Integrand size = 24, antiderivative size = 91 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {a^2}{c x \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-2 a d (b c-a d)\right ) x}{c^2 d \sqrt {c+d x^2}}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{d^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {473, 393, 223, 212} \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {a^2}{c x \sqrt {c+d x^2}}-\frac {x \left (\frac {b^2}{d}-\frac {2 a (b c-a d)}{c^2}\right )}{\sqrt {c+d x^2}}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{d^{3/2}} \]
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Rule 212
Rule 223
Rule 393
Rule 473
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2}{c x \sqrt {c+d x^2}}+\frac {\int \frac {2 a (b c-a d)+b^2 c x^2}{\left (c+d x^2\right )^{3/2}} \, dx}{c} \\ & = -\frac {a^2}{c x \sqrt {c+d x^2}}-\frac {\left (\frac {b^2}{d}-\frac {2 a (b c-a d)}{c^2}\right ) x}{\sqrt {c+d x^2}}+\frac {b^2 \int \frac {1}{\sqrt {c+d x^2}} \, dx}{d} \\ & = -\frac {a^2}{c x \sqrt {c+d x^2}}-\frac {\left (\frac {b^2}{d}-\frac {2 a (b c-a d)}{c^2}\right ) x}{\sqrt {c+d x^2}}+\frac {b^2 \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{d} \\ & = -\frac {a^2}{c x \sqrt {c+d x^2}}-\frac {\left (\frac {b^2}{d}-\frac {2 a (b c-a d)}{c^2}\right ) x}{\sqrt {c+d x^2}}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{d^{3/2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {-b^2 c^2 x^2+2 a b c d x^2-a^2 d \left (c+2 d x^2\right )}{c^2 d x \sqrt {c+d x^2}}-\frac {b^2 \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{d^{3/2}} \]
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Time = 3.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.07
method | result | size |
default | \(b^{2} \left (-\frac {x}{d \sqrt {d \,x^{2}+c}}+\frac {\ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{d^{\frac {3}{2}}}\right )+a^{2} \left (-\frac {1}{c x \sqrt {d \,x^{2}+c}}-\frac {2 d x}{c^{2} \sqrt {d \,x^{2}+c}}\right )+\frac {2 a b x}{c \sqrt {d \,x^{2}+c}}\) | \(97\) |
risch | \(-\frac {a^{2} \sqrt {d \,x^{2}+c}}{c^{2} x}-\frac {a^{2} d x}{c^{2} \sqrt {d \,x^{2}+c}}-\frac {b^{2} x}{d \sqrt {d \,x^{2}+c}}+\frac {b^{2} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{d^{\frac {3}{2}}}+\frac {2 a b x}{c \sqrt {d \,x^{2}+c}}\) | \(99\) |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right ) b^{2} c^{2} x \sqrt {d \,x^{2}+c}-2 a^{2} d^{\frac {5}{2}} x^{2}+2 a b c \,d^{\frac {3}{2}} x^{2}-b^{2} c^{2} x^{2} \sqrt {d}-a^{2} c \,d^{\frac {3}{2}}}{d^{\frac {3}{2}} x \sqrt {d \,x^{2}+c}\, c^{2}}\) | \(100\) |
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Time = 0.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.63 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\left [\frac {{\left (b^{2} c^{2} d x^{3} + b^{2} c^{3} x\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{2 \, {\left (c^{2} d^{3} x^{3} + c^{3} d^{2} x\right )}}, -\frac {{\left (b^{2} c^{2} d x^{3} + b^{2} c^{3} x\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{c^{2} d^{3} x^{3} + c^{3} d^{2} x}\right ] \]
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\[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{x^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {2 \, a b x}{\sqrt {d x^{2} + c} c} - \frac {b^{2} x}{\sqrt {d x^{2} + c} d} - \frac {2 \, a^{2} d x}{\sqrt {d x^{2} + c} c^{2}} + \frac {b^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{d^{\frac {3}{2}}} - \frac {a^{2}}{\sqrt {d x^{2} + c} c x} \]
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Time = 0.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {b^{2} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{2 \, d^{\frac {3}{2}}} + \frac {2 \, a^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )} c} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{\sqrt {d x^{2} + c} c^{2} d} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{x^2\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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