Integrand size = 30, antiderivative size = 293 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2 a^5 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )}+\frac {2 a^4 b (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac {20 a^3 b^2 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^5 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{17/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 d^9 \left (a+b x^2\right )}+\frac {2 b^5 (d x)^{21/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{21 d^{11} \left (a+b x^2\right )} \]
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Time = 0.05 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1126, 276} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2 b^5 (d x)^{21/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{21 d^{11} \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{17/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 d^9 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac {2 a^5 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )}+\frac {2 a^4 b (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac {20 a^3 b^2 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^5 \left (a+b x^2\right )} \]
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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 \frac {\left (a b+b^2 x^2\right )^5}{\sqrt {d x}} \, 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 (\frac {a^5 b^5}{\sqrt {d x}}+\frac {5 a^4 b^6 (d x)^{3/2}}{d^2}+\frac {10 a^3 b^7 (d x)^{7/2}}{d^4}+\frac {10 a^2 b^8 (d x)^{11/2}}{d^6}+\frac {5 a b^9 (d x)^{15/2}}{d^8}+\frac {b^{10} (d x)^{19/2}}{d^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )} \\ & = \frac {2 a^5 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )}+\frac {2 a^4 b (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac {20 a^3 b^2 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^5 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{17/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 d^9 \left (a+b x^2\right )}+\frac {2 b^5 (d x)^{21/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{21 d^{11} \left (a+b x^2\right )} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.30 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2 x \sqrt {\left (a+b x^2\right )^2} \left (13923 a^5+13923 a^4 b x^2+15470 a^3 b^2 x^4+10710 a^2 b^3 x^6+4095 a b^4 x^8+663 b^5 x^{10}\right )}{13923 \sqrt {d x} \left (a+b x^2\right )} \]
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Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.28
method | result | size |
gosper | \(\frac {2 x \left (663 x^{10} b^{5}+4095 a \,x^{8} b^{4}+10710 a^{2} x^{6} b^{3}+15470 a^{3} x^{4} b^{2}+13923 x^{2} a^{4} b +13923 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{13923 \left (b \,x^{2}+a \right )^{5} \sqrt {d x}}\) | \(83\) |
risch | \(\frac {2 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (663 x^{10} b^{5}+4095 a \,x^{8} b^{4}+10710 a^{2} x^{6} b^{3}+15470 a^{3} x^{4} b^{2}+13923 x^{2} a^{4} b +13923 a^{5}\right ) x}{13923 \left (b \,x^{2}+a \right ) \sqrt {d x}}\) | \(83\) |
default | \(\frac {2 {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}} \sqrt {d x}\, \left (663 x^{10} b^{5}+4095 a \,x^{8} b^{4}+10710 a^{2} x^{6} b^{3}+15470 a^{3} x^{4} b^{2}+13923 x^{2} a^{4} b +13923 a^{5}\right )}{13923 d \left (b \,x^{2}+a \right )^{5}}\) | \(85\) |
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Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2 \, {\left (663 \, b^{5} x^{10} + 4095 \, a b^{4} x^{8} + 10710 \, a^{2} b^{3} x^{6} + 15470 \, a^{3} b^{2} x^{4} + 13923 \, a^{4} b x^{2} + 13923 \, a^{5}\right )} \sqrt {d x}}{13923 \, d} \]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{\sqrt {d x}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.52 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2 \, {\left (195 \, {\left (17 \, b^{5} \sqrt {d} x^{3} + 21 \, a b^{4} \sqrt {d} x\right )} x^{\frac {15}{2}} + 1260 \, {\left (13 \, a b^{4} \sqrt {d} x^{3} + 17 \, a^{2} b^{3} \sqrt {d} x\right )} x^{\frac {11}{2}} + 3570 \, {\left (9 \, a^{2} b^{3} \sqrt {d} x^{3} + 13 \, a^{3} b^{2} \sqrt {d} x\right )} x^{\frac {7}{2}} + 6188 \, {\left (5 \, a^{3} b^{2} \sqrt {d} x^{3} + 9 \, a^{4} b \sqrt {d} x\right )} x^{\frac {3}{2}} + \frac {13923 \, {\left (a^{4} b \sqrt {d} x^{3} + 5 \, a^{5} \sqrt {d} x\right )}}{\sqrt {x}}\right )}}{69615 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.47 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2 \, {\left (663 \, \sqrt {d x} b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 4095 \, \sqrt {d x} a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 10710 \, \sqrt {d x} a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 15470 \, \sqrt {d x} a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 13923 \, \sqrt {d x} a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 13923 \, \sqrt {d x} a^{5} \mathrm {sgn}\left (b x^{2} + a\right )\right )}}{13923 \, d} \]
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Time = 13.74 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.38 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2\,x\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (5731\,a^4+8192\,a^3\,b\,x^2+7278\,a^2\,b^2\,x^4+3432\,a\,b^3\,x^6+663\,b^4\,x^8\right )}{13923\,\sqrt {d\,x}}+\frac {16384\,a^5\,x\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{13923\,\sqrt {d\,x}\,\left (b\,x^2+a\right )} \]
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