Integrand size = 30, antiderivative size = 314 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {2 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^{7/2}}-\frac {2 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^{5/2}}+\frac {20 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac {20 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt {d+e x}}-\frac {10 b^4 (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)} \]
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Time = 0.07 (sec) , antiderivative size = 314, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) (d+e x)^{5/2}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^6 (a+b x)}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)}{e^6 (a+b x)}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) \sqrt {d+e x}} \]
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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 {\left (a b+b^2 x\right )^5}{(d+e x)^{9/2}} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)^{9/2}}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^{7/2}}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)^{5/2}}+\frac {10 b^8 (b d-a e)^2}{e^5 (d+e x)^{3/2}}-\frac {5 b^9 (b d-a e)}{e^5 \sqrt {d+e x}}+\frac {b^{10} \sqrt {d+e x}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {2 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^{7/2}}-\frac {2 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^{5/2}}+\frac {20 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac {20 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt {d+e x}}-\frac {10 b^4 (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (3 a^5 e^5+3 a^4 b e^4 (2 d+7 e x)+2 a^3 b^2 e^3 \left (8 d^2+28 d e x+35 e^2 x^2\right )+6 a^2 b^3 e^2 \left (16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right )-3 a b^4 e \left (128 d^4+448 d^3 e x+560 d^2 e^2 x^2+280 d e^3 x^3+35 e^4 x^4\right )+b^5 \left (256 d^5+896 d^4 e x+1120 d^3 e^2 x^2+560 d^2 e^3 x^3+70 d e^4 x^4-7 e^5 x^5\right )\right )}{21 e^6 (a+b x) (d+e x)^{7/2}} \]
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Time = 2.23 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {2 b^{4} \left (b e x +15 a e -14 b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 e^{6} \left (b x +a \right )}-\frac {2 \left (210 e^{3} x^{3} b^{3}+70 x^{2} a \,b^{2} e^{3}+560 x^{2} b^{3} d \,e^{2}+21 a^{2} b \,e^{3} x +98 x a \,b^{2} d \,e^{2}+511 b^{3} d^{2} e x +3 a^{3} e^{3}+12 a^{2} b d \,e^{2}+37 a \,b^{2} d^{2} e +158 b^{3} d^{3}\right ) \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{21 e^{6} \sqrt {e x +d}\, \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right ) \left (b x +a \right )}\) | \(227\) |
gosper | \(-\frac {2 \left (-7 x^{5} e^{5} b^{5}-105 x^{4} a \,b^{4} e^{5}+70 x^{4} b^{5} d \,e^{4}+210 x^{3} a^{2} b^{3} e^{5}-840 x^{3} a \,b^{4} d \,e^{4}+560 x^{3} b^{5} d^{2} e^{3}+70 x^{2} a^{3} b^{2} e^{5}+420 x^{2} a^{2} b^{3} d \,e^{4}-1680 x^{2} a \,b^{4} d^{2} e^{3}+1120 x^{2} b^{5} d^{3} e^{2}+21 a^{4} b \,e^{5} x +56 a^{3} b^{2} d \,e^{4} x +336 x \,a^{2} b^{3} d^{2} e^{3}-1344 x a \,b^{4} d^{3} e^{2}+896 b^{5} d^{4} e x +3 a^{5} e^{5}+6 a^{4} b d \,e^{4}+16 a^{3} b^{2} d^{2} e^{3}+96 a^{2} b^{3} d^{3} e^{2}-384 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{21 \left (e x +d \right )^{\frac {7}{2}} e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
default | \(-\frac {2 \left (-7 x^{5} e^{5} b^{5}-105 x^{4} a \,b^{4} e^{5}+70 x^{4} b^{5} d \,e^{4}+210 x^{3} a^{2} b^{3} e^{5}-840 x^{3} a \,b^{4} d \,e^{4}+560 x^{3} b^{5} d^{2} e^{3}+70 x^{2} a^{3} b^{2} e^{5}+420 x^{2} a^{2} b^{3} d \,e^{4}-1680 x^{2} a \,b^{4} d^{2} e^{3}+1120 x^{2} b^{5} d^{3} e^{2}+21 a^{4} b \,e^{5} x +56 a^{3} b^{2} d \,e^{4} x +336 x \,a^{2} b^{3} d^{2} e^{3}-1344 x a \,b^{4} d^{3} e^{2}+896 b^{5} d^{4} e x +3 a^{5} e^{5}+6 a^{4} b d \,e^{4}+16 a^{3} b^{2} d^{2} e^{3}+96 a^{2} b^{3} d^{3} e^{2}-384 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{21 \left (e x +d \right )^{\frac {7}{2}} e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
