Optimal. Leaf size=155 \[ \frac {6 e^5 (b d-a e) \log (a+b x)}{b^7}-\frac {15 e^4 (b d-a e)^2}{b^7 (a+b x)}-\frac {10 e^3 (b d-a e)^3}{b^7 (a+b x)^2}-\frac {5 e^2 (b d-a e)^4}{b^7 (a+b x)^3}-\frac {3 e (b d-a e)^5}{2 b^7 (a+b x)^4}-\frac {(b d-a e)^6}{5 b^7 (a+b x)^5}+\frac {e^6 x}{b^6} \]
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Rubi [A] time = 0.17, antiderivative size = 155, 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, 43} \[ -\frac {15 e^4 (b d-a e)^2}{b^7 (a+b x)}-\frac {10 e^3 (b d-a e)^3}{b^7 (a+b x)^2}-\frac {5 e^2 (b d-a e)^4}{b^7 (a+b x)^3}+\frac {6 e^5 (b d-a e) \log (a+b x)}{b^7}-\frac {3 e (b d-a e)^5}{2 b^7 (a+b x)^4}-\frac {(b d-a e)^6}{5 b^7 (a+b x)^5}+\frac {e^6 x}{b^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^6}{(a+b x)^6} \, dx\\ &=\int \left (\frac {e^6}{b^6}+\frac {(b d-a e)^6}{b^6 (a+b x)^6}+\frac {6 e (b d-a e)^5}{b^6 (a+b x)^5}+\frac {15 e^2 (b d-a e)^4}{b^6 (a+b x)^4}+\frac {20 e^3 (b d-a e)^3}{b^6 (a+b x)^3}+\frac {15 e^4 (b d-a e)^2}{b^6 (a+b x)^2}+\frac {6 e^5 (b d-a e)}{b^6 (a+b x)}\right ) \, dx\\ &=\frac {e^6 x}{b^6}-\frac {(b d-a e)^6}{5 b^7 (a+b x)^5}-\frac {3 e (b d-a e)^5}{2 b^7 (a+b x)^4}-\frac {5 e^2 (b d-a e)^4}{b^7 (a+b x)^3}-\frac {10 e^3 (b d-a e)^3}{b^7 (a+b x)^2}-\frac {15 e^4 (b d-a e)^2}{b^7 (a+b x)}+\frac {6 e^5 (b d-a e) \log (a+b x)}{b^7}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 300, normalized size = 1.94 \[ -\frac {87 a^6 e^6+a^5 b e^5 (375 e x-137 d)+5 a^4 b^2 e^4 \left (6 d^2-125 d e x+120 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+15 d^2 e x-110 d e^2 x^2+40 e^3 x^3\right )+5 a^2 b^4 e^2 \left (d^4+10 d^3 e x+60 d^2 e^2 x^2-180 d e^3 x^3+10 e^4 x^4\right )+a b^5 e \left (3 d^5+25 d^4 e x+100 d^3 e^2 x^2+300 d^2 e^3 x^3-300 d e^4 x^4-50 e^5 x^5\right )+60 e^5 (a+b x)^5 (a e-b d) \log (a+b x)+b^6 \left (2 d^6+15 d^5 e x+50 d^4 e^2 x^2+100 d^3 e^3 x^3+150 d^2 e^4 x^4-10 e^6 x^6\right )}{10 b^7 (a+b x)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 542, normalized size = 3.50 \[ \frac {10 \, b^{6} e^{6} x^{6} + 50 \, a b^{5} e^{6} x^{5} - 2 \, b^{6} d^{6} - 3 \, a b^{5} d^{5} e - 5 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} - 30 \, a^{4} b^{2} d^{2} e^{4} + 137 \, a^{5} b d e^{5} - 87 \, a^{6} e^{6} - 50 \, {\left (3 \, b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} - 100 \, {\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} - 9 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} - 50 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 22 \, a^{3} b^{3} d e^{5} + 12 \, a^{4} b^{2} e^{6}\right )} x^{2} - 5 \, {\left (3 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 30 \, a^{3} b^{3} d^{2} e^{4} - 125 \, a^{4} b^{2} d e^{5} + 75 \, a^{5} b e^{6}\right )} x + 60 \, {\left (a^{5} b d e^{5} - a^{6} e^{6} + {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 10 \, {\left (a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 10 \, {\left (a^{3} b^{3} d e^{5} - a^{4} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} \log \left (b x + a\right )}{10 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 328, normalized size = 2.12 \[ \frac {x e^{6}}{b^{6}} + \frac {6 \, {\left (b d e^{5} - a e^{6}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {2 \, b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 5 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 30 \, a^{4} b^{2} d^{2} e^{4} - 137 \, a^{5} b d e^{5} + 87 \, a^{6} e^{6} + 150 \, {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \, {\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} - 9 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 50 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 22 \, a^{3} b^{3} d e^{5} + 13 \, a^{4} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (3 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 30 \, a^{3} b^{3} d^{2} e^{4} - 125 \, a^{4} b^{2} d e^{5} + 77 \, a^{5} b e^{6}\right )} x}{10 \, {\left (b x + a\right )}^{5} b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 508, normalized size = 3.28 \[ -\frac {a^{6} e^{6}}{5 \left (b x +a \right )^{5} b^{7}}+\frac {6 a^{5} d \,e^{5}}{5 \left (b x +a \right )^{5} b^{6}}-\frac {3 a^{4} d^{2} e^{4}}{\left (b x +a \right )^{5} b^{5}}+\frac {4 a^{3} d^{3} e^{3}}{\left (b x +a \right )^{5} b^{4}}-\frac {3 a^{2} d^{4} e^{2}}{\left (b x +a \right )^{5} b^{3}}+\frac {6 a \,d^{5} e}{5 \left (b x +a \right )^{5} b^{2}}-\frac {d^{6}}{5 \left (b x +a \right )^{5} b}+\frac {3 a^{5} e^{6}}{2 \left (b x +a \right )^{4} b^{7}}-\frac {15 a^{4} d \,e^{5}}{2 \left (b x +a \right )^{4} b^{6}}+\frac {15 a^{3} d^{2} e^{4}}{\left (b x +a \right )^{4} b^{5}}-\frac {15 a^{2} d^{3} e^{3}}{\left (b x +a \right )^{4} b^{4}}+\frac {15 a \,d^{4} e^{2}}{2 \left (b x +a \right )^{4} b^{3}}-\frac {3 d^{5} e}{2 \left (b x +a \right )^{4} b^{2}}-\frac {5 a^{4} e^{6}}{\left (b x +a \right )^{3} b^{7}}+\frac {20 a^{3} d \,e^{5}}{\left (b x +a \right )^{3} b^{6}}-\frac {30 a^{2} d^{2} e^{4}}{\left (b x +a \right )^{3} b^{5}}+\frac {20 a \,d^{3} e^{3}}{\left (b x +a \right )^{3} b^{4}}-\frac {5 d^{4} e^{2}}{\left (b x +a \right )^{3} b^{3}}+\frac {10 a^{3} e^{6}}{\left (b x +a \right )^{2} b^{7}}-\frac {30 a^{2} d \,e^{5}}{\left (b x +a \right )^{2} b^{6}}+\frac {30 a \,d^{2} e^{4}}{\left (b x +a \right )^{2} b^{5}}-\frac {10 d^{3} e^{3}}{\left (b x +a \right )^{2} b^{4}}-\frac {15 a^{2} e^{6}}{\left (b x +a \right ) b^{7}}+\frac {30 a d \,e^{5}}{\left (b x +a \right ) b^{6}}-\frac {6 a \,e^{6} \ln \left (b x +a \right )}{b^{7}}-\frac {15 d^{2} e^{4}}{\left (b x +a \right ) b^{5}}+\frac {6 d \,e^{5} \ln \left (b x +a \right )}{b^{6}}+\frac {e^{6} x}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.63, size = 397, normalized size = 2.56 \[ \frac {e^{6} x}{b^{6}} - \frac {2 \, b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 5 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 30 \, a^{4} b^{2} d^{2} e^{4} - 137 \, a^{5} b d e^{5} + 87 \, a^{6} e^{6} + 150 \, {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \, {\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} - 9 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 50 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 22 \, a^{3} b^{3} d e^{5} + 13 \, a^{4} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (3 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 30 \, a^{3} b^{3} d^{2} e^{4} - 125 \, a^{4} b^{2} d e^{5} + 77 \, a^{5} b e^{6}\right )} x}{10 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} + \frac {6 \, {\left (b d e^{5} - a e^{6}\right )} \log \left (b x + a\right )}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 399, normalized size = 2.57 \[ \frac {e^6\,x}{b^6}-\frac {\ln \left (a+b\,x\right )\,\left (6\,a\,e^6-6\,b\,d\,e^5\right )}{b^7}-\frac {x^2\,\left (65\,a^4\,b\,e^6-110\,a^3\,b^2\,d\,e^5+30\,a^2\,b^3\,d^2\,e^4+10\,a\,b^4\,d^3\,e^3+5\,b^5\,d^4\,e^2\right )+x^4\,\left (15\,a^2\,b^3\,e^6-30\,a\,b^4\,d\,e^5+15\,b^5\,d^2\,e^4\right )+\frac {87\,a^6\,e^6-137\,a^5\,b\,d\,e^5+30\,a^4\,b^2\,d^2\,e^4+10\,a^3\,b^3\,d^3\,e^3+5\,a^2\,b^4\,d^4\,e^2+3\,a\,b^5\,d^5\,e+2\,b^6\,d^6}{10\,b}+x\,\left (\frac {77\,a^5\,e^6}{2}-\frac {125\,a^4\,b\,d\,e^5}{2}+15\,a^3\,b^2\,d^2\,e^4+5\,a^2\,b^3\,d^3\,e^3+\frac {5\,a\,b^4\,d^4\,e^2}{2}+\frac {3\,b^5\,d^5\,e}{2}\right )+x^3\,\left (50\,a^3\,b^2\,e^6-90\,a^2\,b^3\,d\,e^5+30\,a\,b^4\,d^2\,e^4+10\,b^5\,d^3\,e^3\right )}{a^5\,b^6+5\,a^4\,b^7\,x+10\,a^3\,b^8\,x^2+10\,a^2\,b^9\,x^3+5\,a\,b^{10}\,x^4+b^{11}\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 30.05, size = 420, normalized size = 2.71 \[ \frac {- 87 a^{6} e^{6} + 137 a^{5} b d e^{5} - 30 a^{4} b^{2} d^{2} e^{4} - 10 a^{3} b^{3} d^{3} e^{3} - 5 a^{2} b^{4} d^{4} e^{2} - 3 a b^{5} d^{5} e - 2 b^{6} d^{6} + x^{4} \left (- 150 a^{2} b^{4} e^{6} + 300 a b^{5} d e^{5} - 150 b^{6} d^{2} e^{4}\right ) + x^{3} \left (- 500 a^{3} b^{3} e^{6} + 900 a^{2} b^{4} d e^{5} - 300 a b^{5} d^{2} e^{4} - 100 b^{6} d^{3} e^{3}\right ) + x^{2} \left (- 650 a^{4} b^{2} e^{6} + 1100 a^{3} b^{3} d e^{5} - 300 a^{2} b^{4} d^{2} e^{4} - 100 a b^{5} d^{3} e^{3} - 50 b^{6} d^{4} e^{2}\right ) + x \left (- 385 a^{5} b e^{6} + 625 a^{4} b^{2} d e^{5} - 150 a^{3} b^{3} d^{2} e^{4} - 50 a^{2} b^{4} d^{3} e^{3} - 25 a b^{5} d^{4} e^{2} - 15 b^{6} d^{5} e\right )}{10 a^{5} b^{7} + 50 a^{4} b^{8} x + 100 a^{3} b^{9} x^{2} + 100 a^{2} b^{10} x^{3} + 50 a b^{11} x^{4} + 10 b^{12} x^{5}} + \frac {e^{6} x}{b^{6}} - \frac {6 e^{5} \left (a e - b d\right ) \log {\left (a + b x \right )}}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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