Optimal. Leaf size=119 \[ \frac {(b d-a e)^4 (a+b x)^6}{6 b^5}+\frac {4 e (b d-a e)^3 (a+b x)^7}{7 b^5}+\frac {3 e^2 (b d-a e)^2 (a+b x)^8}{4 b^5}+\frac {4 e^3 (b d-a e) (a+b x)^9}{9 b^5}+\frac {e^4 (a+b x)^{10}}{10 b^5} \]
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Rubi [A]
time = 0.15, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 45}
\begin {gather*} \frac {4 e^3 (a+b x)^9 (b d-a e)}{9 b^5}+\frac {3 e^2 (a+b x)^8 (b d-a e)^2}{4 b^5}+\frac {4 e (a+b x)^7 (b d-a e)^3}{7 b^5}+\frac {(a+b x)^6 (b d-a e)^4}{6 b^5}+\frac {e^4 (a+b x)^{10}}{10 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 (d+e x)^4 \, dx\\ &=\int \left (\frac {(b d-a e)^4 (a+b x)^5}{b^4}+\frac {4 e (b d-a e)^3 (a+b x)^6}{b^4}+\frac {6 e^2 (b d-a e)^2 (a+b x)^7}{b^4}+\frac {4 e^3 (b d-a e) (a+b x)^8}{b^4}+\frac {e^4 (a+b x)^9}{b^4}\right ) \, dx\\ &=\frac {(b d-a e)^4 (a+b x)^6}{6 b^5}+\frac {4 e (b d-a e)^3 (a+b x)^7}{7 b^5}+\frac {3 e^2 (b d-a e)^2 (a+b x)^8}{4 b^5}+\frac {4 e^3 (b d-a e) (a+b x)^9}{9 b^5}+\frac {e^4 (a+b x)^{10}}{10 b^5}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(301\) vs. \(2(119)=238\).
time = 0.06, size = 301, normalized size = 2.53 \begin {gather*} \frac {x \left (252 a^5 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+210 a^4 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+120 a^3 b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+45 a^2 b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+10 a b^4 x^4 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )+b^5 x^5 \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )\right )}{1260} \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. \(558\) vs.
\(2(109)=218\).
time = 1.03, size = 559, normalized size = 4.70
method | result | size |
norman | \(\frac {b^{5} e^{4} x^{10}}{10}+\left (\frac {5}{9} e^{4} a \,b^{4}+\frac {4}{9} d \,e^{3} b^{5}\right ) x^{9}+\left (\frac {5}{4} e^{4} a^{2} b^{3}+\frac {5}{2} d \,e^{3} a \,b^{4}+\frac {3}{4} b^{5} d^{2} e^{2}\right ) x^{8}+\left (\frac {10}{7} e^{4} a^{3} b^{2}+\frac {40}{7} d \,e^{3} a^{2} b^{3}+\frac {30}{7} d^{2} e^{2} a \,b^{4}+\frac {4}{7} d^{3} e \,b^{5}\right ) x^{7}+\left (\frac {5}{6} e^{4} a^{4} b +\frac {20}{3} d \,e^{3} a^{3} b^{2}+10 d^{2} e^{2} a^{2} b^{3}+\frac {10}{3} d^{3} e a \,b^{4}+\frac {1}{6} d^{4} b^{5}\right ) x^{6}+\left (\frac {1}{5} e^{4} a^{5}+4 d \,e^{3} a^{4} b +12 d^{2} e^{2} a^{3} b^{2}+8 d^{3} e \,a^{2} b^{3}+a \,b^{4} d^{4}\right ) x^{5}+\left (d \,e^{3} a^{5}+\frac {15}{2} d^{2} e^{2} a^{4} b +10 d^{3} e \,a^{3} b^{2}+\frac {5}{2} d^{4} a^{2} b^{3}\right ) x^{4}+\left (2 d^{2} e^{2} a^{5}+\frac {20}{3} d^{3} e \,a^{4} b +\frac {10}{3} d^{4} a^{3} b^{2}\right ) x^{3}+\left (2 d^{3} e \,a^{5}+\frac {5}{2} a^{4} b \,d^{4}\right ) x^{2}+d^{4} a^{5} x\) | \(353\) |
