Integrand size = 19, antiderivative size = 47 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {2 A x}{3 a^2 \sqrt {a+b x^2}}+\frac {x \left (A+B x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {386, 197} \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {2 A x}{3 a^2 \sqrt {a+b x^2}}+\frac {x \left (A+B x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \]
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Rule 197
Rule 386
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (A+B x^2\right )}{3 a \left (a+b x^2\right )^{3/2}}+\frac {(2 A) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a} \\ & = \frac {2 A x}{3 a^2 \sqrt {a+b x^2}}+\frac {x \left (A+B x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x \left (3 a A+2 A b x^2+a B x^2\right )}{3 a^2 \left (a+b x^2\right )^{3/2}} \]
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Time = 2.83 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(\frac {x \left (2 A b \,x^{2}+B a \,x^{2}+3 A a \right )}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\) | \(34\) |
trager | \(\frac {x \left (2 A b \,x^{2}+B a \,x^{2}+3 A a \right )}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\) | \(34\) |
pseudoelliptic | \(\frac {x \left (2 A b \,x^{2}+B a \,x^{2}+3 A a \right )}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\) | \(34\) |
default | \(A \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+B \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )\) | \(90\) |
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none
Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (B a + 2 \, A b\right )} x^{3} + 3 \, A a x\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (41) = 82\).
Time = 3.98 (sec) , antiderivative size = 144, normalized size of antiderivative = 3.06 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^{5/2}} \, dx=A \left (\frac {3 a x}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{3}}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + \frac {B x^{3}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {3}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} \]
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Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.45 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {2 \, A x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {A x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {B x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {B x}{3 \, \sqrt {b x^{2} + a} a b} \]
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Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x {\left (\frac {3 \, A}{a} + \frac {{\left (B a b + 2 \, A b^{2}\right )} x^{2}}{a^{2} b}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} \]
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Time = 5.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {3\,A\,a\,x+2\,A\,b\,x^3+B\,a\,x^3}{3\,a^2\,{\left (b\,x^2+a\right )}^{3/2}} \]
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