Optimal. Leaf size=181 \[ -\frac {e^5}{(d+e x) (b d-a e)^6}-\frac {6 b e^5 \log (a+b x)}{(b d-a e)^7}+\frac {6 b e^5 \log (d+e x)}{(b d-a e)^7}-\frac {5 b e^4}{(a+b x) (b d-a e)^6}+\frac {2 b e^3}{(a+b x)^2 (b d-a e)^5}-\frac {b e^2}{(a+b x)^3 (b d-a e)^4}+\frac {b e}{2 (a+b x)^4 (b d-a e)^3}-\frac {b}{5 (a+b x)^5 (b d-a e)^2} \]
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Rubi [A] time = 0.18, antiderivative size = 181, 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} \begin {gather*} -\frac {e^5}{(d+e x) (b d-a e)^6}-\frac {5 b e^4}{(a+b x) (b d-a e)^6}+\frac {2 b e^3}{(a+b x)^2 (b d-a e)^5}-\frac {b e^2}{(a+b x)^3 (b d-a e)^4}-\frac {6 b e^5 \log (a+b x)}{(b d-a e)^7}+\frac {6 b e^5 \log (d+e x)}{(b d-a e)^7}+\frac {b e}{2 (a+b x)^4 (b d-a e)^3}-\frac {b}{5 (a+b x)^5 (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 {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^6 (d+e x)^2} \, dx\\ &=\int \left (\frac {b^2}{(b d-a e)^2 (a+b x)^6}-\frac {2 b^2 e}{(b d-a e)^3 (a+b x)^5}+\frac {3 b^2 e^2}{(b d-a e)^4 (a+b x)^4}-\frac {4 b^2 e^3}{(b d-a e)^5 (a+b x)^3}+\frac {5 b^2 e^4}{(b d-a e)^6 (a+b x)^2}-\frac {6 b^2 e^5}{(b d-a e)^7 (a+b x)}+\frac {e^6}{(b d-a e)^6 (d+e x)^2}+\frac {6 b e^6}{(b d-a e)^7 (d+e x)}\right ) \, dx\\ &=-\frac {b}{5 (b d-a e)^2 (a+b x)^5}+\frac {b e}{2 (b d-a e)^3 (a+b x)^4}-\frac {b e^2}{(b d-a e)^4 (a+b x)^3}+\frac {2 b e^3}{(b d-a e)^5 (a+b x)^2}-\frac {5 b e^4}{(b d-a e)^6 (a+b x)}-\frac {e^5}{(b d-a e)^6 (d+e x)}-\frac {6 b e^5 \log (a+b x)}{(b d-a e)^7}+\frac {6 b e^5 \log (d+e x)}{(b d-a e)^7}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 167, normalized size = 0.92 \begin {gather*} \frac {\frac {10 e^5 (a e-b d)}{d+e x}-\frac {50 b e^4 (b d-a e)}{a+b x}+\frac {20 b e^3 (b d-a e)^2}{(a+b x)^2}-\frac {10 b e^2 (b d-a e)^3}{(a+b x)^3}+\frac {5 b e (b d-a e)^4}{(a+b x)^4}-\frac {2 b (b d-a e)^5}{(a+b x)^5}-60 b e^5 \log (a+b x)+60 b e^5 \log (d+e x)}{10 (b d-a e)^7} \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 {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.44, size = 1420, normalized size = 7.85
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 451, normalized size = 2.49 \begin {gather*} -\frac {6 \, b e^{6} \log \left ({\left | b - \frac {b d}{x e + d} + \frac {a e}{x e + d} \right |}\right )}{b^{7} d^{7} e - 7 \, a b^{6} d^{6} e^{2} + 21 \, a^{2} b^{5} d^{5} e^{3} - 35 \, a^{3} b^{4} d^{4} e^{4} + 35 \, a^{4} b^{3} d^{3} e^{5} - 21 \, a^{5} b^{2} d^{2} e^{6} + 7 \, a^{6} b d e^{7} - a^{7} e^{8}} - \frac {e^{11}}{{\left (b^{6} d^{6} e^{6} - 6 \, a b^{5} d^{5} e^{7} + 15 \, a^{2} b^{4} d^{4} e^{8} - 20 \, a^{3} b^{3} d^{3} e^{9} + 15 \, a^{4} b^{2} d^{2} e^{10} - 6 \, a^{5} b d e^{11} + a^{6} e^{12}\right )} {\left (x e + d\right )}} - \frac {87 \, b^{6} e^{5} - \frac {385 \, {\left (b^{6} d e^{6} - a b^{5} e^{7}\right )} e^{\left (-1\right )}}{x e + d} + \frac {650 \, {\left (b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {500 \, {\left (b^{6} d^{3} e^{8} - 3 \, a b^{5} d^{2} e^{9} + 3 \, a^{2} b^{4} d e^{10} - a^{3} b^{3} e^{11}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {150 \, {\left (b^{6} d^{4} e^{9} - 4 \, a b^{5} d^{3} e^{10} + 6 \, a^{2} b^{4} d^{2} e^{11} - 4 \, a^{3} b^{3} d e^{12} + a^{4} b^{2} e^{13}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}}{10 \, {\left (b d - a e\right )}^{7} {\left (b - \frac {b d}{x e + d} + \frac {a e}{x e + d}\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 178, normalized size = 0.98 \begin {gather*} \frac {6 b \,e^{5} \ln \left (b x +a \right )}{\left (a e -b d \right )^{7}}-\frac {6 b \,e^{5} \ln \left (e x +d \right )}{\left (a e -b d \right )^{7}}-\frac {5 b \,e^{4}}{\left (a e -b d \right )^{6} \left (b x +a \right )}-\frac {e^{5}}{\left (a e -b d \right )^{6} \left (e x +d \right )}-\frac {2 b \,e^{3}}{\left (a e -b d \right )^{5} \left (b x +a \right )^{2}}-\frac {b \,e^{2}}{\left (a e -b d \right )^{4} \left (b x +a \right )^{3}}-\frac {b e}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{4}}-\frac {b}{5 \left (a e -b d \right )^{2} \left (b x +a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.56, size = 1154, normalized size = 6.38 \begin {gather*} -\frac {6 \, b e^{5} \log \left (b x + a\right )}{b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}} + \frac {6 \, b e^{5} \log \left (e x + d\right )}{b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}} - \frac {60 \, b^{5} e^{5} x^{5} + 2 \, b^{5} d^{5} - 13 \, a b^{4} d^{4} e + 37 \, a^{2} b^{3} d^{3} e^{2} - 63 \, a^{3} b^{2} d^{2} e^{3} + 87 \, a^{4} b d e^{4} + 10 \, a^{5} e^{5} + 30 \, {\left (b^{5} d e^{4} + 9 \, a b^{4} e^{5}\right )} x^{4} - 10 \, {\left (b^{5} d^{2} e^{3} - 14 \, a b^{4} d e^{4} - 47 \, a^{2} b^{3} e^{5}\right )} x^{3} + 5 \, {\left (b^{5} d^{3} e^{2} - 9 \, a b^{4} d^{2} e^{3} + 51 \, a^{2} b^{3} d e^{4} + 77 \, a^{3} b^{2} e^{5}\right )} x^{2} - {\left (3 \, b^{5} d^{4} e - 22 \, a b^{4} d^{3} e^{2} + 78 \, a^{2} b^{3} d^{2} e^{3} - 222 \, a^{3} b^{2} d e^{4} - 137 \, a^{4} b e^{5}\right )} x}{10 \, {\left (a^{5} b^{6} d^{7} - 6 \, a^{6} b^{5} d^{6} e + 15 \, a^{7} b^{4} d^{5} e^{2} - 20 \, a^{8} b^{3} d^{4} e^{3} + 15 \, a^{9} b^{2} d^{3} e^{4} - 6 \, a^{10} b d^{2} e^{5} + a^{11} d e^{6} + {\left (b^{11} d^{6} e - 6 \, a b^{10} d^{5} e^{2} + 15 \, a^{2} b^{9} d^{4} e^{3} - 20 \, a^{3} b^{8} d^{3} e^{4} + 15 \, a^{4} b^{7} d^{2} e^{5} - 6 \, a^{5} b^{6} d e^{6} + a^{6} b^{5} e^{7}\right )} x^{6} + {\left (b^{11} d^{7} - a b^{10} d^{6} e - 15 \, a^{2} b^{9} d^{5} e^{2} + 55 \, a^{3} b^{8} d^{4} e^{3} - 85 \, a^{4} b^{7} d^{3} e^{4} + 