Integrand size = 28, antiderivative size = 313 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {a^5 (d x)^{1+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d (1+m) \left (a+b x^2\right )}+\frac {5 a^4 b (d x)^{3+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 (3+m) \left (a+b x^2\right )}+\frac {10 a^3 b^2 (d x)^{5+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^5 (5+m) \left (a+b x^2\right )}+\frac {10 a^2 b^3 (d x)^{7+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^7 (7+m) \left (a+b x^2\right )}+\frac {5 a b^4 (d x)^{9+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^9 (9+m) \left (a+b x^2\right )}+\frac {b^5 (d x)^{11+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^{11} (11+m) \left (a+b x^2\right )} \]
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Time = 0.08 (sec) , antiderivative size = 313, 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 = {1126, 276} \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+11}}{d^{11} (m+11) \left (a+b x^2\right )}+\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+9}}{d^9 (m+9) \left (a+b x^2\right )}+\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^2\right )}+\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left (a+b x^2\right )}+\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left (a+b x^2\right )}+\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+5}}{d^5 (m+5) \left (a+b x^2\right )} \]
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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 (d x)^m \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 (d x)^m+\frac {5 a^4 b^6 (d x)^{2+m}}{d^2}+\frac {10 a^3 b^7 (d x)^{4+m}}{d^4}+\frac {10 a^2 b^8 (d x)^{6+m}}{d^6}+\frac {5 a b^9 (d x)^{8+m}}{d^8}+\frac {b^{10} (d x)^{10+m}}{d^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )} \\ & = \frac {a^5 (d x)^{1+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d (1+m) \left (a+b x^2\right )}+\frac {5 a^4 b (d x)^{3+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 (3+m) \left (a+b x^2\right )}+\frac {10 a^3 b^2 (d x)^{5+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^5 (5+m) \left (a+b x^2\right )}+\frac {10 a^2 b^3 (d x)^{7+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^7 (7+m) \left (a+b x^2\right )}+\frac {5 a b^4 (d x)^{9+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^9 (9+m) \left (a+b x^2\right )}+\frac {b^5 (d x)^{11+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^{11} (11+m) \left (a+b x^2\right )} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.35 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {x (d x)^m \left (\left (a+b x^2\right )^2\right )^{5/2} \left (\frac {a^5}{1+m}+\frac {5 a^4 b x^2}{3+m}+\frac {10 a^3 b^2 x^4}{5+m}+\frac {10 a^2 b^3 x^6}{7+m}+\frac {5 a b^4 x^8}{9+m}+\frac {b^5 x^{10}}{11+m}\right )}{\left (a+b x^2\right )^5} \]
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Time = 0.04 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.45
method | result | size |
gosper | \(\frac {x \left (b^{5} m^{5} x^{10}+25 b^{5} m^{4} x^{10}+5 a \,b^{4} m^{5} x^{8}+230 b^{5} m^{3} x^{10}+135 a \,b^{4} m^{4} x^{8}+950 b^{5} m^{2} x^{10}+10 a^{2} b^{3} m^{5} x^{6}+1310 a \,b^{4} m^{3} x^{8}+1689 m \,x^{10} b^{5}+290 a^{2} b^{3} m^{4} x^{6}+5610 a \,b^{4} m^{2} x^{8}+945 x^{10} b^{5}+10 a^{3} b^{2} m^{5} x^{4}+3020 a^{2} b^{3} m^{3} x^{6}+10205 m \,x^{8} b^{4} a +310 a^{3} b^{2} m^{4} x^{4}+13660 a^{2} b^{3} m^{2} x^{6}+5775 a \,x^{8} b^{4}+5 a^{4} b \,m^{5} x^{2}+3500 a^{3} b^{2} m^{3} x^{4}+25770 m \,x^{6} a^{2} b^{3}+165 a^{4} b \,m^{4} x^{2}+17300 a^{3} b^{2} m^{2} x^{4}+14850 a^{2} x^{6} b^{3}+a^{5} m^{5}+2030 a^{4} b \,m^{3} x^{2}+34890 m \,x^{4} a^{3} b^{2}+35 a^{5} m^{4}+11310 a^{4} b \,m^{2} x^{2}+20790 a^{3} x^{4} b^{2}+470 a^{5} m^{3}+26765 