Integrand size = 28, antiderivative size = 98 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (b d-a e) (d+e x)^5}+\frac {b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{20 (b d-a e)^2 (d+e x)^4} \]
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Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {660, 47, 37} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3}{20 (d+e x)^4 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3}{5 (d+e x)^5 (b d-a e)} \]
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Rule 37
Rule 47
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)^6} \, 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}}{5 (b d-a e) (d+e x)^5}+\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}{5 b (b d-a e) \left (a b+b^2 x\right )} \\ & = \frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (b d-a e) (d+e x)^5}+\frac {b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{20 (b d-a e)^2 (d+e x)^4} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {\sqrt {(a+b x)^2} \left (4 a^3 e^3+3 a^2 b e^2 (d+5 e x)+2 a b^2 e \left (d^2+5 d e x+10 e^2 x^2\right )+b^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )}{20 e^4 (a+b x) (d+e x)^5} \]
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Time = 3.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.29
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{3} x^{3}}{2 e}-\frac {b^{2} \left (2 a e +b d \right ) x^{2}}{2 e^{2}}-\frac {b \left (3 a^{2} e^{2}+2 a b d e +b^{2} d^{2}\right ) x}{4 e^{3}}-\frac {4 a^{3} e^{3}+3 a^{2} b d \,e^{2}+2 a \,b^{2} d^{2} e +b^{3} d^{3}}{20 e^{4}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{5}}\) | \(126\) |
gosper | \(-\frac {\left (10 e^{3} x^{3} b^{3}+20 x^{2} a \,b^{2} e^{3}+10 x^{2} b^{3} d \,e^{2}+15 a^{2} b \,e^{3} x +10 x a \,b^{2} d \,e^{2}+5 b^{3} d^{2} e x +4 a^{3} e^{3}+3 a^{2} b d \,e^{2}+2 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{20 e^{4} \left (e x +d \right )^{5} \left (b x +a \right )^{3}}\) | \(131\) |
default | \(-\frac {\left (10 e^{3} x^{3} b^{3}+20 x^{2} a \,b^{2} e^{3}+10 x^{2} b^{3} d \,e^{2}+15 a^{2} b \,e^{3} x +10 x a \,b^{2} d \,e^{2}+5 b^{3} d^{2} e x +4 a^{3} e^{3}+3 a^{2} b d \,e^{2}+2 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{20 e^{4} \left (e x +d \right )^{5} \left (b x +a \right )^{3}}\) | \(131\) |
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Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (72) = 144\).
Time = 0.33 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {10 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + 2 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 4 \, a^{3} e^{3} + 10 \, {\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + 5 \, {\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x}{20 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{6}}\, dx \]
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Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (72) = 144\).
Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.18 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {b^{5} \mathrm {sgn}\left (b x + a\right )}{20 \, {\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\right )}} - \frac {10 \, b^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, b^{3} d e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, a b^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, b^{3} d^{2} e x \mathrm {sgn}\left (b x + a\right ) + 10 \, a b^{2} d e^{2} x \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b e^{3} x \mathrm {sgn}\left (b x + a\right ) + b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )}{20 \, {\left (e x + d\right )}^{5} e^{4}} \]
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Time = 9.61 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.90 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {\left (\frac {2\,b^3\,d-3\,a\,b^2\,e}{3\,e^4}+\frac {b^3\,d}{3\,e^4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}-\frac {\left (\frac {3\,a^2\,b\,e^2-3\,a\,b^2\,d\,e+b^3\,d^2}{4\,e^4}+\frac {d\,\left (\frac {b^3\,d}{4\,e^3}-\frac {b^2\,\left (3\,a\,e-b\,d\right )}{4\,e^3}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {\left (\frac {a^3}{5\,e}-\frac {d\,\left (\frac {3\,a^2\,b}{5\,e}-\frac {d\,\left (\frac {3\,a\,b^2}{5\,e}-\frac {b^3\,d}{5\,e^2}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,e^4\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2} \]
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