Optimal. Leaf size=56 \[ \frac {1}{(d+e x) (b d-a e)}+\frac {b \log (a+b x)}{(b d-a e)^2}-\frac {b \log (d+e x)}{(b d-a e)^2} \]
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Rubi [A] time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 44} \begin {gather*} \frac {1}{(d+e x) (b d-a e)}+\frac {b \log (a+b x)}{(b d-a e)^2}-\frac {b \log (d+e x)}{(b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {1}{(a+b x) (d+e x)^2} \, dx\\ &=\int \left (\frac {b^2}{(b d-a e)^2 (a+b x)}-\frac {e}{(b d-a e) (d+e x)^2}-\frac {b e}{(b d-a e)^2 (d+e x)}\right ) \, dx\\ &=\frac {1}{(b d-a e) (d+e x)}+\frac {b \log (a+b x)}{(b d-a e)^2}-\frac {b \log (d+e x)}{(b d-a e)^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.95 \begin {gather*} \frac {b (d+e x) \log (a+b x)-a e-b (d+e x) \log (d+e x)+b d}{(d+e x) (b d-a e)^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 {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 92, normalized size = 1.64 \begin {gather*} \frac {b d - a e + {\left (b e x + b d\right )} \log \left (b x + a\right ) - {\left (b e x + b d\right )} \log \left (e x + d\right )}{b^{2} d^{3} - 2 \, a b d^{2} e + a^{2} d e^{2} + {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 82, normalized size = 1.46 \begin {gather*} \frac {b e \log \left ({\left | b - \frac {b d}{x e + d} + \frac {a e}{x e + d} \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} + \frac {e}{{\left (b d e - a e^{2}\right )} {\left (x e + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 58, normalized size = 1.04 \begin {gather*} \frac {b \ln \left (b x +a \right )}{\left (a e -b d \right )^{2}}-\frac {b \ln \left (e x +d \right )}{\left (a e -b d \right )^{2}}-\frac {1}{\left (a e -b d \right ) \left (e x +d \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 90, normalized size = 1.61 \begin {gather*} \frac {b \log \left (b x + a\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} - \frac {b \log \left (e x + d\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} + \frac {1}{b d^{2} - a d e + {\left (b d e - a e^{2}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.32, size = 77, normalized size = 1.38 \begin {gather*} \frac {2\,b\,\mathrm {atanh}\left (\frac {a^2\,e^2-b^2\,d^2}{{\left (a\,e-b\,d\right )}^2}+\frac {2\,b\,e\,x}{a\,e-b\,d}\right )}{{\left (a\,e-b\,d\right )}^2}-\frac {1}{\left (a\,e-b\,d\right )\,\left (d+e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.71, size = 233, normalized size = 4.16 \begin {gather*} - \frac {b \log {\left (x + \frac {- \frac {a^{3} b e^{3}}{\left (a e - b d\right )^{2}} + \frac {3 a^{2} b^{2} d e^{2}}{\left (a e - b d\right )^{2}} - \frac {3 a b^{3} d^{2} e}{\left (a e - b d\right )^{2}} + a b e + \frac {b^{4} d^{3}}{\left (a e - b d\right )^{2}} + b^{2} d}{2 b^{2} e} \right )}}{\left (a e - b d\right )^{2}} + \frac {b \log {\left (x + \frac {\frac {a^{3} b e^{3}}{\left (a e - b d\right )^{2}} - \frac {3 a^{2} b^{2} d e^{2}}{\left (a e - b d\right )^{2}} + \frac {3 a b^{3} d^{2} e}{\left (a e - b d\right )^{2}} + a b e - \frac {b^{4} d^{3}}{\left (a e - b d\right )^{2}} + b^{2} d}{2 b^{2} e} \right )}}{\left (a e - b d\right )^{2}} - \frac {1}{a d e - b d^{2} + x \left (a e^{2} - b d e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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