Optimal. Leaf size=173 \[ \frac{6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac{15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac{20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac{3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac{6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac{(b d-a e)^6}{12 e^7 (d+e x)^{12}}-\frac{b^6}{6 e^7 (d+e x)^6} \]
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Rubi [A] time = 0.118325, antiderivative size = 173, 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} \[ \frac{6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac{15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac{20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac{3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac{6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac{(b d-a e)^6}{12 e^7 (d+e x)^{12}}-\frac{b^6}{6 e^7 (d+e x)^6} \]
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)^{13}} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^{13}} \, dx\\ &=\int \left (\frac{(-b d+a e)^6}{e^6 (d+e x)^{13}}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^{12}}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^{11}}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^{10}}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^9}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)^8}+\frac{b^6}{e^6 (d+e x)^7}\right ) \, dx\\ &=-\frac{(b d-a e)^6}{12 e^7 (d+e x)^{12}}+\frac{6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac{3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac{20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac{15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac{6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac{b^6}{6 e^7 (d+e x)^6}\\ \end{align*}
Mathematica [A] time = 0.0915019, size = 277, normalized size = 1.6 \[ -\frac{21 a^2 b^4 e^2 \left (66 d^2 e^2 x^2+12 d^3 e x+d^4+220 d e^3 x^3+495 e^4 x^4\right )+56 a^3 b^3 e^3 \left (12 d^2 e x+d^3+66 d e^2 x^2+220 e^3 x^3\right )+126 a^4 b^2 e^4 \left (d^2+12 d e x+66 e^2 x^2\right )+252 a^5 b e^5 (d+12 e x)+462 a^6 e^6+6 a b^5 e \left (66 d^3 e^2 x^2+220 d^2 e^3 x^3+12 d^4 e x+d^5+495 d e^4 x^4+792 e^5 x^5\right )+b^6 \left (66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+12 d^5 e x+d^6+792 d e^5 x^5+924 e^6 x^6\right )}{5544 e^7 (d+e x)^{12}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 357, normalized size = 2.1 \begin{align*} -{\frac{20\,{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 ) }{9\,{e}^{7} \left ( ex+d \right ) ^{9}}}-{\frac{{b}^{6}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{3\,{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 ) }{2\,{e}^{7} \left ( ex+d \right ) ^{10}}}-{\frac{6\,{b}^{5} \left ( ae-bd \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}}{12\,{e}^{7} \left ( ex+d \right ) ^{12}}}-{\frac{6\,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 ) }{11\,{e}^{7} \left ( ex+d \right ) ^{11}}}-{\frac{15\,{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{8\,{e}^{7} \left ( ex+d \right ) ^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.27283, size = 640, normalized size = 3.7 \begin{align*} -\frac{924 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \,{\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \,{\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \,{\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \,{\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \,{\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \,{\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76447, size = 1026, normalized size = 5.93 \begin{align*} -\frac{924 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \,{\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \,{\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \,{\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \,{\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \,{\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \,{\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} 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 = 1.17238, size = 475, normalized size = 2.75 \begin{align*} -\frac{{\left (924 \, b^{6} x^{6} e^{6} + 792 \, b^{6} d x^{5} e^{5} + 495 \, b^{6} d^{2} x^{4} e^{4} + 220 \, b^{6} d^{3} x^{3} e^{3} + 66 \, b^{6} d^{4} x^{2} e^{2} + 12 \, b^{6} d^{5} x e + b^{6} d^{6} + 4752 \, a b^{5} x^{5} e^{6} + 2970 \, a b^{5} d x^{4} e^{5} + 1320 \, a b^{5} d^{2} x^{3} e^{4} + 396 \, a b^{5} d^{3} x^{2} e^{3} + 72 \, a b^{5} d^{4} x e^{2} + 6 \, a b^{5} d^{5} e + 10395 \, a^{2} b^{4} x^{4} e^{6} + 4620 \, a^{2} b^{4} d x^{3} e^{5} + 1386 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 252 \, a^{2} b^{4} d^{3} x e^{3} + 21 \, a^{2} b^{4} d^{4} e^{2} + 12320 \, a^{3} b^{3} x^{3} e^{6} + 3696 \, a^{3} b^{3} d x^{2} e^{5} + 672 \, a^{3} b^{3} d^{2} x e^{4} + 56 \, a^{3} b^{3} d^{3} e^{3} + 8316 \, a^{4} b^{2} x^{2} e^{6} + 1512 \, a^{4} b^{2} d x e^{5} + 126 \, a^{4} b^{2} d^{2} e^{4} + 3024 \, a^{5} b x e^{6} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{5544 \,{\left (x e + d\right )}^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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