Optimal. Leaf size=82 \[ \frac {e^2 \log (a+b x)}{(b d-a e)^3}-\frac {e^2 \log (d+e x)}{(b d-a e)^3}+\frac {e}{(a+b x) (b d-a e)^2}-\frac {1}{2 (a+b x)^2 (b d-a e)} \]
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Rubi [A] time = 0.05, antiderivative size = 82, 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} \[ \frac {e^2 \log (a+b x)}{(b d-a e)^3}-\frac {e^2 \log (d+e x)}{(b d-a e)^3}+\frac {e}{(a+b x) (b d-a e)^2}-\frac {1}{2 (a+b x)^2 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a+b x)^3 (d+e x)} \, dx\\ &=\int \left (\frac {b}{(b d-a e) (a+b x)^3}-\frac {b e}{(b d-a e)^2 (a+b x)^2}+\frac {b e^2}{(b d-a e)^3 (a+b x)}-\frac {e^3}{(b d-a e)^3 (d+e x)}\right ) \, dx\\ &=-\frac {1}{2 (b d-a e) (a+b x)^2}+\frac {e}{(b d-a e)^2 (a+b x)}+\frac {e^2 \log (a+b x)}{(b d-a e)^3}-\frac {e^2 \log (d+e x)}{(b d-a e)^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 67, normalized size = 0.82 \[ \frac {\frac {(b d-a e) (3 a e-b d+2 b e x)}{(a+b x)^2}+2 e^2 \log (a+b x)-2 e^2 \log (d+e x)}{2 (b d-a e)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.07, size = 242, normalized size = 2.95 \[ -\frac {b^{2} d^{2} - 4 \, a b d e + 3 \, a^{2} e^{2} - 2 \, {\left (b^{2} d e - a b e^{2}\right )} x - 2 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{3} d^{3} - 3 \, a^{3} b^{2} d^{2} e + 3 \, a^{4} b d e^{2} - a^{5} e^{3} + {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} x^{2} + 2 \, {\left (a b^{4} d^{3} - 3 \, a^{2} b^{3} d^{2} e + 3 \, a^{3} b^{2} d e^{2} - a^{4} b e^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 162, normalized size = 1.98 \[ \frac {b e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} - \frac {e^{3} \log \left ({\left | x e + d \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} - \frac {b^{2} d^{2} - 4 \, a b d e + 3 \, a^{2} e^{2} - 2 \, {\left (b^{2} d e - a b e^{2}\right )} x}{2 \, {\left (b d - a e\right )}^{3} {\left (b x + a\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 81, normalized size = 0.99 \[ -\frac {e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{3}}+\frac {e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{3}}+\frac {e}{\left (a e -b d \right )^{2} \left (b x +a \right )}+\frac {1}{2 \left (a e -b d \right ) \left (b x +a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 202, normalized size = 2.46 \[ \frac {e^{2} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac {e^{2} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} + \frac {2 \, b e x - b d + 3 \, a e}{2 \, {\left (a^{2} b^{2} d^{2} - 2 \, a^{3} b d e + a^{4} e^{2} + {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.16, size = 182, normalized size = 2.22 \[ \frac {\frac {3\,a\,e-b\,d}{2\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}+\frac {b\,e\,x}{a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}}{a^2+2\,a\,b\,x+b^2\,x^2}-\frac {2\,e^2\,\mathrm {atanh}\left (\frac {a^3\,e^3-a^2\,b\,d\,e^2-a\,b^2\,d^2\,e+b^3\,d^3}{{\left (a\,e-b\,d\right )}^3}+\frac {2\,b\,e\,x\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{{\left (a\,e-b\,d\right )}^3}\right )}{{\left (a\,e-b\,d\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.09, size = 381, normalized size = 4.65 \[ \frac {e^{2} \log {\left (x + \frac {- \frac {a^{4} e^{6}}{\left (a e - b d\right )^{3}} + \frac {4 a^{3} b d e^{5}}{\left (a e - b d\right )^{3}} - \frac {6 a^{2} b^{2} d^{2} e^{4}}{\left (a e - b d\right )^{3}} + \frac {4 a b^{3} d^{3} e^{3}}{\left (a e - b d\right )^{3}} + a e^{3} - \frac {b^{4} d^{4} e^{2}}{\left (a e - b d\right )^{3}} + b d e^{2}}{2 b e^{3}} \right )}}{\left (a e - b d\right )^{3}} - \frac {e^{2} \log {\left (x + \frac {\frac {a^{4} e^{6}}{\left (a e - b d\right )^{3}} - \frac {4 a^{3} b d e^{5}}{\left (a e - b d\right )^{3}} + \frac {6 a^{2} b^{2} d^{2} e^{4}}{\left (a e - b d\right )^{3}} - \frac {4 a b^{3} d^{3} e^{3}}{\left (a e - b d\right )^{3}} + a e^{3} + \frac {b^{4} d^{4} e^{2}}{\left (a e - b d\right )^{3}} + b d e^{2}}{2 b e^{3}} \right )}}{\left (a e - b d\right )^{3}} + \frac {3 a e - b d + 2 b e x}{2 a^{4} e^{2} - 4 a^{3} b d e + 2 a^{2} b^{2} d^{2} + x^{2} \left (2 a^{2} b^{2} e^{2} - 4 a b^{3} d e + 2 b^{4} d^{2}\right ) + x \left (4 a^{3} b e^{2} - 8 a^{2} b^{2} d e + 4 a b^{3} d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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