Integrand size = 28, antiderivative size = 48 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (b d-a e) (d+e x)^6} \]
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Time = 0.02 (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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)} \]
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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 )^5}{(d+e x)^7} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (b d-a e) (d+e x)^6} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(218\) vs. \(2(48)=96\).
Time = 1.05 (sec) , antiderivative size = 218, normalized size of antiderivative = 4.54 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=-\frac {\sqrt {(a+b x)^2} \left (a^5 e^5+a^4 b e^4 (d+6 e x)+a^3 b^2 e^3 \left (d^2+6 d e x+15 e^2 x^2\right )+a^2 b^3 e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+a b^4 e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+b^5 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )}{6 e^6 (a+b x) (d+e x)^6} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(246\) vs. \(2(35)=70\).
Time = 3.58 (sec) , antiderivative size = 247, normalized size of antiderivative = 5.15
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} x^{5}}{e}-\frac {5 b^{4} \left (a e +b d \right ) x^{4}}{2 e^{2}}-\frac {10 b^{3} \left (a^{2} e^{2}+a b d e +b^{2} d^{2}\right ) x^{3}}{3 e^{3}}-\frac {5 b^{2} \left (a^{3} e^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{2}}{2 e^{4}}-\frac {b \left (e^{4} a^{4}+b \,e^{3} d \,a^{3}+b^{2} e^{2} d^{2} a^{2}+a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x}{e^{5}}-\frac {a^{5} e^{5}+a^{4} b d \,e^{4}+a^{3} b^{2} d^{2} e^{3}+a^{2} b^{3} d^{3} e^{2}+a \,b^{4} d^{4} e +b^{5} d^{5}}{6 e^{6}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{6}}\) | \(247\) |
gosper | \(-\frac {\left (6 x^{5} e^{5} b^{5}+15 x^{4} a \,b^{4} e^{5}+15 x^{4} b^{5} d \,e^{4}+20 x^{3} a^{2} b^{3} e^{5}+20 x^{3} a \,b^{4} d \,e^{4}+20 x^{3} b^{5} d^{2} e^{3}+15 x^{2} a^{3} b^{2} e^{5}+15 x^{2} a^{2} b^{3} d \,e^{4}+15 x^{2} a \,b^{4} d^{2} e^{3}+15 x^{2} b^{5} d^{3} e^{2}+6 a^{4} b \,e^{5} x +6 a^{3} b^{2} d \,e^{4} x +6 x \,a^{2} b^{3} d^{2} e^{3}+6 x a \,b^{4} d^{3} e^{2}+6 b^{5} d^{4} e x +a^{5} e^{5}+a^{4} b d \,e^{4}+a^{3} b^{2} d^{2} e^{3}+a^{2} b^{3} d^{3} e^{2}+a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{6 \left (e x +d \right )^{6} e^{6} \left (b x +a \right )^{5}}\) | \(283\) |
default | \(-\frac {\left (6 x^{5} e^{5} b^{5}+15 x^{4} a \,b^{4} e^{5}+15 x^{4} b^{5} d \,e^{4}+20 x^{3} a^{2} b^{3} e^{5}+20 x^{3} a \,b^{4} d \,e^{4}+20 x^{3} b^{5} d^{2} e^{3}+15 x^{2} a^{3} b^{2} e^{5}+15 x^{2} a^{2} b^{3} d \,e^{4}+15 x^{2} a \,b^{4} d^{2} e^{3}+15 x^{2} b^{5} d^{3} e^{2}+6 a^{4} b \,e^{5} x +6 a^{3} b^{2} d \,e^{4} x +6 x \,a^{2} b^{3} d^{2} e^{3}+6 x a \,b^{4} d^{3} e^{2}+6 b^{5} d^{4} e x +a^{5} e^{5}+a^{4} b d \,e^{4}+a^{3} b^{2} d^{2} e^{3}+a^{2} b^{3} d^{3} e^{2}+a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{6 \left (e x +d \right )^{6} e^{6} \left (b x +a \right )^{5}}\) | \(283\) |
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Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (35) = 70\).
Time = 0.37 (sec) , antiderivative size = 300, normalized size of antiderivative = 6.25 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=-\frac {6 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + a b^{4} d^{4} e + a^{2} b^{3} d^{3} e^{2} + a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} + a^{5} e^{5} + 15 \, {\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{3} + a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 15 \, {\left (b^{5} d^{3} e^{2} + a b^{4} d^{2} e^{3} + a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 6 \, {\left (b^{5} d^{4} e + a b^{4} d^{3} e^{2} + a^{2} b^{3} d^{2} e^{3} + a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x}{6 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (35) = 70\).
Time = 0.29 (sec) , antiderivative size = 419, normalized size of antiderivative = 8.73 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {b^{6} \mathrm {sgn}\left (b x + a\right )}{6 \, {\left (b d e^{6} - a e^{7}\right )}} - \frac {6 \, b^{5} e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{5} d e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, a b^{4} e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, b^{5} d^{2} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, a b^{4} d e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{2} b^{3} e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{5} d^{3} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a b^{4} d^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{3} d e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{3} b^{2} e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{5} d^{4} e x \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{4} d^{3} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{3} b^{2} d e^{4} x \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{4} b e^{5} x \mathrm {sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )}{6 \, {\left (e x + d\right )}^{6} e^{6}} \]
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Time = 9.63 (sec) , antiderivative size = 687, normalized size of antiderivative = 14.31 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {\left (\frac {4\,b^5\,d-5\,a\,b^4\,e}{2\,e^6}+\frac {b^5\,d}{2\,e^6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2}-\frac {\left (\frac {5\,a^4\,b\,e^4-10\,a^3\,b^2\,d\,e^3+10\,a^2\,b^3\,d^2\,e^2-5\,a\,b^4\,d^3\,e+b^5\,d^4}{5\,e^6}+\frac {d\,\left (\frac {-10\,a^3\,b^2\,e^4+10\,a^2\,b^3\,d\,e^3-5\,a\,b^4\,d^2\,e^2+b^5\,d^3\,e}{5\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{5\,e^3}-\frac {b^4\,\left (5\,a\,e-b\,d\right )}{5\,e^3}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-5\,a\,b\,d\,e+b^2\,d^2\right )}{5\,e^4}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {10\,a^2\,b^3\,e^2-15\,a\,b^4\,d\,e+6\,b^5\,d^2}{3\,e^6}+\frac {d\,\left (\frac {b^5\,d}{3\,e^5}-\frac {b^4\,\left (5\,a\,e-3\,b\,d\right )}{3\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}-\frac {\left (\frac {a^5}{6\,e}-\frac {d\,\left (\frac {5\,a^4\,b}{6\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{6\,e}-\frac {b^5\,d}{6\,e^2}\right )}{e}-\frac {5\,a^2\,b^3}{3\,e}\right )}{e}+\frac {5\,a^3\,b^2}{3\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}+\frac {\left (\frac {-10\,a^3\,b^2\,e^3+20\,a^2\,b^3\,d\,e^2-15\,a\,b^4\,d^2\,e+4\,b^5\,d^3}{4\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{4\,e^4}-\frac {b^4\,\left (5\,a\,e-2\,b\,d\right )}{4\,e^4}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{4\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{e^6\,\left (a+b\,x\right )\,\left (d+e\,x\right )} \]
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