Integrand size = 19, antiderivative size = 47 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {2 c x}{3 a^2 \sqrt {a+b x^2}}+\frac {x \left (c+d x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \]
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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} \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {2 c x}{3 a^2 \sqrt {a+b x^2}}+\frac {x \left (c+d x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \]
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Rule 197
Rule 386
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (c+d x^2\right )}{3 a \left (a+b x^2\right )^{3/2}}+\frac {(2 c) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a} \\ & = \frac {2 c x}{3 a^2 \sqrt {a+b x^2}}+\frac {x \left (c+d x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.79 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x \left (3 a c+2 b c x^2+a d x^2\right )}{3 a^2 \left (a+b x^2\right )^{3/2}} \]
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Time = 2.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(\frac {x \left (a d \,x^{2}+2 c b \,x^{2}+3 a c \right )}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\) | \(34\) |
trager | \(\frac {x \left (a d \,x^{2}+2 c b \,x^{2}+3 a c \right )}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\) | \(34\) |
pseudoelliptic | \(\frac {x \left (a d \,x^{2}+2 c b \,x^{2}+3 a c \right )}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\) | \(34\) |
default | \(c \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+d \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )\) | \(90\) |
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none
Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.15 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (2 \, b c + a d\right )} x^{3} + 3 \, a c x\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (41) = 82\).
Time = 4.02 (sec) , antiderivative size = 144, normalized size of antiderivative = 3.06 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2}} \, dx=c \left (\frac {3 a x}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{3}}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + \frac {d x^{3}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {3}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} \]
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Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.45 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {2 \, c x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {d x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {d x}{3 \, \sqrt {b x^{2} + a} a b} \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.85 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x {\left (\frac {3 \, c}{a} + \frac {{\left (2 \, b^{2} c + a b d\right )} x^{2}}{a^{2} b}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} \]
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Time = 4.55 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {3\,a\,c\,x+a\,d\,x^3+2\,b\,c\,x^3}{3\,a^2\,{\left (b\,x^2+a\right )}^{3/2}} \]
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