Optimal. Leaf size=170 \[ \frac{b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac{15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac{5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac{5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac{3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac{(b d-a e)^6}{11 e^7 (d+e x)^{11}}-\frac{b^6}{5 e^7 (d+e x)^5} \]
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Rubi [A] time = 0.128661, antiderivative size = 170, normalized size of antiderivative = 1., 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} \[ \frac{b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac{15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac{5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac{5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac{3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac{(b d-a e)^6}{11 e^7 (d+e x)^{11}}-\frac{b^6}{5 e^7 (d+e x)^5} \]
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)^{12}} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^{12}} \, dx\\ &=\int \left (\frac{(-b d+a e)^6}{e^6 (d+e x)^{12}}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^{11}}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^{10}}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^9}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^8}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)^7}+\frac{b^6}{e^6 (d+e x)^6}\right ) \, dx\\ &=-\frac{(b d-a e)^6}{11 e^7 (d+e x)^{11}}+\frac{3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac{5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac{5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac{15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac{b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac{b^6}{5 e^7 (d+e x)^5}\\ \end{align*}
Mathematica [A] time = 0.0949331, size = 277, normalized size = 1.63 \[ -\frac{15 a^2 b^4 e^2 \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )+35 a^3 b^3 e^3 \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )+70 a^4 b^2 e^4 \left (d^2+11 d e x+55 e^2 x^2\right )+126 a^5 b e^5 (d+11 e x)+210 a^6 e^6+5 a b^5 e \left (55 d^3 e^2 x^2+165 d^2 e^3 x^3+11 d^4 e x+d^5+330 d e^4 x^4+462 e^5 x^5\right )+b^6 \left (55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+11 d^5 e x+d^6+462 d e^5 x^5+462 e^6 x^6\right )}{2310 e^7 (d+e x)^{11}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 357, normalized size = 2.1 \begin{align*} -{\frac{{b}^{5} \left ( ae-bd \right ) }{{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{3\,b \left ({a}^{5}{e}^{5}-5\,{a}^{4}bd{e}^{4}+10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-10\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+5\,a{b}^{4}{d}^{4}e-{b}^{5}{d}^{5} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{10}}}-{\frac{15\,{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{{e}^{6}{a}^{6}-6\,{a}^{5}bd{e}^{5}+15\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-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+{d}^{6}{b}^{6}}{11\,{e}^{7} \left ( ex+d \right ) ^{11}}}-{\frac{5\,{b}^{2} \left ({a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) }{3\,{e}^{7} \left ( ex+d \right ) ^{9}}}-{\frac{5\,{b}^{3} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{8}}}-{\frac{{b}^{6}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.10339, size = 625, normalized size = 3.68 \begin{align*} -\frac{462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \,{\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \,{\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \,{\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \,{\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \,{\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \,{\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74227, size = 994, normalized size = 5.85 \begin{align*} -\frac{462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \,{\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \,{\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \,{\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \,{\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \,{\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \,{\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16728, size = 475, normalized size = 2.79 \begin{align*} -\frac{{\left (462 \, b^{6} x^{6} e^{6} + 462 \, b^{6} d x^{5} e^{5} + 330 \, b^{6} d^{2} x^{4} e^{4} + 165 \, b^{6} d^{3} x^{3} e^{3} + 55 \, b^{6} d^{4} x^{2} e^{2} + 11 \, b^{6} d^{5} x e + b^{6} d^{6} + 2310 \, a b^{5} x^{5} e^{6} + 1650 \, a b^{5} d x^{4} e^{5} + 825 \, a b^{5} d^{2} x^{3} e^{4} + 275 \, a b^{5} d^{3} x^{2} e^{3} + 55 \, a b^{5} d^{4} x e^{2} + 5 \, a b^{5} d^{5} e + 4950 \, a^{2} b^{4} x^{4} e^{6} + 2475 \, a^{2} b^{4} d x^{3} e^{5} + 825 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 165 \, a^{2} b^{4} d^{3} x e^{3} + 15 \, a^{2} b^{4} d^{4} e^{2} + 5775 \, a^{3} b^{3} x^{3} e^{6} + 1925 \, a^{3} b^{3} d x^{2} e^{5} + 385 \, a^{3} b^{3} d^{2} x e^{4} + 35 \, a^{3} b^{3} d^{3} e^{3} + 3850 \, a^{4} b^{2} x^{2} e^{6} + 770 \, a^{4} b^{2} d x e^{5} + 70 \, a^{4} b^{2} d^{2} e^{4} + 1386 \, a^{5} b x e^{6} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{2310 \,{\left (x e + d\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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