73 \[ \int \frac {x \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {3 \, {\left (b x^{2} + a\right )} B - B a + A b}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} \]
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Time = 5.54 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.73 \[ \int \frac {x \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {A\,b-B\,a+3\,B\,\left (b\,x^2+a\right )}{3\,b^2\,{\left (b\,x^2+a\right )}^{3/2}} \]
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