Optimal. Leaf size=156 \[ \frac {3 e^5 (a+b x)^2 (b d-a e)}{b^7}+\frac {20 e^3 (b d-a e)^3 \log (a+b x)}{b^7}-\frac {15 e^2 (b d-a e)^4}{b^7 (a+b x)}-\frac {3 e (b d-a e)^5}{b^7 (a+b x)^2}-\frac {(b d-a e)^6}{3 b^7 (a+b x)^3}+\frac {e^6 (a+b x)^3}{3 b^7}+\frac {15 e^4 x (b d-a e)^2}{b^6} \]
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Rubi [A] time = 0.18, antiderivative size = 156, 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} \[ \frac {3 e^5 (a+b x)^2 (b d-a e)}{b^7}+\frac {15 e^4 x (b d-a e)^2}{b^6}-\frac {15 e^2 (b d-a e)^4}{b^7 (a+b x)}+\frac {20 e^3 (b d-a e)^3 \log (a+b x)}{b^7}-\frac {3 e (b d-a e)^5}{b^7 (a+b x)^2}-\frac {(b d-a e)^6}{3 b^7 (a+b x)^3}+\frac {e^6 (a+b x)^3}{3 b^7} \]
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 )^2} \, dx &=\int \frac {(d+e x)^6}{(a+b x)^4} \, dx\\ &=\int \left (\frac {15 e^4 (b d-a e)^2}{b^6}+\frac {(b d-a e)^6}{b^6 (a+b x)^4}+\frac {6 e (b d-a e)^5}{b^6 (a+b x)^3}+\frac {15 e^2 (b d-a e)^4}{b^6 (a+b x)^2}+\frac {20 e^3 (b d-a e)^3}{b^6 (a+b x)}+\frac {6 e^5 (b d-a e) (a+b x)}{b^6}+\frac {e^6 (a+b x)^2}{b^6}\right ) \, dx\\ &=\frac {15 e^4 (b d-a e)^2 x}{b^6}-\frac {(b d-a e)^6}{3 b^7 (a+b x)^3}-\frac {3 e (b d-a e)^5}{b^7 (a+b x)^2}-\frac {15 e^2 (b d-a e)^4}{b^7 (a+b x)}+\frac {3 e^5 (b d-a e) (a+b x)^2}{b^7}+\frac {e^6 (a+b x)^3}{3 b^7}+\frac {20 e^3 (b d-a e)^3 \log (a+b x)}{b^7}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 301, normalized size = 1.93 \[ \frac {-37 a^6 e^6+3 a^5 b e^5 (47 d-17 e x)+3 a^4 b^2 e^4 \left (-65 d^2+81 d e x+13 e^2 x^2\right )+a^3 b^3 e^3 \left (110 d^3-405 d^2 e x-27 d e^2 x^2+73 e^3 x^3\right )+3 a^2 b^4 e^2 \left (-5 d^4+90 d^3 e x-45 d^2 e^2 x^2-63 d e^3 x^3+5 e^4 x^4\right )-3 a b^5 e \left (d^5+15 d^4 e x-60 d^3 e^2 x^2-45 d^2 e^3 x^3+15 d e^4 x^4+e^5 x^5\right )-60 e^3 (a+b x)^3 (a e-b d)^3 \log (a+b x)+b^6 \left (-d^6-9 d^5 e x-45 d^4 e^2 x^2+45 d^2 e^4 x^4+9 d e^5 x^5+e^6 x^6\right )}{3 b^7 (a+b x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.00, size = 577, normalized size = 3.70 \[ \frac {b^{6} e^{6} x^{6} - b^{6} d^{6} - 3 \, a b^{5} d^{5} e - 15 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 195 \, a^{4} b^{2} d^{2} e^{4} + 141 \, a^{5} b d e^{5} - 37 \, a^{6} e^{6} + 3 \, {\left (3 \, b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 15 \, {\left (3 \, b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + {\left (135 \, a b^{5} d^{2} e^{4} - 189 \, a^{2} b^{4} d e^{5} + 73 \, a^{3} b^{3} e^{6}\right )} x^{3} - 3 \, {\left (15 \, b^{6} d^{4} e^{2} - 60 \, a b^{5} d^{3} e^{3} + 45 \, a^{2} b^{4} d^{2} e^{4} + 9 \, a^{3} b^{3} d e^{5} - 13 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \, {\left (3 \, b^{6} d^{5} e + 15 \, a b^{5} d^{4} e^{2} - 90 \, a^{2} b^{4} d^{3} e^{3} + 135 \, a^{3} b^{3} d^{2} e^{4} - 81 \, a^{4} b^{2} d e^{5} + 17 \, a^{5} b e^{6}\right )} x + 60 \, {\left (a^{3} b^{3} d^{3} e^{3} - 3 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} - a^{6} e^{6} + {\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (a b^{5} d^{3} e^{3} - 3 \, a^{2} b^{4} d^{2} e^{4} + 3 \, a^{3} b^{3} d e^{5} - a^{4} b^{2} e^{6}\right )} x^{2} + 3 \, {\left (a^{2} b^{4} d^{3} e^{3} - 3 \, a^{3} b^{3} d^{2} e^{4} + 3 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} \log \left (b x + a\right )}{3 \, {\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 333, normalized size = 2.13 \[ \frac {20 \, {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 195 \, a^{4} b^{2} d^{2} e^{4} - 141 \, a^{5} b d e^{5} + 37 \, a^{6} e^{6} + 45 \, {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \, {\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 35 \, a^{4} b^{2} d e^{5} + 9 \, a^{5} b e^{6}\right )} x}{3 \, {\left (b x + a\right )}^{3} b^{7}} + \frac {b^{8} x^{3} e^{6} + 9 \, b^{8} d x^{2} e^{5} + 45 \, b^{8} d^{2} x e^{4} - 6 \, a b^{7} x^{2} e^{6} - 72 \, a b^{7} d x e^{5} + 30 \, a^{2} b^{6} x e^{6}}{3 \, b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 483, normalized size = 3.10 \[ -\frac {a^{6} e^{6}}{3 \left (b x +a \right )^{3} b^{7}}+\frac {2 a^{5} d \,e^{5}}{\left (b x +a \right )^{3} b^{6}}-\frac {5 a^{4} d^{2} e^{4}}{\left (b x +a \right )^{3} b^{5}}+\frac {20 a^{3} d^{3} e^{3}}{3 \left (b x +a \right )^{3} b^{4}}-\frac {5 a^{2} d^{4} e^{2}}{\left (b x +a \right )^{3} b^{3}}+\frac {2 a \,d^{5} e}{\left (b x +a \right )^{3} b^{2}}-\frac {d^{6}}{3 \left (b x +a \right )^{3} b}+\frac {e^{6} x^{3}}{3 b^{4}}+\frac {3 a^{5} e^{6}}{\left (b x +a \right )^{2} b^{7}}-\frac {15 a^{4} d \,e^{5}}{\left (b x +a \right )^{2} b^{6}}+\frac {30 a^{3} d^{2} e^{4}}{\left (b x +a \right )^{2} b^{5}}-\frac {30 a^{2} d^{3} e^{3}}{\left (b x +a \right )^{2} b^{4}}+\frac {15 a \,d^{4} e^{2}}{\left (b x +a \right )^{2} b^{3}}-\frac {2 a \,e^{6} x^{2}}{b^{5}}-\frac {3 d^{5} e}{\left (b x +a \right )^{2} b^{2}}+\frac {3 d \,e^{5} x^{2}}{b^{4}}-\frac {15 a^{4} e^{6}}{\left (b x +a \right ) b^{7}}+\frac {60 a^{3} d \,e^{5}}{\left (b x +a \right ) b^{6}}-\frac {20 a^{3} e^{6} \ln \left (b x +a \right )}{b^{7}}-\frac {90 a^{2} d^{2} e^{4}}{\left (b x +a \right ) b^{5}}+\frac {60 a^{2} d \,e^{5} \ln \left (b x +a \right )}{b^{6}}+\frac {10 a^{2} e^{6} x}{b^{6}}+\frac {60 a \,d^{3} e^{3}}{\left (b x +a \right ) b^{4}}-\frac {60 a \,d^{2} e^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {24 a d \,e^{5} x}{b^{5}}-\frac {15 d^{4} e^{2}}{\left (b x +a \right ) b^{3}}+\frac {20 d^{3} e^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {15 