Optimal. Leaf size=107 \[ -\frac {1}{2 a c x^2}+\frac {b c+a d}{a^2 c^2 x}+\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \log (x)}{a^3 c^3}-\frac {b^3 \log (a+b x)}{a^3 (b c-a d)}+\frac {d^3 \log (c+d x)}{c^3 (b c-a d)} \]
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Rubi [A]
time = 0.06, antiderivative size = 107, 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^3 \log (a+b x)}{a^3 (b c-a d)}+\frac {a d+b c}{a^2 c^2 x}+\frac {\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}+\frac {d^3 \log (c+d x)}{c^3 (b c-a d)}-\frac {1}{2 a c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rubi steps
\begin {align*} \int \frac {1}{x^3 (a+b x) (c+d x)} \, dx &=\int \left (\frac {1}{a c x^3}+\frac {-b c-a d}{a^2 c^2 x^2}+\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {b^4}{a^3 (-b c+a d) (a+b x)}+\frac {d^4}{c^3 (b c-a d) (c+d x)}\right ) \, dx\\ &=-\frac {1}{2 a c x^2}+\frac {b c+a d}{a^2 c^2 x}+\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \log (x)}{a^3 c^3}-\frac {b^3 \log (a+b x)}{a^3 (b c-a d)}+\frac {d^3 \log (c+d x)}{c^3 (b c-a d)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 106, normalized size = 0.99 \begin {gather*} -\frac {1}{2 a c x^2}+\frac {b c+a d}{a^2 c^2 x}+\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \log (x)}{a^3 c^3}+\frac {b^3 \log (a+b x)}{a^3 (-b c+a d)}+\frac {d^3 \log (c+d x)}{c^3 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 109, normalized size = 1.02
method | result | size |
norman | \(\frac {\frac {\left (a d +b c \right ) x}{a^{2} c^{2}}-\frac {1}{2 a c}}{x^{2}}+\frac {b^{3} \ln \left (b x +a \right )}{a^{3} \left (a d -b c \right )}+\frac {\left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{a^{3} c^{3}}-\frac {d^{3} \ln \left (d x +c \right )}{c^{3} \left (a d -b c \right )}\) | \(106\) |
default | \(\frac {b^{3} \ln \left (b x +a \right )}{a^{3} \left (a d -b c \right )}-\frac {d^{3} \ln \left (d x +c \right )}{c^{3} \left (a d -b c \right )}-\frac {1}{2 a c \,x^{2}}-\frac {-a d -b c}{a^{2} c^{2} x}+\frac {\left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{a^{3} c^{3}}\) | \(109\) |
risch | \(\frac {\frac {\left (a d +b c \right ) x}{a^{2} c^{2}}-\frac {1}{2 a c}}{x^{2}}+\frac {\ln \left (-x \right ) d^{2}}{a \,c^{3}}+\frac {\ln \left (-x \right ) b d}{a^{2} c^{2}}+\frac {\ln \left (-x \right ) b^{2}}{a^{3} c}-\frac {d^{3} \ln \left (-d x -c \right )}{c^{3} \left (a d -b c \right )}+\frac {b^{3} \ln \left (b x +a \right )}{a^{3} \left (a d -b c \right )}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 106, normalized size = 0.99 \begin {gather*} -\frac {b^{3} \log \left (b x + a\right )}{a^{3} b c - a^{4} d} + \frac {d^{3} \log \left (d x + c\right )}{b c^{4} - a c^{3} d} + \frac {{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} \log \left (x\right )}{a^{3} c^{3}} - \frac {a c - 2 \, {\left (b c + a d\right )} x}{2 \, a^{2} c^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.55, size = 121, normalized size = 1.13 \begin {gather*} -\frac {2 \, b^{3} c^{3} x^{2} \log \left (b x + a\right ) - 2 \, a^{3} d^{3} x^{2} \log \left (d x + c\right ) + a^{2} b c^{3} - a^{3} c^{2} d - 2 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{2} \log \left (x\right ) - 2 \, {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x}{2 \, {\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{2}} \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 = 2.18, size = 125, normalized size = 1.17 \begin {gather*} -\frac {b^{4} \log \left ({\left | b x + a \right |}\right )}{a^{3} b^{2} c - a^{4} b d} + \frac {d^{4} \log \left ({\left | d x + c \right |}\right )}{b c^{4} d - a c^{3} d^{2}} + \frac {{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{3} c^{3}} - \frac {a^{2} c^{2} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} x}{2 \, a^{3} c^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 107, normalized size = 1.00 \begin {gather*} \frac {b^3\,\ln \left (a+b\,x\right )}{a^3\,\left (a\,d-b\,c\right )}-\frac {\frac {1}{2\,a\,c}-\frac {x\,\left (a\,d+b\,c\right )}{a^2\,c^2}}{x^2}-\frac {d^3\,\ln \left (c+d\,x\right )}{c^3\,\left (a\,d-b\,c\right )}+\frac {\ln \left (x\right )\,\left (a^2\,d^2+a\,b\,c\,d+b^2\,c^2\right )}{a^3\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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