Integrand size = 33, antiderivative size = 146 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^8} \, dx=-\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x) (d+e x)^7}+\frac {b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^6}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5} \]
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Time = 0.05 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45} \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^8} \, dx=-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^3 (a+b x) (d+e x)^6}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^3 (a+b x) (d+e x)^7} \]
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Rule 21
Rule 45
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )}{(d+e x)^8} \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^2}{(d+e x)^8} \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^8}-\frac {2 b (b d-a e)}{e^2 (d+e x)^7}+\frac {b^2}{e^2 (d+e x)^6}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x) (d+e x)^7}+\frac {b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^6}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.50 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^8} \, dx=-\frac {\sqrt {(a+b x)^2} \left (15 a^2 e^2+5 a b e (d+7 e x)+b^2 \left (d^2+7 d e x+21 e^2 x^2\right )\right )}{105 e^3 (a+b x) (d+e x)^7} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 1.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.47
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (21 b^{2} e^{2} x^{2}+35 a b \,e^{2} x +7 b^{2} d e x +15 e^{2} a^{2}+5 a b d e +b^{2} d^{2}\right )}{105 e^{3} \left (e x +d \right )^{7}}\) | \(68\) |
gosper | \(-\frac {\left (21 b^{2} e^{2} x^{2}+35 a b \,e^{2} x +7 b^{2} d e x +15 e^{2} a^{2}+5 a b d e +b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{105 e^{3} \left (e x +d \right )^{7} \left (b x +a \right )}\) | \(78\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{2} x^{2}}{5 e}-\frac {b \left (5 a e +b d \right ) x}{15 e^{2}}-\frac {15 e^{2} a^{2}+5 a b d e +b^{2} d^{2}}{105 e^{3}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{7}}\) | \(79\) |
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Time = 0.31 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^8} \, dx=-\frac {21 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 5 \, a b d e + 15 \, a^{2} e^{2} + 7 \, {\left (b^{2} d e + 5 \, a b e^{2}\right )} x}{105 \, {\left (e^{10} x^{7} + 7 \, d e^{9} x^{6} + 21 \, d^{2} e^{8} x^{5} + 35 \, d^{3} e^{7} x^{4} + 35 \, d^{4} e^{6} x^{3} + 21 \, d^{5} e^{5} x^{2} + 7 \, d^{6} e^{4} x + d^{7} e^{3}\right )}} \]
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Timed out. \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^8} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^8} \, dx=\frac {b^{7} \mathrm {sgn}\left (b x + a\right )}{105 \, {\left (b^{5} d^{5} e^{3} - 5 \, a b^{4} d^{4} e^{4} + 10 \, a^{2} b^{3} d^{3} e^{5} - 10 \, a^{3} b^{2} d^{2} e^{6} + 5 \, a^{4} b d e^{7} - a^{5} e^{8}\right )}} - \frac {21 \, b^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{2} d e x \mathrm {sgn}\left (b x + a\right ) + 35 \, a b e^{2} x \mathrm {sgn}\left (b x + a\right ) + b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b d e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )}{105 \, {\left (e x + d\right )}^{7} e^{3}} \]
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Time = 10.91 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.53 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^8} \, dx=-\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (15\,a^2\,e^2+5\,a\,b\,d\,e+35\,a\,b\,e^2\,x+b^2\,d^2+7\,b^2\,d\,e\,x+21\,b^2\,e^2\,x^2\right )}{105\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7} \]
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