Optimal. Leaf size=70 \[ -\frac {1}{2 \left (a+b x^2\right ) (b c-a d)}-\frac {d \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac {d \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
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Rubi [A] time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {444, 44} \begin {gather*} -\frac {1}{2 \left (a+b x^2\right ) (b c-a d)}-\frac {d \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac {d \log \left (c+d x^2\right )}{2 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 444
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(a+b x)^2 (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 (b c-a d) \left (a+b x^2\right )}-\frac {d \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac {d \log \left (c+d x^2\right )}{2 (b c-a d)^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 0.94 \begin {gather*} \frac {d \left (a+b x^2\right ) \log \left (c+d x^2\right )-d \left (a+b x^2\right ) \log \left (a+b x^2\right )+a d-b c}{2 \left (a+b x^2\right ) (b c-a d)^2} \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 {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 [A] time = 0.57, size = 103, normalized size = 1.47 \begin {gather*} -\frac {b c - a d + {\left (b d x^{2} + a d\right )} \log \left (b x^{2} + a\right ) - {\left (b d x^{2} + a d\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 85, normalized size = 1.21 \begin {gather*} \frac {b d \log \left ({\left | \frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d \right |}\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}} - \frac {b}{2 \, {\left (b^{2} c - a b d\right )} {\left (b x^{2} + a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 1.29 \begin {gather*} \frac {a d}{2 \left (a d -b c \right )^{2} \left (b \,x^{2}+a \right )}-\frac {b c}{2 \left (a d -b c \right )^{2} \left (b \,x^{2}+a \right )}-\frac {d \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{2}}+\frac {d \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 99, normalized size = 1.41 \begin {gather*} -\frac {d \log \left (b x^{2} + a\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac {d \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {1}{2 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 161, normalized size = 2.30 \begin {gather*} -\frac {b\,c-a\,\left (d-d\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}\right )+b\,d\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}}{2\,a^3\,d^2-4\,a^2\,b\,c\,d+2\,a^2\,b\,d^2\,x^2+2\,a\,b^2\,c^2-4\,a\,b^2\,c\,d\,x^2+2\,b^3\,c^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.81, size = 248, normalized size = 3.54 \begin {gather*} \frac {d \log {\left (x^{2} + \frac {- \frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{2 \left (a d - b c\right )^{2}} - \frac {d \log {\left (x^{2} + \frac {\frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{2 \left (a d - b c\right )^{2}} + \frac {1}{2 a^{2} d - 2 a b c + x^{2} \left (2 a b d - 2 b^{2} c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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