Integrand size = 28, antiderivative size = 131 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {2 a^6 (d x)^{5/2}}{5 d}+\frac {4 a^5 b (d x)^{9/2}}{3 d^3}+\frac {30 a^4 b^2 (d x)^{13/2}}{13 d^5}+\frac {40 a^3 b^3 (d x)^{17/2}}{17 d^7}+\frac {10 a^2 b^4 (d x)^{21/2}}{7 d^9}+\frac {12 a b^5 (d x)^{25/2}}{25 d^{11}}+\frac {2 b^6 (d x)^{29/2}}{29 d^{13}} \]
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Time = 0.04 (sec) , antiderivative size = 131, 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} \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {2 a^6 (d x)^{5/2}}{5 d}+\frac {4 a^5 b (d x)^{9/2}}{3 d^3}+\frac {30 a^4 b^2 (d x)^{13/2}}{13 d^5}+\frac {40 a^3 b^3 (d x)^{17/2}}{17 d^7}+\frac {10 a^2 b^4 (d x)^{21/2}}{7 d^9}+\frac {12 a b^5 (d x)^{25/2}}{25 d^{11}}+\frac {2 b^6 (d x)^{29/2}}{29 d^{13}} \]
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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 )^6 \, dx}{b^6} \\ & = \frac {\int \left (a^6 b^6 (d x)^{3/2}+\frac {6 a^5 b^7 (d x)^{7/2}}{d^2}+\frac {15 a^4 b^8 (d x)^{11/2}}{d^4}+\frac {20 a^3 b^9 (d x)^{15/2}}{d^6}+\frac {15 a^2 b^{10} (d x)^{19/2}}{d^8}+\frac {6 a b^{11} (d x)^{23/2}}{d^{10}}+\frac {b^{12} (d x)^{27/2}}{d^{12}}\right ) \, dx}{b^6} \\ & = \frac {2 a^6 (d x)^{5/2}}{5 d}+\frac {4 a^5 b (d x)^{9/2}}{3 d^3}+\frac {30 a^4 b^2 (d x)^{13/2}}{13 d^5}+\frac {40 a^3 b^3 (d x)^{17/2}}{17 d^7}+\frac {10 a^2 b^4 (d x)^{21/2}}{7 d^9}+\frac {12 a b^5 (d x)^{25/2}}{25 d^{11}}+\frac {2 b^6 (d x)^{29/2}}{29 d^{13}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.59 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {2 x (d x)^{3/2} \left (672945 a^6+2243150 a^5 b x^2+3882375 a^4 b^2 x^4+3958500 a^3 b^3 x^6+2403375 a^2 b^4 x^8+807534 a b^5 x^{10}+116025 b^6 x^{12}\right )}{3364725} \]
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Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.56
method | result | size |
gosper | \(\frac {2 x \left (116025 b^{6} x^{12}+807534 a \,b^{5} x^{10}+2403375 a^{2} b^{4} x^{8}+3958500 a^{3} b^{3} x^{6}+3882375 a^{4} b^{2} x^{4}+2243150 a^{5} b \,x^{2}+672945 a^{6}\right ) \left (d x \right )^{\frac {3}{2}}}{3364725}\) | \(74\) |
pseudoelliptic | \(\frac {2 \left (\frac {5}{29} b^{6} x^{12}+\frac {6}{5} a \,b^{5} x^{10}+\frac {25}{7} a^{2} b^{4} x^{8}+\frac {100}{17} a^{3} b^{3} x^{6}+\frac {75}{13} a^{4} b^{2} x^{4}+\frac {10}{3} a^{5} b \,x^{2}+a^{6}\right ) \sqrt {d x}\, d \,x^{2}}{5}\) | \(75\) |
trager | \(\frac {2 d \,x^{2} \left (116025 b^{6} x^{12}+807534 a \,b^{5} x^{10}+2403375 a^{2} b^{4} x^{8}+3958500 a^{3} b^{3} x^{6}+3882375 a^{4} b^{2} x^{4}+2243150 a^{5} b \,x^{2}+672945 a^{6}\right ) \sqrt {d x}}{3364725}\) | \(77\) |
risch | \(\frac {2 d^{2} x^{3} \left (116025 b^{6} x^{12}+807534 a \,b^{5} x^{10}+2403375 a^{2} b^{4} x^{8}+3958500 a^{3} b^{3} x^{6}+3882375 a^{4} b^{2} x^{4}+2243150 a^{5} b \,x^{2}+672945 a^{6}\right )}{3364725 \sqrt {d x}}\) | \(79\) |
derivativedivides | \(\frac {\frac {2 b^{6} \left (d x \right )^{\frac {29}{2}}}{29}+\frac {12 a \,b^{5} d^{2} \left (d x \right )^{\frac {25}{2}}}{25}+\frac {10 a^{2} d^{4} b^{4} \left (d x \right )^{\frac {21}{2}}}{7}+\frac {40 a^{3} d^{6} b^{3} \left (d x \right )^{\frac {17}{2}}}{17}+\frac {30 a^{4} d^{8} b^{2} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {4 a^{5} d^{10} b \left (d x \right )^{\frac {9}{2}}}{3}+\frac {2 a^{6} d^{12} \left (d x \right )^{\frac {5}{2}}}{5}}{d^{13}}\) | \(106\) |
