Integrand size = 28, antiderivative size = 365 \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {10 b^2 e^3}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2}{4 (b d-a e)^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 e}{(b d-a e)^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b^2 e^2}{(b d-a e)^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x)}{2 (b d-a e)^5 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b e^4 (a+b x)}{(b d-a e)^6 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 b^2 e^4 (a+b x) \log (a+b x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 b^2 e^4 (a+b x) \log (d+e x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.17 (sec) , antiderivative size = 365, 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} \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {5 b e^4 (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^6}+\frac {e^4 (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^5}+\frac {15 b^2 e^4 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^7}-\frac {15 b^2 e^4 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac {10 b^2 e^3}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac {3 b^2 e^2}{(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac {b^2 e}{(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {b^2}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]
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Rule 46
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^5 (d+e x)^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {1}{b^2 (b d-a e)^3 (a+b x)^5}-\frac {3 e}{b^2 (b d-a e)^4 (a+b x)^4}+\frac {6 e^2}{b^2 (b d-a e)^5 (a+b x)^3}-\frac {10 e^3}{b^2 (b d-a e)^6 (a+b x)^2}+\frac {15 e^4}{b^2 (b d-a e)^7 (a+b x)}-\frac {e^5}{b^5 (b d-a e)^5 (d+e x)^3}-\frac {5 e^5}{b^4 (b d-a e)^6 (d+e x)^2}-\frac {15 e^5}{b^3 (b d-a e)^7 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {10 b^2 e^3}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2}{4 (b d-a e)^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 e}{(b d-a e)^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b^2 e^2}{(b d-a e)^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x)}{2 (b d-a e)^5 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b e^4 (a+b x)}{(b d-a e)^6 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 b^2 e^4 (a+b x) \log (a+b x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 b^2 e^4 (a+b x) \log (d+e x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.09 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {4 b^2 e (b d-a e)^3-\frac {b^2 (b d-a e)^4}{a+b x}-12 b^2 e^2 (b d-a e)^2 (a+b x)+40 b^2 e^3 (b d-a e) (a+b x)^2+\frac {2 e^4 (b d-a e)^2 (a+b x)^3}{(d+e x)^2}+\frac {20 b e^4 (b d-a e) (a+b x)^3}{d+e x}+60 b^2 e^4 (a+b x)^3 \log (a+b x)-60 b^2 e^4 (a+b x)^3 \log (d+e x)}{4 (b d-a e)^7 \left ((a+b x)^2\right )^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(982\) vs. \(2(273)=546\).
Time = 2.60 (sec) , antiderivative size = 983, normalized size of antiderivative = 2.69
method | result | size |
default | \(-\frac {\left (-12 x \,a^{5} b \,e^{6}+2 x \,b^{6} d^{5} e -60 x^{5} a \,b^{5} e^{6}+60 x^{5} b^{6} d \,e^{5}-210 x^{4} a^{2} b^{4} e^{6}+90 x^{4} b^{6} d^{2} e^{4}-260 x^{3} a^{3} b^{3} e^{6}+20 x^{3} b^{6} d^{3} e^{3}-125 x^{2} a^{4} b^{2} e^{6}+480 \ln \left (b x +a \right ) a \,b^{5} d \,e^{5} x^{4}+60 \ln \left (b x +a \right ) b^{6} d^{2} e^{4} x^{4}+2 a^{6} e^{6}-b^{6} d^{6}+8 a \,b^{5} d^{5} e -35 a^{4} b^{2} d^{2} e^{4}+80 a^{3} b^{3} d^{3} e^{3}-30 a^{2} b^{4} d^{4} e^{2}-24 a^{5} b d \,e^{5}-360 \ln \left (e x +d \right ) x^{2} a^{2} b^{4} d^{2} e^{4}-60 \ln \left (e x +d \right ) a^{4} b^{2} e^{6} x^{2}+60 \ln \left (b x +a \right ) a^{4} b^{2} e^{6} x^{2}-720 \ln \left (e x +d \right ) x^{3} a^{2} b^{4} d \,e^{5}-480 \ln \left (e x +d \right ) x^{4} a \,b^{5} d \,e^{5}+360 \ln \left (b x +a \right ) a^{2} b^{4} e^{6} x^{4}+60 \ln \left (b x +a \right ) b^{6} e^{6} x^{6}-480 \ln \left (e x +d \right ) a^{3} b^{3} d \,e^{5} x^{2}+240 \ln \left (b x +a \right ) a^{3} b^{3} e^{6} x^{3}+120 \ln \left (b x +a \right ) a^{4} b^{2} d \,e^{5} x +240 \ln \left (b x +a \right ) a^{3} b^{3} d^{2} e^{4} x -240 \ln \left (e x +d \right ) a \,b^{5} e^{6} x^{5}-120 \ln \left (e x +d \right ) a^{4} b^{2} d \,e^{5} x -240 \ln \left (e x +d \right ) a^{3} b^{3} d^{2} e^{4} x +720 \ln \left (b x +a \right ) a^{2} b^{4} d \,e^{5} x^{3}+240 \ln \left (b x +a \right ) a \,b^{5} d^{2} e^{4} x^{3}+480 \ln \left (b x +a \right ) a^{3} b^{3} d \,e^{5} x^{2}+360 \ln \left (b x +a \right ) a^{2} b^{4} d^{2} e^{4} x^{2}-240 \ln \left (e x +d \right ) x^{3} a \,b^{5} d^{2} e^{4}+60 \ln \left (b x +a \right ) a^{4} b^{2} d^{2} e^{4}+240 \ln \left (b x +a \right ) a \,b^{5} e^{6} x^{5}+120 \ln \left (b x +a \right ) b^{6} d \,e^{5} x^{5}-240 \ln \left (e x +d \right ) a^{3} b^{3} e^{6} x^{3}-60 \ln \left (e x +d \right ) x^{6} b^{6} e^{6}-60 \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{4}-360 \ln \left (e x +d \right ) x^{4} a^{2} b^{4} e^{6}-60 \ln \left (e x +d \right ) x^{4} b^{6} d^{2} e^{4}-120 \ln \left (e x +d \right ) x^{5} b^{6} d \,e^{5}-5 x^{2} b^{6} d^{4} e^{2}+120 x^{4} a \,b^{5} d \,e^{5}-60 x^{3} a^{2} b^{4} d \,e^{5}+300 x^{3} a \,b^{5} d^{2} e^{4}-280 x^{2} a^{3} b^{3} d \,e^{5}+330 x^{2} a^{2} b^{4} d^{2} e^{4}+80 x^{2} a \,b^{5} d^{3} e^{3}-190 x \,a^{4} b^{2} d \,e^{5}+100 x \,a^{3} b^{3} d^{2} e^{4}+120 x \,a^{2} b^{4} d^{3} e^{3}-20 x a \,b^{5} d^{4} e^{2}\right ) \left (b x +a \right )}{4 \left (e x +d \right )^{2} \left (a e -b d \right )^{7} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(983\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {15 b^{5} e^{5} x^{5}}{a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}+\frac {15 b^{4} e^{4} \left (7 a e +3 b d \right ) x^{4}}{2 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}+\frac {5 b^{3} e^{3} \left (13 a^{2} e^{2}+16 a b d e +b^{2} d^{2}\right ) x^{3}}{a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}+\frac {5 b^{2} e^{2} \left (25 a^{3} e^{3}+81 a^{2} b d \,e^{2}+15 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{2}}{4 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}+\frac {\left (6 e^{4} a^{4}+101 b \,e^{3} d \,a^{3}+51 b^{2} e^{2} d^{2} a^{2}-9 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) b e x}{2 a^{6} e^{6}-12 a^{5} b d \,e^{5}+30 a^{4} b^{2} d^{2} e^{4}-40 a^{3} b^{3} d^{3} e^{3}+30 a^{2} b^{4} d^{4} e^{2}-12 a \,b^{5} d^{5} e +2 b^{6} d^{6}}-\frac {2 a^{5} e^{5}-22 a^{4} b d \,e^{4}-57 a^{3} b^{2} d^{2} e^{3}+23 a^{2} b^{3} d^{3} e^{2}-7 a \,b^{4} d^{4} e +b^{5} d^{5}}{4 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}\right )}{\left (b x +a \right )^{5} \left (e x +d \right )^{2}}+\frac {15 \sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{4} \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (a^{7} e^{7}-7 a^{6} b d \,e^{6}+21 a^{5} b^{2} d^{2} e^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}-\frac {15 \sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{4} \ln \left (b x +a \right )}{\left (b x +a \right ) \left (a^{7} e^{7}-7 a^{6} b d \,e^{6}+21 a^{5} b^{2} d^{2} e^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}\) | \(985\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1565 vs. \(2 (273) = 546\).
Time = 0.44 (sec) , antiderivative size = 1565, normalized size of antiderivative = 4.29 \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (273) = 546\).
Time = 0.28 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.87 \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {15 \, b^{3} e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{8} d^{7} \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{7} d^{6} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{6} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{5} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{4} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{3} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b^{2} d e^{6} \mathrm {sgn}\left (b x + a\right ) - a^{7} b e^{7} \mathrm {sgn}\left (b x + a\right )} - \frac {15 \, b^{2} e^{5} \log \left ({\left | e x + d \right |}\right )}{b^{7} d^{7} e \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{6} d^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{5} d^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{4} d^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{3} d^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{2} d^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b d e^{7} \mathrm {sgn}\left (b x + a\right ) - a^{7} e^{8} \mathrm {sgn}\left (b x + a\right )} - \frac {b^{6} d^{6} - 8 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} - 80 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 24 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 60 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} - 30 \, {\left (3 \, b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} - 7 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} + 15 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 13 \, a^{3} b^{3} e^{6}\right )} x^{3} + 5 \, {\left (b^{6} d^{4} e^{2} - 16 \, a b^{5} d^{3} e^{3} - 66 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 25 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (b^{6} d^{5} e - 10 \, a b^{5} d^{4} e^{2} + 60 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 95 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x}{4 \, {\left (b d - a e\right )}^{7} {\left (b x + a\right )}^{4} {\left (e x + d\right )}^{2} \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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