Optimal. Leaf size=82 \[ \frac {b^2 \log (a+b x)}{(b d-a e)^3}-\frac {b^2 \log (d+e x)}{(b d-a e)^3}+\frac {b}{(d+e x) (b d-a e)^2}+\frac {1}{2 (d+e x)^2 (b d-a e)} \]
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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} \[ \frac {b^2 \log (a+b x)}{(b d-a e)^3}-\frac {b^2 \log (d+e x)}{(b d-a e)^3}+\frac {b}{(d+e x) (b d-a e)^2}+\frac {1}{2 (d+e x)^2 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {1}{(a+b x) (d+e x)^3} \, dx\\ &=\int \left (\frac {b^3}{(b d-a e)^3 (a+b x)}-\frac {e}{(b d-a e) (d+e x)^3}-\frac {b e}{(b d-a e)^2 (d+e x)^2}-\frac {b^2 e}{(b d-a e)^3 (d+e x)}\right ) \, dx\\ &=\frac {1}{2 (b d-a e) (d+e x)^2}+\frac {b}{(b d-a e)^2 (d+e x)}+\frac {b^2 \log (a+b x)}{(b d-a e)^3}-\frac {b^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 \[ \frac {2 b^2 \log (a+b x)+\frac {(b d-a e) (-a e+3 b d+2 b e x)}{(d+e x)^2}-2 b^2 \log (d+e x)}{2 (b d-a e)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.24, size = 242, normalized size = 2.95 \[ \frac {3 \, b^{2} d^{2} - 4 \, a b d e + a^{2} e^{2} + 2 \, {\left (b^{2} d e - a b e^{2}\right )} x + 2 \, {\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (e x + d\right )}{2 \, {\left (b^{3} d^{5} - 3 \, a b^{2} d^{4} e + 3 \, a^{2} b d^{3} e^{2} - a^{3} d^{2} e^{3} + {\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} x^{2} + 2 \, {\left (b^{3} d^{4} e - 3 \, a b^{2} d^{3} e^{2} + 3 \, a^{2} b d^{2} e^{3} - a^{3} d e^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 166, normalized size = 2.02 \[ \frac {b^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} - \frac {b^{2} e \log \left ({\left | x e + d \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} + \frac {3 \, b^{2} d^{2} - 4 \, a b d e + a^{2} e^{2} + 2 \, {\left (b^{2} d e - a b e^{2}\right )} x}{2 \, {\left (b d - a e\right )}^{3} {\left (x e + d\right )}^{2}} \]
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 \[ -\frac {b^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{3}}+\frac {b^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{3}}+\frac {b}{\left (a e -b d \right )^{2} \left (e x +d \right )}-\frac {1}{2 \left (a e -b d \right ) \left (e x +d \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 202, normalized size = 2.46 \[ \frac {b^{2} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac {b^{2} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} + \frac {2 \, b e x + 3 \, b d - a e}{2 \, {\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.26, size = 183, normalized size = 2.23 \[ -\frac {\frac {a\,e-3\,b\,d}{2\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}-\frac {b\,e\,x}{a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}}{d^2+2\,d\,e\,x+e^2\,x^2}-\frac {2\,b^2\,\mathrm {atanh}\left (\frac {a^3\,e^3-a^2\,b\,d\,e^2-a\,b^2\,d^2\,e+b^3\,d^3}{{\left (a\,e-b\,d\right )}^3}+\frac {2\,b\,e\,x\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{{\left (a\,e-b\,d\right )}^3}\right )}{{\left (a\,e-b\,d\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.08, size = 381, normalized size = 4.65 \[ \frac {b^{2} \log {\left (x + \frac {- \frac {a^{4} b^{2} e^{4}}{\left (a e - b d\right )^{3}} + \frac {4 a^{3} b^{3} d e^{3}}{\left (a e - b d\right )^{3}} - \frac {6 a^{2} b^{4} d^{2} e^{2}}{\left (a e - b d\right )^{3}} + \frac {4 a b^{5} d^{3} e}{\left (a e - b d\right )^{3}} + a b^{2} e - \frac {b^{6} d^{4}}{\left (a e - b d\right )^{3}} + b^{3} d}{2 b^{3} e} \right )}}{\left (a e - b d\right )^{3}} - \frac {b^{2} \log {\left (x + \frac {\frac {a^{4} b^{2} e^{4}}{\left (a e - b d\right )^{3}} - \frac {4 a^{3} b^{3} d e^{3}}{\left (a e - b d\right )^{3}} + \frac {6 a^{2} b^{4} d^{2} e^{2}}{\left (a e - b d\right )^{3}} - \frac {4 a b^{5} d^{3} e}{\left (a e - b d\right )^{3}} + a b^{2} e + \frac {b^{6} d^{4}}{\left (a e - b d\right )^{3}} + b^{3} d}{2 b^{3} e} \right )}}{\left (a e - b d\right )^{3}} + \frac {- a e + 3 b d + 2 b e x}{2 a^{2} d^{2} e^{2} - 4 a b d^{3} e + 2 b^{2} d^{4} + x^{2} \left (2 a^{2} e^{4} - 4 a b d e^{3} + 2 b^{2} d^{2} e^{2}\right ) + x \left (4 a^{2} d e^{3} - 8 a b d^{2} e^{2} + 4 b^{2} d^{3} e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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