Integrand size = 28, antiderivative size = 253 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {10 e^3 (b d-a e)^2}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^5}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (b d-a e)^4}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e^2 (b d-a e)^3}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^5 x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^4 (b d-a e) (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.12 (sec) , antiderivative size = 253, 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, 45} \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {5 e^4 (a+b x) (b d-a e) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {10 e^3 (b d-a e)^2}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e^2 (b d-a e)^3}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (b d-a e)^4}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^5}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^5 x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^5}{\left (a b+b^2 x\right )^5} \, 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 {e^5}{b^{10}}+\frac {(b d-a e)^5}{b^{10} (a+b x)^5}+\frac {5 e (b d-a e)^4}{b^{10} (a+b x)^4}+\frac {10 e^2 (b d-a e)^3}{b^{10} (a+b x)^3}+\frac {10 e^3 (b d-a e)^2}{b^{10} (a+b x)^2}+\frac {5 e^4 (b d-a e)}{b^{10} (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {10 e^3 (b d-a e)^2}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^5}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (b d-a e)^4}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e^2 (b d-a e)^3}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^5 x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^4 (b d-a e) (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {-77 a^5 e^5+a^4 b e^4 (125 d-248 e x)-2 a^3 b^2 e^3 \left (15 d^2-220 d e x+126 e^2 x^2\right )-2 a^2 b^3 e^2 \left (5 d^3+60 d^2 e x-270 d e^2 x^2+24 e^3 x^3\right )+a b^4 e \left (-5 d^4-40 d^3 e x-180 d^2 e^2 x^2+240 d e^3 x^3+48 e^4 x^4\right )-b^5 \left (3 d^5+20 d^4 e x+60 d^3 e^2 x^2+120 d^2 e^3 x^3-12 e^5 x^5\right )-60 e^4 (-b d+a e) (a+b x)^4 \log (a+b x)}{12 b^6 (a+b x)^3 \sqrt {(a+b x)^2}} \]
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Time = 2.87 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.15
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{5} x}{\left (b x +a \right ) b^{5}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-10 a^{2} b^{2} e^{5}+20 a \,b^{3} d \,e^{4}-10 d^{2} e^{3} b^{4}\right ) x^{3}-5 b \,e^{2} \left (5 a^{3} e^{3}-9 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{2}-\frac {5 e \left (13 e^{4} a^{4}-22 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}+2 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x}{3}-\frac {77 a^{5} e^{5}-125 a^{4} b d \,e^{4}+30 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +3 b^{5} d^{5}}{12 b}\right )}{\left (b x +a \right )^{5} b^{5}}-\frac {5 \sqrt {\left (b x +a \right )^{2}}\, e^{4} \left (a e -b d \right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{6}}\) | \(291\) |
default | \(-\frac {\left (3 b^{5} d^{5}+77 a^{5} e^{5}-125 a^{4} b d \,e^{4}+30 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -240 \ln \left (b x +a \right ) a \,b^{4} d \,e^{4} x^{3}-360 \ln \left (b x +a \right ) a^{2} b^{3} d \,e^{4} x^{2}-440 a^{3} b^{2} d \,e^{4} x -60 \ln \left (b x +a \right ) b^{5} d \,e^{4} x^{4}+248 a^{4} b \,e^{5} x -240 x^{3} a \,b^{4} d \,e^{4}-540 x^{2} a^{2} b^{3} d \,e^{4}+180 x^{2} a \,b^{4} d^{2} e^{3}+240 \ln \left (b x +a \right ) x \,a^{4} b \,e^{5}-60 \ln \left (b x +a \right ) a^{4} b d \,e^{4}-240 \ln \left (b x +a \right ) x \,a^{3} b^{2} d \,e^{4}+48 x^{3} a^{2} b^{3} e^{5}+120 x^{3} b^{5} d^{2} e^{3}+60 \ln \left (b x +a \right ) x^{4} a \,b^{4} e^{5}+20 b^{5} d^{4} e x -12 x^{5} e^{5} b^{5}+60 \ln \left (b x +a \right ) a^{5} e^{5}-48 x^{4} a \,b^{4} e^{5}+240 \ln \left (b x +a \right ) x^{3} a^{2} b^{3} e^{5}+360 \ln \left (b x +a \right ) x^{2} a^{3} b^{2} e^{5}+252 x^{2} a^{3} b^{2} e^{5}+60 x^{2} b^{5} d^{3} e^{2}+120 x \,a^{2} b^{3} d^{2} e^{3}+40 x a \,b^{4} d^{3} e^{2}\right ) \left (b x +a \right )}{12 b^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(449\) |
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Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (183) = 366\).
Time = 0.28 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.63 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {12 \, b^{5} e^{5} x^{5} + 48 \, a b^{4} e^{5} x^{4} - 3 \, b^{5} d^{5} - 5 \, a b^{4} d^{4} e - 10 \, a^{2} b^{3} d^{3} e^{2} - 30 \, a^{3} b^{2} d^{2} e^{3} + 125 \, a^{4} b d e^{4} - 77 \, a^{5} e^{5} - 24 \, {\left (5 \, b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4} + 2 \, a^{2} b^{3} e^{5}\right )} x^{3} - 12 \, {\left (5 \, b^{5} d^{3} e^{2} + 15 \, a b^{4} d^{2} e^{3} - 45 \, a^{2} b^{3} d e^{4} + 21 \, a^{3} b^{2} e^{5}\right )} x^{2} - 4 \, {\left (5 \, b^{5} d^{4} e + 10 \, a b^{4} d^{3} e^{2} + 30 \, a^{2} b^{3} d^{2} e^{3} - 110 \, a^{3} b^{2} d e^{4} + 62 \, a^{4} b e^{5}\right )} x + 60 \, {\left (a^{4} b d e^{4} - a^{5} e^{5} + {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 4 \, {\left (a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 6 \, {\left (a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{2} d e^{4} - a^{4} b e^{5}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} \]
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\[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{5}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (183) = 366\).
Time = 0.23 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.75 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {1}{12} \, e^{5} {\left (\frac {12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}} - \frac {60 \, a \log \left (b x + a\right )}{b^{6}}\right )} + \frac {5}{12} \, d e^{4} {\left (\frac {48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}} + \frac {12 \, \log \left (b x + a\right )}{b^{5}}\right )} - \frac {5}{6} \, d^{2} e^{3} {\left (\frac {12 \, x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}} + \frac {6 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {3 \, a^{3}}{b^{8} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {5}{12} \, d^{4} e {\left (\frac {4}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {3 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {5}{6} \, d^{3} e^{2} {\left (\frac {6}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{3}} + \frac {3 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {d^{5}}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^{5} x}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {5 \, {\left (b d e^{4} - a e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {3 \, b^{5} d^{5} + 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} - 125 \, a^{4} b d e^{4} + 77 \, a^{5} e^{5} + 120 \, {\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 60 \, {\left (b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} - 9 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 20 \, {\left (b^{5} d^{4} e + 2 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 22 \, a^{3} b^{2} d e^{4} + 13 \, a^{4} b e^{5}\right )} x}{12 \, {\left (b x + a\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^5}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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