Integrand size = 30, antiderivative size = 94 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=\frac {2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt {d+e x}} \]
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Time = 0.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {660, 45} \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt {d+e x}} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{(d+e x)^{5/2}} \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e)}{e (d+e x)^{5/2}}+\frac {b^2}{e (d+e x)^{3/2}}\right ) \, dx}{a b+b^2 x} \\ & = \frac {2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt {d+e x}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} (2 b d+a e+3 b e x)}{3 e^2 (a+b x) (d+e x)^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 2.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.34
method | result | size |
default | \(-\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (3 b e x +a e +2 b d \right )}{3 e^{2} \left (e x +d \right )^{\frac {3}{2}}}\) | \(32\) |
gosper | \(-\frac {2 \left (3 b e x +a e +2 b d \right ) \sqrt {\left (b x +a \right )^{2}}}{3 \left (e x +d \right )^{\frac {3}{2}} e^{2} \left (b x +a \right )}\) | \(42\) |
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Time = 0.36 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (3 \, b e x + 2 \, b d + a e\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]
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\[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\sqrt {\left (a + b x\right )^{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (3 \, b e x + 2 \, b d + a e\right )}}{3 \, {\left (e^{3} x + d e^{2}\right )} \sqrt {e x + d}} \]
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Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (e x + d\right )} b \mathrm {sgn}\left (b x + a\right ) - b d \mathrm {sgn}\left (b x + a\right ) + a e \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{2}} \]
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Time = 9.93 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx=-\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,x}{e^2}+\frac {2\,a\,e+4\,b\,d}{3\,b\,e^3}\right )}{x^2\,\sqrt {d+e\,x}+\frac {a\,d\,\sqrt {d+e\,x}}{b\,e}+\frac {x\,\left (3\,a\,e^3+3\,b\,d\,e^2\right )\,\sqrt {d+e\,x}}{3\,b\,e^3}} \]
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