Optimal. Leaf size=92 \[ -\frac {b}{2 \left (a+b x^2\right ) (b c-a d)^2}-\frac {d}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac {b d \log \left (c+d x^2\right )}{(b c-a d)^3} \]
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Rubi [A] time = 0.08, antiderivative size = 92, 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 {b}{2 \left (a+b x^2\right ) (b c-a d)^2}-\frac {d}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac {b d \log \left (c+d x^2\right )}{(b c-a d)^3} \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 )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(a+b x)^2 (c+d x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b^2}{(b c-a d)^2 (a+b x)^2}-\frac {2 b^2 d}{(b c-a d)^3 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)^2}+\frac {2 b d^2}{(b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b}{2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {d}{2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac {b d \log \left (c+d x^2\right )}{(b c-a d)^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 77, normalized size = 0.84 \begin {gather*} \frac {\frac {b (a d-b c)}{a+b x^2}+\frac {d (a d-b c)}{c+d x^2}-2 b d \log \left (a+b x^2\right )+2 b d \log \left (c+d x^2\right )}{2 (b c-a d)^3} \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 )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.05, size = 253, normalized size = 2.75 \begin {gather*} -\frac {b^{2} c^{2} - a^{2} d^{2} + 2 \, {\left (b^{2} c d - a b d^{2}\right )} x^{2} + 2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{4} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 163, normalized size = 1.77 \begin {gather*} \frac {b^{2} d \log \left ({\left | \frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac {b^{3}}{2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (b x^{2} + a\right )}} + \frac {b d^{2}}{2 \, {\left (b c - a d\right )}^{3} {\left (\frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 143, normalized size = 1.55 \begin {gather*} -\frac {a b d}{2 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )}-\frac {a \,d^{2}}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )}+\frac {b^{2} c}{2 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )}+\frac {b c d}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )}+\frac {b d \ln \left (b \,x^{2}+a \right )}{\left (a d -b c \right )^{3}}-\frac {b d \ln \left (d \,x^{2}+c \right )}{\left (a d -b c \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.06, size = 215, normalized size = 2.34 \begin {gather*} -\frac {b d \log \left (b x^{2} + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {b d \log \left (d x^{2} + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac {2 \, b d x^{2} + b c + a d}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 378, normalized size = 4.11 \begin {gather*} -\frac {b^2\,c^2-a^2\,d^2+b^2\,d^2\,x^4\,\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 )\,4{}\mathrm {i}-2\,a\,b\,d^2\,x^2+2\,b^2\,c\,d\,x^2+a\,b\,d^2\,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 )\,4{}\mathrm {i}+b^2\,c\,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 )\,4{}\mathrm {i}+a\,b\,c\,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 )\,4{}\mathrm {i}}{-2\,a^4\,c\,d^3-2\,a^4\,d^4\,x^2+6\,a^3\,b\,c^2\,d^2+4\,a^3\,b\,c\,d^3\,x^2-2\,a^3\,b\,d^4\,x^4-6\,a^2\,b^2\,c^3\,d+6\,a^2\,b^2\,c\,d^3\,x^4+2\,a\,b^3\,c^4-4\,a\,b^3\,c^3\,d\,x^2-6\,a\,b^3\,c^2\,d^2\,x^4+2\,b^4\,c^4\,x^2+2\,b^4\,c^3\,d\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.76, size = 410, normalized size = 4.46 \begin {gather*} - \frac {b d \log {\left (x^{2} + \frac {- \frac {a^{4} b d^{5}}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac {4 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + a b d^{2} - \frac {b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + b^{2} c d}{2 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} + \frac {b d \log {\left (x^{2} + \frac {\frac {a^{4} b d^{5}}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac {4 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + a b d^{2} + \frac {b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + b^{2} c d}{2 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} + \frac {- a d - b c - 2 b d x^{2}}{2 a^{3} c d^{2} - 4 a^{2} b c^{2} d + 2 a b^{2} c^{3} + x^{4} \left (2 a^{2} b d^{3} - 4 a b^{2} c d^{2} + 2 b^{3} c^{2} d\right ) + x^{2} \left (2 a^{3} d^{3} - 2 a^{2} b c d^{2} - 2 a b^{2} c^{2} d + 2 b^{3} c^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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