Optimal. Leaf size=109 \[ -\frac {(-b-2 c x+1)^{11}}{22528 c^6}+\frac {(-b-2 c x+1)^{10}}{2048 c^6}-\frac {5 (-b-2 c x+1)^9}{2304 c^6}+\frac {5 (-b-2 c x+1)^8}{1024 c^6}-\frac {5 (-b-2 c x+1)^7}{896 c^6}+\frac {(-b-2 c x+1)^6}{384 c^6} \]
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Rubi [A] time = 0.14, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {610, 43} \[ -\frac {(-b-2 c x+1)^{11}}{22528 c^6}+\frac {(-b-2 c x+1)^{10}}{2048 c^6}-\frac {5 (-b-2 c x+1)^9}{2304 c^6}+\frac {5 (-b-2 c x+1)^8}{1024 c^6}-\frac {5 (-b-2 c x+1)^7}{896 c^6}+\frac {(-b-2 c x+1)^6}{384 c^6} \]
Antiderivative was successfully verified.
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Rule 43
Rule 610
Rubi steps
\begin {align*} \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx &=\frac {\int \left (\frac {1}{2} (-1+b)+c x\right )^5 \left (\frac {1+b}{2}+c x\right )^5 \, dx}{c^5}\\ &=\frac {\int \left (\left (\frac {1}{2} (-1+b)+c x\right )^5+5 \left (\frac {1}{2} (-1+b)+c x\right )^6+10 \left (\frac {1}{2} (-1+b)+c x\right )^7+10 \left (\frac {1}{2} (-1+b)+c x\right )^8+5 \left (\frac {1}{2} (-1+b)+c x\right )^9+\left (\frac {1}{2} (-1+b)+c x\right )^{10}\right ) \, dx}{c^5}\\ &=\frac {(1-b-2 c x)^6}{384 c^6}-\frac {5 (1-b-2 c x)^7}{896 c^6}+\frac {5 (1-b-2 c x)^8}{1024 c^6}-\frac {5 (1-b-2 c x)^9}{2304 c^6}+\frac {(1-b-2 c x)^{10}}{2048 c^6}-\frac {(1-b-2 c x)^{11}}{22528 c^6}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 207, normalized size = 1.90 \[ \frac {5}{8} \left (3 b^3-b\right ) c^2 x^8+\frac {\left (b^2-1\right )^5 x}{1024 c^5}+\frac {5 b \left (b^2-1\right )^4 x^2}{512 c^4}+\frac {5}{36} \left (9 b^2-1\right ) c^3 x^9+\frac {5 \left (b^2-1\right )^3 \left (9 b^2-1\right ) x^3}{768 c^3}+\frac {5 b \left (b^2-1\right )^2 \left (3 b^2-1\right ) x^4}{64 c^2}+\frac {5}{56} \left (21 b^4-14 b^2+1\right ) c x^7+\frac {\left (b^2-1\right ) \left (21 b^4-14 b^2+1\right ) x^5}{32 c}+\frac {1}{48} b \left (63 b^4-70 b^2+15\right ) x^6+\frac {1}{2} b c^4 x^{10}+\frac {c^5 x^{11}}{11} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 233, normalized size = 2.14 \[ \frac {64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 98560 \, {\left (9 \, b^{2} - 1\right )} c^{8} x^{9} + 443520 \, {\left (3 \, b^{3} - b\right )} c^{7} x^{8} + 63360 \, {\left (21 \, b^{4} - 14 \, b^{2} + 1\right )} c^{6} x^{7} + 14784 \, {\left (63 \, b^{5} - 70 \, b^{3} + 15 \, b\right )} c^{5} x^{6} + 22176 \, {\left (21 \, b^{6} - 35 \, b^{4} + 15 \, b^{2} - 1\right )} c^{4} x^{5} + 55440 \, {\left (3 \, b^{7} - 7 \, b^{5} + 5 \, b^{3} - b\right )} c^{3} x^{4} + 4620 \, {\left (9 \, b^{8} - 28 \, b^{6} + 30 \, b^{4} - 12 \, b^{2} + 1\right )} c^{2} x^{3} + 6930 \, {\left (b^{9} - 4 \, b^{7} + 6 \, b^{5} - 4 \, b^{3} + b\right )} c x^{2} + 693 \, {\left (b^{10} - 5 \, b^{8} + 10 \, b^{6} - 10 \, b^{4} + 5 \, b^{2} - 1\right )} x}{709632 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 334, normalized size = 3.06 \[ \frac {64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 887040 \, b^{2} c^{8} x^{9} + 1330560 \, b^{3} c^{7} x^{8} + 1330560 \, b^{4} c^{6} x^{7} - 98560 \, c^{8} x^{9} + 931392 \, b^{5} c^{5} x^{6} - 443520 \, b c^{7} x^{8} + 465696 \, b^{6} c^{4} x^{5} - 887040 \, b^{2} c^{6} x^{7} + 166320 \, b^{7} c^{3} x^{4} - 1034880 \, b^{3} c^{5} x^{6} + 41580 \, b^{8} c^{2} x^{3} - 776160 \, b^{4} c^{4} x^{5} + 63360 \, c^{6} x^{7} + 6930 \, b^{9} c x^{2} - 388080 \, b^{5} c^{3} x^{4} + 221760 \, b c^{5} x^{6} + 693 \, b^{10} x - 129360 \, b^{6} c^{2} x^{3} + 332640 \, b^{2} c^{4} x^{5} - 27720 \, b^{7} c x^{2} + 277200 \, b^{3} c^{3} x^{4} - 3465 \, b^{8} x + 138600 \, b^{4} c^{2} x^{3} - 22176 \, c^{4} x^{5} + 41580 \, b^{5} c x^{2} - 55440 \, b c^{3} x^{4} + 6930 \, b^{6} x - 55440 \, b^{2} c^{2} x^{3} - 27720 \, b^{3} c x^{2} - 6930 \, b^{4} x + 4620 \, c^{2} x^{3} + 6930 \, b c x^{2} + 3465 \, b^{2} x - 693 \, x}{709632 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 636, normalized size = 5.83 \[ \frac {c^{5} x^{11}}{11}+\frac {b \,c^{4} x^{10}}{2}+\frac {\left (4 b^{2} c^{3}+\frac {\left (b^{2}-1\right ) c^{3}}{4}+\left (4 b^{2} c^{2}+2 \left (\frac {3 b^{2}}{2}-\frac {1}{2}\right ) c^{2}\right ) c \right ) x^{9}}{9}+\frac {\left (\left (b^{2}-1\right ) b \,c^{2}+\left (4 b^{2} c^{2}+2 \left (\frac {3 b^{2}}{2}-\frac {1}{2}\right ) c^{2}\right ) b +\left (\left (b^{2}-1\right ) b c +4 \left (\frac {3 b^{2}}{2}-\frac {1}{2}\right ) b c \right ) c \right ) x^{8}}{8}+\frac {\left (\left (\left (b^{2}-1\right ) b c +4 \left (\frac {3 b^{2}}{2}-\frac {1}{2}\right ) b c \right ) b +\left (2 \left (b^{2}-1\right ) b^{2}+\frac {\left (b^{2}-1\right )^{2}}{8}+\left (\frac {3 b^{2}}{2}-\frac {1}{2}\right )^{2}\right ) c +\frac {\left (b^{2}-1\right ) \left (4 b^{2} c^{2}+2 \left (\frac {3 b^{2}}{2}-\frac {1}{2}\right ) c^{2}\right )}{4 c}\right ) x^{7}}{7}+\frac {\left (\left (2 \left (b^{2}-1\right ) b^{2}+\frac {\left (b^{2}-1\right )^{2}}{8}+\left (\frac {3 b^{2}}{2}-\frac {1}{2}\right )^{2}\right ) b +\left (\frac {\left (b^{2}-1\right )^{2} b}{4 c}+\frac {\left (b^{2}-1\right ) \left (\frac {3 b^{2}}{2}-\frac {1}{2}\right ) b}{c}\right ) c +\frac {\left (b^{2}-1\right ) \left (\left (b^{2}-1\right ) b c +4 \left (\frac {3 b^{2}}{2}-\frac {1}{2}\right ) b c \right )}{4 c}\right ) x^{6}}{6}+\frac {\left (\left (\frac {\left (b^{2}-1\right )^{2} b}{4 c}+\frac {\left (b^{2}-1\right ) \left (\frac {3 b^{2}}{2}-\frac {1}{2}\right ) b}{c}\right ) b +\left (\frac {\left (b^{2}-1\right )^{2} b^{2}}{4 c^{2}}+\frac {\left (b^{2}-1\right )^{2} \left (\frac {3 b^{2}}{2}-\frac {1}{2}\right )}{8 c^{2}}\right ) c +\frac {\left (b^{2}-1\right ) \left (2 \left (b^{2}-1\right ) b^{2}+\frac {\left (b^{2}-1\right )^{2}}{8}+\left (\frac {3 b^{2}}{2}-\frac {1}{2}\right )^{2}\right )}{4 c}\right ) x^{5}}{5}+\frac {\left (\left (\frac {\left (b^{2}-1\right )^{2} b^{2}}{4 c^{2}}+\frac {\left (b^{2}-1\right )^{2} \left (\frac {3 b^{2}}{2}-\frac {1}{2}\right )}{8 c^{2}}\right ) b +\frac {\left (b^{2}-1\right )^{3} b}{16 c^{2}}+\frac {\left (b^{2}-1\right ) \left (\frac {\left (b^{2}-1\right )^{2} b}{4 c}+\frac {\left (b^{2}-1\right ) \left (\frac {3 b^{2}}{2}-\frac {1}{2}\right ) b}{c}\right )}{4 c}\right ) x^{4}}{4}+\frac {\left (\frac {\left (b^{2}-1\right )^{3} b^{2}}{16 c^{3}}+\frac {\left (b^{2}-1\right ) \left (\frac {\left (b^{2}-1\right )^{2} b^{2}}{4 c^{2}}+\frac {\left (b^{2}-1\right )^{2} \left (\frac {3 b^{2}}{2}-\frac {1}{2}\right )}{8 c^{2}}\right )}{4 c}+\frac {\left (b^{2}-1\right )^{4}}{256 c^{3}}\right ) x^{3}}{3}+\frac {5 \left (b^{2}-1\right )^{4} b \,x^{2}}{512 c^{4}}+\frac {\left (b^{2}-1\right )^{5} x}{1024 c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.41, size = 234, normalized size = 2.15 \[ \frac {1}{11} \, c^{5} x^{11} + \frac {1}{2} \, b c^{4} x^{10} + \frac {10}{9} \, b^{2} c^{3} x^{9} + \frac {5}{4} \, b^{3} c^{2} x^{8} + \frac {5}{7} \, b^{4} c x^{7} + \frac {1}{6} \, b^{5} x^{6} + \frac {5 \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} {\left (b^{2} - 1\right )}^{4}}{1536 \, c^{4}} + \frac {{\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} {\left (b^{2} - 1\right )}^{3}}{192 \, c^{3}} + \frac {{\left (20 \, c^{3} x^{7} + 70 \, b c^{2} x^{6} + 84 \, b^{2} c x^{5} + 35 \, b^{3} x^{4}\right )} {\left (b^{2} - 1\right )}^{2}}{224 \, c^{2}} + \frac {{\left (70 \, c^{4} x^{9} + 315 \, b c^{3} x^{8} + 540 \, b^{2} c^{2} x^{7} + 420 \, b^{3} c x^{6} + 126 \, b^{4} x^{5}\right )} {\left (b^{2} - 1\right )}}{504 \, c} + \frac {{\left (b^{2} - 1\right )}^{5} x}{1024 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 184, normalized size = 1.69 \[ \frac {c^5\,x^{11}}{11}+\frac {x\,{\left (b^2-1\right )}^5}{1024\,c^5}+\frac {b\,x^6\,\left (63\,b^4-70\,b^2+15\right )}{48}+\frac {5\,c\,x^7\,\left (21\,b^4-14\,b^2+1\right )}{56}+\frac {b\,c^4\,x^{10}}{2}+\frac {5\,c^3\,x^9\,\left (9\,b^2-1\right )}{36}+\frac {x^5\,\left (21\,b^6-35\,b^4+15\,b^2-1\right )}{32\,c}+\frac {5\,b\,c^2\,x^8\,\left (3\,b^2-1\right )}{8}+\frac {5\,b\,x^2\,{\left (b^2-1\right )}^4}{512\,c^4}+\frac {5\,x^3\,{\left (b^2-1\right )}^3\,\left (9\,b^2-1\right )}{768\,c^3}+\frac {5\,b\,x^4\,{\left (b^2-1\right )}^2\,\left (3\,b^2-1\right )}{64\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.19, size = 253, normalized size = 2.32 \[ \frac {b c^{4} x^{10}}{2} + \frac {c^{5} x^{11}}{11} + x^{9} \left (\frac {5 b^{2} c^{3}}{4} - \frac {5 c^{3}}{36}\right ) + x^{8} \left (\frac {15 b^{3} c^{2}}{8} - \frac {5 b c^{2}}{8}\right ) + x^{7} \left (\frac {15 b^{4} c}{8} - \frac {5 b^{2} c}{4} + \frac {5 c}{56}\right ) + x^{6} \left (\frac {21 b^{5}}{16} - \frac {35 b^{3}}{24} + \frac {5 b}{16}\right ) + \frac {x^{5} \left (21 b^{6} - 35 b^{4} + 15 b^{2} - 1\right )}{32 c} + \frac {x^{4} \left (15 b^{7} - 35 b^{5} + 25 b^{3} - 5 b\right )}{64 c^{2}} + \frac {x^{3} \left (45 b^{8} - 140 b^{6} + 150 b^{4} - 60 b^{2} + 5\right )}{768 c^{3}} + \frac {x^{2} \left (5 b^{9} - 20 b^{7} + 30 b^{5} - 20 b^{3} + 5 b\right )}{512 c^{4}} + \frac {x \left (b^{10} - 5 b^{8} + 10 b^{6} - 10 b^{4} + 5 b^{2} - 1\right )}{1024 c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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