Optimal. Leaf size=120 \[ \frac {(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^7}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 (a+b x)^7}{120 (b d-a e)^3 (d+e x)^8}+\frac {b^3 (a+b x)^7}{840 (b d-a e)^4 (d+e x)^7} \]
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Rubi [A]
time = 0.02, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {27, 47, 37}
\begin {gather*} \frac {b^3 (a+b x)^7}{840 (d+e x)^7 (b d-a e)^4}+\frac {b^2 (a+b x)^7}{120 (d+e x)^8 (b d-a e)^3}+\frac {b (a+b x)^7}{30 (d+e x)^9 (b d-a e)^2}+\frac {(a+b x)^7}{10 (d+e x)^{10} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{11}} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^{11}} \, dx\\ &=\frac {(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac {(3 b) \int \frac {(a+b x)^6}{(d+e x)^{10}} \, dx}{10 (b d-a e)}\\ &=\frac {(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^7}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{15 (b d-a e)^2}\\ &=\frac {(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^7}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 (a+b x)^7}{120 (b d-a e)^3 (d+e x)^8}+\frac {b^3 \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{120 (b d-a e)^3}\\ &=\frac {(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^7}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 (a+b x)^7}{120 (b d-a e)^3 (d+e x)^8}+\frac {b^3 (a+b x)^7}{840 (b d-a e)^4 (d+e x)^7}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(277\) vs. \(2(120)=240\).
time = 0.06, size = 277, normalized size = 2.31 \begin {gather*} -\frac {84 a^6 e^6+56 a^5 b e^5 (d+10 e x)+35 a^4 b^2 e^4 \left (d^2+10 d e x+45 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+10 a^2 b^4 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+4 a b^5 e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+b^6 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )}{840 e^7 (d+e x)^{10}} \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.
\(2(112)=224\).
time = 0.72, size = 357, normalized size = 2.98
method | result | size |
risch | \(\frac {-\frac {b^{6} x^{6}}{4 e}-\frac {3 b^{5} \left (4 a e +b d \right ) x^{5}}{10 e^{2}}-\frac {b^{4} \left (10 a^{2} e^{2}+4 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}-\frac {b^{3} \left (20 e^{3} a^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{7 e^{4}}-\frac {3 b^{2} \left (35 e^{4} a^{4}+20 a^{3} b d \,e^{3}+10 a^{2} b^{2} d^{2} e^{2}+4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{56 e^{5}}-\frac {b \left (56 a^{5} e^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{84 e^{6}}-\frac {84 a^{6} e^{6}+56 a^{5} b d \,e^{5}+35 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+10 a^{2} b^{4} d^{4} e^{2}+4 a \,b^{5} d^{5} e +b^{6} d^{6}}{840 e^{7}}}{\left (e x +d \right )^{10}}\) | \(335\) |
default | \(-\frac {20 b^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {2 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{3 e^{7} \left (e x +d \right )^{9}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{10 e^{7} \left (e x +d \right )^{10}}-\frac {6 b^{5} \left (a e -b d \right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {5 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{2 e^{7} \left (e x +d \right )^{6}}-\frac {15 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{8 e^{7} \left (e x +d \right )^{8}}-\frac {b^{6}}{4 e^{7} \left (e x +d \right )^{4}}\) | \(357\) |
