Optimal. Leaf size=109 \[ -\frac {(-b-2 c x+4)^{11}}{22528 c^6}+\frac {(-b-2 c x+4)^{10}}{512 c^6}-\frac {5 (-b-2 c x+4)^9}{144 c^6}+\frac {5 (-b-2 c x+4)^8}{16 c^6}-\frac {10 (-b-2 c x+4)^7}{7 c^6}+\frac {8 (-b-2 c x+4)^6}{3 c^6} \]
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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} \[ -\frac {(-b-2 c x+4)^{11}}{22528 c^6}+\frac {(-b-2 c x+4)^{10}}{512 c^6}-\frac {5 (-b-2 c x+4)^9}{144 c^6}+\frac {5 (-b-2 c x+4)^8}{16 c^6}-\frac {10 (-b-2 c x+4)^7}{7 c^6}+\frac {8 (-b-2 c x+4)^6}{3 c^6} \]
Antiderivative was successfully verified.
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Rule 43
Rule 610
Rubi steps
\begin {align*} \int \left (\frac {-16+b^2}{4 c}+b x+c x^2\right )^5 \, dx &=\frac {\int \left (\frac {1}{2} (-4+b)+c x\right )^5 \left (\frac {4+b}{2}+c x\right )^5 \, dx}{c^5}\\ &=\frac {\int \left (1024 \left (\frac {1}{2} (-4+b)+c x\right )^5+1280 \left (\frac {1}{2} (-4+b)+c x\right )^6+640 \left (\frac {1}{2} (-4+b)+c x\right )^7+160 \left (\frac {1}{2} (-4+b)+c x\right )^8+20 \left (\frac {1}{2} (-4+b)+c x\right )^9+\left (\frac {1}{2} (-4+b)+c x\right )^{10}\right ) \, dx}{c^5}\\ &=\frac {8 (4-b-2 c x)^6}{3 c^6}-\frac {10 (4-b-2 c x)^7}{7 c^6}+\frac {5 (4-b-2 c x)^8}{16 c^6}-\frac {5 (4-b-2 c x)^9}{144 c^6}+\frac {(4-b-2 c x)^{10}}{512 c^6}-\frac {(4-b-2 c x)^{11}}{22528 c^6}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 207, normalized size = 1.90 \[ \frac {5}{8} \left (3 b^3-16 b\right ) c^2 x^8+\frac {\left (b^2-16\right )^5 x}{1024 c^5}+\frac {5 b \left (b^2-16\right )^4 x^2}{512 c^4}+\frac {5}{36} \left (9 b^2-16\right ) c^3 x^9+\frac {5 \left (b^2-16\right )^3 \left (9 b^2-16\right ) x^3}{768 c^3}+\frac {5 b \left (b^2-16\right )^2 \left (3 b^2-16\right ) x^4}{64 c^2}+\frac {5}{56} \left (21 b^4-224 b^2+256\right ) c x^7+\frac {\left (b^2-16\right ) \left (21 b^4-224 b^2+256\right ) x^5}{32 c}+\frac {1}{48} b \left (63 b^4-1120 b^2+3840\right ) x^6+\frac {1}{2} b c^4 x^{10}+\frac {c^5 x^{11}}{11} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 235, normalized size = 2.16 \[ \frac {64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 98560 \, {\left (9 \, b^{2} - 16\right )} c^{8} x^{9} + 443520 \, {\left (3 \, b^{3} - 16 \, b\right )} c^{7} x^{8} + 63360 \, {\left (21 \, b^{4} - 224 \, b^{2} + 256\right )} c^{6} x^{7} + 14784 \, {\left (63 \, b^{5} - 1120 \, b^{3} + 3840 \, b\right )} c^{5} x^{6} + 22176 \, {\left (21 \, b^{6} - 560 \, b^{4} + 3840 \, b^{2} - 4096\right )} c^{4} x^{5} + 55440 \, {\left (3 \, b^{7} - 112 \, b^{5} + 1280 \, b^{3} - 4096 \, b\right )} c^{3} x^{4} + 4620 \, {\left (9 \, b^{8} - 448 \, b^{6} + 7680 \, b^{4} - 49152 \, b^{2} + 65536\right )} c^{2} x^{3} + 6930 \, {\left (b^{9} - 64 \, b^{7} + 1536 \, b^{5} - 16384 \, b^{3} + 65536 \, b\right )} c x^{2} + 693 \, {\left (b^{10} - 80 \, b^{8} + 2560 \, b^{6} - 40960 \, b^{4} + 327680 \, b^{2} - 1048576\right )} x}{709632 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.53, size = 334, normalized size = 3.06 \[ \frac {64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 887040 \, b^{2} c^{8} x^{9} + 1330560 \, b^{3} c^{7} x^{8} + 1330560 \, b^{4} c^{6} x^{7} - 1576960 \, c^{8} x^{9} + 931392 \, b^{5} c^{5} x^{6} - 7096320 \, b c^{7} x^{8} + 465696 \, b^{6} c^{4} x^{5} - 14192640 \, b^{2} c^{6} x^{7} + 166320 \, b^{7} c^{3} x^{4} - 16558080 \, b^{3} c^{5} x^{6} + 41580 \, b^{8} c^{2} x^{3} - 12418560 \, b^{4} c^{4} x^{5} + 16220160 \, c^{6} x^{7} + 6930 \, b^{9} c x^{2} - 6209280 \, b^{5} c^{3} x^{4} + 56770560 \, b c^{5} x^{6} + 693 \, b^{10} x - 2069760 \, b^{6} c^{2} x^{3} + 85155840 \, b^{2} c^{4} x^{5} - 443520 \, b^{7} c x^{2} + 70963200 \, b^{3} c^{3} x^{4} - 55440 \, b^{8} x + 35481600 \, b^{4} c^{2} x^{3} - 90832896 \, c^{4} x^{5} + 10644480 \, b^{5} c x^{2} - 227082240 \, b c^{3} x^{4} + 1774080 \, b^{6} x - 227082240 \, b^{2} c^{2} x^{3} - 113541120 \, b^{3} c x^{2} - 28385280 \, b^{4} x + 302776320 \, c^{2} x^{3} + 454164480 \, b c x^{2} + 227082240 \, b^{2} x - 726663168 \, x}{709632 \, c^{5}} \]
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}-16\right ) c^{3}}{4}+\left (4 b^{2} c^{2}+2 \left (\frac {3 b^{2}}{2}-8\right ) c^{2}\right ) c \right ) x^{9}}{9}+\frac {\left (\left (b^{2}-16\right ) b \,c^{2}+\left (4 b^{2} c^{2}+2 \left (\frac {3 b^{2}}{2}-8\right ) c^{2}\right ) b +\left (\left (b^{2}-16\right ) b c +4 \left (\frac {3 b^{2}}{2}-8\right ) b c \right ) c \right ) x^{8}}{8}+\frac {\left (\left (\left (b^{2}-16\right ) b c +4 \left (\frac {3 b^{2}}{2}-8\right ) b c \right ) b +\left (2 \left (b^{2}-16\right ) b^{2}+\frac {\left (b^{2}-16\right )^{2}}{8}+\left (\frac {3 b^{2}}{2}-8\right )^{2}\right ) c +\frac {\left (b^{2}-16\right ) \left (4 b^{2} c^{2}+2 \left (\frac {3 b^{2}}{2}-8\right ) c^{2}\right )}{4 c}\right ) x^{7}}{7}+\frac {\left (\left (2 \left (b^{2}-16\right ) b^{2}+\frac {\left (b^{2}-16\right )^{2}}{8}+\left (\frac {3 b^{2}}{2}-8\right )^{2}\right ) b +\left (\frac {\left (b^{2}-16\right )^{2} b}{4 c}+\frac {\left (b^{2}-16\right ) \left (\frac {3 b^{2}}{2}-8\right ) b}{c}\right ) c +\frac {\left (b^{2}-16\right ) \left (\left (b^{2}-16\right ) b c +4 \left (\frac {3 b^{2}}{2}-8\right ) b c \right )}{4 c}\right ) x^{6}}{6}+\frac {\left (\left (\frac {\left (b^{2}-16\right )^{2} b}{4 c}+\frac {\left (b^{2}-16\right ) \left (\frac {3 b^{2}}{2}-8\right ) b}{c}\right ) b +\left (\frac {\left (b^{2}-16\right )^{2} b^{2}}{4 c^{2}}+\frac {\left (b^{2}-16\right )^{2} \left (\frac {3 b^{2}}{2}-8\right )}{8 c^{2}}\right ) c +\frac {\left (b^{2}-16\right ) \left (2 \left (b^{2}-16\right ) b^{2}+\frac {\left (b^{2}-16\right )^{2}}{8}+\left (\frac {3 b^{2}}{2}-8\right )^{2}\right )}{4 c}\right ) x^{5}}{5}+\frac {\left (\left (\frac {\left (b^{2}-16\right )^{2} b^{2}}{4 c^{2}}+\frac {\left (b^{2}-16\right )^{2} \left (\frac {3 b^{2}}{2}-8\right )}{8 c^{2}}\right ) b +\frac {\left (b^{2}-16\right )^{3} b}{16 c^{2}}+\frac {\left (b^{2}-16\right ) \left (\frac {\left (b^{2}-16\right )^{2} b}{4 c}+\frac {\left (b^{2}-16\right ) \left (\frac {3 b^{2}}{2}-8\right ) b}{c}\right )}{4 c}\right ) x^{4}}{4}+\frac {\left (\frac {\left (b^{2}-16\right )^{3} b^{2}}{16 c^{3}}+\frac {\left (b^{2}-16\right ) \left (\frac {\left (b^{2}-16\right )^{2} b^{2}}{4 c^{2}}+\frac {\left (b^{2}-16\right )^{2} \left (\frac {3 b^{2}}{2}-8\right )}{8 c^{2}}\right )}{4 c}+\frac {\left (b^{2}-16\right )^{4}}{256 c^{3}}\right ) x^{3}}{3}+\frac {5 \left (b^{2}-16\right )^{4} b \,x^{2}}{512 c^{4}}+\frac {\left (b^{2}-16\right )^{5} x}{1024 c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.40, size = 234, normalized size = 2.15 \[ \frac {1}{11} \, c^{5} x^{11} + \frac {1}{2} \, b c^{4} x^{10} + \frac {10}{9} \, b^{2} c^{3} x^{9} + \frac {5}{4} \, b^{3} c^{2} x^{8} + \frac {5}{7} \, b^{4} c x^{7} + \frac {1}{6} \, b^{5} x^{6} + \frac {5 \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} {\left (b^{2} - 16\right )}^{4}}{1536 \, c^{4}} + \frac {{\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} {\left (b^{2} - 16\right )}^{3}}{192 \, c^{3}} + \frac {{\left (20 \, c^{3} x^{7} + 70 \, b c^{2} x^{6} + 84 \, b^{2} c x^{5} + 35 \, b^{3} x^{4}\right )} {\left (b^{2} - 16\right )}^{2}}{224 \, c^{2}} + \frac {{\left (70 \, c^{4} x^{9} + 315 \, b c^{3} x^{8} + 540 \, b^{2} c^{2} x^{7} + 420 \, b^{3} c x^{6} + 126 \, b^{4} x^{5}\right )} {\left (b^{2} - 16\right )}}{504 \, c} + \frac {{\left (b^{2} - 16\right )}^{5} x}{1024 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 184, normalized size = 1.69 \[ \frac {c^5\,x^{11}}{11}+\frac {x\,{\left (b^2-16\right )}^5}{1024\,c^5}+\frac {b\,x^6\,\left (63\,b^4-1120\,b^2+3840\right )}{48}+\frac {5\,c\,x^7\,\left (21\,b^4-224\,b^2+256\right )}{56}+\frac {b\,c^4\,x^{10}}{2}+\frac {5\,c^3\,x^9\,\left (9\,b^2-16\right )}{36}+\frac {x^5\,\left (21\,b^6-560\,b^4+3840\,b^2-4096\right )}{32\,c}+\frac {5\,b\,c^2\,x^8\,\left (3\,b^2-16\right )}{8}+\frac {5\,b\,x^2\,{\left (b^2-16\right )}^4}{512\,c^4}+\frac {5\,x^3\,{\left (b^2-16\right )}^3\,\left (9\,b^2-16\right )}{768\,c^3}+\frac {5\,b\,x^4\,{\left (b^2-16\right )}^2\,\left (3\,b^2-16\right )}{64\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.20, size = 248, normalized size = 2.28 \[ \frac {b c^{4} x^{10}}{2} + \frac {c^{5} x^{11}}{11} + x^{9} \left (\frac {5 b^{2} c^{3}}{4} - \frac {20 c^{3}}{9}\right ) + x^{8} \left (\frac {15 b^{3} c^{2}}{8} - 10 b c^{2}\right ) + x^{7} \left (\frac {15 b^{4} c}{8} - 20 b^{2} c + \frac {160 c}{7}\right ) + x^{6} \left (\frac {21 b^{5}}{16} - \frac {70 b^{3}}{3} + 80 b\right ) + \frac {x^{5} \left (21 b^{6} - 560 b^{4} + 3840 b^{2} - 4096\right )}{32 c} + \frac {x^{4} \left (15 b^{7} - 560 b^{5} + 6400 b^{3} - 20480 b\right )}{64 c^{2}} + \frac {x^{3} \left (45 b^{8} - 2240 b^{6} + 38400 b^{4} - 245760 b^{2} + 327680\right )}{768 c^{3}} + \frac {x^{2} \left (5 b^{9} - 320 b^{7} + 7680 b^{5} - 81920 b^{3} + 327680 b\right )}{512 c^{4}} + \frac {x \left (b^{10} - 80 b^{8} + 2560 b^{6} - 40960 b^{4} + 327680 b^{2} - 1048576\right )}{1024 c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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