Integrand size = 28, antiderivative size = 200 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 (b d-a e)^2 (d+e x)^8}+\frac {b^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{84 (b d-a e)^3 (d+e x)^7}+\frac {b^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{504 (b d-a e)^4 (d+e x)^6} \]
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Time = 0.05 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {660, 47, 37} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{84 (d+e x)^7 (b d-a e)^3}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{24 (d+e x)^8 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{9 (d+e x)^9 (b d-a e)}+\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{504 (d+e x)^6 (b d-a e)^4} \]
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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 )^5}{(d+e x)^{10}} \, 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}}{9 (b d-a e) (d+e x)^9}+\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)^9} \, dx}{3 b^3 (b d-a e) \left (a b+b^2 x\right )} \\ & = \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 (b d-a e)^2 (d+e x)^8}+\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)^8} \, dx}{12 b^2 (b d-a e)^2 \left (a b+b^2 x\right )} \\ & = \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 (b d-a e)^2 (d+e x)^8}+\frac {b^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{84 (b d-a e)^3 (d+e x)^7}+\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}{84 b (b d-a e)^3 \left (a b+b^2 x\right )} \\ & = \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 (b d-a e)^2 (d+e x)^8}+\frac {b^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{84 (b d-a e)^3 (d+e x)^7}+\frac {b^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{504 (b d-a e)^4 (d+e x)^6} \\ \end{align*}
Time = 1.06 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (56 a^5 e^5+35 a^4 b e^4 (d+9 e x)+20 a^3 b^2 e^3 \left (d^2+9 d e x+36 e^2 x^2\right )+10 a^2 b^3 e^2 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 a b^4 e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+b^5 \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )}{504 e^6 (a+b x) (d+e x)^9} \]
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Time = 6.89 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.31
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} x^{5}}{4 e}-\frac {b^{4} \left (4 a e +b d \right ) x^{4}}{4 e^{2}}-\frac {b^{3} \left (10 a^{2} e^{2}+4 a b d e +b^{2} d^{2}\right ) x^{3}}{6 e^{3}}-\frac {b^{2} \left (20 a^{3} e^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{2}}{14 e^{4}}-\frac {b \left (35 e^{4} a^{4}+20 b \,e^{3} d \,a^{3}+10 b^{2} e^{2} d^{2} a^{2}+4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x}{56 e^{5}}-\frac {56 a^{5} e^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}}{504 e^{6}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{9}}\) | \(262\) |
gosper | \(-\frac {\left (126 x^{5} e^{5} b^{5}+504 x^{4} a \,b^{4} e^{5}+126 x^{4} b^{5} d \,e^{4}+840 x^{3} a^{2} b^{3} e^{5}+336 x^{3} a \,b^{4} d \,e^{4}+84 x^{3} b^{5} d^{2} e^{3}+720 x^{2} a^{3} b^{2} e^{5}+360 x^{2} a^{2} b^{3} d \,e^{4}+144 x^{2} a \,b^{4} d^{2} e^{3}+36 x^{2} b^{5} d^{3} e^{2}+315 a^{4} b \,e^{5} x +180 a^{3} b^{2} d \,e^{4} x +90 x \,a^{2} b^{3} d^{2} e^{3}+36 x a \,b^{4} d^{3} e^{2}+9 b^{5} d^{4} e x +56 a^{5} e^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 e^{6} \left (e x +d \right )^{9} \left (b x +a \right )^{5}}\) | \(288\) |
default | \(-\frac {\left (126 x^{5} e^{5} b^{5}+504 x^{4} a \,b^{4} e^{5}+126 x^{4} b^{5} d \,e^{4}+840 x^{3} a^{2} b^{3} e^{5}+336 x^{3} a \,b^{4} d \,e^{4}+84 x^{3} b^{5} d^{2} e^{3}+720 x^{2} a^{3} b^{2} e^{5}+360 x^{2} a^{2} b^{3} d \,e^{4}+144 x^{2} a \,b^{4} d^{2} e^{3}+36 x^{2} b^{5} d^{3} e^{2}+315 a^{4} b \,e^{5} x +180 a^{3} b^{2} d \,e^{4} x +90 x \,a^{2} b^{3} d^{2} e^{3}+36 x a \,b^{4} d^{3} e^{2}+9 b^{5} d^{4} e x +56 a^{5} e^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 e^{6} \left (e x +d \right )^{9} \left (b x +a \right )^{5}}\) | \(288\) |
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Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (148) = 296\).