norman | \(\frac {-\frac {b^{6} x^{6}}{4 e}-\frac {3 \left (4 e^{4} a \,b^{5}+d \,e^{3} b^{6}\right ) x^{5}}{10 e^{5}}-\frac {\left (10 e^{5} a^{2} b^{4}+4 d \,e^{4} a \,b^{5}+d^{2} e^{3} b^{6}\right ) x^{4}}{4 e^{6}}-\frac {\left (20 a^{3} b^{3} e^{6}+10 a^{2} b^{4} d \,e^{5}+4 a \,b^{5} d^{2} e^{4}+b^{6} d^{3} e^{3}\right ) x^{3}}{7 e^{7}}-\frac {3 \left (35 a^{4} b^{2} e^{7}+20 a^{3} b^{3} d \,e^{6}+10 a^{2} b^{4} d^{2} e^{5}+4 a \,b^{5} d^{3} e^{4}+b^{6} e^{3} d^{4}\right ) x^{2}}{56 e^{8}}-\frac {\left (56 a^{5} b \,e^{8}+35 a^{4} b^{2} d \,e^{7}+20 a^{3} b^{3} d^{2} e^{6}+10 a^{2} b^{4} e^{5} d^{3}+4 a \,b^{5} d^{4} e^{4}+b^{6} d^{5} e^{3}\right ) x}{84 e^{9}}-\frac {84 a^{6} e^{9}+56 a^{5} b d \,e^{8}+35 a^{4} b^{2} d^{2} e^{7}+20 a^{3} b^{3} d^{3} e^{6}+10 a^{2} b^{4} d^{4} e^{5}+4 a \,b^{5} d^{5} e^{4}+b^{6} d^{6} e^{3}}{840 e^{10}}}{\left (e x +d \right )^{10}}\) | \(375\) |
gosper | \(-\frac {210 b^{6} x^{6} e^{6}+1008 a \,b^{5} e^{6} x^{5}+252 b^{6} d \,e^{5} x^{5}+2100 a^{2} b^{4} e^{6} x^{4}+840 a \,b^{5} d \,e^{5} x^{4}+210 b^{6} d^{2} e^{4} x^{4}+2400 a^{3} b^{3} e^{6} x^{3}+1200 a^{2} b^{4} d \,e^{5} x^{3}+480 a \,b^{5} d^{2} e^{4} x^{3}+120 b^{6} d^{3} e^{3} x^{3}+1575 a^{4} b^{2} e^{6} x^{2}+900 a^{3} b^{3} d \,e^{5} x^{2}+450 a^{2} b^{4} d^{2} e^{4} x^{2}+180 a \,b^{5} d^{3} e^{3} x^{2}+45 b^{6} d^{4} e^{2} x^{2}+560 a^{5} b \,e^{6} x +350 a^{4} b^{2} d \,e^{5} x +200 a^{3} b^{3} d^{2} e^{4} x +100 a^{2} b^{4} d^{3} e^{3} x +40 a \,b^{5} d^{4} e^{2} x +10 b^{6} d^{5} e x +84 a^{6} e^{6}+56 a^{5} b d \,e^{5}+35 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+10 a^{2} b^{4} d^{4} e^{2}+4 a \,b^{5} d^{5} e +b^{6} d^{6}}{840 e^{7} \left (e x +d \right )^{10}}\) | \(376\) |
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. 418 vs.
\(2 (120) = 240\).
time = 0.30, size = 418, normalized size = 3.48 \begin {gather*} -\frac {210 \, b^{6} x^{6} e^{6} + b^{6} d^{6} + 4 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6} + 252 \, {\left (b^{6} d e^{5} + 4 \, a b^{5} e^{6}\right )} x^{5} + 210 \, {\left (b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} + 120 \, {\left (b^{6} d^{3} e^{3} + 4 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} + 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 45 \, {\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \, {\left (b^{6} d^{5} e + 4 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 20 \, a^{3} b^{3} d^{2} e^{4} + 35 \, a^{4} b^{2} d e^{5} + 56 \, a^{5} b e^{6}\right )} x}{840 \, {\left (x^{10} e^{17} + 10 \, d x^{9} e^{16} + 45 \, d^{2} x^{8} e^{15} + 120 \, d^{3} x^{7} e^{14} + 210 \, d^{4} x^{6} e^{13} + 252 \, d^{5} x^{5} e^{12} + 210 \, d^{6} x^{4} e^{11} + 120 \, d^{7} x^{3} e^{10} + 45 \, d^{8} x^{2} e^{9} + 10 \, d^{9} x e^{8} + d^{10} 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. 417 vs.
\(2 (120) = 240\).
