Integrand size = 24, antiderivative size = 90 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {a^2}{c x \left (c+d x^2\right )^{3/2}}+\frac {x \left (2 a (b c-2 a d)+b^2 c x^2\right )}{3 c^2 \left (c+d x^2\right )^{3/2}}+\frac {4 a (b c-2 a d) x}{3 c^3 \sqrt {c+d x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {473, 386, 197} \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {a^2}{c x \left (c+d x^2\right )^{3/2}}+\frac {x \left (2 a (b c-2 a d)+b^2 c x^2\right )}{3 c^2 \left (c+d x^2\right )^{3/2}}+\frac {4 a x (b c-2 a d)}{3 c^3 \sqrt {c+d x^2}} \]
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Rule 197
Rule 386
Rule 473
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2}{c x \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {2 a (b c-2 a d)+b^2 c x^2}{\left (c+d x^2\right )^{5/2}} \, dx}{c} \\ & = -\frac {a^2}{c x \left (c+d x^2\right )^{3/2}}+\frac {x \left (2 a (b c-2 a d)+b^2 c x^2\right )}{3 c^2 \left (c+d x^2\right )^{3/2}}+\frac {(4 a (b c-2 a d)) \int \frac {1}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c^2} \\ & = -\frac {a^2}{c x \left (c+d x^2\right )^{3/2}}+\frac {x \left (2 a (b c-2 a d)+b^2 c x^2\right )}{3 c^2 \left (c+d x^2\right )^{3/2}}+\frac {4 a (b c-2 a d) x}{3 c^3 \sqrt {c+d x^2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {b^2 c^2 x^4+2 a b c x^2 \left (3 c+2 d x^2\right )-a^2 \left (3 c^2+12 c d x^2+8 d^2 x^4\right )}{3 c^3 x \left (c+d x^2\right )^{3/2}} \]
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Time = 2.90 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.78
method | result | size |
pseudoelliptic | \(\frac {\left (-8 d^{2} x^{4}-12 c d \,x^{2}-3 c^{2}\right ) a^{2}+6 x^{2} b \left (\frac {2 d \,x^{2}}{3}+c \right ) c a +b^{2} c^{2} x^{4}}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} x \,c^{3}}\) | \(70\) |
gosper | \(-\frac {8 a^{2} d^{2} x^{4}-4 x^{4} a b c d -b^{2} c^{2} x^{4}+12 a^{2} c d \,x^{2}-6 a b \,c^{2} x^{2}+3 a^{2} c^{2}}{3 x \left (d \,x^{2}+c \right )^{\frac {3}{2}} c^{3}}\) | \(78\) |
trager | \(-\frac {8 a^{2} d^{2} x^{4}-4 x^{4} a b c d -b^{2} c^{2} x^{4}+12 a^{2} c d \,x^{2}-6 a b \,c^{2} x^{2}+3 a^{2} c^{2}}{3 x \left (d \,x^{2}+c \right )^{\frac {3}{2}} c^{3}}\) | \(78\) |
risch | \(-\frac {a^{2} \sqrt {d \,x^{2}+c}}{c^{3} x}-\frac {\left (a d -b c \right ) \left (5 a d \,x^{2}+c b \,x^{2}+6 a c \right ) x \sqrt {d \,x^{2}+c}}{3 \left (d^{2} x^{4}+2 c d \,x^{2}+c^{2}\right ) c^{3}}\) | \(83\) |
default | \(b^{2} \left (-\frac {x}{2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {c \left (\frac {x}{3 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{2} \sqrt {d \,x^{2}+c}}\right )}{2 d}\right )+a^{2} \left (-\frac {1}{c x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {4 d \left (\frac {x}{3 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{2} \sqrt {d \,x^{2}+c}}\right )}{c}\right )+2 a b \left (\frac {x}{3 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{2} \sqrt {d \,x^{2}+c}}\right )\) | \(153\) |
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Time = 0.25 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (b^{2} c^{2} + 4 \, a b c d - 8 \, a^{2} d^{2}\right )} x^{4} - 3 \, a^{2} c^{2} + 6 \, {\left (a b c^{2} - 2 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{3 \, {\left (c^{3} d^{2} x^{5} + 2 \, c^{4} d x^{3} + c^{5} x\right )}} \]
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\[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{x^{2} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {4 \, a b x}{3 \, \sqrt {d x^{2} + c} c^{2}} + \frac {2 \, a b x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c} - \frac {b^{2} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {b^{2} x}{3 \, \sqrt {d x^{2} + c} c d} - \frac {8 \, a^{2} d x}{3 \, \sqrt {d x^{2} + c} c^{3}} - \frac {4 \, a^{2} d x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{2}} - \frac {a^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} c x} \]
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Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {x {\left (\frac {{\left (b^{2} c^{4} d + 4 \, a b c^{3} d^{2} - 5 \, a^{2} c^{2} d^{3}\right )} x^{2}}{c^{5} d} + \frac {6 \, {\left (a b c^{4} d - a^{2} c^{3} d^{2}\right )}}{c^{5} d}\right )}}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {2 \, a^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )} c^{2}} \]
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Time = 5.63 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {3\,a^2\,c^2+12\,a^2\,c\,d\,x^2+8\,a^2\,d^2\,x^4-6\,a\,b\,c^2\,x^2-4\,a\,b\,c\,d\,x^4-b^2\,c^2\,x^4}{3\,c^3\,x\,{\left (d\,x^2+c\right )}^{3/2}} \]
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