Integrand size = 26, antiderivative size = 67 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{5/2}} \, dx=-\frac {2 (b d-a e)^2}{3 e^3 (d+e x)^{3/2}}+\frac {4 b (b d-a e)}{e^3 \sqrt {d+e x}}+\frac {2 b^2 \sqrt {d+e x}}{e^3} \]
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Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{5/2}} \, dx=\frac {4 b (b d-a e)}{e^3 \sqrt {d+e x}}-\frac {2 (b d-a e)^2}{3 e^3 (d+e x)^{3/2}}+\frac {2 b^2 \sqrt {d+e x}}{e^3} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^2}{(d+e x)^{5/2}} \, dx \\ & = \int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^{5/2}}-\frac {2 b (b d-a e)}{e^2 (d+e x)^{3/2}}+\frac {b^2}{e^2 \sqrt {d+e x}}\right ) \, dx \\ & = -\frac {2 (b d-a e)^2}{3 e^3 (d+e x)^{3/2}}+\frac {4 b (b d-a e)}{e^3 \sqrt {d+e x}}+\frac {2 b^2 \sqrt {d+e x}}{e^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{5/2}} \, dx=\frac {-2 a^2 e^2-4 a b e (2 d+3 e x)+2 b^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^{3/2}} \]
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Time = 2.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {2 b^{2} \sqrt {e x +d}}{e^{3}}-\frac {2 \left (6 b e x +a e +5 b d \right ) \left (a e -b d \right )}{3 e^{3} \left (e x +d \right )^{\frac {3}{2}}}\) | \(50\) |
pseudoelliptic | \(-\frac {2 \left (\left (-3 b^{2} x^{2}+6 a b x +a^{2}\right ) e^{2}+4 b d \left (-3 b x +a \right ) e -8 b^{2} d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\) | \(54\) |
gosper | \(-\frac {2 \left (-3 x^{2} b^{2} e^{2}+6 x a b \,e^{2}-12 b^{2} d e x +a^{2} e^{2}+4 a b d e -8 b^{2} d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\) | \(62\) |
trager | \(-\frac {2 \left (-3 x^{2} b^{2} e^{2}+6 x a b \,e^{2}-12 b^{2} d e x +a^{2} e^{2}+4 a b d e -8 b^{2} d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\) | \(62\) |
derivativedivides | \(\frac {2 b^{2} \sqrt {e x +d}-\frac {4 b \left (a e -b d \right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{3}}\) | \(66\) |
default | \(\frac {2 b^{2} \sqrt {e x +d}-\frac {4 b \left (a e -b d \right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{3}}\) | \(66\) |
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Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.27 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 4 \, a b d e - a^{2} e^{2} + 6 \, {\left (2 \, b^{2} d e - a b e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (61) = 122\).
Time = 0.35 (sec) , antiderivative size = 265, normalized size of antiderivative = 3.96 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{5/2}} \, dx=\begin {cases} - \frac {2 a^{2} e^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} - \frac {8 a b d e}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} - \frac {12 a b e^{2} x}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {16 b^{2} d^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {24 b^{2} d e x}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {6 b^{2} e^{2} x^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.07 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, \sqrt {e x + d} b^{2}}{e^{2}} - \frac {b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2} - 6 \, {\left (b^{2} d - a b e\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{2}}\right )}}{3 \, e} \]
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Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.07 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, \sqrt {e x + d} b^{2}}{e^{3}} + \frac {2 \, {\left (6 \, {\left (e x + d\right )} b^{2} d - b^{2} d^{2} - 6 \, {\left (e x + d\right )} a b e + 2 \, a b d e - a^{2} e^{2}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{3}} \]
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Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.01 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{5/2}} \, dx=\frac {6\,b^2\,{\left (d+e\,x\right )}^2-2\,a^2\,e^2-2\,b^2\,d^2+12\,b^2\,d\,\left (d+e\,x\right )-12\,a\,b\,e\,\left (d+e\,x\right )+4\,a\,b\,d\,e}{3\,e^3\,{\left (d+e\,x\right )}^{3/2}} \]
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