Optimal. Leaf size=100 \[ -\frac {b^2 \log \left (a+b x^2\right )}{2 a (b c-a d)^2}+\frac {d (2 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2}-\frac {d}{2 c \left (c+d x^2\right ) (b c-a d)}+\frac {\log (x)}{a c^2} \]
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Rubi [A] time = 0.10, antiderivative size = 100, 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} \[ -\frac {b^2 \log \left (a+b x^2\right )}{2 a (b c-a d)^2}+\frac {d (2 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2}-\frac {d}{2 c \left (c+d x^2\right ) (b c-a d)}+\frac {\log (x)}{a c^2} \]
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 ) \left (c+d x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (a+b x) (c+d x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a c^2 x}-\frac {b^3}{a (-b c+a d)^2 (a+b x)}+\frac {d^2}{c (b c-a d) (c+d x)^2}+\frac {d^2 (2 b c-a d)}{c^2 (b c-a d)^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {d}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {\log (x)}{a c^2}-\frac {b^2 \log \left (a+b x^2\right )}{2 a (b c-a d)^2}+\frac {d (2 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 98, normalized size = 0.98 \[ \frac {1}{2} \left (-\frac {b^2 \log \left (a+b x^2\right )}{a (b c-a d)^2}+\frac {d (2 b c-a d) \log \left (c+d x^2\right )}{c^2 (b c-a d)^2}-\frac {d}{c \left (c+d x^2\right ) (b c-a d)}+\frac {2 \log (x)}{a c^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.36, size = 219, normalized size = 2.19 \[ -\frac {a b c^{2} d - a^{2} c d^{2} + {\left (b^{2} c^{2} d x^{2} + b^{2} c^{3}\right )} \log \left (b x^{2} + a\right ) - {\left (2 \, a b c^{2} d - a^{2} c d^{2} + {\left (2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (a b^{2} c^{5} - 2 \, a^{2} b c^{4} d + a^{3} c^{3} d^{2} + {\left (a b^{2} c^{4} d - 2 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 185, normalized size = 1.85 \[ -\frac {b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )}} + \frac {{\left (2 \, b c d^{2} - a d^{3}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )}} - \frac {2 \, b c d^{2} x^{2} - a d^{3} x^{2} + 3 \, b c^{2} d - 2 \, a c d^{2}}{2 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} {\left (d x^{2} + c\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 139, normalized size = 1.39 \[ \frac {a \,d^{2}}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right ) c}-\frac {a \,d^{2} \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{2} c^{2}}-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{2} a}+\frac {b d \ln \left (d \,x^{2}+c \right )}{\left (a d -b c \right )^{2} c}-\frac {b d}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}+\frac {\ln \relax (x )}{a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 138, normalized size = 1.38 \[ -\frac {b^{2} \log \left (b x^{2} + a\right )}{2 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac {{\left (2 \, b c d - a d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} - \frac {d}{2 \, {\left (b c^{3} - a c^{2} d + {\left (b c^{2} d - a c d^{2}\right )} x^{2}\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 127, normalized size = 1.27 \[ \frac {\ln \relax (x)}{a\,c^2}-\frac {\ln \left (d\,x^2+c\right )\,\left (a\,d^2-2\,b\,c\,d\right )}{2\,a^2\,c^2\,d^2-4\,a\,b\,c^3\,d+2\,b^2\,c^4}-\frac {b^2\,\ln \left (b\,x^2+a\right )}{2\,a^3\,d^2-4\,a^2\,b\,c\,d+2\,a\,b^2\,c^2}+\frac {d}{2\,c\,\left (d\,x^2+c\right )\,\left (a\,d-b\,c\right )} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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