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 = 0.12, antiderivative size = 173, 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} \[ \frac {6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac {15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac {20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac {3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac {6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac {(b d-a e)^6}{12 e^7 (d+e x)^{12}}-\frac {b^6}{6 e^7 (d+e x)^6} \]
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*}
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Mathematica [A] time = 0.09, size = 277, normalized size = 1.60 \[ -\frac {462 a^6 e^6+252 a^5 b e^5 (d+12 e x)+126 a^4 b^2 e^4 \left (d^2+12 d e x+66 e^2 x^2\right )+56 a^3 b^3 e^3 \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+21 a^2 b^4 e^2 \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+6 a b^5 e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+b^6 \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )}{5544 e^7 (d+e x)^{12}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 474, normalized size = 2.74 \[ -\frac {924 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \, {\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \, {\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \, {\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \, {\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \, {\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \, {\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 352, normalized size = 2.03 \[ -\frac {{\left (924 \, b^{6} x^{6} e^{6} + 792 \, b^{6} d x^{5} e^{5} + 495 \, b^{6} d^{2} x^{4} e^{4} + 220 \, b^{6} d^{3} x^{3} e^{3} + 66 \, b^{6} d^{4} x^{2} e^{2} + 12 \, b^{6} d^{5} x e + b^{6} d^{6} + 4752 \, a b^{5} x^{5} e^{6} + 2970 \, a b^{5} d x^{4} e^{5} + 1320 \, a b^{5} d^{2} x^{3} e^{4} + 396 \, a b^{5} d^{3} x^{2} e^{3} + 72 \, a b^{5} d^{4} x e^{2} + 6 \, a b^{5} d^{5} e + 10395 \, a^{2} b^{4} x^{4} e^{6} + 4620 \, a^{2} b^{4} d x^{3} e^{5} + 1386 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 252 \, a^{2} b^{4} d^{3} x e^{3} + 21 \, a^{2} b^{4} d^{4} e^{2} + 12320 \, a^{3} b^{3} x^{3} e^{6} + 3696 \, a^{3} b^{3} d x^{2} e^{5} + 672 \, a^{3} b^{3} d^{2} x e^{4} + 56 \, a^{3} b^{3} d^{3} e^{3} + 8316 \, a^{4} b^{2} x^{2} e^{6} + 1512 \, a^{4} b^{2} d x e^{5} + 126 \, a^{4} b^{2} d^{2} e^{4} + 3024 \, a^{5} b x e^{6} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{5544 \, {\left (x e + d\right )}^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 357, normalized size = 2.06 \[ -\frac {b^{6}}{6 \left (e x +d \right )^{6} e^{7}}-\frac {6 \left (a e -b d \right ) b^{5}}{7 \left (e x +d \right )^{7} e^{7}}-\frac {15 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}}{8 \left (e x +d \right )^{8} e^{7}}-\frac {20 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) b^{3}}{9 \left (e x +d \right )^{9} e^{7}}-\frac {3 \left (e^{4} a^{4}-4 d \,e^{3} a^{3} b +6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) b^{2}}{2 \left (e x +d \right )^{10} e^{7}}-\frac {6 \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) b}{11 \left (e x +d \right )^{11} e^{7}}-\frac {a^{6} e^{6}-6 d \,e^{5} a^{5} b +15 d^{2} e^{4} a^{4} b^{2}-20 d^{3} e^{3} a^{3} b^{3}+15 d^{4} a^{2} b^{4} e^{2}-6 d^{5} e a \,b^{5}+b^{6} d^{6}}{12 \left (e x +d \right )^{12} e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.76, size = 474, normalized size = 2.74 \[ -\frac {924 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \, {\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \, {\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \, {\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \, {\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \, {\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \, {\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 456, normalized size = 2.64 \[ -\frac {\frac {462\,a^6\,e^6+252\,a^5\,b\,d\,e^5+126\,a^4\,b^2\,d^2\,e^4+56\,a^3\,b^3\,d^3\,e^3+21\,a^2\,b^4\,d^4\,e^2+6\,a\,b^5\,d^5\,e+b^6\,d^6}{5544\,e^7}+\frac {b^6\,x^6}{6\,e}+\frac {5\,b^3\,x^3\,\left (56\,a^3\,e^3+21\,a^2\,b\,d\,e^2+6\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{126\,e^4}+\frac {b\,x\,\left (252\,a^5\,e^5+126\,a^4\,b\,d\,e^4+56\,a^3\,b^2\,d^2\,e^3+21\,a^2\,b^3\,d^3\,e^2+6\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{462\,e^6}+\frac {b^5\,x^5\,\left (6\,a\,e+b\,d\right )}{7\,e^2}+\frac {b^2\,x^2\,\left (126\,a^4\,e^4+56\,a^3\,b\,d\,e^3+21\,a^2\,b^2\,d^2\,e^2+6\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{84\,e^5}+\frac {5\,b^4\,x^4\,\left (21\,a^2\,e^2+6\,a\,b\,d\,e+b^2\,d^2\right )}{56\,e^3}}{d^{12}+12\,d^{11}\,e\,x+66\,d^{10}\,e^2\,x^2+220\,d^9\,e^3\,x^3+495\,d^8\,e^4\,x^4+792\,d^7\,e^5\,x^5+924\,d^6\,e^6\,x^6+792\,d^5\,e^7\,x^7+495\,d^4\,e^8\,x^8+220\,d^3\,e^9\,x^9+66\,d^2\,e^{10}\,x^{10}+12\,d\,e^{11}\,x^{11}+e^{12}\,x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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