Optimal. Leaf size=132 \[ -\frac {e^3}{(d+e x) (b d-a e)^4}-\frac {4 b e^3 \log (a+b x)}{(b d-a e)^5}+\frac {4 b e^3 \log (d+e x)}{(b d-a e)^5}-\frac {3 b e^2}{(a+b x) (b d-a e)^4}+\frac {b e}{(a+b x)^2 (b d-a e)^3}-\frac {b}{3 (a+b x)^3 (b d-a e)^2} \]
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Rubi [A] time = 0.10, antiderivative size = 132, 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} \[ -\frac {e^3}{(d+e x) (b d-a e)^4}-\frac {3 b e^2}{(a+b x) (b d-a e)^4}-\frac {4 b e^3 \log (a+b x)}{(b d-a e)^5}+\frac {4 b e^3 \log (d+e x)}{(b d-a e)^5}+\frac {b e}{(a+b x)^2 (b d-a e)^3}-\frac {b}{3 (a+b x)^3 (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a+b x)^4 (d+e x)^2} \, dx\\ &=\int \left (\frac {b^2}{(b d-a e)^2 (a+b x)^4}-\frac {2 b^2 e}{(b d-a e)^3 (a+b x)^3}+\frac {3 b^2 e^2}{(b d-a e)^4 (a+b x)^2}-\frac {4 b^2 e^3}{(b d-a e)^5 (a+b x)}+\frac {e^4}{(b d-a e)^4 (d+e x)^2}+\frac {4 b e^4}{(b d-a e)^5 (d+e x)}\right ) \, dx\\ &=-\frac {b}{3 (b d-a e)^2 (a+b x)^3}+\frac {b e}{(b d-a e)^3 (a+b x)^2}-\frac {3 b e^2}{(b d-a e)^4 (a+b x)}-\frac {e^3}{(b d-a e)^4 (d+e x)}-\frac {4 b e^3 \log (a+b x)}{(b d-a e)^5}+\frac {4 b e^3 \log (d+e x)}{(b d-a e)^5}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 121, normalized size = 0.92 \[ \frac {\frac {3 e^3 (a e-b d)}{d+e x}-\frac {9 b e^2 (b d-a e)}{a+b x}+\frac {3 b e (b d-a e)^2}{(a+b x)^2}-\frac {b (b d-a e)^3}{(a+b x)^3}-12 b e^3 \log (a+b x)+12 b e^3 \log (d+e x)}{3 (b d-a e)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 751, normalized size = 5.69 \[ -\frac {b^{4} d^{4} - 6 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 10 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} - 5 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \, {\left (b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} + 11 \, a^{3} b e^{4}\right )} x + 12 \, {\left (b^{4} e^{4} x^{4} + a^{3} b d e^{3} + {\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 3 \, {\left (a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + {\left (3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right ) - 12 \, {\left (b^{4} e^{4} x^{4} + a^{3} b d e^{3} + {\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 3 \, {\left (a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + {\left (3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (a^{3} b^{5} d^{6} - 5 \, a^{4} b^{4} d^{5} e + 10 \, a^{5} b^{3} d^{4} e^{2} - 10 \, a^{6} b^{2} d^{3} e^{3} + 5 \, a^{7} b d^{2} e^{4} - a^{8} d e^{5} + {\left (b^{8} d^{5} e - 5 \, a b^{7} d^{4} e^{2} + 10 \, a^{2} b^{6} d^{3} e^{3} - 10 \, a^{3} b^{5} d^{2} e^{4} + 5 \, a^{4} b^{4} d e^{5} - a^{5} b^{3} e^{6}\right )} x^{4} + {\left (b^{8} d^{6} - 2 \, a b^{7} d^{5} e - 5 \, a^{2} b^{6} d^{4} e^{2} + 20 \, a^{3} b^{5} d^{3} e^{3} - 25 \, a^{4} b^{4} d^{2} e^{4} + 14 \, a^{5} b^{3} d e^{5} - 3 \, a^{6} b^{2} e^{6}\right )} x^{3} + 3 \, {\left (a b^{7} d^{6} - 4 \, a^{2} b^{6} d^{5} e + 5 \, a^{3} b^{5} d^{4} e^{2} - 5 \, a^{5} b^{3} d^{2} e^{4} + 4 \, a^{6} b^{2} d e^{5} - a^{7} b e^{6}\right )} x^{2} + {\left (3 \, a^{2} b^{6} d^{6} - 14 \, a^{3} b^{5} d^{5} e + 25 \, a^{4} b^{4} d^{4} e^{2} - 20 \, a^{5} b^{3} d^{3} e^{3} + 5 \, a^{6} b^{2} d^{2} e^{4} + 2 \, a^{7} b d e^{5} - a^{8} e^{6}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 279, normalized size = 2.11 \[ -\frac {4 \, b e^{4} \log \left ({\left | b - \frac {b d}{x e + d} + \frac {a e}{x e + d} \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac {e^{7}}{{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} {\left (x e + d\right )}} - \frac {13 \, b^{4} e^{3} - \frac {30 \, {\left (b^{4} d e^{4} - a b^{3} e^{5}\right )} e^{\left (-1\right )}}{x e + d} + \frac {18 \, {\left (b^{4} d^{2} e^{5} - 2 \, a b^{3} d e^{6} + a^{2} b^{2} e^{7}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}}{3 \, {\left (b d - a e\right )}^{5} {\left (b - \frac {b d}{x e + d} + \frac {a e}{x e + d}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 132, normalized size = 1.00 \[ \frac {4 b \,e^{3} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}-\frac {4 b \,e^{3} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}-\frac {3 b \,e^{2}}{\left (a e -b d \right )^{4} \left (b x +a \right )}-\frac {e^{3}}{\left (a e -b d \right )^{4} \left (e x +d \right )}-\frac {b e}{\left (a e -b d \right )^{3} \left (b x +a \right )^{2}}-\frac {b}{3 \left (a e -b d \right )^{2} \left (b x +a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.78, size = 598, normalized size = 4.53 \[ -\frac {4 \, b e^{3} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac {4 \, b e^{3} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac {12 \, b^{3} e^{3} x^{3} + b^{3} d^{3} - 5 \, a b^{2} d^{2} e + 13 \, a^{2} b d e^{2} + 3 \, a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} + 5 \, a b^{2} e^{3}\right )} x^{2} - 2 \, {\left (b^{3} d^{2} e - 8 \, a b^{2} d e^{2} - 11 \, a^{2} b e^{3}\right )} x}{3 \, {\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} + {\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} + {\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \, {\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 534, normalized size = 4.05 \[ \frac {8\,b\,e^3\,\mathrm {atanh}\left (\frac {a^5\,e^5-3\,a^4\,b\,d\,e^4+2\,a^3\,b^2\,d^2\,e^3+2\,a^2\,b^3\,d^3\,e^2-3\,a\,b^4\,d^4\,e+b^5\,d^5}{{\left (a\,e-b\,d\right )}^5}+\frac {2\,b\,e\,x\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^5}\right )}{{\left (a\,e-b\,d\right )}^5}-\frac {\frac {3\,a^3\,e^3+13\,a^2\,b\,d\,e^2-5\,a\,b^2\,d^2\,e+b^3\,d^3}{3\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {4\,b^3\,e^3\,x^3}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+\frac {2\,e^2\,x^2\,\left (d\,b^3+5\,a\,e\,b^2\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+\frac {2\,e\,x\,\left (11\,a^2\,b\,e^2+8\,a\,b^2\,d\,e-b^3\,d^2\right )}{3\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}}{x^3\,\left (d\,b^3+3\,a\,e\,b^2\right )+x^2\,\left (3\,e\,a^2\,b+3\,d\,a\,b^2\right )+a^3\,d+x\,\left (e\,a^3+3\,b\,d\,a^2\right )+b^3\,e\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.48, size = 882, normalized size = 6.68 \[ - \frac {4 b e^{3} \log {\left (x + \frac {- \frac {4 a^{6} b e^{9}}{\left (a e - b d\right )^{5}} + \frac {24 a^{5} b^{2} d e^{8}}{\left (a e - b d\right )^{5}} - \frac {60 a^{4} b^{3} d^{2} e^{7}}{\left (a e - b d\right )^{5}} + \frac {80 a^{3} b^{4} d^{3} e^{6}}{\left (a e - b d\right )^{5}} - \frac {60 a^{2} b^{5} d^{4} e^{5}}{\left (a e - b d\right )^{5}} + \frac {24 a b^{6} d^{5} e^{4}}{\left (a e - b d\right )^{5}} + 4 a b e^{4} - \frac {4 b^{7} d^{6} e^{3}}{\left (a e - b d\right )^{5}} + 4 b^{2} d e^{3}}{8 b^{2} e^{4}} \right )}}{\left (a e - b d\right )^{5}} + \frac {4 b e^{3} \log {\left (x + \frac {\frac {4 a^{6} b e^{9}}{\left (a e - b d\right )^{5}} - \frac {24 a^{5} b^{2} d e^{8}}{\left (a e - b d\right )^{5}} + \frac {60 a^{4} b^{3} d^{2} e^{7}}{\left (a e - b d\right )^{5}} - \frac {80 a^{3} b^{4} d^{3} e^{6}}{\left (a e - b d\right )^{5}} + \frac {60 a^{2} b^{5} d^{4} e^{5}}{\left (a e - b d\right )^{5}} - \frac {24 a b^{6} d^{5} e^{4}}{\left (a e - b d\right )^{5}} + 4 a b e^{4} + \frac {4 b^{7} d^{6} e^{3}}{\left (a e - b d\right )^{5}} + 4 b^{2} d e^{3}}{8 b^{2} e^{4}} \right )}}{\left (a e - b d\right )^{5}} + \frac {- 3 a^{3} e^{3} - 13 a^{2} b d e^{2} + 5 a b^{2} d^{2} e - b^{3} d^{3} - 12 b^{3} e^{3} x^{3} + x^{2} \left (- 30 a b^{2} e^{3} - 6 b^{3} d e^{2}\right ) + x \left (- 22 a^{2} b e^{3} - 16 a b^{2} d e^{2} + 2 b^{3} d^{2} e\right )}{3 a^{7} d e^{4} - 12 a^{6} b d^{2} e^{3} + 18 a^{5} b^{2} d^{3} e^{2} - 12 a^{4} b^{3} d^{4} e + 3 a^{3} b^{4} d^{5} + x^{4} \left (3 a^{4} b^{3} e^{5} - 12 a^{3} b^{4} d e^{4} + 18 a^{2} b^{5} d^{2} e^{3} - 12 a b^{6} d^{3} e^{2} + 3 b^{7} d^{4} e\right ) + x^{3} \left (9 a^{5} b^{2} e^{5} - 33 a^{4} b^{3} d e^{4} + 42 a^{3} b^{4} d^{2} e^{3} - 18 a^{2} b^{5} d^{3} e^{2} - 3 a b^{6} d^{4} e + 3 b^{7} d^{5}\right ) + x^{2} \left (9 a^{6} b e^{5} - 27 a^{5} b^{2} d e^{4} + 18 a^{4} b^{3} d^{2} e^{3} + 18 a^{3} b^{4} d^{3} e^{2} - 27 a^{2} b^{5} d^{4} e + 9 a b^{6} d^{5}\right ) + x \left (3 a^{7} e^{5} - 3 a^{6} b d e^{4} - 18 a^{5} b^{2} d^{2} e^{3} + 42 a^{4} b^{3} d^{3} e^{2} - 33 a^{3} b^{4} d^{4} e + 9 a^{2} b^{5} d^{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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