Integrand size = 28, antiderivative size = 200 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=\frac {(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x) (d+e x)^7}-\frac {b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x) (d+e x)^6}+\frac {3 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x) (d+e x)^5}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^4 (a+b x) (d+e x)^4} \]
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Time = 0.06 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^5}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^4 (a+b x) (d+e x)^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^4 (a+b x) (d+e x)^7}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^4 (a+b x) (d+e x)^4} \]
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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 )^3}{(d+e x)^8} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3}{e^3 (d+e x)^8}+\frac {3 b^4 (b d-a e)^2}{e^3 (d+e x)^7}-\frac {3 b^5 (b d-a e)}{e^3 (d+e x)^6}+\frac {b^6}{e^3 (d+e x)^5}\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x) (d+e x)^7}-\frac {b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x) (d+e x)^6}+\frac {3 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x) (d+e x)^5}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^4 (a+b x) (d+e x)^4} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.56 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=-\frac {\sqrt {(a+b x)^2} \left (20 a^3 e^3+10 a^2 b e^2 (d+7 e x)+4 a b^2 e \left (d^2+7 d e x+21 e^2 x^2\right )+b^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )}{140 e^4 (a+b x) (d+e x)^7} \]
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Time = 3.62 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.63
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{3} x^{3}}{4 e}-\frac {3 b^{2} \left (4 a e +b d \right ) x^{2}}{20 e^{2}}-\frac {b \left (10 a^{2} e^{2}+4 a b d e +b^{2} d^{2}\right ) x}{20 e^{3}}-\frac {20 a^{3} e^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}}{140 e^{4}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{7}}\) | \(126\) |
gosper | \(-\frac {\left (35 e^{3} x^{3} b^{3}+84 x^{2} a \,b^{2} e^{3}+21 x^{2} b^{3} d \,e^{2}+70 a^{2} b \,e^{3} x +28 x a \,b^{2} d \,e^{2}+7 b^{3} d^{2} e x +20 a^{3} e^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{140 e^{4} \left (e x +d \right )^{7} \left (b x +a \right )^{3}}\) | \(131\) |
default | \(-\frac {\left (35 e^{3} x^{3} b^{3}+84 x^{2} a \,b^{2} e^{3}+21 x^{2} b^{3} d \,e^{2}+70 a^{2} b \,e^{3} x +28 x a \,b^{2} d \,e^{2}+7 b^{3} d^{2} e x +20 a^{3} e^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{140 e^{4} \left (e x +d \right )^{7} \left (b x +a \right )^{3}}\) | \(131\) |
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Time = 0.30 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=-\frac {35 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + 4 \, a b^{2} d^{2} e + 10 \, a^{2} b d e^{2} + 20 \, a^{3} e^{3} + 21 \, {\left (b^{3} d e^{2} + 4 \, a b^{2} e^{3}\right )} x^{2} + 7 \, {\left (b^{3} d^{2} e + 4 \, a b^{2} d e^{2} + 10 \, a^{2} b e^{3}\right )} x}{140 \, {\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=\frac {b^{7} \mathrm {sgn}\left (b x + a\right )}{140 \, {\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )}} - \frac {35 \, b^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, b^{3} d e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 84 \, a b^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{3} d^{2} e x \mathrm {sgn}\left (b x + a\right ) + 28 \, a b^{2} d e^{2} x \mathrm {sgn}\left (b x + a\right ) + 70 \, a^{2} b e^{3} x \mathrm {sgn}\left (b x + a\right ) + b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )}{140 \, {\left (e x + d\right )}^{7} e^{4}} \]
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Time = 9.56 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=\frac {\left (\frac {2\,b^3\,d-3\,a\,b^2\,e}{5\,e^4}+\frac {b^3\,d}{5\,e^4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {3\,a^2\,b\,e^2-3\,a\,b^2\,d\,e+b^3\,d^2}{6\,e^4}+\frac {d\,\left (\frac {b^3\,d}{6\,e^3}-\frac {b^2\,\left (3\,a\,e-b\,d\right )}{6\,e^3}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {a^3}{7\,e}-\frac {d\,\left (\frac {3\,a^2\,b}{7\,e}-\frac {d\,\left (\frac {3\,a\,b^2}{7\,e}-\frac {b^3\,d}{7\,e^2}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,e^4\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \]
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