risch | \(\frac {1}{5} x^{5} e^{4} a^{5}+\frac {5}{9} x^{9} e^{4} a \,b^{4}+\frac {4}{9} x^{9} d \,e^{3} b^{5}+\frac {5}{4} x^{8} e^{4} a^{2} b^{3}+\frac {3}{4} x^{8} b^{5} d^{2} e^{2}+\frac {10}{7} x^{7} e^{4} a^{3} b^{2}+\frac {4}{7} x^{7} d^{3} e \,b^{5}+\frac {5}{6} x^{6} e^{4} a^{4} b +x^{5} a \,b^{4} d^{4}+x^{4} d \,e^{3} a^{5}+\frac {5}{2} x^{4} d^{4} a^{2} b^{3}+2 x^{3} d^{2} e^{2} a^{5}+\frac {10}{3} x^{3} d^{4} a^{3} b^{2}+2 x^{2} d^{3} e \,a^{5}+\frac {5}{2} x^{2} a^{4} b \,d^{4}+\frac {5}{2} x^{8} d \,e^{3} a \,b^{4}+\frac {40}{7} x^{7} d \,e^{3} a^{2} b^{3}+\frac {30}{7} x^{7} d^{2} e^{2} a \,b^{4}+\frac {20}{3} x^{6} d \,e^{3} a^{3} b^{2}+10 x^{6} d^{2} e^{2} a^{2} b^{3}+\frac {10}{3} x^{6} d^{3} e a \,b^{4}+4 x^{5} d \,e^{3} a^{4} b +12 x^{5} d^{2} e^{2} a^{3} b^{2}+8 x^{5} d^{3} e \,a^{2} b^{3}+\frac {15}{2} x^{4} d^{2} e^{2} a^{4} b +10 x^{4} d^{3} e \,a^{3} b^{2}+\frac {1}{6} x^{6} d^{4} b^{5}+\frac {1}{10} b^{5} e^{4} x^{10}+d^{4} a^{5} x +\frac {20}{3} x^{3} d^{3} e \,a^{4} b\) | \(397\) |
gosper | \(\frac {x \left (126 b^{5} e^{4} x^{9}+700 x^{8} e^{4} a \,b^{4}+560 x^{8} d \,e^{3} b^{5}+1575 x^{7} e^{4} a^{2} b^{3}+3150 x^{7} d \,e^{3} a \,b^{4}+945 x^{7} b^{5} d^{2} e^{2}+1800 x^{6} e^{4} a^{3} b^{2}+7200 x^{6} d \,e^{3} a^{2} b^{3}+5400 x^{6} d^{2} e^{2} a \,b^{4}+720 x^{6} d^{3} e \,b^{5}+1050 x^{5} e^{4} a^{4} b +8400 x^{5} d \,e^{3} a^{3} b^{2}+12600 x^{5} d^{2} e^{2} a^{2} b^{3}+4200 x^{5} d^{3} e a \,b^{4}+210 x^{5} d^{4} b^{5}+252 x^{4} e^{4} a^{5}+5040 x^{4} d \,e^{3} a^{4} b +15120 x^{4} d^{2} e^{2} a^{3} b^{2}+10080 x^{4} d^{3} e \,a^{2} b^{3}+1260 x^{4} a \,b^{4} d^{4}+1260 x^{3} d \,e^{3} a^{5}+9450 x^{3} d^{2} e^{2} a^{4} b +12600 x^{3} d^{3} e \,a^{3} b^{2}+3150 x^{3} d^{4} a^{2} b^{3}+2520 x^{2} d^{2} e^{2} a^{5}+8400 x^{2} d^{3} e \,a^{4} b +4200 x^{2} d^{4} a^{3} b^{2}+2520 x \,d^{3} e \,a^{5}+3150 x \,a^{4} b \,d^{4}+1260 d^{4} a^{5}\right )}{1260}\) | \(398\) |
