Integrand size = 28, antiderivative size = 300 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x) (d+e x)^5}-\frac {5 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^4}+\frac {10 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^3}-\frac {5 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^2}+\frac {5 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)} \]
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Time = 0.10 (sec) , antiderivative size = 300, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^3}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^6 (a+b x) (d+e x)^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^5}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)}+\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) (d+e x)}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^2} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^6} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)^6}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^5}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)^4}+\frac {10 b^8 (b d-a e)^2}{e^5 (d+e x)^3}-\frac {5 b^9 (b d-a e)}{e^5 (d+e x)^2}+\frac {b^{10}}{e^5 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x) (d+e x)^5}-\frac {5 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^4}+\frac {10 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^3}-\frac {5 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^2}+\frac {5 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.65 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {\sqrt {(a+b x)^2} \left ((b d-a e) \left (12 a^4 e^4+3 a^3 b e^3 (9 d+25 e x)+a^2 b^2 e^2 \left (47 d^2+175 d e x+200 e^2 x^2\right )+a b^3 e \left (77 d^3+325 d^2 e x+500 d e^2 x^2+300 e^3 x^3\right )+b^4 \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 b^5 (d+e x)^5 \log (d+e x)\right )}{60 e^6 (a+b x) (d+e x)^5} \]
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Time = 3.22 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {5 b^{4} \left (a e -b d \right ) x^{4}}{e^{2}}-\frac {5 b^{3} \left (a^{2} e^{2}+2 a b d e -3 b^{2} d^{2}\right ) x^{3}}{e^{3}}-\frac {5 b^{2} \left (2 a^{3} e^{3}+3 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -11 b^{3} d^{3}\right ) x^{2}}{3 e^{4}}-\frac {5 b \left (3 e^{4} a^{4}+4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}+12 a \,b^{3} d^{3} e -25 b^{4} d^{4}\right ) x}{12 e^{5}}-\frac {12 a^{5} e^{5}+15 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+30 a^{2} b^{3} d^{3} e^{2}+60 a \,b^{4} d^{4} e -137 b^{5} d^{5}}{60 e^{6}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{5}}+\frac {b^{5} \ln \left (e x +d \right ) \sqrt {\left (b x +a \right )^{2}}}{e^{6} \left (b x +a \right )}\) | \(284\) |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 \ln \left (e x +d \right ) b^{5} e^{5} x^{5}+300 \ln \left (e x +d \right ) b^{5} d^{4} e x +137 b^{5} d^{5}-12 a^{5} e^{5}-15 a^{4} b d \,e^{4}-20 a^{3} b^{2} d^{2} e^{3}-30 a^{2} b^{3} d^{3} e^{2}-60 a \,b^{4} d^{4} e -100 a^{3} b^{2} d \,e^{4} x -75 a^{4} b \,e^{5} x +600 \ln \left (e x +d \right ) b^{5} d^{3} e^{2} x^{2}+300 \ln \left (e x +d \right ) b^{5} d \,e^{4} x^{4}-600 x^{3} a \,b^{4} d \,e^{4}-300 x^{2} a^{2} b^{3} d \,e^{4}-600 x^{2} a \,b^{4} d^{2} e^{3}+600 \ln \left (e x +d \right ) b^{5} d^{2} e^{3} x^{3}+300 x^{4} b^{5} d \,e^{4}-300 x^{3} a^{2} b^{3} e^{5}+900 x^{3} b^{5} d^{2} e^{3}+60 \ln \left (e x +d \right ) b^{5} d^{5}+625 b^{5} d^{4} e x -300 x^{4} a \,b^{4} e^{5}-200 x^{2} a^{3} b^{2} e^{5}+1100 x^{2} b^{5} d^{3} e^{2}-150 x \,a^{2} b^{3} d^{2} e^{3}-300 x a \,b^{4} d^{3} e^{2}\right )}{60 \left (b x +a \right )^{5} e^{6} \left (e x +d \right )^{5}}\) | \(383\) |
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Time = 0.40 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {137 \, b^{5} d^{5} - 60 \, a b^{4} d^{4} e - 30 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 15 \, a^{4} b d e^{4} - 12 \, a^{5} e^{5} + 300 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \, {\left (3 \, b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 100 \, {\left (11 \, b^{5} d^{3} e^{2} - 6 \, a b^{4} d^{2} e^{3} - 3 \, a^{2} b^{3} d e^{4} - 2 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \, {\left (25 \, b^{5} d^{4} e - 12 \, a b^{4} d^{3} e^{2} - 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - 3 \, a^{4} b e^{5}\right )} x + 60 \, {\left (b^{5} e^{5} x^{5} + 5 \, b^{5} d e^{4} x^{4} + 10 \, b^{5} d^{2} e^{3} x^{3} + 10 \, b^{5} d^{3} e^{2} x^{2} + 5 \, b^{5} d^{4} e x + b^{5} d^{5}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{6}}\, dx \]
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Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {b^{5} \log \left ({\left | e x + d \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{e^{6}} + \frac {300 \, {\left (b^{5} d e^{3} \mathrm {sgn}\left (b x + a\right ) - a b^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x^{4} + 300 \, {\left (3 \, b^{5} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{4} d e^{3} \mathrm {sgn}\left (b x + a\right ) - a^{2} b^{3} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 100 \, {\left (11 \, b^{5} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{2} b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 2 \, a^{3} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 25 \, {\left (25 \, b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - 12 \, a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x + \frac {137 \, b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 60 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 15 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - 12 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )}{e}}{60 \, {\left (e x + d\right )}^{5} e^{5}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^6} \,d x \]
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