m \,x^{2} b \,a^{4}+3010 a^{5} m^{2}+17325 x^{2} a^{4} b +9129 m \,a^{5}+10395 a^{5}\right ) \left (d x \right )^{m} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right ) \left (b \,x^{2}+a \right )^{5}}\) | \(453\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (b^{5} m^{5} x^{10}+25 b^{5} m^{4} x^{10}+5 a \,b^{4} m^{5} x^{8}+230 b^{5} m^{3} x^{10}+135 a \,b^{4} m^{4} x^{8}+950 b^{5} m^{2} x^{10}+10 a^{2} b^{3} m^{5} x^{6}+1310 a \,b^{4} m^{3} x^{8}+1689 m \,x^{10} b^{5}+290 a^{2} b^{3} m^{4} x^{6}+5610 a \,b^{4} m^{2} x^{8}+945 x^{10} b^{5}+10 a^{3} b^{2} m^{5} x^{4}+3020 a^{2} b^{3} m^{3} x^{6}+10205 m \,x^{8} b^{4} a +310 a^{3} b^{2} m^{4} x^{4}+13660 a^{2} b^{3} m^{2} x^{6}+5775 a \,x^{8} b^{4}+5 a^{4} b \,m^{5} x^{2}+3500 a^{3} b^{2} m^{3} x^{4}+25770 m \,x^{6} a^{2} b^{3}+165 a^{4} b \,m^{4} x^{2}+17300 a^{3} b^{2} m^{2} x^{4}+14850 a^{2} x^{6} b^{3}+a^{5} m^{5}+2030 a^{4} b \,m^{3} x^{2}+34890 m \,x^{4} a^{3} b^{2}+35 a^{5} m^{4}+11310 a^{4} b \,m^{2} x^{2}+20790 a^{3} x^{4} b^{2}+470 a^{5} m^{3}+26765 m \,x^{2} b \,a^{4}+3010 a^{5} m^{2}+17325 x^{2} a^{4} b +9129 m \,a^{5}+10395 a^{5}\right ) x \left (d x \right )^{m}}{\left (b \,x^{2}+a \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(453\) |
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Time = 0.26 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.18 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {{\left ({\left (b^{5} m^{5} + 25 \, b^{5} m^{4} + 230 \, b^{5} m^{3} + 950 \, b^{5} m^{2} + 1689 \, b^{5} m + 945 \, b^{5}\right )} x^{11} + 5 \, {\left (a b^{4} m^{5} + 27 \, a b^{4} m^{4} + 262 \, a b^{4} m^{3} + 1122 \, a b^{4} m^{2} + 2041 \, a b^{4} m + 1155 \, a b^{4}\right )} x^{9} + 10 \, {\left (a^{2} b^{3} m^{5} + 29 \, a^{2} b^{3} m^{4} + 302 \, a^{2} b^{3} m^{3} + 1366 \, a^{2} b^{3} m^{2} + 2577 \, a^{2} b^{3} m + 1485 \, a^{2} b^{3}\right )} x^{7} + 10 \, {\left (a^{3} b^{2} m^{5} + 31 \, a^{3} b^{2} m^{4} + 350 \, a^{3} b^{2} m^{3} + 1730 \, a^{3} b^{2} m^{2} + 3489 \, a^{3} b^{2} m + 2079 \, a^{3} b^{2}\right )} x^{5} + 5 \, {\left (a^{4} b m^{5} + 33 \, a^{4} b m^{4} + 406 \, a^{4} b m^{3} + 2262 \, a^{4} b m^{2} + 5353 \, a^{4} b m + 3465 \, a^{4} b\right )} x^{3} + {\left (a^{5} m^{5} + 35 \, a^{5} m^{4} + 470 \, a^{5} m^{3} + 3010 \, a^{5} m^{2} + 9129 \, a^{5} m + 10395 \, a^{5}\right )} x\right )} \left (d x\right )^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \]
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\[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\int \left (d x\right )^{m} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.78 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {{\left ({\left (m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945\right )} b^{5} d^{m} x^{11} + 5 \, {\left (m^{5} + 27 \, m^{4} + 262 \, m^{3} + 1122 \, m^{2} + 2041 \, m + 1155\right )} a b^{4} d^{m} x^{9} + 10 \, {\left (m^{5} + 29 \, m^{4} + 302 \, m^{3} + 1366 \, m^{2} + 2577 \, m + 1485\right )} a^{2} b^{3} d^{m} x^{7} + 10 \, {\left (m^{5} + 31 \, m^{4} + 350 \, m^{3} + 1730 \, m^{2} + 3489 \, m + 2079\right )} a^{3} b^{2} d^{m} x^{5} + 5 \, {\left (m^{5} + 33 \, m^{4} + 406 \, m^{3} + 2262 \, m^{2} + 5353 \, m + 3465\right )} a^{4} b d^{m} x^{3} + {\left (m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395\right )} a^{5} d^{m} x\right )} x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \]
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Leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (247) = 494\).
Time = 0.35 (sec) , antiderivative size = 900, normalized size of antiderivative = 2.88 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\int {\left (d\,x\right )}^m\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \]
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