Optimal. Leaf size=107 \[ -\frac {e^3 \log (a+b x)}{(b d-a e)^4}+\frac {e^3 \log (d+e x)}{(b d-a e)^4}-\frac {e^2}{(a+b x) (b d-a e)^3}+\frac {e}{2 (a+b x)^2 (b d-a e)^2}-\frac {1}{3 (a+b x)^3 (b d-a e)} \]
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Rubi [A] time = 0.07, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 44} \[ -\frac {e^2}{(a+b x) (b d-a e)^3}-\frac {e^3 \log (a+b x)}{(b d-a e)^4}+\frac {e^3 \log (d+e x)}{(b d-a e)^4}+\frac {e}{2 (a+b x)^2 (b d-a e)^2}-\frac {1}{3 (a+b x)^3 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a+b x)^4 (d+e x)} \, dx\\ &=\int \left (\frac {b}{(b d-a e) (a+b x)^4}-\frac {b e}{(b d-a e)^2 (a+b x)^3}+\frac {b e^2}{(b d-a e)^3 (a+b x)^2}-\frac {b e^3}{(b d-a e)^4 (a+b x)}+\frac {e^4}{(b d-a e)^4 (d+e x)}\right ) \, dx\\ &=-\frac {1}{3 (b d-a e) (a+b x)^3}+\frac {e}{2 (b d-a e)^2 (a+b x)^2}-\frac {e^2}{(b d-a e)^3 (a+b x)}-\frac {e^3 \log (a+b x)}{(b d-a e)^4}+\frac {e^3 \log (d+e x)}{(b d-a e)^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 107, normalized size = 1.00 \[ -\frac {e^3 \log (a+b x)}{(b d-a e)^4}+\frac {e^3 \log (d+e x)}{(b d-a e)^4}-\frac {e^2}{(a+b x) (b d-a e)^3}+\frac {e}{2 (a+b x)^2 (b d-a e)^2}+\frac {1}{3 (a+b x)^3 (a e-b d)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 425, normalized size = 3.97 \[ -\frac {2 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \, {\left (b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 5 \, a^{2} b e^{3}\right )} x + 6 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (e x + d\right )}{6 \, {\left (a^{3} b^{4} d^{4} - 4 \, a^{4} b^{3} d^{3} e + 6 \, a^{5} b^{2} d^{2} e^{2} - 4 \, a^{6} b d e^{3} + a^{7} e^{4} + {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} x^{3} + 3 \, {\left (a b^{6} d^{4} - 4 \, a^{2} b^{5} d^{3} e + 6 \, a^{3} b^{4} d^{2} e^{2} - 4 \, a^{4} b^{3} d e^{3} + a^{5} b^{2} e^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} d^{4} - 4 \, a^{3} b^{4} d^{3} e + 6 \, a^{4} b^{3} d^{2} e^{2} - 4 \, a^{5} b^{2} d e^{3} + a^{6} b e^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 234, normalized size = 2.19 \[ -\frac {b e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac {e^{4} \log \left ({\left | x e + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac {2 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \, {\left (b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 5 \, a^{2} b e^{3}\right )} x}{6 \, {\left (b d - a e\right )}^{4} {\left (b x + a\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 103, normalized size = 0.96 \[ -\frac {e^{3} \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}+\frac {e^{3} \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}+\frac {e^{2}}{\left (a e -b d \right )^{3} \left (b x +a \right )}+\frac {e}{2 \left (a e -b d \right )^{2} \left (b x +a \right )^{2}}+\frac {1}{3 \left (a e -b d \right ) \left (b x +a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.54, size = 361, normalized size = 3.37 \[ -\frac {e^{3} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac {e^{3} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac {6 \, b^{2} e^{2} x^{2} + 2 \, b^{2} d^{2} - 7 \, a b d e + 11 \, a^{2} e^{2} - 3 \, {\left (b^{2} d e - 5 \, a b e^{2}\right )} x}{6 \, {\left (a^{3} b^{3} d^{3} - 3 \, a^{4} b^{2} d^{2} e + 3 \, a^{5} b d e^{2} - a^{6} e^{3} + {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{3} + 3 \, {\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 312, normalized size = 2.92 \[ \frac {\frac {11\,a^2\,e^2-7\,a\,b\,d\,e+2\,b^2\,d^2}{6\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}-\frac {e\,x\,\left (b^2\,d-5\,a\,b\,e\right )}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {b^2\,e^2\,x^2}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3}-\frac {2\,e^3\,\mathrm {atanh}\left (\frac {a^4\,e^4-2\,a^3\,b\,d\,e^3+2\,a\,b^3\,d^3\,e-b^4\,d^4}{{\left (a\,e-b\,d\right )}^4}+\frac {2\,b\,e\,x\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^4}\right )}{{\left (a\,e-b\,d\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.58, size = 570, normalized size = 5.33 \[ \frac {e^{3} \log {\left (x + \frac {- \frac {a^{5} e^{8}}{\left (a e - b d\right )^{4}} + \frac {5 a^{4} b d e^{7}}{\left (a e - b d\right )^{4}} - \frac {10 a^{3} b^{2} d^{2} e^{6}}{\left (a e - b d\right )^{4}} + \frac {10 a^{2} b^{3} d^{3} e^{5}}{\left (a e - b d\right )^{4}} - \frac {5 a b^{4} d^{4} e^{4}}{\left (a e - b d\right )^{4}} + a e^{4} + \frac {b^{5} d^{5} e^{3}}{\left (a e - b d\right )^{4}} + b d e^{3}}{2 b e^{4}} \right )}}{\left (a e - b d\right )^{4}} - \frac {e^{3} \log {\left (x + \frac {\frac {a^{5} e^{8}}{\left (a e - b d\right )^{4}} - \frac {5 a^{4} b d e^{7}}{\left (a e - b d\right )^{4}} + \frac {10 a^{3} b^{2} d^{2} e^{6}}{\left (a e - b d\right )^{4}} - \frac {10 a^{2} b^{3} d^{3} e^{5}}{\left (a e - b d\right )^{4}} + \frac {5 a b^{4} d^{4} e^{4}}{\left (a e - b d\right )^{4}} + a e^{4} - \frac {b^{5} d^{5} e^{3}}{\left (a e - b d\right )^{4}} + b d e^{3}}{2 b e^{4}} \right )}}{\left (a e - b d\right )^{4}} + \frac {11 a^{2} e^{2} - 7 a b d e + 2 b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (15 a b e^{2} - 3 b^{2} d e\right )}{6 a^{6} e^{3} - 18 a^{5} b d e^{2} + 18 a^{4} b^{2} d^{2} e - 6 a^{3} b^{3} d^{3} + x^{3} \left (6 a^{3} b^{3} e^{3} - 18 a^{2} b^{4} d e^{2} + 18 a b^{5} d^{2} e - 6 b^{6} d^{3}\right ) + x^{2} \left (18 a^{4} b^{2} e^{3} - 54 a^{3} b^{3} d e^{2} + 54 a^{2} b^{4} d^{2} e - 18 a b^{5} d^{3}\right ) + x \left (18 a^{5} b e^{3} - 54 a^{4} b^{2} d e^{2} + 54 a^{3} b^{3} d^{2} e - 18 a^{2} b^{4} d^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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