default | \(\frac {b^{5} e^{4} x^{10}}{10}+\frac {\left (\left (e^{4} a +4 b d \,e^{3}\right ) b^{4}+4 e^{4} a \,b^{4}\right ) x^{9}}{9}+\frac {\left (\left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) b^{4}+4 \left (e^{4} a +4 b d \,e^{3}\right ) a \,b^{3}+6 e^{4} a^{2} b^{3}\right ) x^{8}}{8}+\frac {\left (\left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) b^{4}+4 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a \,b^{3}+6 \left (e^{4} a +4 b d \,e^{3}\right ) a^{2} b^{2}+4 e^{4} a^{3} b^{2}\right ) x^{7}}{7}+\frac {\left (\left (4 a \,d^{3} e +b \,d^{4}\right ) b^{4}+4 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a \,b^{3}+6 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{2} b^{2}+4 \left (e^{4} a +4 b d \,e^{3}\right ) a^{3} b +e^{4} a^{4} b \right ) x^{6}}{6}+\frac {\left (a \,b^{4} d^{4}+4 \left (4 a \,d^{3} e +b \,d^{4}\right ) a \,b^{3}+6 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{2} b^{2}+4 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{3} b +\left (e^{4} a +4 b d \,e^{3}\right ) a^{4}\right ) x^{5}}{5}+\frac {\left (4 d^{4} a^{2} b^{3}+6 \left (4 a \,d^{3} e +b \,d^{4}\right ) a^{2} b^{2}+4 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{3} b +\left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{4}\right ) x^{4}}{4}+\frac {\left (6 d^{4} a^{3} b^{2}+4 \left (4 a \,d^{3} e +b \,d^{4}\right ) a^{3} b +\left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{4}\right ) x^{3}}{3}+\frac {\left (4 a^{4} b \,d^{4}+\left (4 a \,d^{3} e +b \,d^{4}\right ) a^{4}\right ) x^{2}}{2}+d^{4} a^{5} x\) | \(559\) |
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. 348 vs.
\(2 (111) = 222\).
time = 0.30, size = 348, normalized size = 2.92 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} e^{4} + a^{5} d^{4} x + \frac {1}{9} \, {\left (4 \, b^{5} d e^{3} + 5 \, a b^{4} e^{4}\right )} x^{9} + \frac {1}{4} \, {\left (3 \, b^{5} d^{2} e^{2} + 10 \, a b^{4} d e^{3} + 5 \, a^{2} b^{3} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left (2 \, b^{5} d^{3} e + 15 \, a b^{4} d^{2} e^{2} + 20 \, a^{2} b^{3} d e^{3} + 5 \, a^{3} b^{2} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{4} + 20 \, a b^{4} d^{3} e + 60 \, a^{2} b^{3} d^{2} e^{2} + 40 \, a^{3} b^{2} d e^{3} + 5 \, a^{4} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, a b^{4} d^{4} + 40 \, a^{2} b^{3} d^{3} e + 60 \, a^{3} b^{2} d^{2} e^{2} + 20 \, a^{4} b d e^{3} + a^{5} e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (5 \, a^{2} b^{3} d^{4} + 20 \, a^{3} b^{2} d^{3} e + 15 \, a^{4} b d^{2} e^{2} + 2 \, a^{5} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (5 \, a^{3} b^{2} d^{4} + 10 \, a^{4} b d^{3} e + 3 \, a^{5} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{4} + 4 \, a^{5} d^{3} e\right )} x^{2} \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. 357 vs.
