Integrand size = 33, antiderivative size = 146 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=-\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x) (d+e x)^6}+\frac {2 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4} \]
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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} \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^3 (a+b x) (d+e x)^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{6 e^3 (a+b x) (d+e x)^6} \]
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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)^7} \, 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)^7} \, 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)^7}-\frac {2 b (b d-a e)}{e^2 (d+e x)^6}+\frac {b^2}{e^2 (d+e x)^5}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x) (d+e x)^6}+\frac {2 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.50 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=-\frac {\sqrt {(a+b x)^2} \left (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )}{60 e^3 (a+b x) (d+e x)^6} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.94 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.47
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (15 b^{2} e^{2} x^{2}+24 a b \,e^{2} x +6 b^{2} d e x +10 e^{2} a^{2}+4 a b d e +b^{2} d^{2}\right )}{60 e^{3} \left (e x +d \right )^{6}}\) | \(68\) |
gosper | \(-\frac {\left (15 b^{2} e^{2} x^{2}+24 a b \,e^{2} x +6 b^{2} d e x +10 e^{2} a^{2}+4 a b d e +b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{60 e^{3} \left (e x +d \right )^{6} \left (b x +a \right )}\) | \(78\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{2} x^{2}}{4 e}-\frac {b \left (4 a e +b d \right ) x}{10 e^{2}}-\frac {10 e^{2} a^{2}+4 a b d e +b^{2} d^{2}}{60 e^{3}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{6}}\) | \(79\) |
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Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=-\frac {15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \, {\left (b^{2} d e + 4 \, a b e^{2}\right )} x}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]
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Timed out. \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=\frac {b^{6} \mathrm {sgn}\left (b x + a\right )}{60 \, {\left (b^{4} d^{4} e^{3} - 4 \, a b^{3} d^{3} e^{4} + 6 \, a^{2} b^{2} d^{2} e^{5} - 4 \, a^{3} b d e^{6} + a^{4} e^{7}\right )}} - \frac {15 \, b^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{2} d e x \mathrm {sgn}\left (b x + a\right ) + 24 \, a b e^{2} x \mathrm {sgn}\left (b x + a\right ) + b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b d e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )}{60 \, {\left (e x + d\right )}^{6} e^{3}} \]
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Time = 10.94 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.53 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx=-\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (10\,a^2\,e^2+4\,a\,b\,d\,e+24\,a\,b\,e^2\,x+b^2\,d^2+6\,b^2\,d\,e\,x+15\,b^2\,e^2\,x^2\right )}{60\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6} \]
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