default | \(\frac {\frac {2 b^{6} \left (d x \right )^{\frac {29}{2}}}{29}+\frac {12 a \,b^{5} d^{2} \left (d x \right )^{\frac {25}{2}}}{25}+\frac {10 a^{2} d^{4} b^{4} \left (d x \right )^{\frac {21}{2}}}{7}+\frac {40 a^{3} d^{6} b^{3} \left (d x \right )^{\frac {17}{2}}}{17}+\frac {30 a^{4} d^{8} b^{2} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {4 a^{5} d^{10} b \left (d x \right )^{\frac {9}{2}}}{3}+\frac {2 a^{6} d^{12} \left (d x \right )^{\frac {5}{2}}}{5}}{d^{13}}\) | \(106\) |
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Time = 0.24 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.63 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {2}{3364725} \, {\left (116025 \, b^{6} d x^{14} + 807534 \, a b^{5} d x^{12} + 2403375 \, a^{2} b^{4} d x^{10} + 3958500 \, a^{3} b^{3} d x^{8} + 3882375 \, a^{4} b^{2} d x^{6} + 2243150 \, a^{5} b d x^{4} + 672945 \, a^{6} d x^{2}\right )} \sqrt {d x} \]
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Time = 0.53 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {2 a^{6} x \left (d x\right )^{\frac {3}{2}}}{5} + \frac {4 a^{5} b x^{3} \left (d x\right )^{\frac {3}{2}}}{3} + \frac {30 a^{4} b^{2} x^{5} \left (d x\right )^{\frac {3}{2}}}{13} + \frac {40 a^{3} b^{3} x^{7} \left (d x\right )^{\frac {3}{2}}}{17} + \frac {10 a^{2} b^{4} x^{9} \left (d x\right )^{\frac {3}{2}}}{7} + \frac {12 a b^{5} x^{11} \left (d x\right )^{\frac {3}{2}}}{25} + \frac {2 b^{6} x^{13} \left (d x\right )^{\frac {3}{2}}}{29} \]
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Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.80 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {2 \, {\left (116025 \, \left (d x\right )^{\frac {29}{2}} b^{6} + 807534 \, \left (d x\right )^{\frac {25}{2}} a b^{5} d^{2} + 2403375 \, \left (d x\right )^{\frac {21}{2}} a^{2} b^{4} d^{4} + 3958500 \, \left (d x\right )^{\frac {17}{2}} a^{3} b^{3} d^{6} + 3882375 \, \left (d x\right )^{\frac {13}{2}} a^{4} b^{2} d^{8} + 2243150 \, \left (d x\right )^{\frac {9}{2}} a^{5} b d^{10} + 672945 \, \left (d x\right )^{\frac {5}{2}} a^{6} d^{12}\right )}}{3364725 \, d^{13}} \]
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Time = 0.31 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.81 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {2}{3364725} \, {\left (116025 \, \sqrt {d x} b^{6} x^{14} + 807534 \, \sqrt {d x} a b^{5} x^{12} + 2403375 \, \sqrt {d x} a^{2} b^{4} x^{10} + 3958500 \, \sqrt {d x} a^{3} b^{3} x^{8} + 3882375 \, \sqrt {d x} a^{4} b^{2} x^{6} + 2243150 \, \sqrt {d x} a^{5} b x^{4} + 672945 \, \sqrt {d x} a^{6} x^{2}\right )} d \]
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Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {2\,a^6\,{\left (d\,x\right )}^{5/2}}{5\,d}+\frac {2\,b^6\,{\left (d\,x\right )}^{29/2}}{29\,d^{13}}+\frac {30\,a^4\,b^2\,{\left (d\,x\right )}^{13/2}}{13\,d^5}+\frac {40\,a^3\,b^3\,{\left (d\,x\right )}^{17/2}}{17\,d^7}+\frac {10\,a^2\,b^4\,{\left (d\,x\right )}^{21/2}}{7\,d^9}+\frac {4\,a^5\,b\,{\left (d\,x\right )}^{9/2}}{3\,d^3}+\frac {12\,a\,b^5\,{\left (d\,x\right )}^{25/2}}{25\,d^{11}} \]
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