Integrand size = 22, antiderivative size = 81 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=-\frac {a^2}{2 c^2 x^2}-\frac {(b c-a d)^2}{2 c^2 d \left (c+d x^2\right )}+\frac {2 a (b c-a d) \log (x)}{c^3}-\frac {a (b c-a d) \log \left (c+d x^2\right )}{c^3} \]
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Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 90} \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=-\frac {a^2}{2 c^2 x^2}-\frac {a (b c-a d) \log \left (c+d x^2\right )}{c^3}+\frac {2 a \log (x) (b c-a d)}{c^3}-\frac {(b c-a d)^2}{2 c^2 d \left (c+d x^2\right )} \]
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Rule 90
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2}{x^2 (c+d x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2}{c^2 x^2}-\frac {2 a (-b c+a d)}{c^3 x}+\frac {(b c-a d)^2}{c^2 (c+d x)^2}+\frac {2 a d (-b c+a d)}{c^3 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {a^2}{2 c^2 x^2}-\frac {(b c-a d)^2}{2 c^2 d \left (c+d x^2\right )}+\frac {2 a (b c-a d) \log (x)}{c^3}-\frac {a (b c-a d) \log \left (c+d x^2\right )}{c^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=-\frac {\frac {a^2 c}{x^2}+\frac {c (b c-a d)^2}{d \left (c+d x^2\right )}+4 a (-b c+a d) \log (x)-2 a (-b c+a d) \log \left (c+d x^2\right )}{2 c^3} \]
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Time = 2.66 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {a^{2}}{2 c^{2} x^{2}}-\frac {2 \left (a d -b c \right ) a \ln \left (x \right )}{c^{3}}+\frac {\left (a d -b c \right ) \left (2 a \ln \left (d \,x^{2}+c \right )-\frac {\left (a d -b c \right ) c}{d \left (d \,x^{2}+c \right )}\right )}{2 c^{3}}\) | \(77\) |
norman | \(\frac {-\frac {a^{2}}{2 c}+\frac {\left (2 a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{4}}{2 c^{3}}}{x^{2} \left (d \,x^{2}+c \right )}+\frac {\left (a d -b c \right ) a \ln \left (d \,x^{2}+c \right )}{c^{3}}-\frac {2 \left (a d -b c \right ) a \ln \left (x \right )}{c^{3}}\) | \(91\) |
risch | \(\frac {-\frac {\left (2 a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{2}}{2 c^{2} d}-\frac {a^{2}}{2 c}}{x^{2} \left (d \,x^{2}+c \right )}-\frac {2 a^{2} \ln \left (x \right ) d}{c^{3}}+\frac {2 a \ln \left (x \right ) b}{c^{2}}+\frac {a^{2} \ln \left (-d \,x^{2}-c \right ) d}{c^{3}}-\frac {a \ln \left (-d \,x^{2}-c \right ) b}{c^{2}}\) | \(114\) |
parallelrisch | \(-\frac {4 \ln \left (x \right ) x^{4} a^{2} d^{2}-4 \ln \left (x \right ) x^{4} a b c d -2 \ln \left (d \,x^{2}+c \right ) x^{4} a^{2} d^{2}+2 \ln \left (d \,x^{2}+c \right ) x^{4} a b c d -2 a^{2} d^{2} x^{4}+2 x^{4} a b c d -b^{2} c^{2} x^{4}+4 \ln \left (x \right ) x^{2} a^{2} c d -4 \ln \left (x \right ) x^{2} a b \,c^{2}-2 \ln \left (d \,x^{2}+c \right ) x^{2} a^{2} c d +2 \ln \left (d \,x^{2}+c \right ) x^{2} a b \,c^{2}+a^{2} c^{2}}{2 c^{3} x^{2} \left (d \,x^{2}+c \right )}\) | \(177\) |
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Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (77) = 154\).
Time = 0.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=-\frac {a^{2} c^{2} d + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x^{2} + 2 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{4} + {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 4 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{4} + {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (c^{3} d^{2} x^{4} + c^{4} d x^{2}\right )}} \]
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Time = 0.76 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=- \frac {2 a \left (a d - b c\right ) \log {\left (x \right )}}{c^{3}} + \frac {a \left (a d - b c\right ) \log {\left (\frac {c}{d} + x^{2} \right )}}{c^{3}} + \frac {- a^{2} c d + x^{2} \left (- 2 a^{2} d^{2} + 2 a b c d - b^{2} c^{2}\right )}{2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=-\frac {a^{2} c d + {\left (b^{2} c^{2} - 2 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}}{2 \, {\left (c^{2} d^{2} x^{4} + c^{3} d x^{2}\right )}} - \frac {{\left (a b c - a^{2} d\right )} \log \left (d x^{2} + c\right )}{c^{3}} + \frac {{\left (a b c - a^{2} d\right )} \log \left (x^{2}\right )}{c^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=\frac {{\left (a b c - a^{2} d\right )} \log \left (x^{2}\right )}{c^{3}} - \frac {{\left (a b c d - a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{c^{3} d} - \frac {b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + 2 \, a^{2} d^{2} x^{2} + a^{2} c d}{2 \, {\left (d x^{4} + c x^{2}\right )} c^{2} d} \]
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Time = 5.12 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d-a\,b\,c\right )}{c^3}-\frac {\frac {a^2}{2\,c}+\frac {x^2\,\left (2\,a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^2\,d}}{d\,x^4+c\,x^2}-\frac {\ln \left (x\right )\,\left (2\,a^2\,d-2\,a\,b\,c\right )}{c^3} \]
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