69 \, a^{5} b^{6} d^{2} e^{5} - 29 \, a^{6} b^{5} d e^{6} + 5 \, a^{7} b^{4} e^{7}\right )} x^{5} + 5 \, {\left (a b^{10} d^{7} - 4 \, a^{2} b^{9} d^{6} e + 3 \, a^{3} b^{8} d^{5} e^{2} + 10 \, a^{4} b^{7} d^{4} e^{3} - 25 \, a^{5} b^{6} d^{3} e^{4} + 24 \, a^{6} b^{5} d^{2} e^{5} - 11 \, a^{7} b^{4} d e^{6} + 2 \, a^{8} b^{3} e^{7}\right )} x^{4} + 10 \, {\left (a^{2} b^{9} d^{7} - 5 \, a^{3} b^{8} d^{6} e + 9 \, a^{4} b^{7} d^{5} e^{2} - 5 \, a^{5} b^{6} d^{4} e^{3} - 5 \, a^{6} b^{5} d^{3} e^{4} + 9 \, a^{7} b^{4} d^{2} e^{5} - 5 \, a^{8} b^{3} d e^{6} + a^{9} b^{2} e^{7}\right )} x^{3} + 5 \, {\left (2 \, a^{3} b^{8} d^{7} - 11 \, a^{4} b^{7} d^{6} e + 24 \, a^{5} b^{6} d^{5} e^{2} - 25 \, a^{6} b^{5} d^{4} e^{3} + 10 \, a^{7} b^{4} d^{3} e^{4} + 3 \, a^{8} b^{3} d^{2} e^{5} - 4 \, a^{9} b^{2} d e^{6} + a^{10} b e^{7}\right )} x^{2} + {\left (5 \, a^{4} b^{7} d^{7} - 29 \, a^{5} b^{6} d^{6} e + 69 \, a^{6} b^{5} d^{5} e^{2} - 85 \, a^{7} b^{4} d^{4} e^{3} + 55 \, a^{8} b^{3} d^{3} e^{4} - 15 \, a^{9} b^{2} d^{2} e^{5} - a^{10} b d e^{6} + a^{11} e^{7}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 1047, normalized size = 5.78 \begin {gather*} \frac {12\,b\,e^5\,\mathrm {atanh}\left (\frac {a^7\,e^7-5\,a^6\,b\,d\,e^6+9\,a^5\,b^2\,d^2\,e^5-5\,a^4\,b^3\,d^3\,e^4-5\,a^3\,b^4\,d^4\,e^3+9\,a^2\,b^5\,d^5\,e^2-5\,a\,b^6\,d^6\,e+b^7\,d^7}{{\left (a\,e-b\,d\right )}^7}+\frac {2\,b\,e\,x\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^7}\right )}{{\left (a\,e-b\,d\right )}^7}-\frac {\frac {10\,a^5\,e^5+87\,a^4\,b\,d\,e^4-63\,a^3\,b^2\,d^2\,e^3+37\,a^2\,b^3\,d^3\,e^2-13\,a\,b^4\,d^4\,e+2\,b^5\,d^5}{10\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}+\frac {e^3\,x^3\,\left (47\,a^2\,b^3\,e^2+14\,a\,b^4\,d\,e-b^5\,d^2\right )}{a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6}+\frac {e^2\,x^2\,\left (77\,a^3\,b^2\,e^3+51\,a^2\,b^3\,d\,e^2-9\,a\,b^4\,d^2\,e+b^5\,d^3\right )}{2\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}+\frac {6\,b^5\,e^5\,x^5}{a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6}+\frac {3\,e^4\,x^4\,\left (d\,b^5+9\,a\,e\,b^4\right )}{a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6}+\frac {e\,x\,\left (137\,a^4\,b\,e^4+222\,a^3\,b^2\,d\,e^3-78\,a^2\,b^3\,d^2\,e^2+22\,a\,b^4\,d^3\,e-3\,b^5\,d^4\right )}{10\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}}{x^5\,\left (d\,b^5+5\,a\,e\,b^4\right )+x^3\,\left (10\,e\,a^3\,b^2+10\,d\,a^2\,b^3\right )+a^5\,d+x\,\left (e\,a^5+5\,b\,d\,a^4\right )+x^2\,\left (5\,e\,a^4\,b+10\,d\,a^3\,b^2\right )+x^4\,\left (10\,e\,a^2\,b^3+5\,d\,a\,b^4\right )+b^5\,e\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.18, size = 1518, normalized size = 8.39
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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