Integrand size = 28, antiderivative size = 48 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (b d-a e) (d+e x)^4} \]
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Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 37} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (d+e x)^4 (b d-a e)} \]
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Rule 37
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{(d+e x)^5} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (b d-a e) (d+e x)^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(48)=96\).
Time = 1.06 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.27 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {\sqrt {(a+b x)^2} \left (a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )}{4 e^4 (a+b x) (d+e x)^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(119\) vs. \(2(35)=70\).
Time = 2.98 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.50
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{3} x^{3}}{e}-\frac {3 b^{2} \left (a e +b d \right ) x^{2}}{2 e^{2}}-\frac {b \left (a^{2} e^{2}+a b d e +b^{2} d^{2}\right ) x}{e^{3}}-\frac {a^{3} e^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}}{4 e^{4}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{4}}\) | \(120\) |
gosper | \(-\frac {\left (4 e^{3} x^{3} b^{3}+6 x^{2} a \,b^{2} e^{3}+6 x^{2} b^{3} d \,e^{2}+4 a^{2} b \,e^{3} x +4 x a \,b^{2} d \,e^{2}+4 b^{3} d^{2} e x +a^{3} e^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{4 \left (e x +d \right )^{4} e^{4} \left (b x +a \right )^{3}}\) | \(128\) |
default | \(-\frac {\left (4 e^{3} x^{3} b^{3}+6 x^{2} a \,b^{2} e^{3}+6 x^{2} b^{3} d \,e^{2}+4 a^{2} b \,e^{3} x +4 x a \,b^{2} d \,e^{2}+4 b^{3} d^{2} e x +a^{3} e^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{4 \left (e x +d \right )^{4} e^{4} \left (b x +a \right )^{3}}\) | \(128\) |
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Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (35) = 70\).
Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.98 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {4 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + a b^{2} d^{2} e + a^{2} b d e^{2} + a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \, {\left (b^{3} d^{2} e + a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{4 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{5}}\, dx \]
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Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (35) = 70\).
Time = 0.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 4.12 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {b^{4} \mathrm {sgn}\left (b x + a\right )}{4 \, {\left (b d e^{4} - a e^{5}\right )}} - \frac {4 \, b^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{3} d e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, b^{3} d^{2} e x \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{2} d e^{2} x \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{2} b e^{3} x \mathrm {sgn}\left (b x + a\right ) + b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) + a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )}{4 \, {\left (e x + d\right )}^{4} e^{4}} \]
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Time = 9.68 (sec) , antiderivative size = 284, normalized size of antiderivative = 5.92 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {\left (\frac {2\,b^3\,d-3\,a\,b^2\,e}{2\,e^4}+\frac {b^3\,d}{2\,e^4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2}-\frac {\left (\frac {3\,a^2\,b\,e^2-3\,a\,b^2\,d\,e+b^3\,d^2}{3\,e^4}+\frac {d\,\left (\frac {b^3\,d}{3\,e^3}-\frac {b^2\,\left (3\,a\,e-b\,d\right )}{3\,e^3}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}-\frac {\left (\frac {a^3}{4\,e}-\frac {d\,\left (\frac {3\,a^2\,b}{4\,e}-\frac {d\,\left (\frac {3\,a\,b^2}{4\,e}-\frac {b^3\,d}{4\,e^2}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{e^4\,\left (a+b\,x\right )\,\left (d+e\,x\right )} \]
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