Integrand size = 24, antiderivative size = 19 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {643} \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rule 643
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d}{3 \left (a+b x+c x^2\right )^{3/2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d}{3 (a+x (b+c x))^{3/2}} \]
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Time = 2.44 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(-\frac {2 d}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(16\) |
default | \(-\frac {2 d}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(16\) |
trager | \(-\frac {2 d}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(16\) |
pseudoelliptic | \(-\frac {2 d}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(16\) |
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Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (15) = 30\).
Time = 0.39 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.74 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, \sqrt {c x^{2} + b x + a} d}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (19) = 38\).
Time = 0.51 (sec) , antiderivative size = 82, normalized size of antiderivative = 4.32 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\begin {cases} - \frac {2 d}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} & \text {for}\: a \neq - x \left (b + c x\right ) \\\tilde {\infty } b d x + \tilde {\infty } c d x^{2} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, d}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]
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none
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, d^{5}}{3 \, {\left (a d^{2} + {\left (c d x^{2} + b d x\right )} d\right )}^{\frac {3}{2}} {\left | d \right |}} \]
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Time = 9.51 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2\,d}{3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
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