Integrand size = 28, antiderivative size = 217 \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {2 b e}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{2 (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (a+b x)}{(b d-a e)^3 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b e^2 (a+b x) \log (a+b x)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b e^2 (a+b x) \log (d+e x)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 217, 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, 46} \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e^2 (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac {3 b e^2 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {3 b e^2 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac {2 b e}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {b}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]
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Rule 46
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 (d+e x)^2} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {1}{b (b d-a e)^2 (a+b x)^3}-\frac {2 e}{b (b d-a e)^3 (a+b x)^2}+\frac {3 e^2}{b (b d-a e)^4 (a+b x)}-\frac {e^3}{b^3 (b d-a e)^3 (d+e x)^2}-\frac {3 e^3}{b^2 (b d-a e)^4 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 b e}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{2 (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (a+b x)}{(b d-a e)^3 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b e^2 (a+b x) \log (a+b x)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b e^2 (a+b x) \log (d+e x)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.06 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {-\left ((b d-a e) \left (-2 a^2 e^2-a b e (5 d+9 e x)+b^2 \left (d^2-3 d e x-6 e^2 x^2\right )\right )\right )+6 b e^2 (a+b x)^2 (d+e x) \log (a+b x)-6 b e^2 (a+b x)^2 (d+e x) \log (d+e x)}{2 (b d-a e)^4 (a+b x) \sqrt {(a+b x)^2} (d+e x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(329\) vs. \(2(160)=320\).
Time = 2.49 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.52
method | result | size |
default | \(\frac {\left (6 \ln \left (b x +a \right ) x^{3} b^{3} e^{3}-6 \ln \left (e x +d \right ) b^{3} e^{3} x^{3}+12 \ln \left (b x +a \right ) x^{2} a \,b^{2} e^{3}+6 \ln \left (b x +a \right ) b^{3} d \,e^{2} x^{2}-12 \ln \left (e x +d \right ) a \,b^{2} e^{3} x^{2}-6 \ln \left (e x +d \right ) b^{3} d \,e^{2} x^{2}+6 \ln \left (b x +a \right ) x \,a^{2} b \,e^{3}+12 \ln \left (b x +a \right ) x a \,b^{2} d \,e^{2}-6 \ln \left (e x +d \right ) a^{2} b \,e^{3} x -12 \ln \left (e x +d \right ) a \,b^{2} d \,e^{2} x -6 x^{2} a \,b^{2} e^{3}+6 x^{2} b^{3} d \,e^{2}+6 \ln \left (b x +a \right ) a^{2} b d \,e^{2}-6 \ln \left (e x +d \right ) a^{2} b d \,e^{2}-9 a^{2} b \,e^{3} x +6 x a \,b^{2} d \,e^{2}+3 b^{3} d^{2} e x -2 a^{3} e^{3}-3 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (b x +a \right )}{2 \left (e x +d \right ) \left (a e -b d \right )^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(330\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {3 b^{2} e^{2} x^{2}}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}-\frac {3 \left (3 a e +b d \right ) e b x}{2 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {2 a^{2} e^{2}+5 a b d e -b^{2} d^{2}}{2 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\right )}{\left (b x +a \right )^{3} \left (e x +d \right )}-\frac {3 \sqrt {\left (b x +a \right )^{2}}\, e^{2} b \ln \left (e x +d \right )}{\left (b x +a \right ) \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {3 \sqrt {\left (b x +a \right )^{2}}\, e^{2} b \ln \left (-b x -a \right )}{\left (b x +a \right ) \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\) | \(351\) |
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Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (160) = 320\).
Time = 0.25 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.28 \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} - 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \, {\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x - 6 \, {\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} + {\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + {\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} + {\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + {\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{4} d^{5} - 4 \, a^{3} b^{3} d^{4} e + 6 \, a^{4} b^{2} d^{3} e^{2} - 4 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4} + {\left (b^{6} d^{4} e - 4 \, a b^{5} d^{3} e^{2} + 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} + a^{4} b^{2} e^{5}\right )} x^{3} + {\left (b^{6} d^{5} - 2 \, a b^{5} d^{4} e - 2 \, a^{2} b^{4} d^{3} e^{2} + 8 \, a^{3} b^{3} d^{2} e^{3} - 7 \, a^{4} b^{2} d e^{4} + 2 \, a^{5} b e^{5}\right )} x^{2} + {\left (2 \, a b^{5} d^{5} - 7 \, a^{2} b^{4} d^{4} e + 8 \, a^{3} b^{3} d^{3} e^{2} - 2 \, a^{4} b^{2} d^{2} e^{3} - 2 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x\right )}} \]
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\[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (160) = 320\).
Time = 0.28 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {3 \, b^{2} e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {3 \, b e^{3} \log \left ({\left | e x + d \right |}\right )}{b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} - 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \, {\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x}{2 \, {\left (b d - a e\right )}^{4} {\left (b x + a\right )}^{2} {\left (e x + d\right )} \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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