Integrand size = 28, antiderivative size = 87 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=-\frac {2 a^4}{5 d (d x)^{5/2}}-\frac {8 a^3 b}{d^3 \sqrt {d x}}+\frac {4 a^2 b^2 (d x)^{3/2}}{d^5}+\frac {8 a b^3 (d x)^{7/2}}{7 d^7}+\frac {2 b^4 (d x)^{11/2}}{11 d^9} \]
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Time = 0.03 (sec) , antiderivative size = 87, 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} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=-\frac {2 a^4}{5 d (d x)^{5/2}}-\frac {8 a^3 b}{d^3 \sqrt {d x}}+\frac {4 a^2 b^2 (d x)^{3/2}}{d^5}+\frac {8 a b^3 (d x)^{7/2}}{7 d^7}+\frac {2 b^4 (d x)^{11/2}}{11 d^9} \]
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Rule 28
Rule 276
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\left (a b+b^2 x^2\right )^4}{(d x)^{7/2}} \, dx}{b^4} \\ & = \frac {\int \left (\frac {a^4 b^4}{(d x)^{7/2}}+\frac {4 a^3 b^5}{d^2 (d x)^{3/2}}+\frac {6 a^2 b^6 \sqrt {d x}}{d^4}+\frac {4 a b^7 (d x)^{5/2}}{d^6}+\frac {b^8 (d x)^{9/2}}{d^8}\right ) \, dx}{b^4} \\ & = -\frac {2 a^4}{5 d (d x)^{5/2}}-\frac {8 a^3 b}{d^3 \sqrt {d x}}+\frac {4 a^2 b^2 (d x)^{3/2}}{d^5}+\frac {8 a b^3 (d x)^{7/2}}{7 d^7}+\frac {2 b^4 (d x)^{11/2}}{11 d^9} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=-\frac {2 \sqrt {d x} \left (77 a^4+1540 a^3 b x^2-770 a^2 b^2 x^4-220 a b^3 x^6-35 b^4 x^8\right )}{385 d^4 x^3} \]
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Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(-\frac {2 \left (-35 b^{4} x^{8}-220 a \,b^{3} x^{6}-770 a^{2} b^{2} x^{4}+1540 a^{3} b \,x^{2}+77 a^{4}\right ) x}{385 \left (d x \right )^{\frac {7}{2}}}\) | \(52\) |
pseudoelliptic | \(-\frac {2 \left (-\frac {5}{11} b^{4} x^{8}-\frac {20}{7} a \,b^{3} x^{6}-10 a^{2} b^{2} x^{4}+20 a^{3} b \,x^{2}+a^{4}\right )}{5 \sqrt {d x}\, d^{3} x^{2}}\) | \(55\) |
trager | \(-\frac {2 \left (-35 b^{4} x^{8}-220 a \,b^{3} x^{6}-770 a^{2} b^{2} x^{4}+1540 a^{3} b \,x^{2}+77 a^{4}\right ) \sqrt {d x}}{385 d^{4} x^{3}}\) | \(57\) |
risch | \(-\frac {2 \left (-35 b^{4} x^{8}-220 a \,b^{3} x^{6}-770 a^{2} b^{2} x^{4}+1540 a^{3} b \,x^{2}+77 a^{4}\right )}{385 d^{3} x^{2} \sqrt {d x}}\) | \(57\) |
derivativedivides | \(\frac {\frac {2 b^{4} \left (d x \right )^{\frac {11}{2}}}{11}+\frac {8 a \,b^{3} d^{2} \left (d x \right )^{\frac {7}{2}}}{7}+4 a^{2} b^{2} d^{4} \left (d x \right )^{\frac {3}{2}}-\frac {8 a^{3} b \,d^{6}}{\sqrt {d x}}-\frac {2 a^{4} d^{8}}{5 \left (d x \right )^{\frac {5}{2}}}}{d^{9}}\) | \(74\) |
default | \(\frac {\frac {2 b^{4} \left (d x \right )^{\frac {11}{2}}}{11}+\frac {8 a \,b^{3} d^{2} \left (d x \right )^{\frac {7}{2}}}{7}+4 a^{2} b^{2} d^{4} \left (d x \right )^{\frac {3}{2}}-\frac {8 a^{3} b \,d^{6}}{\sqrt {d x}}-\frac {2 a^{4} d^{8}}{5 \left (d x \right )^{\frac {5}{2}}}}{d^{9}}\) | \(74\) |
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Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=\frac {2 \, {\left (35 \, b^{4} x^{8} + 220 \, a b^{3} x^{6} + 770 \, a^{2} b^{2} x^{4} - 1540 \, a^{3} b x^{2} - 77 \, a^{4}\right )} \sqrt {d x}}{385 \, d^{4} x^{3}} \]
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Time = 0.38 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=- \frac {2 a^{4} x}{5 \left (d x\right )^{\frac {7}{2}}} - \frac {8 a^{3} b x^{3}}{\left (d x\right )^{\frac {7}{2}}} + \frac {4 a^{2} b^{2} x^{5}}{\left (d x\right )^{\frac {7}{2}}} + \frac {8 a b^{3} x^{7}}{7 \left (d x\right )^{\frac {7}{2}}} + \frac {2 b^{4} x^{9}}{11 \left (d x\right )^{\frac {7}{2}}} \]
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Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=-\frac {2 \, {\left (\frac {77 \, {\left (20 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )}}{\left (d x\right )^{\frac {5}{2}} d^{2}} - \frac {5 \, {\left (7 \, \left (d x\right )^{\frac {11}{2}} b^{4} + 44 \, \left (d x\right )^{\frac {7}{2}} a b^{3} d^{2} + 154 \, \left (d x\right )^{\frac {3}{2}} a^{2} b^{2} d^{4}\right )}}{d^{8}}\right )}}{385 \, d} \]
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Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=-\frac {2 \, {\left (\frac {77 \, {\left (20 \, a^{3} b d^{3} x^{2} + a^{4} d^{3}\right )}}{\sqrt {d x} d^{2} x^{2}} - \frac {5 \, {\left (7 \, \sqrt {d x} b^{4} d^{55} x^{5} + 44 \, \sqrt {d x} a b^{3} d^{55} x^{3} + 154 \, \sqrt {d x} a^{2} b^{2} d^{55} x\right )}}{d^{55}}\right )}}{385 \, d^{4}} \]
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Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=\frac {2\,b^4\,{\left (d\,x\right )}^{11/2}}{11\,d^9}-\frac {\frac {2\,a^4\,d^2}{5}+8\,b\,a^3\,d^2\,x^2}{d^3\,{\left (d\,x\right )}^{5/2}}+\frac {4\,a^2\,b^2\,{\left (d\,x\right )}^{3/2}}{d^5}+\frac {8\,a\,b^3\,{\left (d\,x\right )}^{7/2}}{7\,d^7} \]
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