Optimal. Leaf size=87 \[ -\frac {b^2 \log (a+b x)}{a (b c-a d)^2}+\frac {d (2 b c-a d) \log (c+d x)}{c^2 (b c-a d)^2}-\frac {d}{c (c+d x) (b c-a d)}+\frac {\log (x)}{a c^2} \]
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Rubi [A] time = 0.07, antiderivative size = 87, 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 = {72} \begin {gather*} -\frac {b^2 \log (a+b x)}{a (b c-a d)^2}+\frac {d (2 b c-a d) \log (c+d x)}{c^2 (b c-a d)^2}-\frac {d}{c (c+d x) (b c-a d)}+\frac {\log (x)}{a c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x) (c+d x)^2} \, dx &=\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\\ &=-\frac {d}{c (b c-a d) (c+d x)}+\frac {\log (x)}{a c^2}-\frac {b^2 \log (a+b x)}{a (b c-a d)^2}+\frac {d (2 b c-a d) \log (c+d x)}{c^2 (b c-a d)^2}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 83, normalized size = 0.95 \begin {gather*} \frac {\frac {a d ((c+d x) (2 b c-a d) \log (c+d x)+c (a d-b c))-b^2 c^2 (c+d x) \log (a+b x)}{(c+d x) (b c-a d)^2}+\log (x)}{a c^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 {1}{x (a+b x) (c+d x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 3.46, size = 207, normalized size = 2.38 \begin {gather*} -\frac {a b c^{2} d - a^{2} c d^{2} + {\left (b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \left (b x + 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\right )} \log \left (d x + c\right ) - {\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\right )} \log \relax (x)}{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.06, size = 282, normalized size = 3.24 \begin {gather*} -\frac {1}{2} \, d {\left (\frac {{\left (2 \, b c - a d\right )} \log \left ({\left | -b + \frac {2 \, b c}{d x + c} - \frac {b c^{2}}{{\left (d x + c\right )}^{2}} - \frac {a d}{d x + c} + \frac {a c d}{{\left (d x + c\right )}^{2}} \right |}\right )}{b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}} + \frac {2 \, d^{2}}{{\left (b c^{2} d^{2} - a c d^{3}\right )} {\left (d x + c\right )}} + \frac {{\left (2 \, b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (\frac {{\left | -2 \, b c d + \frac {2 \, b c^{2} d}{d x + c} + a d^{2} - \frac {2 \, a c d^{2}}{d x + c} - d^{2} {\left | a \right |} \right |}}{{\left | -2 \, b c d + \frac {2 \, b c^{2} d}{d x + c} + a d^{2} - \frac {2 \, a c d^{2}}{d x + c} + d^{2} {\left | a \right |} \right |}}\right )}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} d^{2} {\left | a \right |}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 105, normalized size = 1.21 \begin {gather*} -\frac {a \,d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{2} c^{2}}-\frac {b^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{2} a}+\frac {2 b d \ln \left (d x +c \right )}{\left (a d -b c \right )^{2} c}+\frac {d}{\left (a d -b c \right ) \left (d x +c \right ) c}+\frac {\ln \relax (x )}{a \,c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 128, normalized size = 1.47 \begin {gather*} -\frac {b^{2} \log \left (b x + a\right )}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}} + \frac {{\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}} - \frac {d}{b c^{3} - a c^{2} d + {\left (b c^{2} d - a c d^{2}\right )} x} + \frac {\log \relax (x)}{a c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 116, normalized size = 1.33 \begin {gather*} \frac {\ln \relax (x)}{a\,c^2}-\frac {\ln \left (c+d\,x\right )\,\left (a\,d^2-2\,b\,c\,d\right )}{a^2\,c^2\,d^2-2\,a\,b\,c^3\,d+b^2\,c^4}-\frac {b^2\,\ln \left (a+b\,x\right )}{a^3\,d^2-2\,a^2\,b\,c\,d+a\,b^2\,c^2}+\frac {d}{c\,\left (a\,d-b\,c\right )\,\left (c+d\,x\right )} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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