Optimal. Leaf size=155 \[ -\frac {6 b^5 (b d-a e) \log (d+e x)}{e^7}-\frac {15 b^4 (b d-a e)^2}{e^7 (d+e x)}+\frac {10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac {5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}+\frac {3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac {(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac {b^6 x}{e^6} \]
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Rubi [A] time = 0.14, 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} \[ -\frac {15 b^4 (b d-a e)^2}{e^7 (d+e x)}+\frac {10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac {5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}-\frac {6 b^5 (b d-a e) \log (d+e x)}{e^7}+\frac {3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac {(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac {b^6 x}{e^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^6} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^6} \, dx\\ &=\int \left (\frac {b^6}{e^6}+\frac {(-b d+a e)^6}{e^6 (d+e x)^6}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^5}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^4}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^3}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^2}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)}\right ) \, dx\\ &=\frac {b^6 x}{e^6}-\frac {(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac {3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac {5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}+\frac {10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac {15 b^4 (b d-a e)^2}{e^7 (d+e x)}-\frac {6 b^5 (b d-a e) \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 297, normalized size = 1.92 \[ -\frac {2 a^6 e^6+3 a^5 b e^5 (d+5 e x)+5 a^4 b^2 e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+30 a^2 b^4 e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-a b^5 d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+60 b^5 (d+e x)^5 (b d-a e) \log (d+e x)+b^6 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )}{10 e^7 (d+e x)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 542, normalized size = 3.50 \[ \frac {10 \, b^{6} e^{6} x^{6} + 50 \, b^{6} d e^{5} x^{5} - 87 \, b^{6} d^{6} + 137 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 50 \, {\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} - 100 \, {\left (4 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} - 50 \, {\left (12 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} - 5 \, {\left (75 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \, {\left (b^{6} d^{6} - a b^{5} d^{5} e + {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - a b^{5} d e^{5}\right )} x^{4} + 10 \, {\left (b^{6} d^{3} e^{3} - a b^{5} d^{2} e^{4}\right )} x^{3} + 10 \, {\left (b^{6} d^{4} e^{2} - a b^{5} d^{3} e^{3}\right )} x^{2} + 5 \, {\left (b^{6} d^{5} e - a b^{5} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{10 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 331, normalized size = 2.14 \[ b^{6} x e^{\left (-6\right )} - 6 \, {\left (b^{6} d - a b^{5} e\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (87 \, b^{6} d^{6} - 137 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + 2 \, a^{6} e^{6} + 150 \, {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \, {\left (5 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 50 \, {\left (13 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (77 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{10 \, {\left (x e + d\right )}^{5}} \]
