Optimal. Leaf size=67 \[ -\frac {1}{2} \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \log \left (c+d x^2\right )+\frac {a^2 \log (x)}{c^2}+\frac {(b c-a d)^2}{2 c d^2 \left (c+d x^2\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 88} \[ -\frac {1}{2} \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \log \left (c+d x^2\right )+\frac {a^2 \log (x)}{c^2}+\frac {(b c-a d)^2}{2 c d^2 \left (c+d x^2\right )} \]
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x (c+d x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{c^2 x}-\frac {(b c-a d)^2}{c d (c+d x)^2}+\frac {b^2 c^2-a^2 d^2}{c^2 d (c+d x)}\right ) \, dx,x,x^2\right )\\ &=\frac {(b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {a^2 \log (x)}{c^2}-\frac {1}{2} \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \log \left (c+d x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 70, normalized size = 1.04 \[ \frac {2 a^2 \log (x)+\frac {(b c-a d) \left (\left (c+d x^2\right ) (a d+b c) \log \left (c+d x^2\right )+c (b c-a d)\right )}{d^2 \left (c+d x^2\right )}}{2 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 116, normalized size = 1.73 \[ \frac {b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{3} - a^{2} c d^{2} + {\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \, {\left (a^{2} d^{3} x^{2} + a^{2} c d^{2}\right )} \log \relax (x)}{2 \, {\left (c^{2} d^{3} x^{2} + c^{3} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 99, normalized size = 1.48 \[ \frac {a^{2} \log \left (x^{2}\right )}{2 \, c^{2}} + \frac {{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{2} d^{2}} - \frac {b^{2} c^{2} x^{2} - a^{2} d^{2} x^{2} + 2 \, a b c^{2} - 2 \, a^{2} c d}{2 \, {\left (d x^{2} + c\right )} c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 94, normalized size = 1.40 \[ \frac {a^{2}}{2 \left (d \,x^{2}+c \right ) c}+\frac {a^{2} \ln \relax (x )}{c^{2}}-\frac {a^{2} \ln \left (d \,x^{2}+c \right )}{2 c^{2}}-\frac {a b}{\left (d \,x^{2}+c \right ) d}+\frac {b^{2} c}{2 \left (d \,x^{2}+c \right ) d^{2}}+\frac {b^{2} \ln \left (d \,x^{2}+c \right )}{2 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 86, normalized size = 1.28 \[ \frac {a^{2} \log \left (x^{2}\right )}{2 \, c^{2}} + \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}} + \frac {{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 80, normalized size = 1.19 \[ \frac {a^2\,\ln \relax (x)}{c^2}+\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{2\,c\,d^2\,\left (d\,x^2+c\right )}-\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^2-b^2\,c^2\right )}{2\,c^2\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.25, size = 80, normalized size = 1.19 \[ \frac {a^{2} \log {\relax (x )}}{c^{2}} + \frac {a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{2 c^{2} d^{2} + 2 c d^{3} x^{2}} - \frac {\left (a d - b c\right ) \left (a d + b c\right ) \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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