Optimal. Leaf size=170 \[ -\frac {(b d-a e)^6}{11 e^7 (d+e x)^{11}}+\frac {3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac {5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac {5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac {15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac {b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac {b^6}{5 e^7 (d+e x)^5} \]
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Rubi [A]
time = 0.09, antiderivative size = 170, 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, 45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
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*}
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Mathematica [A]
time = 0.06, size = 277, normalized size = 1.63 \begin {gather*} -\frac {210 a^6 e^6+126 a^5 b e^5 (d+11 e x)+70 a^4 b^2 e^4 \left (d^2+11 d e x+55 e^2 x^2\right )+35 a^3 b^3 e^3 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+15 a^2 b^4 e^2 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 a b^5 e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+b^6 \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )}{2310 e^7 (d+e x)^{11}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(356\) vs.
\(2(158)=316\).
time = 0.63, size = 357, normalized size = 2.10
method | result | size |
risch | \(\frac {-\frac {b^{6} x^{6}}{5 e}-\frac {b^{5} \left (5 a e +b d \right ) x^{5}}{5 e^{2}}-\frac {b^{4} \left (15 a^{2} e^{2}+5 a b d e +b^{2} d^{2}\right ) x^{4}}{7 e^{3}}-\frac {b^{3} \left (35 e^{3} a^{3}+15 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{14 e^{4}}-\frac {b^{2} \left (70 e^{4} a^{4}+35 a^{3} b d \,e^{3}+15 a^{2} b^{2} d^{2} e^{2}+5 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{42 e^{5}}-\frac {b \left (126 a^{5} e^{5}+70 a^{4} b d \,e^{4}+35 a^{3} b^{2} d^{2} e^{3}+15 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{210 e^{6}}-\frac {210 a^{6} e^{6}+126 a^{5} b d \,e^{5}+70 a^{4} b^{2} d^{2} e^{4}+35 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+5 a \,b^{5} d^{5} e +b^{6} d^{6}}{2310 e^{7}}}{\left (e x +d \right )^{11}}\) | \(335\) |
default | \(-\frac {15 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{11 e^{7} \left (e x +d \right )^{11}}-\frac {5 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{3 e^{7} \left (e x +d \right )^{9}}-\frac {3 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{5 e^{7} \left (e x +d \right )^{10}}-\frac {b^{6}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {b^{5} \left (a e -b d \right )}{e^{7} \left (e x +d \right )^{6}}-\frac {5 b^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{2 e^{7} \left (e x +d \right )^{8}}\) | \(357\) |
norman | \(\frac {-\frac {b^{6} x^{6}}{5 e}-\frac {\left (5 e^{5} a \,b^{5}+d \,e^{4} b^{6}\right ) x^{5}}{5 e^{6}}-\frac {\left (15 a^{2} b^{4} e^{6}+5 a \,b^{5} d \,e^{5}+b^{6} d^{2} e^{4}\right ) x^{4}}{7 e^{7}}-\frac {\left (35 a^{3} b^{3} e^{7}+15 a^{2} b^{4} d \,e^{6}+5 a \,b^{5} d^{2} e^{5}+b^{6} d^{3} e^{4}\right ) x^{3}}{14 e^{8}}-\frac {\left (70 a^{4} b^{2} e^{8}+35 a^{3} b^{3} d \,e^{7}+15 a^{2} b^{4} d^{2} e^{6}+5 a \,b^{5} d^{3} e^{5}+b^{6} d^{4} e^{4}\right ) x^{2}}{42 e^{9}}-\frac {\left (126 a^{5} b \,e^{9}+70 a^{4} b^{2} d \,e^{8}+35 a^{3} b^{3} d^{2} e^{7}+15 a^{2} b^{4} d^{3} e^{6}+5 a \,b^{5} d^{4} e^{5}+b^{6} d^{5} e^{4}\right ) x}{210 e^{10}}-\frac {210 a^{6} e^{10}+126 a^{5} b d \,e^{9}+70 a^{4} b^{2} d^{2} e^{8}+35 a^{3} b^{3} d^{3} e^{7}+15 a^{2} b^{4} d^{4} e^{6}+5 a \,b^{5} d^{5} e^{5}+b^{6} d^{6} e^{4}}{2310 e^{11}}}{\left (e x +d \right )^{11}}\) | \(375\) |
gosper | \(-\frac {462 b^{6} x^{6} e^{6}+2310 a \,b^{5} e^{6} x^{5}+462 b^{6} d \,e^{5} x^{5}+4950 a^{2} b^{4} e^{6} x^{4}+1650 a \,b^{5} d \,e^{5} x^{4}+330 b^{6} d^{2} e^{4} x^{4}+5775 a^{3} b^{3} e^{6} x^{3}+2475 a^{2} b^{4} d \,e^{5} x^{3}+825 a \,b^{5} d^{2} e^{4} x^{3}+165 b^{6} d^{3} e^{3} x^{3}+3850 a^{4} b^{2} e^{6} x^{2}+1925 a^{3} b^{3} d \,e^{5} x^{2}+825 a^{2} b^{4} d^{2} e^{4} x^{2}+275 a \,b^{5} d^{3} e^{3} x^{2}+55 b^{6} d^{4} e^{2} x^{2}+1386 a^{5} b \,e^{6} x +770 a^{4} b^{2} d \,e^{5} x +385 a^{3} b^{3} d^{2} e^{4} x +165 a^{2} b^{4} d^{3} e^{3} x +55 a \,b^{5} d^{4} e^{2} x +11 b^{6} d^{5} e x +210 a^{6} e^{6}+126 a^{5} b d \,e^{5}+70 a^{4} b^{2} d^{2} e^{4}+35 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+5 a \,b^{5} d^{5} e +b^{6} d^{6}}{2310 e^{7} \left (e x +d \right )^{11}}\) | \(376\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 428 vs.