\(2 (111) = 222\).
time = 2.32, size = 357, normalized size = 3.00 \begin {gather*} \frac {1}{6} \, b^{5} d^{4} x^{6} + a b^{4} d^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} d^{4} x^{4} + \frac {10}{3} \, a^{3} b^{2} d^{4} x^{3} + \frac {5}{2} \, a^{4} b d^{4} x^{2} + a^{5} d^{4} x + \frac {1}{1260} \, {\left (126 \, b^{5} x^{10} + 700 \, a b^{4} x^{9} + 1575 \, a^{2} b^{3} x^{8} + 1800 \, a^{3} b^{2} x^{7} + 1050 \, a^{4} b x^{6} + 252 \, a^{5} x^{5}\right )} e^{4} + \frac {1}{126} \, {\left (56 \, b^{5} d x^{9} + 315 \, a b^{4} d x^{8} + 720 \, a^{2} b^{3} d x^{7} + 840 \, a^{3} b^{2} d x^{6} + 504 \, a^{4} b d x^{5} + 126 \, a^{5} d x^{4}\right )} e^{3} + \frac {1}{28} \, {\left (21 \, b^{5} d^{2} x^{8} + 120 \, a b^{4} d^{2} x^{7} + 280 \, a^{2} b^{3} d^{2} x^{6} + 336 \, a^{3} b^{2} d^{2} x^{5} + 210 \, a^{4} b d^{2} x^{4} + 56 \, a^{5} d^{2} x^{3}\right )} e^{2} + \frac {2}{21} \, {\left (6 \, b^{5} d^{3} x^{7} + 35 \, a b^{4} d^{3} x^{6} + 84 \, a^{2} b^{3} d^{3} x^{5} + 105 \, a^{3} b^{2} d^{3} x^{4} + 70 \, a^{4} b d^{3} x^{3} + 21 \, a^{5} d^{3} x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 401 vs.
\(2 (107) = 214\).
time = 0.04, size = 401, normalized size = 3.37 \begin {gather*} a^{5} d^{4} x + \frac {b^{5} e^{4} x^{10}}{10} + x^{9} \cdot \left (\frac {5 a b^{4} e^{4}}{9} + \frac {4 b^{5} d e^{3}}{9}\right ) + x^{8} \cdot \left (\frac {5 a^{2} b^{3} e^{4}}{4} + \frac {5 a b^{4} d e^{3}}{2} + \frac {3 b^{5} d^{2} e^{2}}{4}\right ) + x^{7} \cdot \left (\frac {10 a^{3} b^{2} e^{4}}{7} + \frac {40 a^{2} b^{3} d e^{3}}{7} + \frac {30 a b^{4} d^{2} e^{2}}{7} + \frac {4 b^{5} d^{3} e}{7}\right ) + x^{6} \cdot \left (\frac {5 a^{4} b e^{4}}{6} + \frac {20 a^{3} b^{2} d e^{3}}{3} + 10 a^{2} b^{3} d^{2} e^{2} + \frac {10 a b^{4} d^{3} e}{3} + \frac {b^{5} d^{4}}{6}\right ) + x^{5} \left (\frac {a^{5} e^{4}}{5} + 4 a^{4} b d e^{3} + 12 a^{3} b^{2} d^{2} e^{2} + 8 a^{2} b^{3} d^{3} e + a b^{4} d^{4}\right ) + x^{4} \left (a^{5} d e^{3} + \frac {15 a^{4} b d^{2} e^{2}}{2} + 10 a^{3} b^{2} d^{3} e + \frac {5 a^{2} b^{3} d^{4}}{2}\right ) + x^{3} \cdot \left (2 a^{5} d^{2} e^{2} + \frac {20 a^{4} b d^{3} e}{3} + \frac {10 a^{3} b^{2} d^{4}}{3}\right ) + x^{2} \cdot \left (2 a^{5} d^{3} e + \frac {5 a^{4} b d^{4}}{2}\right ) \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. 384 vs.
