Integrand size = 28, antiderivative size = 91 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 a^4 (d x)^{5/2}}{5 d}+\frac {8 a^3 b (d x)^{9/2}}{9 d^3}+\frac {12 a^2 b^2 (d x)^{13/2}}{13 d^5}+\frac {8 a b^3 (d x)^{17/2}}{17 d^7}+\frac {2 b^4 (d x)^{21/2}}{21 d^9} \]
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Time = 0.03 (sec) , antiderivative size = 91, 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 = {28, 276} \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 a^4 (d x)^{5/2}}{5 d}+\frac {8 a^3 b (d x)^{9/2}}{9 d^3}+\frac {12 a^2 b^2 (d x)^{13/2}}{13 d^5}+\frac {8 a b^3 (d x)^{17/2}}{17 d^7}+\frac {2 b^4 (d x)^{21/2}}{21 d^9} \]
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Rule 28
Rule 276
Rubi steps \begin{align*} \text {integral}& = \frac {\int (d x)^{3/2} \left (a b+b^2 x^2\right )^4 \, dx}{b^4} \\ & = \frac {\int \left (a^4 b^4 (d x)^{3/2}+\frac {4 a^3 b^5 (d x)^{7/2}}{d^2}+\frac {6 a^2 b^6 (d x)^{11/2}}{d^4}+\frac {4 a b^7 (d x)^{15/2}}{d^6}+\frac {b^8 (d x)^{19/2}}{d^8}\right ) \, dx}{b^4} \\ & = \frac {2 a^4 (d x)^{5/2}}{5 d}+\frac {8 a^3 b (d x)^{9/2}}{9 d^3}+\frac {12 a^2 b^2 (d x)^{13/2}}{13 d^5}+\frac {8 a b^3 (d x)^{17/2}}{17 d^7}+\frac {2 b^4 (d x)^{21/2}}{21 d^9} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.60 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 x (d x)^{3/2} \left (13923 a^4+30940 a^3 b x^2+32130 a^2 b^2 x^4+16380 a b^3 x^6+3315 b^4 x^8\right )}{69615} \]
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Time = 0.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(\frac {2 x \left (3315 b^{4} x^{8}+16380 a \,b^{3} x^{6}+32130 a^{2} b^{2} x^{4}+30940 a^{3} b \,x^{2}+13923 a^{4}\right ) \left (d x \right )^{\frac {3}{2}}}{69615}\) | \(52\) |
pseudoelliptic | \(\frac {2 \left (\frac {5}{21} b^{4} x^{8}+\frac {20}{17} a \,b^{3} x^{6}+\frac {30}{13} a^{2} b^{2} x^{4}+\frac {20}{9} a^{3} b \,x^{2}+a^{4}\right ) \sqrt {d x}\, d \,x^{2}}{5}\) | \(53\) |
trager | \(\frac {2 d \,x^{2} \left (3315 b^{4} x^{8}+16380 a \,b^{3} x^{6}+32130 a^{2} b^{2} x^{4}+30940 a^{3} b \,x^{2}+13923 a^{4}\right ) \sqrt {d x}}{69615}\) | \(55\) |
risch | \(\frac {2 d^{2} x^{3} \left (3315 b^{4} x^{8}+16380 a \,b^{3} x^{6}+32130 a^{2} b^{2} x^{4}+30940 a^{3} b \,x^{2}+13923 a^{4}\right )}{69615 \sqrt {d x}}\) | \(57\) |
derivativedivides | \(\frac {\frac {2 b^{4} \left (d x \right )^{\frac {21}{2}}}{21}+\frac {8 a \,b^{3} d^{2} \left (d x \right )^{\frac {17}{2}}}{17}+\frac {12 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {8 a^{3} d^{6} b \left (d x \right )^{\frac {9}{2}}}{9}+\frac {2 a^{4} d^{8} \left (d x \right )^{\frac {5}{2}}}{5}}{d^{9}}\) | \(74\) |
default | \(\frac {\frac {2 b^{4} \left (d x \right )^{\frac {21}{2}}}{21}+\frac {8 a \,b^{3} d^{2} \left (d x \right )^{\frac {17}{2}}}{17}+\frac {12 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {8 a^{3} d^{6} b \left (d x \right )^{\frac {9}{2}}}{9}+\frac {2 a^{4} d^{8} \left (d x \right )^{\frac {5}{2}}}{5}}{d^{9}}\) | \(74\) |
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Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2}{69615} \, {\left (3315 \, b^{4} d x^{10} + 16380 \, a b^{3} d x^{8} + 32130 \, a^{2} b^{2} d x^{6} + 30940 \, a^{3} b d x^{4} + 13923 \, a^{4} d x^{2}\right )} \sqrt {d x} \]
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Time = 0.33 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 a^{4} x \left (d x\right )^{\frac {3}{2}}}{5} + \frac {8 a^{3} b x^{3} \left (d x\right )^{\frac {3}{2}}}{9} + \frac {12 a^{2} b^{2} x^{5} \left (d x\right )^{\frac {3}{2}}}{13} + \frac {8 a b^{3} x^{7} \left (d x\right )^{\frac {3}{2}}}{17} + \frac {2 b^{4} x^{9} \left (d x\right )^{\frac {3}{2}}}{21} \]
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Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.80 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 \, {\left (3315 \, \left (d x\right )^{\frac {21}{2}} b^{4} + 16380 \, \left (d x\right )^{\frac {17}{2}} a b^{3} d^{2} + 32130 \, \left (d x\right )^{\frac {13}{2}} a^{2} b^{2} d^{4} + 30940 \, \left (d x\right )^{\frac {9}{2}} a^{3} b d^{6} + 13923 \, \left (d x\right )^{\frac {5}{2}} a^{4} d^{8}\right )}}{69615 \, d^{9}} \]
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Time = 0.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.81 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2}{69615} \, {\left (3315 \, \sqrt {d x} b^{4} x^{10} + 16380 \, \sqrt {d x} a b^{3} x^{8} + 32130 \, \sqrt {d x} a^{2} b^{2} x^{6} + 30940 \, \sqrt {d x} a^{3} b x^{4} + 13923 \, \sqrt {d x} a^{4} x^{2}\right )} d \]
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Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.78 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2\,a^4\,{\left (d\,x\right )}^{5/2}}{5\,d}+\frac {2\,b^4\,{\left (d\,x\right )}^{21/2}}{21\,d^9}+\frac {12\,a^2\,b^2\,{\left (d\,x\right )}^{13/2}}{13\,d^5}+\frac {8\,a^3\,b\,{\left (d\,x\right )}^{9/2}}{9\,d^3}+\frac {8\,a\,b^3\,{\left (d\,x\right )}^{17/2}}{17\,d^7} \]
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