Optimal. Leaf size=99 \[ -\frac {b (b c-2 a d) \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^2}+\frac {\log (x)}{a^2 c}-\frac {d^2 \log \left (c+d x^2\right )}{2 c (b c-a d)^2}+\frac {b}{2 a \left (a+b x^2\right ) (b c-a d)} \]
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Rubi [A] time = 0.11, antiderivative size = 99, 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, 72} \begin {gather*} -\frac {b (b c-2 a d) \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^2}+\frac {\log (x)}{a^2 c}-\frac {d^2 \log \left (c+d x^2\right )}{2 c (b c-a d)^2}+\frac {b}{2 a \left (a+b x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^2 (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^2 c x}+\frac {b^2}{a (-b c+a d) (a+b x)^2}+\frac {b^2 (-b c+2 a d)}{a^2 (-b c+a d)^2 (a+b x)}-\frac {d^3}{c (b c-a d)^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=\frac {b}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {\log (x)}{a^2 c}-\frac {b (b c-2 a d) \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^2}-\frac {d^2 \log \left (c+d x^2\right )}{2 c (b c-a d)^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 97, normalized size = 0.98 \begin {gather*} \frac {2 \log (x)-\frac {a \left (a d^2 \left (a+b x^2\right ) \log \left (c+d x^2\right )+b c (a d-b c)\right )+b c \left (a+b x^2\right ) (b c-2 a d) \log \left (a+b x^2\right )}{\left (a+b x^2\right ) (b c-a d)^2}}{2 a^2 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 2.62, size = 218, normalized size = 2.20 \begin {gather*} \frac {a b^{2} c^{2} - a^{2} b c d - {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + {\left (b^{3} c^{2} - 2 \, a b^{2} c d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - {\left (a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (a^{3} b^{2} c^{3} - 2 \, a^{4} b c^{2} d + a^{5} c d^{2} + {\left (a^{2} b^{3} c^{3} - 2 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 183, normalized size = 1.85 \begin {gather*} -\frac {d^{3} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )}} - \frac {{\left (b^{3} c - 2 \, a b^{2} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )}} + \frac {b^{3} c x^{2} - 2 \, a b^{2} d x^{2} + 2 \, a b^{2} c - 3 \, a^{2} b d}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} {\left (b x^{2} + a\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 139, normalized size = 1.40 \begin {gather*} \frac {b^{2} c}{2 \left (a d -b c \right )^{2} \left (b \,x^{2}+a \right ) a}+\frac {b d \ln \left (b \,x^{2}+a \right )}{\left (a d -b c \right )^{2} a}-\frac {b^{2} c \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{2} a^{2}}-\frac {b d}{2 \left (a d -b c \right )^{2} \left (b \,x^{2}+a \right )}-\frac {d^{2} \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{2} c}+\frac {\ln \relax (x )}{a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 137, normalized size = 1.38 \begin {gather*} -\frac {d^{2} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} - \frac {{\left (b^{2} c - 2 \, a b d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} + \frac {b}{2 \, {\left (a^{2} b c - a^{3} d + {\left (a b^{2} c - a^{2} b d\right )} x^{2}\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 127, normalized size = 1.28 \begin {gather*} \frac {\ln \relax (x)}{a^2\,c}-\frac {d^2\,\ln \left (d\,x^2+c\right )}{2\,a^2\,c\,d^2-4\,a\,b\,c^2\,d+2\,b^2\,c^3}-\frac {\ln \left (b\,x^2+a\right )\,\left (b^2\,c-2\,a\,b\,d\right )}{2\,a^4\,d^2-4\,a^3\,b\,c\,d+2\,a^2\,b^2\,c^2}-\frac {b}{2\,a\,\left (b\,x^2+a\right )\,\left (a\,d-b\,c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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