Optimal. Leaf size=119 \[ -\frac {(b d-a e)^4}{10 e^5 (d+e x)^{10}}+\frac {4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac {3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac {4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac {b^4}{6 e^5 (d+e x)^6} \]
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Rubi [A]
time = 0.05, antiderivative size = 119, 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 {4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac {3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac {4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac {(b d-a e)^4}{10 e^5 (d+e x)^{10}}-\frac {b^4}{6 e^5 (d+e x)^6} \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 )^2}{(d+e x)^{11}} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^{11}} \, dx\\ &=\int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^{11}}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^{10}}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^9}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)^8}+\frac {b^4}{e^4 (d+e x)^7}\right ) \, dx\\ &=-\frac {(b d-a e)^4}{10 e^5 (d+e x)^{10}}+\frac {4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac {3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac {4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac {b^4}{6 e^5 (d+e x)^6}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 144, normalized size = 1.21 \begin {gather*} -\frac {126 a^4 e^4+56 a^3 b e^3 (d+10 e x)+21 a^2 b^2 e^2 \left (d^2+10 d e x+45 e^2 x^2\right )+6 a b^3 e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+b^4 \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 )}{1260 e^5 (d+e x)^{10}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.63, size = 186, normalized size = 1.56
method | result | size |
risch | \(\frac {-\frac {b^{4} x^{4}}{6 e}-\frac {2 b^{3} \left (6 a e +b d \right ) x^{3}}{21 e^{2}}-\frac {b^{2} \left (21 a^{2} e^{2}+6 a b d e +b^{2} d^{2}\right ) x^{2}}{28 e^{3}}-\frac {b \left (56 e^{3} a^{3}+21 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{126 e^{4}}-\frac {126 e^{4} a^{4}+56 a^{3} b d \,e^{3}+21 a^{2} b^{2} d^{2} e^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}}{1260 e^{5}}}{\left (e x +d \right )^{10}}\) | \(171\) |
gosper | \(-\frac {210 b^{4} x^{4} e^{4}+720 a \,b^{3} e^{4} x^{3}+120 b^{4} d \,e^{3} x^{3}+945 a^{2} b^{2} e^{4} x^{2}+270 a \,b^{3} d \,e^{3} x^{2}+45 b^{4} d^{2} e^{2} x^{2}+560 a^{3} b \,e^{4} x +210 a^{2} b^{2} d \,e^{3} x +60 a \,b^{3} d^{2} e^{2} x +10 b^{4} d^{3} e x +126 e^{4} a^{4}+56 a^{3} b d \,e^{3}+21 a^{2} b^{2} d^{2} e^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}}{1260 e^{5} \left (e x +d \right )^{10}}\) | \(185\) |
default | \(-\frac {4 b^{3} \left (a e -b d \right )}{7 e^{5} \left (e x +d \right )^{7}}-\frac {e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{10 e^{5} \left (e x +d \right )^{10}}-\frac {b^{4}}{6 e^{5} \left (e x +d \right )^{6}}-\frac {4 b \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{9 e^{5} \left (e x +d \right )^{9}}-\frac {3 b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{4 e^{5} \left (e x +d \right )^{8}}\) | \(186\) |
norman | \(\frac {-\frac {b^{4} x^{4}}{6 e}-\frac {2 \left (6 a \,b^{3} e^{6}+b^{4} d \,e^{5}\right ) x^{3}}{21 e^{7}}-\frac {\left (21 a^{2} b^{2} e^{7}+6 a \,b^{3} d \,e^{6}+b^{4} d^{2} e^{5}\right ) x^{2}}{28 e^{8}}-\frac {\left (56 a^{3} b \,e^{8}+21 a^{2} b^{2} d \,e^{7}+6 a \,b^{3} d^{2} e^{6}+b^{4} d^{3} e^{5}\right ) x}{126 e^{9}}-\frac {126 a^{4} e^{9}+56 a^{3} b d \,e^{8}+21 a^{2} b^{2} d^{2} e^{7}+6 a \,b^{3} d^{3} e^{6}+b^{4} d^{4} e^{5}}{1260 e^{10}}}{\left (e x +d \right )^{10}}\) | \(197\) |
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. 259 vs.