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 \[ -\frac {a^{6}}{5 \left (e x +d \right )^{5} e}+\frac {6 a^{5} b d}{5 \left (e x +d \right )^{5} e^{2}}-\frac {3 a^{4} b^{2} d^{2}}{\left (e x +d \right )^{5} e^{3}}+\frac {4 a^{3} b^{3} d^{3}}{\left (e x +d \right )^{5} e^{4}}-\frac {3 a^{2} b^{4} d^{4}}{\left (e x +d \right )^{5} e^{5}}+\frac {6 a \,b^{5} d^{5}}{5 \left (e x +d \right )^{5} e^{6}}-\frac {b^{6} d^{6}}{5 \left (e x +d \right )^{5} e^{7}}-\frac {3 a^{5} b}{2 \left (e x +d \right )^{4} e^{2}}+\frac {15 a^{4} b^{2} d}{2 \left (e x +d \right )^{4} e^{3}}-\frac {15 a^{3} b^{3} d^{2}}{\left (e x +d \right )^{4} e^{4}}+\frac {15 a^{2} b^{4} d^{3}}{\left (e x +d \right )^{4} e^{5}}-\frac {15 a \,b^{5} d^{4}}{2 \left (e x +d \right )^{4} e^{6}}+\frac {3 b^{6} d^{5}}{2 \left (e x +d \right )^{4} e^{7}}-\frac {5 a^{4} b^{2}}{\left (e x +d \right )^{3} e^{3}}+\frac {20 a^{3} b^{3} d}{\left (e x +d \right )^{3} e^{4}}-\frac {30 a^{2} b^{4} d^{2}}{\left (e x +d \right )^{3} e^{5}}+\frac {20 a \,b^{5} d^{3}}{\left (e x +d \right )^{3} e^{6}}-\frac {5 b^{6} d^{4}}{\left (e x +d \right )^{3} e^{7}}-\frac {10 a^{3} b^{3}}{\left (e x +d \right )^{2} e^{4}}+\frac {30 a^{2} b^{4} d}{\left (e x +d \right )^{2} e^{5}}-\frac {30 a \,b^{5} d^{2}}{\left (e x +d \right )^{2} e^{6}}+\frac {10 b^{6} d^{3}}{\left (e x +d \right )^{2} e^{7}}-\frac {15 a^{2} b^{4}}{\left (e x +d \right ) e^{5}}+\frac {30 a \,b^{5} d}{\left (e x +d \right ) e^{6}}+\frac {6 a \,b^{5} \ln \left (e x +d \right )}{e^{6}}-\frac {15 b^{6} d^{2}}{\left (e x +d \right ) e^{7}}-\frac {6 b^{6} d \ln \left (e x +d \right )}{e^{7}}+\frac {b^{6} x}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.68, size = 397, normalized size = 2.56 \[ \frac {b^{6} x}{e^{6}} - \frac {87 \, b^{6} d^{6} - 137 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + 2 \, a^{6} e^{6} + 150 \, {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \, {\left (5 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 50 \, {\left (13 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (77 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x}{10 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} - \frac {6 \, {\left (b^{6} d - a b^{5} e\right )} \log \left (e x + d\right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 399, normalized size = 2.57 \[ \frac {b^6\,x}{e^6}-\frac {\ln \left (d+e\,x\right )\,\left (6\,b^6\,d-6\,a\,b^5\,e\right )}{e^7}-\frac {x^2\,\left (5\,a^4\,b^2\,e^5+10\,a^3\,b^3\,d\,e^4+30\,a^2\,b^4\,d^2\,e^3-110\,a\,b^5\,d^3\,e^2+65\,b^6\,d^4\,e\right )+x^4\,\left (15\,a^2\,b^4\,e^5-30\,a\,b^5\,d\,e^4+15\,b^6\,d^2\,e^3\right )+\frac {2\,a^6\,e^6+3\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4+10\,a^3\,b^3\,d^3\,e^3+30\,a^2\,b^4\,d^4\,e^2-137\,a\,b^5\,d^5\,e+87\,b^6\,d^6}{10\,e}+x\,\left (\frac {3\,a^5\,b\,e^5}{2}+\frac {5\,a^4\,b^2\,d\,e^4}{2}+5\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-\frac {125\,a\,b^5\,d^4\,e}{2}+\frac {77\,b^6\,d^5}{2}\right )+x^3\,\left (10\,a^3\,b^3\,e^5+30\,a^2\,b^4\,d\,e^4-90\,a\,b^5\,d^2\,e^3+50\,b^6\,d^3\,e^2\right )}{d^5\,e^6+5\,d^4\,e^7\,x+10\,d^3\,e^8\,x^2+10\,d^2\,e^9\,x^3+5\,d\,e^{10}\,x^4+e^{11}\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 30.17, size = 420, normalized size = 2.71 \[ \frac {b^{6} x}{e^{6}} + \frac {6 b^{5} \left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{7}} + \frac {- 2 a^{6} e^{6} - 3 a^{5} b d e^{5} - 5 a^{4} b^{2} d^{2} e^{4} - 10 a^{3} b^{3} d^{3} e^{3} - 30 a^{2} b^{4} d^{4} e^{2} + 137 a b^{5} d^{5} e - 87 b^{6} d^{6} + x^{4} \left (- 150 a^{2} b^{4} e^{6} + 300 a b^{5} d e^{5} - 150 b^{6} d^{2} e^{4}\right ) + x^{3} \left (- 100 a^{3} b^{3} e^{6} - 300 a^{2} b^{4} d e^{5} + 900 a b^{5} d^{2} e^{4} - 500 b^{6} d^{3} e^{3}\right ) + x^{2} \left (- 50 a^{4} b^{2} e^{6} - 100 a^{3} b^{3} d e^{5} - 300 a^{2} b^{4} d^{2} e^{4} + 1100 a b^{5} d^{3} e^{3} - 650 b^{6} d^{4} e^{2}\right ) + x \left (- 15 a^{5} b e^{6} - 25 a^{4} b^{2} d e^{5} - 50 a^{3} b^{3} d^{2} e^{4} - 150 a^{2} b^{4} d^{3} e^{3} + 625 a b^{5} d^{4} e^{2} - 385 b^{6} d^{5} e\right )}{10 d^{5} e^{7} + 50 d^{4} e^{8} x + 100 d^{3} e^{9} x^{2} + 100 d^{2} e^{10} x^{3} + 50 d e^{11} x^{4} + 10 e^{12} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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