Integrand size = 28, antiderivative size = 254 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {b (b d-a e)^4 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac {(b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^4}+\frac {(b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3}-\frac {(b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^2}+\frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e}-\frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)} \]
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Time = 0.08 (sec) , antiderivative size = 254, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 \log (d+e x)}{e^6 (a+b x)}+\frac {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x)}-\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^4}+\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^3}-\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2}+\frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e} \]
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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} \, 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^6 (b d-a e)^4}{e^5}-\frac {b^5 (b d-a e)^3 \left (a b+b^2 x\right )}{e^4}+\frac {b^4 (b d-a e)^2 \left (a b+b^2 x\right )^2}{e^3}-\frac {b^3 (b d-a e) \left (a b+b^2 x\right )^3}{e^2}+\frac {b^2 \left (a b+b^2 x\right )^4}{e}-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {b (b d-a e)^4 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac {(b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^4}+\frac {(b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3}-\frac {(b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^2}+\frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e}-\frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)} \\ \end{align*}
Time = 1.06 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {\sqrt {(a+b x)^2} \left (b e x \left (300 a^4 e^4+300 a^3 b e^3 (-2 d+e x)+100 a^2 b^2 e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+25 a b^3 e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )-60 (b d-a e)^5 \log (d+e x)\right )}{60 e^6 (a+b x)} \]
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Time = 3.02 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.17
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b \left (\frac {1}{5} b^{4} x^{5} e^{4}+\frac {5}{4} x^{4} a \,b^{3} e^{4}-\frac {1}{4} x^{4} b^{4} d \,e^{3}+\frac {10}{3} x^{3} a^{2} b^{2} e^{4}-\frac {5}{3} x^{3} a \,b^{3} d \,e^{3}+\frac {1}{3} x^{3} b^{4} d^{2} e^{2}+5 x^{2} a^{3} b \,e^{4}-5 x^{2} a^{2} b^{2} d \,e^{3}+\frac {5}{2} x^{2} a \,b^{3} d^{2} e^{2}-\frac {1}{2} x^{2} b^{4} d^{3} e +5 e^{4} a^{4} x -10 b \,e^{3} d \,a^{3} x +10 b^{2} e^{2} d^{2} a^{2} x -5 a \,b^{3} d^{3} e x +b^{4} d^{4} x \right )}{\left (b x +a \right ) e^{5}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{6}}\) | \(298\) |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (12 x^{5} e^{5} b^{5}+75 x^{4} a \,b^{4} e^{5}-15 x^{4} b^{5} d \,e^{4}+200 x^{3} a^{2} b^{3} e^{5}-100 x^{3} a \,b^{4} d \,e^{4}+20 x^{3} b^{5} d^{2} e^{3}+300 x^{2} a^{3} b^{2} e^{5}-300 x^{2} a^{2} b^{3} d \,e^{4}+150 x^{2} a \,b^{4} d^{2} e^{3}-30 x^{2} b^{5} d^{3} e^{2}+60 \ln \left (e x +d \right ) a^{5} e^{5}-300 \ln \left (e x +d \right ) a^{4} b d \,e^{4}+600 \ln \left (e x +d \right ) a^{3} b^{2} d^{2} e^{3}-600 \ln \left (e x +d \right ) a^{2} b^{3} d^{3} e^{2}+300 \ln \left (e x +d \right ) a \,b^{4} d^{4} e -60 \ln \left (e x +d \right ) b^{5} d^{5}+300 a^{4} b \,e^{5} x -600 a^{3} b^{2} d \,e^{4} x +600 x \,a^{2} b^{3} d^{2} e^{3}-300 x a \,b^{4} d^{3} e^{2}+60 b^{5} d^{4} e x \right )}{60 \left (b x +a \right )^{5} e^{6}}\) | \(318\) |
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Time = 0.27 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {12 \, b^{5} e^{5} x^{5} - 15 \, {\left (b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{3} - 5 \, a b^{4} d e^{4} + 10 \, a^{2} b^{3} e^{5}\right )} x^{3} - 30 \, {\left (b^{5} d^{3} e^{2} - 5 \, a b^{4} d^{2} e^{3} + 10 \, a^{2} b^{3} d e^{4} - 10 \, a^{3} b^{2} e^{5}\right )} x^{2} + 60 \, {\left (b^{5} d^{4} e - 5 \, a b^{4} d^{3} e^{2} + 10 \, a^{2} b^{3} d^{2} e^{3} - 10 \, a^{3} b^{2} d e^{4} + 5 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \log \left (e x + d\right )}{60 \, e^{6}} \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{d + e x}\, dx \]
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Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (180) = 360\).
Time = 0.27 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.57 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {12 \, b^{5} e^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) - 15 \, b^{5} d e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 75 \, a b^{4} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, b^{5} d^{2} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) - 100 \, a b^{4} d e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 200 \, a^{2} b^{3} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) - 30 \, b^{5} d^{3} e x^{2} \mathrm {sgn}\left (b x + a\right ) + 150 \, a b^{4} d^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) - 300 \, a^{2} b^{3} d e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 300 \, a^{3} b^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 60 \, b^{5} d^{4} x \mathrm {sgn}\left (b x + a\right ) - 300 \, a b^{4} d^{3} e x \mathrm {sgn}\left (b x + a\right ) + 600 \, a^{2} b^{3} d^{2} e^{2} x \mathrm {sgn}\left (b x + a\right ) - 600 \, a^{3} b^{2} d e^{3} x \mathrm {sgn}\left (b x + a\right ) + 300 \, a^{4} b e^{4} x \mathrm {sgn}\left (b x + a\right )}{60 \, e^{5}} - \frac {{\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | e x + d \right |}\right )}{e^{6}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{d+e\,x} \,d x \]
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