Integrand size = 24, antiderivative size = 231 \[ \int x^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {a^5 x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac {5 a^4 b x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {5 a^3 b^2 x^8 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {10 a^2 b^3 x^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {a b^4 x^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^5 x^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)} \]
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Time = 0.04 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45} \[ \int x^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {b^5 x^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {a b^4 x^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {10 a^2 b^3 x^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {a^5 x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac {5 a^4 b x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {5 a^3 b^2 x^8 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]
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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 x^5 \left (a b+b^2 x\right )^5 \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a^5 b^5 x^5+5 a^4 b^6 x^6+10 a^3 b^7 x^7+10 a^2 b^8 x^8+5 a b^9 x^9+b^{10} x^{10}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {a^5 x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac {5 a^4 b x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {5 a^3 b^2 x^8 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {10 a^2 b^3 x^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {a b^4 x^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^5 x^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.33 \[ \int x^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x^6 \sqrt {(a+b x)^2} \left (462 a^5+1980 a^4 b x+3465 a^3 b^2 x^2+3080 a^2 b^3 x^3+1386 a b^4 x^4+252 b^5 x^5\right )}{2772 (a+b x)} \]
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Time = 2.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.32
method | result | size |
gosper | \(\frac {x^{6} \left (252 b^{5} x^{5}+1386 a \,b^{4} x^{4}+3080 a^{2} b^{3} x^{3}+3465 a^{3} b^{2} x^{2}+1980 a^{4} b x +462 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2772 \left (b x +a \right )^{5}}\) | \(74\) |
default | \(\frac {x^{6} \left (252 b^{5} x^{5}+1386 a \,b^{4} x^{4}+3080 a^{2} b^{3} x^{3}+3465 a^{3} b^{2} x^{2}+1980 a^{4} b x +462 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2772 \left (b x +a \right )^{5}}\) | \(74\) |
risch | \(\frac {a^{5} x^{6} \sqrt {\left (b x +a \right )^{2}}}{6 b x +6 a}+\frac {5 a^{4} b \,x^{7} \sqrt {\left (b x +a \right )^{2}}}{7 \left (b x +a \right )}+\frac {5 a^{3} b^{2} x^{8} \sqrt {\left (b x +a \right )^{2}}}{4 \left (b x +a \right )}+\frac {10 a^{2} b^{3} x^{9} \sqrt {\left (b x +a \right )^{2}}}{9 \left (b x +a \right )}+\frac {a \,b^{4} x^{10} \sqrt {\left (b x +a \right )^{2}}}{2 b x +2 a}+\frac {b^{5} x^{11} \sqrt {\left (b x +a \right )^{2}}}{11 b x +11 a}\) | \(154\) |
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Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.25 \[ \int x^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{11} \, b^{5} x^{11} + \frac {1}{2} \, a b^{4} x^{10} + \frac {10}{9} \, a^{2} b^{3} x^{9} + \frac {5}{4} \, a^{3} b^{2} x^{8} + \frac {5}{7} \, a^{4} b x^{7} + \frac {1}{6} \, a^{5} x^{6} \]
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Time = 0.87 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.19 \[ \int x^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{10}}{2772 b^{6}} + \frac {a^{9} x}{2772 b^{5}} - \frac {a^{8} x^{2}}{2772 b^{4}} + \frac {a^{7} x^{3}}{2772 b^{3}} - \frac {a^{6} x^{4}}{2772 b^{2}} + \frac {a^{5} x^{5}}{2772 b} + \frac {461 a^{4} x^{6}}{2772} + \frac {217 a^{3} b x^{7}}{396} + \frac {139 a^{2} b^{2} x^{8}}{198} + \frac {9 a b^{3} x^{9}}{22} + \frac {b^{4} x^{10}}{11}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{10} \left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7} + \frac {5 a^{8} \left (a^{2} + 2 a b x\right )^{\frac {9}{2}}}{9} - \frac {10 a^{6} \left (a^{2} + 2 a b x\right )^{\frac {11}{2}}}{11} + \frac {10 a^{4} \left (a^{2} + 2 a b x\right )^{\frac {13}{2}}}{13} - \frac {a^{2} \left (a^{2} + 2 a b x\right )^{\frac {15}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {17}{2}}}{17}}{32 a^{6} b^{6}} & \text {for}\: a b \neq 0 \\\frac {x^{6} \left (a^{2}\right )^{\frac {5}{2}}}{6} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.82 \[ \int x^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x^{4}}{11 \, b^{2}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a x^{3}}{22 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5} x}{6 \, b^{5}} + \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} x^{2}}{198 \, b^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{6}}{6 \, b^{6}} - \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{3} x}{396 \, b^{5}} + \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{4}}{2772 \, b^{6}} \]
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Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.46 \[ \int x^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{11} \, b^{5} x^{11} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a b^{4} x^{10} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, a^{3} b^{2} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, a^{4} b x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) - \frac {a^{11} \mathrm {sgn}\left (b x + a\right )}{2772 \, b^{6}} \]
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Timed out. \[ \int x^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int x^5\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]
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