d^{2} e^{4} x}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.59, size = 374, normalized size = 2.40 \[ -\frac {b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 195 \, a^{4} b^{2} d^{2} e^{4} - 141 \, a^{5} b d e^{5} + 37 \, a^{6} e^{6} + 45 \, {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \, {\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 35 \, a^{4} b^{2} d e^{5} + 9 \, a^{5} b e^{6}\right )} x}{3 \, {\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} + \frac {b^{2} e^{6} x^{3} + 3 \, {\left (3 \, b^{2} d e^{5} - 2 \, a b e^{6}\right )} x^{2} + 3 \, {\left (15 \, b^{2} d^{2} e^{4} - 24 \, a b d e^{5} + 10 \, a^{2} e^{6}\right )} x}{3 \, b^{6}} + \frac {20 \, {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} \log \left (b x + a\right )}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 393, normalized size = 2.52 \[ x\,\left (\frac {4\,a\,\left (\frac {4\,a\,e^6}{b^5}-\frac {6\,d\,e^5}{b^4}\right )}{b}-\frac {6\,a^2\,e^6}{b^6}+\frac {15\,d^2\,e^4}{b^4}\right )-\frac {x^2\,\left (15\,a^4\,b\,e^6-60\,a^3\,b^2\,d\,e^5+90\,a^2\,b^3\,d^2\,e^4-60\,a\,b^4\,d^3\,e^3+15\,b^5\,d^4\,e^2\right )+\frac {37\,a^6\,e^6-141\,a^5\,b\,d\,e^5+195\,a^4\,b^2\,d^2\,e^4-110\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2+3\,a\,b^5\,d^5\,e+b^6\,d^6}{3\,b}+x\,\left (27\,a^5\,e^6-105\,a^4\,b\,d\,e^5+150\,a^3\,b^2\,d^2\,e^4-90\,a^2\,b^3\,d^3\,e^3+15\,a\,b^4\,d^4\,e^2+3\,b^5\,d^5\,e\right )}{a^3\,b^6+3\,a^2\,b^7\,x+3\,a\,b^8\,x^2+b^9\,x^3}-x^2\,\left (\frac {2\,a\,e^6}{b^5}-\frac {3\,d\,e^5}{b^4}\right )-\frac {\ln \left (a+b\,x\right )\,\left (20\,a^3\,e^6-60\,a^2\,b\,d\,e^5+60\,a\,b^2\,d^2\,e^4-20\,b^3\,d^3\,e^3\right )}{b^7}+\frac {e^6\,x^3}{3\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.37, size = 367, normalized size = 2.35 \[ x^{2} \left (- \frac {2 a e^{6}}{b^{5}} + \frac {3 d e^{5}}{b^{4}}\right ) + x \left (\frac {10 a^{2} e^{6}}{b^{6}} - \frac {24 a d e^{5}}{b^{5}} + \frac {15 d^{2} e^{4}}{b^{4}}\right ) + \frac {- 37 a^{6} e^{6} + 141 a^{5} b d e^{5} - 195 a^{4} b^{2} d^{2} e^{4} + 110 a^{3} b^{3} d^{3} e^{3} - 15 a^{2} b^{4} d^{4} e^{2} - 3 a b^{5} d^{5} e - b^{6} d^{6} + x^{2} \left (- 45 a^{4} b^{2} e^{6} + 180 a^{3} b^{3} d e^{5} - 270 a^{2} b^{4} d^{2} e^{4} + 180 a b^{5} d^{3} e^{3} - 45 b^{6} d^{4} e^{2}\right ) + x \left (- 81 a^{5} b e^{6} + 315 a^{4} b^{2} d e^{5} - 450 a^{3} b^{3} d^{2} e^{4} + 270 a^{2} b^{4} d^{3} e^{3} - 45 a b^{5} d^{4} e^{2} - 9 b^{6} d^{5} e\right )}{3 a^{3} b^{7} + 9 a^{2} b^{8} x + 9 a b^{9} x^{2} + 3 b^{10} x^{3}} + \frac {e^{6} x^{3}}{3 b^{4}} - \frac {20 e^{3} \left (a e - b d\right )^{3} \log {\left (a + b x \right )}}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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