Integrand size = 22, antiderivative size = 61 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{3/2}} \, dx=-\frac {2 a^2 c}{\sqrt {x}}+\frac {2}{3} a (2 b c+a d) x^{3/2}+\frac {2}{7} b (b c+2 a d) x^{7/2}+\frac {2}{11} b^2 d x^{11/2} \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{3/2}} \, dx=-\frac {2 a^2 c}{\sqrt {x}}+\frac {2}{7} b x^{7/2} (2 a d+b c)+\frac {2}{3} a x^{3/2} (a d+2 b c)+\frac {2}{11} b^2 d x^{11/2} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 c}{x^{3/2}}+a (2 b c+a d) \sqrt {x}+b (b c+2 a d) x^{5/2}+b^2 d x^{9/2}\right ) \, dx \\ & = -\frac {2 a^2 c}{\sqrt {x}}+\frac {2}{3} a (2 b c+a d) x^{3/2}+\frac {2}{7} b (b c+2 a d) x^{7/2}+\frac {2}{11} b^2 d x^{11/2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{3/2}} \, dx=-\frac {2 \left (231 a^2 c-154 a b c x^2-77 a^2 d x^2-33 b^2 c x^4-66 a b d x^4-21 b^2 d x^6\right )}{231 \sqrt {x}} \]
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Time = 2.67 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {2 b^{2} d \,x^{\frac {11}{2}}}{11}+\frac {4 a b d \,x^{\frac {7}{2}}}{7}+\frac {2 b^{2} c \,x^{\frac {7}{2}}}{7}+\frac {2 a^{2} d \,x^{\frac {3}{2}}}{3}+\frac {4 a b c \,x^{\frac {3}{2}}}{3}-\frac {2 a^{2} c}{\sqrt {x}}\) | \(54\) |
default | \(\frac {2 b^{2} d \,x^{\frac {11}{2}}}{11}+\frac {4 a b d \,x^{\frac {7}{2}}}{7}+\frac {2 b^{2} c \,x^{\frac {7}{2}}}{7}+\frac {2 a^{2} d \,x^{\frac {3}{2}}}{3}+\frac {4 a b c \,x^{\frac {3}{2}}}{3}-\frac {2 a^{2} c}{\sqrt {x}}\) | \(54\) |
gosper | \(-\frac {2 \left (-21 b^{2} d \,x^{6}-66 a b d \,x^{4}-33 b^{2} c \,x^{4}-77 a^{2} d \,x^{2}-154 a b c \,x^{2}+231 a^{2} c \right )}{231 \sqrt {x}}\) | \(56\) |
trager | \(-\frac {2 \left (-21 b^{2} d \,x^{6}-66 a b d \,x^{4}-33 b^{2} c \,x^{4}-77 a^{2} d \,x^{2}-154 a b c \,x^{2}+231 a^{2} c \right )}{231 \sqrt {x}}\) | \(56\) |
risch | \(-\frac {2 \left (-21 b^{2} d \,x^{6}-66 a b d \,x^{4}-33 b^{2} c \,x^{4}-77 a^{2} d \,x^{2}-154 a b c \,x^{2}+231 a^{2} c \right )}{231 \sqrt {x}}\) | \(56\) |
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Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{3/2}} \, dx=\frac {2 \, {\left (21 \, b^{2} d x^{6} + 33 \, {\left (b^{2} c + 2 \, a b d\right )} x^{4} - 231 \, a^{2} c + 77 \, {\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{231 \, \sqrt {x}} \]
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Time = 0.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{3/2}} \, dx=- \frac {2 a^{2} c}{\sqrt {x}} + \frac {2 a^{2} d x^{\frac {3}{2}}}{3} + \frac {4 a b c x^{\frac {3}{2}}}{3} + \frac {4 a b d x^{\frac {7}{2}}}{7} + \frac {2 b^{2} c x^{\frac {7}{2}}}{7} + \frac {2 b^{2} d x^{\frac {11}{2}}}{11} \]
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Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{3/2}} \, dx=\frac {2}{11} \, b^{2} d x^{\frac {11}{2}} + \frac {2}{7} \, {\left (b^{2} c + 2 \, a b d\right )} x^{\frac {7}{2}} - \frac {2 \, a^{2} c}{\sqrt {x}} + \frac {2}{3} \, {\left (2 \, a b c + a^{2} d\right )} x^{\frac {3}{2}} \]
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Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{3/2}} \, dx=\frac {2}{11} \, b^{2} d x^{\frac {11}{2}} + \frac {2}{7} \, b^{2} c x^{\frac {7}{2}} + \frac {4}{7} \, a b d x^{\frac {7}{2}} + \frac {4}{3} \, a b c x^{\frac {3}{2}} + \frac {2}{3} \, a^{2} d x^{\frac {3}{2}} - \frac {2 \, a^{2} c}{\sqrt {x}} \]
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Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{3/2}} \, dx=x^{3/2}\,\left (\frac {2\,d\,a^2}{3}+\frac {4\,b\,c\,a}{3}\right )+x^{7/2}\,\left (\frac {2\,c\,b^2}{7}+\frac {4\,a\,d\,b}{7}\right )-\frac {2\,a^2\,c}{\sqrt {x}}+\frac {2\,b^2\,d\,x^{11/2}}{11} \]
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