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Time = 0.36 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e - 96 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 6 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 35 \, {\left (2 \, b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} - 70 \, {\left (8 \, b^{5} d^{2} e^{3} - 12 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} - 70 \, {\left (16 \, b^{5} d^{3} e^{2} - 24 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} - 7 \, {\left (128 \, b^{5} d^{4} e - 192 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 8 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{21 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e - 96 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 6 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 35 \, {\left (2 \, b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} - 70 \, {\left (8 \, b^{5} d^{2} e^{3} - 12 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} - 70 \, {\left (16 \, b^{5} d^{3} e^{2} - 24 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} - 7 \, {\left (128 \, b^{5} d^{4} e - 192 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 8 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x\right )}}{21 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )} \sqrt {e x + d}} \]
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Time = 0.30 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {2 \, {\left (210 \, {\left (e x + d\right )}^{3} b^{5} d^{2} \mathrm {sgn}\left (b x + a\right ) - 70 \, {\left (e x + d\right )}^{2} b^{5} d^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (e x + d\right )} b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - 3 \, b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 420 \, {\left (e x + d\right )}^{3} a b^{4} d e \mathrm {sgn}\left (b x + a\right ) + 210 \, {\left (e x + d\right )}^{2} a b^{4} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 84 \, {\left (e x + d\right )} a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 210 \, {\left (e x + d\right )}^{3} a^{2} b^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 210 \, {\left (e x + d\right )}^{2} a^{2} b^{3} d e^{2} \mathrm {sgn}\left (b x + a\right ) + 126 \, {\left (e x + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 70 \, {\left (e x + d\right )}^{2} a^{3} b^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 84 \, {\left (e x + d\right )} a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (e x + d\right )} a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right ) - 15 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )}}{21 \, {\left (e x + d\right )}^{\frac {7}{2}} e^{6}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} b^{5} e^{12} \mathrm {sgn}\left (b x + a\right ) - 15 \, \sqrt {e x + d} b^{5} d e^{12} \mathrm {sgn}\left (b x + a\right ) + 15 \, \sqrt {e x + d} a b^{4} e^{13} \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, e^{18}} \]
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Time = 10.57 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {6\,a^5\,e^5+12\,a^4\,b\,d\,e^4+32\,a^3\,b^2\,d^2\,e^3+192\,a^2\,b^3\,d^3\,e^2-768\,a\,b^4\,d^4\,e+512\,b^5\,d^5}{21\,b\,e^9}-\frac {2\,b^4\,x^5}{3\,e^4}-\frac {10\,b^3\,x^4\,\left (3\,a\,e-2\,b\,d\right )}{3\,e^5}+\frac {x\,\left (42\,a^4\,b\,e^5+112\,a^3\,b^2\,d\,e^4+672\,a^2\,b^3\,d^2\,e^3-2688\,a\,b^4\,d^3\,e^2+1792\,b^5\,d^4\,e\right )}{21\,b\,e^9}+\frac {20\,b^2\,x^3\,\left (3\,a^2\,e^2-12\,a\,b\,d\,e+8\,b^2\,d^2\right )}{3\,e^6}+\frac {20\,b\,x^2\,\left (a^3\,e^3+6\,a^2\,b\,d\,e^2-24\,a\,b^2\,d^2\,e+16\,b^3\,d^3\right )}{3\,e^7}\right )}{x^4\,\sqrt {d+e\,x}+\frac {a\,d^3\,\sqrt {d+e\,x}}{b\,e^3}+\frac {x^3\,\left (21\,a\,e^9+63\,b\,d\,e^8\right )\,\sqrt {d+e\,x}}{21\,b\,e^9}+\frac {3\,d\,x^2\,\left (a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^2\,x\,\left (3\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}} \]
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