\(2 (111) = 222\).
time = 2.58, size = 384, normalized size = 3.23 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} e^{4} + \frac {4}{9} \, b^{5} d x^{9} e^{3} + \frac {3}{4} \, b^{5} d^{2} x^{8} e^{2} + \frac {4}{7} \, b^{5} d^{3} x^{7} e + \frac {1}{6} \, b^{5} d^{4} x^{6} + \frac {5}{9} \, a b^{4} x^{9} e^{4} + \frac {5}{2} \, a b^{4} d x^{8} e^{3} + \frac {30}{7} \, a b^{4} d^{2} x^{7} e^{2} + \frac {10}{3} \, a b^{4} d^{3} x^{6} e + a b^{4} d^{4} x^{5} + \frac {5}{4} \, a^{2} b^{3} x^{8} e^{4} + \frac {40}{7} \, a^{2} b^{3} d x^{7} e^{3} + 10 \, a^{2} b^{3} d^{2} x^{6} e^{2} + 8 \, a^{2} b^{3} d^{3} x^{5} e + \frac {5}{2} \, a^{2} b^{3} d^{4} x^{4} + \frac {10}{7} \, a^{3} b^{2} x^{7} e^{4} + \frac {20}{3} \, a^{3} b^{2} d x^{6} e^{3} + 12 \, a^{3} b^{2} d^{2} x^{5} e^{2} + 10 \, a^{3} b^{2} d^{3} x^{4} e + \frac {10}{3} \, a^{3} b^{2} d^{4} x^{3} + \frac {5}{6} \, a^{4} b x^{6} e^{4} + 4 \, a^{4} b d x^{5} e^{3} + \frac {15}{2} \, a^{4} b d^{2} x^{4} e^{2} + \frac {20}{3} \, a^{4} b d^{3} x^{3} e + \frac {5}{2} \, a^{4} b d^{4} x^{2} + \frac {1}{5} \, a^{5} x^{5} e^{4} + a^{5} d x^{4} e^{3} + 2 \, a^{5} d^{2} x^{3} e^{2} + 2 \, a^{5} d^{3} x^{2} e + a^{5} d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.07, size = 340, normalized size = 2.86 \begin {gather*} x^4\,\left (a^5\,d\,e^3+\frac {15\,a^4\,b\,d^2\,e^2}{2}+10\,a^3\,b^2\,d^3\,e+\frac {5\,a^2\,b^3\,d^4}{2}\right )+x^7\,\left (\frac {10\,a^3\,b^2\,e^4}{7}+\frac {40\,a^2\,b^3\,d\,e^3}{7}+\frac {30\,a\,b^4\,d^2\,e^2}{7}+\frac {4\,b^5\,d^3\,e}{7}\right )+x^5\,\left (\frac {a^5\,e^4}{5}+4\,a^4\,b\,d\,e^3+12\,a^3\,b^2\,d^2\,e^2+8\,a^2\,b^3\,d^3\,e+a\,b^4\,d^4\right )+x^6\,\left (\frac {5\,a^4\,b\,e^4}{6}+\frac {20\,a^3\,b^2\,d\,e^3}{3}+10\,a^2\,b^3\,d^2\,e^2+\frac {10\,a\,b^4\,d^3\,e}{3}+\frac {b^5\,d^4}{6}\right )+a^5\,d^4\,x+\frac {b^5\,e^4\,x^{10}}{10}+\frac {a^4\,d^3\,x^2\,\left (4\,a\,e+5\,b\,d\right )}{2}+\frac {b^4\,e^3\,x^9\,\left (5\,a\,e+4\,b\,d\right )}{9}+\frac {2\,a^3\,d^2\,x^3\,\left (3\,a^2\,e^2+10\,a\,b\,d\,e+5\,b^2\,d^2\right )}{3}+\frac {b^3\,e^2\,x^8\,\left (5\,a^2\,e^2+10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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