time = 3.15, size = 417, normalized size = 3.48 \begin {gather*} -\frac {b^{6} d^{6} + {\left (210 \, b^{6} x^{6} + 1008 \, a b^{5} x^{5} + 2100 \, a^{2} b^{4} x^{4} + 2400 \, a^{3} b^{3} x^{3} + 1575 \, a^{4} b^{2} x^{2} + 560 \, a^{5} b x + 84 \, a^{6}\right )} e^{6} + 2 \, {\left (126 \, b^{6} d x^{5} + 420 \, a b^{5} d x^{4} + 600 \, a^{2} b^{4} d x^{3} + 450 \, a^{3} b^{3} d x^{2} + 175 \, a^{4} b^{2} d x + 28 \, a^{5} b d\right )} e^{5} + 5 \, {\left (42 \, b^{6} d^{2} x^{4} + 96 \, a b^{5} d^{2} x^{3} + 90 \, a^{2} b^{4} d^{2} x^{2} + 40 \, a^{3} b^{3} d^{2} x + 7 \, a^{4} b^{2} d^{2}\right )} e^{4} + 20 \, {\left (6 \, b^{6} d^{3} x^{3} + 9 \, a b^{5} d^{3} x^{2} + 5 \, a^{2} b^{4} d^{3} x + a^{3} b^{3} d^{3}\right )} e^{3} + 5 \, {\left (9 \, b^{6} d^{4} x^{2} + 8 \, a b^{5} d^{4} x + 2 \, a^{2} b^{4} d^{4}\right )} e^{2} + 2 \, {\left (5 \, b^{6} d^{5} x + 2 \, a b^{5} d^{5}\right )} e}{840 \, {\left (x^{10} e^{17} + 10 \, d x^{9} e^{16} + 45 \, d^{2} x^{8} e^{15} + 120 \, d^{3} x^{7} e^{14} + 210 \, d^{4} x^{6} e^{13} + 252 \, d^{5} x^{5} e^{12} + 210 \, d^{6} x^{4} e^{11} + 120 \, d^{7} x^{3} e^{10} + 45 \, d^{8} x^{2} e^{9} + 10 \, d^{9} x e^{8} + d^{10} 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.
\(2 (120) = 240\).
time = 2.65, size = 352, normalized size = 2.93 \begin {gather*} -\frac {{\left (210 \, b^{6} x^{6} e^{6} + 252 \, b^{6} d x^{5} e^{5} + 210 \, b^{6} d^{2} x^{4} e^{4} + 120 \, b^{6} d^{3} x^{3} e^{3} + 45 \, b^{6} d^{4} x^{2} e^{2} + 10 \, b^{6} d^{5} x e + b^{6} d^{6} + 1008 \, a b^{5} x^{5} e^{6} + 840 \, a b^{5} d x^{4} e^{5} + 480 \, a b^{5} d^{2} x^{3} e^{4} + 180 \, a b^{5} d^{3} x^{2} e^{3} + 40 \, a b^{5} d^{4} x e^{2} + 4 \, a b^{5} d^{5} e + 2100 \, a^{2} b^{4} x^{4} e^{6} + 1200 \, a^{2} b^{4} d x^{3} e^{5} + 450 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 100 \, a^{2} b^{4} d^{3} x e^{3} + 10 \, a^{2} b^{4} d^{4} e^{2} + 2400 \, a^{3} b^{3} x^{3} e^{6} + 900 \, a^{3} b^{3} d x^{2} e^{5} + 200 \, a^{3} b^{3} d^{2} x e^{4} + 20 \, a^{3} b^{3} d^{3} e^{3} + 1575 \, a^{4} b^{2} x^{2} e^{6} + 350 \, a^{4} b^{2} d x e^{5} + 35 \, a^{4} b^{2} d^{2} e^{4} + 560 \, a^{5} b x e^{6} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{840 \, {\left (x e + d\right )}^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 434, normalized size = 3.62 \begin {gather*} -\frac {\frac {84\,a^6\,e^6+56\,a^5\,b\,d\,e^5+35\,a^4\,b^2\,d^2\,e^4+20\,a^3\,b^3\,d^3\,e^3+10\,a^2\,b^4\,d^4\,e^2+4\,a\,b^5\,d^5\,e+b^6\,d^6}{840\,e^7}+\frac {b^6\,x^6}{4\,e}+\frac {b^3\,x^3\,\left (20\,a^3\,e^3+10\,a^2\,b\,d\,e^2+4\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{7\,e^4}+\frac {b\,x\,\left (56\,a^5\,e^5+35\,a^4\,b\,d\,e^4+20\,a^3\,b^2\,d^2\,e^3+10\,a^2\,b^3\,d^3\,e^2+4\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{84\,e^6}+\frac {3\,b^5\,x^5\,\left (4\,a\,e+b\,d\right )}{10\,e^2}+\frac {3\,b^2\,x^2\,\left (35\,a^4\,e^4+20\,a^3\,b\,d\,e^3+10\,a^2\,b^2\,d^2\,e^2+4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{56\,e^5}+\frac {b^4\,x^4\,\left (10\,a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )}{4\,e^3}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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