Optimal. Leaf size=61 \[ -\frac {c}{d (c+d x) (b c-a d)}-\frac {a \log (a+b x)}{(b c-a d)^2}+\frac {a \log (c+d x)}{(b c-a d)^2} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \begin {gather*} -\frac {c}{d (c+d x) (b c-a d)}-\frac {a \log (a+b x)}{(b c-a d)^2}+\frac {a \log (c+d x)}{(b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x}{(a+b x) (c+d x)^2} \, dx &=\int \left (-\frac {a b}{(b c-a d)^2 (a+b x)}+\frac {c}{(b c-a d) (c+d x)^2}+\frac {a d}{(-b c+a d)^2 (c+d x)}\right ) \, dx\\ &=-\frac {c}{d (b c-a d) (c+d x)}-\frac {a \log (a+b x)}{(b c-a d)^2}+\frac {a \log (c+d x)}{(b c-a d)^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 0.98 \begin {gather*} \frac {c}{d (c+d x) (a d-b c)}-\frac {a \log (a+b x)}{(b c-a d)^2}+\frac {a \log (c+d x)}{(b c-a d)^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 {x}{(a+b x) (c+d x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.72, size = 107, normalized size = 1.75 \begin {gather*} -\frac {b c^{2} - a c d + {\left (a d^{2} x + a c d\right )} \log \left (b x + a\right ) - {\left (a d^{2} x + a c d\right )} \log \left (d x + c\right )}{b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 85, normalized size = 1.39 \begin {gather*} -\frac {\frac {a d^{2} \log \left ({\left | b - \frac {b c}{d x + c} + \frac {a d}{d x + c} \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} + \frac {c d}{{\left (b c d - a d^{2}\right )} {\left (d x + c\right )}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 1.00 \begin {gather*} -\frac {a \ln \left (b x +a \right )}{\left (a d -b c \right )^{2}}+\frac {a \ln \left (d x +c \right )}{\left (a d -b c \right )^{2}}+\frac {c}{\left (a d -b c \right ) \left (d x +c \right ) d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 98, normalized size = 1.61 \begin {gather*} -\frac {a \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac {a \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac {c}{b c^{2} d - a c d^{2} + {\left (b c d^{2} - a d^{3}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 49, normalized size = 0.80 \begin {gather*} \frac {a\,\ln \left (\frac {c+d\,x}{a+b\,x}\right )}{{\left (a\,d-b\,c\right )}^2}+\frac {c}{d\,\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 [B] time = 0.92, size = 238, normalized size = 3.90 \begin {gather*} \frac {a \log {\left (x + \frac {- \frac {a^{4} d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d + \frac {a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{\left (a d - b c\right )^{2}} - \frac {a \log {\left (x + \frac {\frac {a^{4} d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d - \frac {a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{\left (a d - b c\right )^{2}} + \frac {c}{a c d^{2} - b c^{2} d + x \left (a d^{3} - b c d^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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