Integrand size = 30, antiderivative size = 195 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {2 a^3 (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}+\frac {2 a^2 b (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}+\frac {6 a b^2 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^5 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{17/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 d^7 \left (a+b x^2\right )} \]
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Time = 0.03 (sec) , antiderivative size = 195, 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} \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {6 a b^2 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^5 \left (a+b x^2\right )}+\frac {2 a^2 b (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{17/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 d^7 \left (a+b x^2\right )}+\frac {2 a^3 (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )} \]
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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)^{3/2} \left (a b+b^2 x^2\right )^3 \, dx}{b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^3 b^3 (d x)^{3/2}+\frac {3 a^2 b^4 (d x)^{7/2}}{d^2}+\frac {3 a b^5 (d x)^{11/2}}{d^4}+\frac {b^6 (d x)^{15/2}}{d^6}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {2 a^3 (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}+\frac {2 a^2 b (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}+\frac {6 a b^2 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^5 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{17/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 d^7 \left (a+b x^2\right )} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.34 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {2 x (d x)^{3/2} \sqrt {\left (a+b x^2\right )^2} \left (663 a^3+1105 a^2 b x^2+765 a b^2 x^4+195 b^3 x^6\right )}{3315 \left (a+b x^2\right )} \]
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Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.31
method | result | size |
gosper | \(\frac {2 x \left (195 b^{3} x^{6}+765 b^{2} x^{4} a +1105 a^{2} b \,x^{2}+663 a^{3}\right ) \left (d x \right )^{\frac {3}{2}} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}{3315 \left (b \,x^{2}+a \right )^{3}}\) | \(61\) |
default | \(\frac {2 {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}} \left (d x \right )^{\frac {5}{2}} \left (195 b^{3} x^{6}+765 b^{2} x^{4} a +1105 a^{2} b \,x^{2}+663 a^{3}\right )}{3315 d \left (b \,x^{2}+a \right )^{3}}\) | \(63\) |
risch | \(\frac {2 d^{2} \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x^{3} \left (195 b^{3} x^{6}+765 b^{2} x^{4} a +1105 a^{2} b \,x^{2}+663 a^{3}\right )}{3315 \left (b \,x^{2}+a \right ) \sqrt {d x}}\) | \(66\) |
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Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.24 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {2}{3315} \, {\left (195 \, b^{3} d x^{8} + 765 \, a b^{2} d x^{6} + 1105 \, a^{2} b d x^{4} + 663 \, a^{3} d x^{2}\right )} \sqrt {d x} \]
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\[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\int \left (d x\right )^{\frac {3}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.43 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {2}{221} \, {\left (13 \, b^{3} d^{\frac {3}{2}} x^{3} + 17 \, a b^{2} d^{\frac {3}{2}} x\right )} x^{\frac {11}{2}} + \frac {4}{117} \, {\left (9 \, a b^{2} d^{\frac {3}{2}} x^{3} + 13 \, a^{2} b d^{\frac {3}{2}} x\right )} x^{\frac {7}{2}} + \frac {2}{45} \, {\left (5 \, a^{2} b d^{\frac {3}{2}} x^{3} + 9 \, a^{3} d^{\frac {3}{2}} x\right )} x^{\frac {3}{2}} \]
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Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.46 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {2}{3315} \, {\left (195 \, \sqrt {d x} b^{3} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 765 \, \sqrt {d x} a b^{2} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 1105 \, \sqrt {d x} a^{2} b x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 663 \, \sqrt {d x} a^{3} x^{2} \mathrm {sgn}\left (b x^{2} + a\right )\right )} d \]
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Timed out. \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\int {\left (d\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2} \,d x \]
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