Time = 0.31 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.74 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=-\frac {126 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 4 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} + 20 \, a^{3} b^{2} d^{2} e^{3} + 35 \, a^{4} b d e^{4} + 56 \, a^{5} e^{5} + 126 \, {\left (b^{5} d e^{4} + 4 \, a b^{4} e^{5}\right )} x^{4} + 84 \, {\left (b^{5} d^{2} e^{3} + 4 \, a b^{4} d e^{4} + 10 \, a^{2} b^{3} e^{5}\right )} x^{3} + 36 \, {\left (b^{5} d^{3} e^{2} + 4 \, a b^{4} d^{2} e^{3} + 10 \, a^{2} b^{3} d e^{4} + 20 \, a^{3} b^{2} e^{5}\right )} x^{2} + 9 \, {\left (b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 10 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x}{504 \, {\left (e^{15} x^{9} + 9 \, d e^{14} x^{8} + 36 \, d^{2} e^{13} x^{7} + 84 \, d^{3} e^{12} x^{6} + 126 \, d^{4} e^{11} x^{5} + 126 \, d^{5} e^{10} x^{4} + 84 \, d^{6} e^{9} x^{3} + 36 \, d^{7} e^{8} x^{2} + 9 \, d^{8} e^{7} x + d^{9} e^{6}\right )}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (148) = 296\).
Time = 0.28 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.32 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\frac {b^{9} \mathrm {sgn}\left (b x + a\right )}{504 \, {\left (b^{4} d^{4} e^{6} - 4 \, a b^{3} d^{3} e^{7} + 6 \, a^{2} b^{2} d^{2} e^{8} - 4 \, a^{3} b d e^{9} + a^{4} e^{10}\right )}} - \frac {126 \, b^{5} e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 126 \, b^{5} d e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 504 \, a b^{4} e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 84 \, b^{5} d^{2} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 336 \, a b^{4} d e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 840 \, a^{2} b^{3} e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, b^{5} d^{3} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 144 \, a b^{4} d^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 360 \, a^{2} b^{3} d e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 720 \, a^{3} b^{2} e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, b^{5} d^{4} e x \mathrm {sgn}\left (b x + a\right ) + 36 \, a b^{4} d^{3} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 90 \, a^{2} b^{3} d^{2} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 180 \, a^{3} b^{2} d e^{4} x \mathrm {sgn}\left (b x + a\right ) + 315 \, a^{4} b e^{5} x \mathrm {sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )}{504 \, {\left (e x + d\right )}^{9} e^{6}} \]
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Time = 9.57 (sec) , antiderivative size = 687, normalized size of antiderivative = 3.44 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\frac {\left (\frac {4\,b^5\,d-5\,a\,b^4\,e}{5\,e^6}+\frac {b^5\,d}{5\,e^6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {5\,a^4\,b\,e^4-10\,a^3\,b^2\,d\,e^3+10\,a^2\,b^3\,d^2\,e^2-5\,a\,b^4\,d^3\,e+b^5\,d^4}{8\,e^6}+\frac {d\,\left (\frac {-10\,a^3\,b^2\,e^4+10\,a^2\,b^3\,d\,e^3-5\,a\,b^4\,d^2\,e^2+b^5\,d^3\,e}{8\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{8\,e^3}-\frac {b^4\,\left (5\,a\,e-b\,d\right )}{8\,e^3}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-5\,a\,b\,d\,e+b^2\,d^2\right )}{8\,e^4}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {\left (\frac {10\,a^2\,b^3\,e^2-15\,a\,b^4\,d\,e+6\,b^5\,d^2}{6\,e^6}+\frac {d\,\left (\frac {b^5\,d}{6\,e^5}-\frac {b^4\,\left (5\,a\,e-3\,b\,d\right )}{6\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {a^5}{9\,e}-\frac {d\,\left (\frac {5\,a^4\,b}{9\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{9\,e}-\frac {b^5\,d}{9\,e^2}\right )}{e}-\frac {10\,a^2\,b^3}{9\,e}\right )}{e}+\frac {10\,a^3\,b^2}{9\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}+\frac {\left (\frac {-10\,a^3\,b^2\,e^3+20\,a^2\,b^3\,d\,e^2-15\,a\,b^4\,d^2\,e+4\,b^5\,d^3}{7\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{7\,e^4}-\frac {b^4\,\left (5\,a\,e-2\,b\,d\right )}{7\,e^4}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{7\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,e^6\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \]
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