Integrand size = 26, antiderivative size = 252 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {a^5 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {a^4 b x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {10 a^3 b^2 x^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac {10 a^2 b^3 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac {5 a b^4 x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac {b^5 x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )} \]
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Time = 0.04 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1126, 276} \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {b^5 x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac {5 a b^4 x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac {10 a^2 b^3 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac {a^5 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {a^4 b x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {10 a^3 b^2 x^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )} \]
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Rule 276
Rule 1126
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int x^2 \left (a b+b^2 x^2\right )^5 \, dx}{b^4 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^5 b^5 x^2+5 a^4 b^6 x^4+10 a^3 b^7 x^6+10 a^2 b^8 x^8+5 a b^9 x^{10}+b^{10} x^{12}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )} \\ & = \frac {a^5 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {a^4 b x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {10 a^3 b^2 x^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac {10 a^2 b^3 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac {5 a b^4 x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac {b^5 x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.33 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {x^3 \sqrt {\left (a+b x^2\right )^2} \left (3003 a^5+9009 a^4 b x^2+12870 a^3 b^2 x^4+10010 a^2 b^3 x^6+4095 a b^4 x^8+693 b^5 x^{10}\right )}{9009 \left (a+b x^2\right )} \]
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Time = 0.78 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.32
method | result | size |
gosper | \(\frac {x^{3} \left (693 x^{10} b^{5}+4095 a \,x^{8} b^{4}+10010 a^{2} x^{6} b^{3}+12870 a^{3} x^{4} b^{2}+9009 x^{2} a^{4} b +3003 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{9009 \left (b \,x^{2}+a \right )^{5}}\) | \(80\) |
default | \(\frac {x^{3} \left (693 x^{10} b^{5}+4095 a \,x^{8} b^{4}+10010 a^{2} x^{6} b^{3}+12870 a^{3} x^{4} b^{2}+9009 x^{2} a^{4} b +3003 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{9009 \left (b \,x^{2}+a \right )^{5}}\) | \(80\) |
risch | \(\frac {a^{5} x^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{3 b \,x^{2}+3 a}+\frac {a^{4} b \,x^{5} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b \,x^{2}+a}+\frac {10 a^{3} b^{2} x^{7} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{7 \left (b \,x^{2}+a \right )}+\frac {10 a^{2} b^{3} x^{9} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{9 \left (b \,x^{2}+a \right )}+\frac {5 a \,b^{4} x^{11} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{11 \left (b \,x^{2}+a \right )}+\frac {b^{5} x^{13} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{13 b \,x^{2}+13 a}\) | \(177\) |
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Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.22 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {1}{13} \, b^{5} x^{13} + \frac {5}{11} \, a b^{4} x^{11} + \frac {10}{9} \, a^{2} b^{3} x^{9} + \frac {10}{7} \, a^{3} b^{2} x^{7} + a^{4} b x^{5} + \frac {1}{3} \, a^{5} x^{3} \]
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\[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\int x^{2} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.22 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {1}{13} \, b^{5} x^{13} + \frac {5}{11} \, a b^{4} x^{11} + \frac {10}{9} \, a^{2} b^{3} x^{9} + \frac {10}{7} \, a^{3} b^{2} x^{7} + a^{4} b x^{5} + \frac {1}{3} \, a^{5} x^{3} \]
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Time = 0.44 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.41 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {1}{13} \, b^{5} x^{13} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{11} \, a b^{4} x^{11} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {10}{9} \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {10}{7} \, a^{3} b^{2} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{4} b x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{3} \, a^{5} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) \]
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Timed out. \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\int x^2\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \]
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