Optimal. Leaf size=76 \[ -\frac {1}{a c x}-\frac {(b c+a d) \log (x)}{a^2 c^2}+\frac {b^2 \log (a+b x)}{a^2 (b c-a d)}-\frac {d^2 \log (c+d x)}{c^2 (b c-a d)} \]
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Rubi [A]
time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {84}
\begin {gather*} \frac {b^2 \log (a+b x)}{a^2 (b c-a d)}-\frac {\log (x) (a d+b c)}{a^2 c^2}-\frac {d^2 \log (c+d x)}{c^2 (b c-a d)}-\frac {1}{a c x} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x) (c+d x)} \, dx &=\int \left (\frac {1}{a c x^2}+\frac {-b c-a d}{a^2 c^2 x}-\frac {b^3}{a^2 (-b c+a d) (a+b x)}-\frac {d^3}{c^2 (b c-a d) (c+d x)}\right ) \, dx\\ &=-\frac {1}{a c x}-\frac {(b c+a d) \log (x)}{a^2 c^2}+\frac {b^2 \log (a+b x)}{a^2 (b c-a d)}-\frac {d^2 \log (c+d x)}{c^2 (b c-a d)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 78, normalized size = 1.03 \begin {gather*} -\frac {1}{a c x}+\frac {(-b c-a d) \log (x)}{a^2 c^2}-\frac {b^2 \log (a+b x)}{a^2 (-b c+a d)}-\frac {d^2 \log (c+d x)}{c^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 78, normalized size = 1.03
method | result | size |
norman | \(-\frac {1}{a c x}+\frac {d^{2} \ln \left (d x +c \right )}{c^{2} \left (a d -b c \right )}-\frac {b^{2} \ln \left (b x +a \right )}{a^{2} \left (a d -b c \right )}-\frac {\left (a d +b c \right ) \ln \left (x \right )}{a^{2} c^{2}}\) | \(77\) |
default | \(-\frac {b^{2} \ln \left (b x +a \right )}{a^{2} \left (a d -b c \right )}+\frac {d^{2} \ln \left (d x +c \right )}{c^{2} \left (a d -b c \right )}-\frac {1}{a c x}+\frac {\left (-a d -b c \right ) \ln \left (x \right )}{a^{2} c^{2}}\) | \(78\) |
risch | \(-\frac {1}{a c x}-\frac {\ln \left (-x \right ) d}{a \,c^{2}}-\frac {\ln \left (-x \right ) b}{a^{2} c}+\frac {d^{2} \ln \left (-d x -c \right )}{c^{2} \left (a d -b c \right )}-\frac {b^{2} \ln \left (b x +a \right )}{a^{2} \left (a d -b c \right )}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 80, normalized size = 1.05 \begin {gather*} \frac {b^{2} \log \left (b x + a\right )}{a^{2} b c - a^{3} d} - \frac {d^{2} \log \left (d x + c\right )}{b c^{3} - a c^{2} d} - \frac {{\left (b c + a d\right )} \log \left (x\right )}{a^{2} c^{2}} - \frac {1}{a c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.09, size = 88, normalized size = 1.16 \begin {gather*} \frac {b^{2} c^{2} x \log \left (b x + a\right ) - a^{2} d^{2} x \log \left (d x + c\right ) - a b c^{2} + a^{2} c d - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x \log \left (x\right )}{{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 2.02, size = 89, normalized size = 1.17 \begin {gather*} \frac {b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{2} b^{2} c - a^{3} b d} - \frac {d^{3} \log \left ({\left | d x + c \right |}\right )}{b c^{3} d - a c^{2} d^{2}} - \frac {{\left (b c + a d\right )} \log \left ({\left | x \right |}\right )}{a^{2} c^{2}} - \frac {1}{a c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 78, normalized size = 1.03 \begin {gather*} \frac {d^2\,\ln \left (c+d\,x\right )}{c^2\,\left (a\,d-b\,c\right )}-\frac {1}{a\,c\,x}-\frac {b^2\,\ln \left (a+b\,x\right )}{a^3\,d-a^2\,b\,c}-\frac {\ln \left (x\right )\,\left (a\,d+b\,c\right )}{a^2\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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