Integrand size = 28, antiderivative size = 276 \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {3 b^2 e}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2}{2 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (a+b x)}{2 (b d-a e)^3 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b e^2 (a+b x)}{(b d-a e)^4 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {6 b^2 e^2 (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {6 b^2 e^2 (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 276, 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} \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {3 b e^2 (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}+\frac {e^2 (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}+\frac {6 b^2 e^2 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac {6 b^2 e^2 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac {b^2}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac {3 b^2 e}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]
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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)^3} \, 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 d-a e)^3 (a+b x)^3}-\frac {3 e}{(b d-a e)^4 (a+b x)^2}+\frac {6 e^2}{(b d-a e)^5 (a+b x)}-\frac {e^3}{b^3 (b d-a e)^3 (d+e x)^3}-\frac {3 e^3}{b^2 (b d-a e)^4 (d+e x)^2}-\frac {6 e^3}{b (b d-a e)^5 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {3 b^2 e}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2}{2 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (a+b x)}{2 (b d-a e)^3 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b e^2 (a+b x)}{(b d-a e)^4 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {6 b^2 e^2 (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {6 b^2 e^2 (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.06 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.59 \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {(a+b x) \left (-b^2 (b d-a e)^2+6 b^2 e (b d-a e) (a+b x)+\frac {e^2 (b d-a e)^2 (a+b x)^2}{(d+e x)^2}+\frac {6 b e^2 (b d-a e) (a+b x)^2}{d+e x}+12 b^2 e^2 (a+b x)^2 \log (a+b x)-12 b^2 e^2 (a+b x)^2 \log (d+e x)\right )}{2 (b d-a e)^5 \left ((a+b x)^2\right )^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(507\) vs. \(2(206)=412\).
Time = 2.61 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.84
method | result | size |
default | \(-\frac {\left (48 \ln \left (b x +a \right ) x^{2} a \,b^{3} d \,e^{3}+e^{4} a^{4}-b^{4} d^{4}+8 a \,b^{3} d^{3} e -8 b \,e^{3} d \,a^{3}+24 \ln \left (b x +a \right ) x \,a^{2} b^{2} d \,e^{3}+24 \ln \left (b x +a \right ) x a \,b^{3} d^{2} e^{2}-48 \ln \left (e x +d \right ) x^{2} a \,b^{3} d \,e^{3}-12 \ln \left (e x +d \right ) x^{2} a^{2} b^{2} e^{4}+12 \ln \left (b x +a \right ) b^{4} d^{2} e^{2} x^{2}+4 x \,b^{4} d^{3} e -24 \ln \left (e x +d \right ) x a \,b^{3} d^{2} e^{2}-24 \ln \left (e x +d \right ) a \,b^{3} e^{4} x^{3}-24 \ln \left (e x +d \right ) x \,a^{2} b^{2} d \,e^{3}-12 \ln \left (e x +d \right ) x^{4} b^{4} e^{4}-12 \ln \left (e x +d \right ) x^{2} b^{4} d^{2} e^{2}-24 \ln \left (e x +d \right ) x^{3} b^{4} d \,e^{3}+12 \ln \left (b x +a \right ) x^{2} a^{2} b^{2} e^{4}+24 \ln \left (b x +a \right ) x^{3} a \,b^{3} e^{4}+12 \ln \left (b x +a \right ) b^{4} e^{4} x^{4}-12 x^{3} a \,b^{3} e^{4}+12 x^{3} b^{4} d \,e^{3}-18 x^{2} a^{2} b^{2} e^{4}+18 x^{2} b^{4} d^{2} e^{2}-4 x \,a^{3} b \,e^{4}-12 \ln \left (e x +d \right ) a^{2} b^{2} d^{2} e^{2}+24 \ln \left (b x +a \right ) x^{3} b^{4} d \,e^{3}-24 x \,a^{2} b^{2} d \,e^{3}+24 x a \,b^{3} d^{2} e^{2}+12 \ln \left (b x +a \right ) a^{2} b^{2} d^{2} e^{2}\right ) \left (b x +a \right )}{2 \left (e x +d \right )^{2} \left (a e -b d \right )^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(508\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {6 b^{3} e^{3} x^{3}}{e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {9 b^{2} e^{2} \left (a e +b d \right ) x^{2}}{e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {2 \left (a^{2} e^{2}+7 a b d e +b^{2} d^{2}\right ) b e x}{e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {a^{3} e^{3}-7 a^{2} b d \,e^{2}-7 a \,b^{2} d^{2} e +b^{3} d^{3}}{2 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\right )}{\left (b x +a \right )^{3} \left (e x +d \right )^{2}}-\frac {6 \sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{2} \ln \left (b x +a \right )}{\left (b x +a \right ) \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}+\frac {6 \sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{2} \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}\) | \(518\) |
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Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (206) = 412\).
Time = 0.28 (sec) , antiderivative size = 760, normalized size of antiderivative = 2.75 \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {b^{4} d^{4} - 8 \, a b^{3} d^{3} e + 8 \, a^{3} b d e^{3} - a^{4} e^{4} - 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} - 18 \, {\left (b^{4} d^{2} e^{2} - a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} - 6 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x - 12 \, {\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{5} d^{7} - 5 \, a^{3} b^{4} d^{6} e + 10 \, a^{4} b^{3} d^{5} e^{2} - 10 \, a^{5} b^{2} d^{4} e^{3} + 5 \, a^{6} b d^{3} e^{4} - a^{7} d^{2} e^{5} + {\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{4} + 2 \, {\left (b^{7} d^{6} e - 4 \, a b^{6} d^{5} e^{2} + 5 \, a^{2} b^{5} d^{4} e^{3} - 5 \, a^{4} b^{3} d^{2} e^{5} + 4 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x^{3} + {\left (b^{7} d^{7} - a b^{6} d^{6} e - 9 \, a^{2} b^{5} d^{5} e^{2} + 25 \, a^{3} b^{4} d^{4} e^{3} - 25 \, a^{4} b^{3} d^{3} e^{4} + 9 \, a^{5} b^{2} d^{2} e^{5} + a^{6} b d e^{6} - a^{7} e^{7}\right )} x^{2} + 2 \, {\left (a b^{6} d^{7} - 4 \, a^{2} b^{5} d^{6} e + 5 \, a^{3} b^{4} d^{5} e^{2} - 5 \, a^{5} b^{2} d^{3} e^{4} + 4 \, a^{6} b d^{2} e^{5} - a^{7} d e^{6}\right )} x\right )}} \]
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\[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (206) = 412\).
Time = 0.28 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {6 \, b^{3} e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {6 \, b^{2} e^{3} \log \left ({\left | e x + d \right |}\right )}{b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {12 \, b^{3} e^{3} x^{3} + 18 \, b^{3} d e^{2} x^{2} + 18 \, a b^{2} e^{3} x^{2} + 4 \, b^{3} d^{2} e x + 28 \, a b^{2} d e^{2} x + 4 \, a^{2} b e^{3} x - b^{3} d^{3} + 7 \, a b^{2} d^{2} e + 7 \, a^{2} b d e^{2} - a^{3} e^{3}}{2 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} {\left (b e x^{2} + b d x + a e x + a d\right )}^{2}} \]
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Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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