Optimal. Leaf size=81 \[ -\frac {a^2}{2 c^2 x^2}-\frac {a (b c-a d) \log \left (c+d x^2\right )}{c^3}+\frac {2 a \log (x) (b c-a d)}{c^3}-\frac {(b c-a d)^2}{2 c^2 d \left (c+d x^2\right )} \]
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Rubi [A] time = 0.08, antiderivative size = 81, 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} \[ -\frac {a^2}{2 c^2 x^2}-\frac {(b c-a d)^2}{2 c^2 d \left (c+d x^2\right )}-\frac {a (b c-a d) \log \left (c+d x^2\right )}{c^3}+\frac {2 a \log (x) (b c-a d)}{c^3} \]
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^3 \left (c+d x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^2 (c+d x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {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*}
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Mathematica [A] time = 0.10, size = 72, normalized size = 0.89 \[ -\frac {\frac {a^2 c}{x^2}+\frac {c (b c-a d)^2}{d \left (c+d x^2\right )}-2 a (a d-b c) \log \left (c+d x^2\right )+4 a \log (x) (a d-b c)}{2 c^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 159, normalized size = 1.96 \[ -\frac {a^{2} c^{2} d + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x^{2} + 2 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{4} + {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 4 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{4} + {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (c^{3} d^{2} x^{4} + c^{4} d x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 109, normalized size = 1.35 \[ \frac {{\left (a b c - a^{2} d\right )} \log \left (x^{2}\right )}{c^{3}} - \frac {{\left (a b c d - a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{c^{3} d} - \frac {b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + 2 \, a^{2} d^{2} x^{2} + a^{2} c d}{2 \, {\left (d x^{4} + c x^{2}\right )} c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 114, normalized size = 1.41 \[ -\frac {a^{2} d}{2 \left (d \,x^{2}+c \right ) c^{2}}-\frac {2 a^{2} d \ln \relax (x )}{c^{3}}+\frac {a^{2} d \ln \left (d \,x^{2}+c \right )}{c^{3}}+\frac {a b}{\left (d \,x^{2}+c \right ) c}+\frac {2 a b \ln \relax (x )}{c^{2}}-\frac {a b \ln \left (d \,x^{2}+c \right )}{c^{2}}-\frac {b^{2}}{2 \left (d \,x^{2}+c \right ) d}-\frac {a^{2}}{2 c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 100, normalized size = 1.23 \[ -\frac {a^{2} c d + {\left (b^{2} c^{2} - 2 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}}{2 \, {\left (c^{2} d^{2} x^{4} + c^{3} d x^{2}\right )}} - \frac {{\left (a b c - a^{2} d\right )} \log \left (d x^{2} + c\right )}{c^{3}} + \frac {{\left (a b c - a^{2} d\right )} \log \left (x^{2}\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 100, normalized size = 1.23 \[ \frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d-a\,b\,c\right )}{c^3}-\frac {\frac {a^2}{2\,c}+\frac {x^2\,\left (2\,a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^2\,d}}{d\,x^4+c\,x^2}-\frac {\ln \relax (x)\,\left (2\,a^2\,d-2\,a\,b\,c\right )}{c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.36, size = 92, normalized size = 1.14 \[ - \frac {2 a \left (a d - b c\right ) \log {\relax (x )}}{c^{3}} + \frac {a \left (a d - b c\right ) \log {\left (\frac {c}{d} + x^{2} \right )}}{c^{3}} + \frac {- a^{2} c d + x^{2} \left (- 2 a^{2} d^{2} + 2 a b c d - b^{2} c^{2}\right )}{2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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