Integrand size = 28, antiderivative size = 143 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{6 (b d-a e) (d+e x)^6}+\frac {b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{15 (b d-a e)^2 (d+e x)^5}+\frac {b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{60 (b d-a e)^3 (d+e x)^4} \]
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Time = 0.06 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.40, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^4 (a+b x) (d+e x)^4}-\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^4 (a+b x) (d+e x)^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{6 e^4 (a+b x) (d+e x)^6}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^3} \]
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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)^7} \, 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)^7}+\frac {3 b^4 (b d-a e)^2}{e^3 (d+e x)^6}-\frac {3 b^5 (b d-a e)}{e^3 (d+e x)^5}+\frac {b^6}{e^3 (d+e x)^4}\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}}{6 e^4 (a+b x) (d+e x)^6}-\frac {3 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x) (d+e x)^5}+\frac {3 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^4 (a+b x) (d+e x)^4}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^3} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {\sqrt {(a+b x)^2} \left (10 a^3 e^3+6 a^2 b e^2 (d+6 e x)+3 a b^2 e \left (d^2+6 d e x+15 e^2 x^2\right )+b^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )}{60 e^4 (a+b x) (d+e x)^6} \]
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Time = 3.48 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{3} x^{3}}{3 e}-\frac {b^{2} \left (3 a e +b d \right ) x^{2}}{4 e^{2}}-\frac {b \left (6 a^{2} e^{2}+3 a b d e +b^{2} d^{2}\right ) x}{10 e^{3}}-\frac {10 a^{3} e^{3}+6 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}}{60 e^{4}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{6}}\) | \(126\) |
gosper | \(-\frac {\left (20 e^{3} x^{3} b^{3}+45 x^{2} a \,b^{2} e^{3}+15 x^{2} b^{3} d \,e^{2}+36 a^{2} b \,e^{3} x +18 x a \,b^{2} d \,e^{2}+6 b^{3} d^{2} e x +10 a^{3} e^{3}+6 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 e^{4} \left (e x +d \right )^{6} \left (b x +a \right )^{3}}\) | \(131\) |
default | \(-\frac {\left (20 e^{3} x^{3} b^{3}+45 x^{2} a \,b^{2} e^{3}+15 x^{2} b^{3} d \,e^{2}+36 a^{2} b \,e^{3} x +18 x a \,b^{2} d \,e^{2}+6 b^{3} d^{2} e x +10 a^{3} e^{3}+6 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 e^{4} \left (e x +d \right )^{6} \left (b x +a \right )^{3}}\) | \(131\) |
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Time = 0.31 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {20 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} + 10 \, a^{3} e^{3} + 15 \, {\left (b^{3} d e^{2} + 3 \, a b^{2} e^{3}\right )} x^{2} + 6 \, {\left (b^{3} d^{2} e + 3 \, a b^{2} d e^{2} + 6 \, a^{2} b e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{7}}\, dx \]
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Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {b^{6} \mathrm {sgn}\left (b x + a\right )}{60 \, {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )}} - \frac {20 \, b^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{3} d e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 45 \, a b^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{3} d^{2} e x \mathrm {sgn}\left (b x + a\right ) + 18 \, a b^{2} d e^{2} x \mathrm {sgn}\left (b x + a\right ) + 36 \, a^{2} b e^{3} x \mathrm {sgn}\left (b x + a\right ) + b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )}{60 \, {\left (e x + d\right )}^{6} e^{4}} \]
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Time = 9.50 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {\left (\frac {2\,b^3\,d-3\,a\,b^2\,e}{4\,e^4}+\frac {b^3\,d}{4\,e^4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {\left (\frac {3\,a^2\,b\,e^2-3\,a\,b^2\,d\,e+b^3\,d^2}{5\,e^4}+\frac {d\,\left (\frac {b^3\,d}{5\,e^3}-\frac {b^2\,\left (3\,a\,e-b\,d\right )}{5\,e^3}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {a^3}{6\,e}-\frac {d\,\left (\frac {a^2\,b}{2\,e}-\frac {d\,\left (\frac {a\,b^2}{2\,e}-\frac {b^3\,d}{6\,e^2}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,e^4\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3} \]
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