Integrand size = 20, antiderivative size = 71 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (c d^2-b d e+a e^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac {2 (2 c d-b e)}{e^3 \sqrt {d+e x}}+\frac {2 c \sqrt {d+e x}}{e^3} \]
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Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (a e^2-b d e+c d^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac {2 (2 c d-b e)}{e^3 \sqrt {d+e x}}+\frac {2 c \sqrt {d+e x}}{e^3} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c d^2-b d e+a e^2}{e^2 (d+e x)^{5/2}}+\frac {-2 c d+b e}{e^2 (d+e x)^{3/2}}+\frac {c}{e^2 \sqrt {d+e x}}\right ) \, dx \\ & = -\frac {2 \left (c d^2-b d e+a e^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac {2 (2 c d-b e)}{e^3 \sqrt {d+e x}}+\frac {2 c \sqrt {d+e x}}{e^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.77 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2}} \, dx=\frac {-2 e (2 b d+a e+3 b e x)+2 c \left (8 d^2+12 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^{3/2}} \]
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Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (-3 c \,x^{2}+3 b x +a \right ) e^{2}+2 d \left (-6 c x +b \right ) e -8 c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\) | \(46\) |
gosper | \(-\frac {2 \left (-3 c \,x^{2} e^{2}+3 b \,e^{2} x -12 c d e x +a \,e^{2}+2 b d e -8 c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\) | \(52\) |
trager | \(-\frac {2 \left (-3 c \,x^{2} e^{2}+3 b \,e^{2} x -12 c d e x +a \,e^{2}+2 b d e -8 c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\) | \(52\) |
risch | \(\frac {2 c \sqrt {e x +d}}{e^{3}}-\frac {2 \left (3 b \,e^{2} x -6 c d e x +a \,e^{2}+2 b d e -5 c \,d^{2}\right )}{3 e^{3} \left (e x +d \right )^{\frac {3}{2}}}\) | \(57\) |
derivativedivides | \(\frac {2 c \sqrt {e x +d}-\frac {2 \left (b e -2 c d \right )}{\sqrt {e x +d}}-\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{3}}\) | \(58\) |
default | \(\frac {2 c \sqrt {e x +d}-\frac {2 \left (b e -2 c d \right )}{\sqrt {e x +d}}-\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{3}}\) | \(58\) |
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Time = 0.33 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.06 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, c e^{2} x^{2} + 8 \, c d^{2} - 2 \, b d e - a e^{2} + 3 \, {\left (4 \, c d e - b e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (73) = 146\).
Time = 0.34 (sec) , antiderivative size = 252, normalized size of antiderivative = 3.55 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2}} \, dx=\begin {cases} - \frac {2 a e^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} - \frac {4 b d e}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} - \frac {6 b e^{2} x}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {16 c d^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {24 c d e x}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {6 c e^{2} x^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a x + \frac {b x^{2}}{2} + \frac {c x^{3}}{3}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, \sqrt {e x + d} c}{e^{2}} - \frac {c d^{2} - b d e + a e^{2} - 3 \, {\left (2 \, c d - b e\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{2}}\right )}}{3 \, e} \]
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Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, \sqrt {e x + d} c}{e^{3}} + \frac {2 \, {\left (6 \, {\left (e x + d\right )} c d - c d^{2} - 3 \, {\left (e x + d\right )} b e + b d e - a e^{2}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{3}} \]
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Time = 9.77 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2}} \, dx=\frac {6\,c\,{\left (d+e\,x\right )}^2-2\,a\,e^2-2\,c\,d^2-6\,b\,e\,\left (d+e\,x\right )+12\,c\,d\,\left (d+e\,x\right )+2\,b\,d\,e}{3\,e^3\,{\left (d+e\,x\right )}^{3/2}} \]
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