\(2 (113) = 226\).
time = 0.30, size = 259, normalized size = 2.18 \begin {gather*} -\frac {210 \, b^{4} x^{4} e^{4} + b^{4} d^{4} + 6 \, a b^{3} d^{3} e + 21 \, a^{2} b^{2} d^{2} e^{2} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4} + 120 \, {\left (b^{4} d e^{3} + 6 \, a b^{3} e^{4}\right )} x^{3} + 45 \, {\left (b^{4} d^{2} e^{2} + 6 \, a b^{3} d e^{3} + 21 \, a^{2} b^{2} e^{4}\right )} x^{2} + 10 \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 21 \, a^{2} b^{2} d e^{3} + 56 \, a^{3} b e^{4}\right )} x}{1260 \, {\left (x^{10} e^{15} + 10 \, d x^{9} e^{14} + 45 \, d^{2} x^{8} e^{13} + 120 \, d^{3} x^{7} e^{12} + 210 \, d^{4} x^{6} e^{11} + 252 \, d^{5} x^{5} e^{10} + 210 \, d^{6} x^{4} e^{9} + 120 \, d^{7} x^{3} e^{8} + 45 \, d^{8} x^{2} e^{7} + 10 \, d^{9} x e^{6} + d^{10} e^{5}\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. 256 vs.
\(2 (113) = 226\).
time = 3.06, size = 256, normalized size = 2.15 \begin {gather*} -\frac {b^{4} d^{4} + {\left (210 \, b^{4} x^{4} + 720 \, a b^{3} x^{3} + 945 \, a^{2} b^{2} x^{2} + 560 \, a^{3} b x + 126 \, a^{4}\right )} e^{4} + 2 \, {\left (60 \, b^{4} d x^{3} + 135 \, a b^{3} d x^{2} + 105 \, a^{2} b^{2} d x + 28 \, a^{3} b d\right )} e^{3} + 3 \, {\left (15 \, b^{4} d^{2} x^{2} + 20 \, a b^{3} d^{2} x + 7 \, a^{2} b^{2} d^{2}\right )} e^{2} + 2 \, {\left (5 \, b^{4} d^{3} x + 3 \, a b^{3} d^{3}\right )} e}{1260 \, {\left (x^{10} e^{15} + 10 \, d x^{9} e^{14} + 45 \, d^{2} x^{8} e^{13} + 120 \, d^{3} x^{7} e^{12} + 210 \, d^{4} x^{6} e^{11} + 252 \, d^{5} x^{5} e^{10} + 210 \, d^{6} x^{4} e^{9} + 120 \, d^{7} x^{3} e^{8} + 45 \, d^{8} x^{2} e^{7} + 10 \, d^{9} x e^{6} + d^{10} e^{5}\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 [A]
time = 1.54, size = 174, normalized size = 1.46 \begin {gather*} -\frac {{\left (210 \, b^{4} x^{4} e^{4} + 120 \, b^{4} d x^{3} e^{3} + 45 \, b^{4} d^{2} x^{2} e^{2} + 10 \, b^{4} d^{3} x e + b^{4} d^{4} + 720 \, a b^{3} x^{3} e^{4} + 270 \, a b^{3} d x^{2} e^{3} + 60 \, a b^{3} d^{2} x e^{2} + 6 \, a b^{3} d^{3} e + 945 \, a^{2} b^{2} x^{2} e^{4} + 210 \, a^{2} b^{2} d x e^{3} + 21 \, a^{2} b^{2} d^{2} e^{2} + 560 \, a^{3} b x e^{4} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{1260 \, {\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 = 0.24, size = 270, normalized size = 2.27 \begin {gather*} -\frac {\frac {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}{1260\,e^5}+\frac {b^4\,x^4}{6\,e}+\frac {2\,b^3\,x^3\,\left (6\,a\,e+b\,d\right )}{21\,e^2}+\frac {b\,x\,\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^2\,x^2\,\left (21\,a^2\,e^2+6\,a\,b\,d\,e+b^2\,d^2\right )}{28\,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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