Integrand size = 26, antiderivative size = 67 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{3/2}} \, dx=-\frac {2 (b d-a e)^2}{e^3 \sqrt {d+e x}}-\frac {4 b (b d-a e) \sqrt {d+e x}}{e^3}+\frac {2 b^2 (d+e x)^{3/2}}{3 e^3} \]
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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} \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{3/2}} \, dx=-\frac {4 b \sqrt {d+e x} (b d-a e)}{e^3}-\frac {2 (b d-a e)^2}{e^3 \sqrt {d+e x}}+\frac {2 b^2 (d+e x)^{3/2}}{3 e^3} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^2}{(d+e x)^{3/2}} \, dx \\ & = \int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^{3/2}}-\frac {2 b (b d-a e)}{e^2 \sqrt {d+e x}}+\frac {b^2 \sqrt {d+e x}}{e^2}\right ) \, dx \\ & = -\frac {2 (b d-a e)^2}{e^3 \sqrt {d+e x}}-\frac {4 b (b d-a e) \sqrt {d+e x}}{e^3}+\frac {2 b^2 (d+e x)^{3/2}}{3 e^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{3/2}} \, dx=\frac {2 \left (-3 a^2 e^2+6 a b e (2 d+e x)+b^2 \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 \sqrt {d+e x}} \]
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Time = 2.33 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.82
method | result | size |
pseudoelliptic | \(\frac {\frac {2 \left (x^{2} e^{2}-4 d e x -8 d^{2}\right ) b^{2}}{3}+8 \left (\frac {e x}{2}+d \right ) e a b -2 a^{2} e^{2}}{\sqrt {e x +d}\, e^{3}}\) | \(55\) |
risch | \(\frac {2 b \left (b e x +6 a e -5 b d \right ) \sqrt {e x +d}}{3 e^{3}}-\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{3} \sqrt {e x +d}}\) | \(61\) |
gosper | \(-\frac {2 \left (-x^{2} b^{2} e^{2}-6 x a b \,e^{2}+4 b^{2} d e x +3 a^{2} e^{2}-12 a b d e +8 b^{2} d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}}\) | \(63\) |
trager | \(-\frac {2 \left (-x^{2} b^{2} e^{2}-6 x a b \,e^{2}+4 b^{2} d e x +3 a^{2} e^{2}-12 a b d e +8 b^{2} d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}}\) | \(63\) |
derivativedivides | \(\frac {\frac {2 \left (e x +d \right )^{\frac {3}{2}} b^{2}}{3}+4 a b e \sqrt {e x +d}-4 b^{2} d \sqrt {e x +d}-\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\) | \(74\) |
default | \(\frac {\frac {2 \left (e x +d \right )^{\frac {3}{2}} b^{2}}{3}+4 a b e \sqrt {e x +d}-4 b^{2} d \sqrt {e x +d}-\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\) | \(74\) |
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Time = 0.34 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (b^{2} e^{2} x^{2} - 8 \, b^{2} d^{2} + 12 \, a b d e - 3 \, a^{2} e^{2} - 2 \, {\left (2 \, b^{2} d e - 3 \, a b e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x + d e^{3}\right )}} \]
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Time = 1.35 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.40 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{2} \left (d + e x\right )^{\frac {3}{2}}}{3 e^{2}} + \frac {\sqrt {d + e x} \left (2 a b e - 2 b^{2} d\right )}{e^{2}} - \frac {\left (a e - b d\right )^{2}}{e^{2} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} b^{2} - 6 \, {\left (b^{2} d - a b e\right )} \sqrt {e x + d}}{e^{2}} - \frac {3 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}}{\sqrt {e x + d} e^{2}}\right )}}{3 \, e} \]
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Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.25 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}}{\sqrt {e x + d} e^{3}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} b^{2} e^{6} - 6 \, \sqrt {e x + d} b^{2} d e^{6} + 6 \, \sqrt {e x + d} a b e^{7}\right )}}{3 \, e^{9}} \]
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Time = 9.55 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{3/2}} \, dx=\frac {\frac {2\,b^2\,{\left (d+e\,x\right )}^2}{3}-2\,a^2\,e^2-2\,b^2\,d^2-4\,b^2\,d\,\left (d+e\,x\right )+4\,a\,b\,e\,\left (d+e\,x\right )+4\,a\,b\,d\,e}{e^3\,\sqrt {d+e\,x}} \]
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