\(2 (164) = 328\).
time = 0.32, size = 428, normalized size = 2.52 \begin {gather*} -\frac {462 \, b^{6} x^{6} e^{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 (x^{11} e^{18} + 11 \, d x^{10} e^{17} + 55 \, d^{2} x^{9} e^{16} + 165 \, d^{3} x^{8} e^{15} + 330 \, d^{4} x^{7} e^{14} + 462 \, d^{5} x^{6} e^{13} + 462 \, d^{6} x^{5} e^{12} + 330 \, d^{7} x^{4} e^{11} + 165 \, d^{8} x^{3} e^{10} + 55 \, d^{9} x^{2} e^{9} + 11 \, d^{10} x e^{8} + d^{11} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 426 vs.
\(2 (164) = 328\).
time = 3.14, size = 426, normalized size = 2.51 \begin {gather*} -\frac {b^{6} d^{6} + {\left (462 \, b^{6} x^{6} + 2310 \, a b^{5} x^{5} + 4950 \, a^{2} b^{4} x^{4} + 5775 \, a^{3} b^{3} x^{3} + 3850 \, a^{4} b^{2} x^{2} + 1386 \, a^{5} b x + 210 \, a^{6}\right )} e^{6} + {\left (462 \, b^{6} d x^{5} + 1650 \, a b^{5} d x^{4} + 2475 \, a^{2} b^{4} d x^{3} + 1925 \, a^{3} b^{3} d x^{2} + 770 \, a^{4} b^{2} d x + 126 \, a^{5} b d\right )} e^{5} + 5 \, {\left (66 \, b^{6} d^{2} x^{4} + 165 \, a b^{5} d^{2} x^{3} + 165 \, a^{2} b^{4} d^{2} x^{2} + 77 \, a^{3} b^{3} d^{2} x + 14 \, a^{4} b^{2} d^{2}\right )} e^{4} + 5 \, {\left (33 \, b^{6} d^{3} x^{3} + 55 \, a b^{5} d^{3} x^{2} + 33 \, a^{2} b^{4} d^{3} x + 7 \, a^{3} b^{3} d^{3}\right )} e^{3} + 5 \, {\left (11 \, b^{6} d^{4} x^{2} + 11 \, a b^{5} d^{4} x + 3 \, a^{2} b^{4} d^{4}\right )} e^{2} + {\left (11 \, b^{6} d^{5} x + 5 \, a b^{5} d^{5}\right )} e}{2310 \, {\left (x^{11} e^{18} + 11 \, d x^{10} e^{17} + 55 \, d^{2} x^{9} e^{16} + 165 \, d^{3} x^{8} e^{15} + 330 \, d^{4} x^{7} e^{14} + 462 \, d^{5} x^{6} e^{13} + 462 \, d^{6} x^{5} e^{12} + 330 \, d^{7} x^{4} e^{11} + 165 \, d^{8} x^{3} e^{10} + 55 \, d^{9} x^{2} e^{9} + 11 \, d^{10} x e^{8} + d^{11} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs.
\(2 (164) = 328\).
time = 1.51, size = 352, normalized size = 2.07 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.66, size = 445, normalized size = 2.62 \begin {gather*} -\frac {\frac {210\,a^6\,e^6+126\,a^5\,b\,d\,e^5+70\,a^4\,b^2\,d^2\,e^4+35\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2+5\,a\,b^5\,d^5\,e+b^6\,d^6}{2310\,e^7}+\frac {b^6\,x^6}{5\,e}+\frac {b^3\,x^3\,\left (35\,a^3\,e^3+15\,a^2\,b\,d\,e^2+5\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{14\,e^4}+\frac {b\,x\,\left (126\,a^5\,e^5+70\,a^4\,b\,d\,e^4+35\,a^3\,b^2\,d^2\,e^3+15\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{210\,e^6}+\frac {b^5\,x^5\,\left (5\,a\,e+b\,d\right )}{5\,e^2}+\frac {b^2\,x^2\,\left (70\,a^4\,e^4+35\,a^3\,b\,d\,e^3+15\,a^2\,b^2\,d^2\,e^2+5\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{42\,e^5}+\frac {b^4\,x^4\,\left (15\,a^2\,e^2+5\,a\,b\,d\,e+b^2\,d^2\right )}{7\,e^3}}{d^{11}+11\,d^{10}\,e\,x+55\,d^9\,e^2\,x^2+165\,d^8\,e^3\,x^3+330\,d^7\,e^4\,x^4+462\,d^6\,e^5\,x^5+462\,d^5\,e^6\,x^6+330\,d^4\,e^7\,x^7+165\,d^3\,e^8\,x^8+55\,d^2\,e^9\,x^9+11\,d\,e^{10}\,x^{10}+e^{11}\,x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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