Optimal. Leaf size=109 \[ -\frac {(-b-2 c x+3)^{11}}{22528 c^6}+\frac {3 (-b-2 c x+3)^{10}}{2048 c^6}-\frac {5 (-b-2 c x+3)^9}{256 c^6}+\frac {135 (-b-2 c x+3)^8}{1024 c^6}-\frac {405 (-b-2 c x+3)^7}{896 c^6}+\frac {81 (-b-2 c x+3)^6}{128 c^6} \]
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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} \[ -\frac {(-b-2 c x+3)^{11}}{22528 c^6}+\frac {3 (-b-2 c x+3)^{10}}{2048 c^6}-\frac {5 (-b-2 c x+3)^9}{256 c^6}+\frac {135 (-b-2 c x+3)^8}{1024 c^6}-\frac {405 (-b-2 c x+3)^7}{896 c^6}+\frac {81 (-b-2 c x+3)^6}{128 c^6} \]
Antiderivative was successfully verified.
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Rule 43
Rule 610
Rubi steps
\begin {align*} \int \left (\frac {-9+b^2}{4 c}+b x+c x^2\right )^5 \, dx &=\frac {\int \left (\frac {1}{2} (-3+b)+c x\right )^5 \left (\frac {3+b}{2}+c x\right )^5 \, dx}{c^5}\\ &=\frac {\int \left (243 \left (\frac {1}{2} (-3+b)+c x\right )^5+405 \left (\frac {1}{2} (-3+b)+c x\right )^6+270 \left (\frac {1}{2} (-3+b)+c x\right )^7+90 \left (\frac {1}{2} (-3+b)+c x\right )^8+15 \left (\frac {1}{2} (-3+b)+c x\right )^9+\left (\frac {1}{2} (-3+b)+c x\right )^{10}\right ) \, dx}{c^5}\\ &=\frac {81 (3-b-2 c x)^6}{128 c^6}-\frac {405 (3-b-2 c x)^7}{896 c^6}+\frac {135 (3-b-2 c x)^8}{1024 c^6}-\frac {5 (3-b-2 c x)^9}{256 c^6}+\frac {3 (3-b-2 c x)^{10}}{2048 c^6}-\frac {(3-b-2 c x)^{11}}{22528 c^6}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 199, normalized size = 1.83 \[ \frac {15}{8} \left (b^3-3 b\right ) c^2 x^8+\frac {\left (b^2-9\right )^5 x}{1024 c^5}+\frac {5 b \left (b^2-9\right )^4 x^2}{512 c^4}+\frac {5}{4} \left (b^2-1\right ) c^3 x^9+\frac {15 \left (b^2-9\right )^3 \left (b^2-1\right ) x^3}{256 c^3}+\frac {15 b \left (b^2-9\right )^2 \left (b^2-3\right ) x^4}{64 c^2}+\frac {15}{56} \left (7 b^4-42 b^2+27\right ) c x^7+\frac {3 \left (b^2-9\right ) \left (7 b^4-42 b^2+27\right ) x^5}{32 c}+\frac {3}{16} b \left (7 b^4-70 b^2+135\right ) x^6+\frac {1}{2} b c^4 x^{10}+\frac {c^5 x^{11}}{11} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 227, normalized size = 2.08 \[ \frac {7168 \, c^{10} x^{11} + 39424 \, b c^{9} x^{10} + 98560 \, {\left (b^{2} - 1\right )} c^{8} x^{9} + 147840 \, {\left (b^{3} - 3 \, b\right )} c^{7} x^{8} + 21120 \, {\left (7 \, b^{4} - 42 \, b^{2} + 27\right )} c^{6} x^{7} + 14784 \, {\left (7 \, b^{5} - 70 \, b^{3} + 135 \, b\right )} c^{5} x^{6} + 7392 \, {\left (7 \, b^{6} - 105 \, b^{4} + 405 \, b^{2} - 243\right )} c^{4} x^{5} + 18480 \, {\left (b^{7} - 21 \, b^{5} + 135 \, b^{3} - 243 \, b\right )} c^{3} x^{4} + 4620 \, {\left (b^{8} - 28 \, b^{6} + 270 \, b^{4} - 972 \, b^{2} + 729\right )} c^{2} x^{3} + 770 \, {\left (b^{9} - 36 \, b^{7} + 486 \, b^{5} - 2916 \, b^{3} + 6561 \, b\right )} c x^{2} + 77 \, {\left (b^{10} - 45 \, b^{8} + 810 \, b^{6} - 7290 \, b^{4} + 32805 \, b^{2} - 59049\right )} x}{78848 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 334, normalized size = 3.06 \[ \frac {7168 \, c^{10} x^{11} + 39424 \, b c^{9} x^{10} + 98560 \, b^{2} c^{8} x^{9} + 147840 \, b^{3} c^{7} x^{8} + 147840 \, b^{4} c^{6} x^{7} - 98560 \, c^{8} x^{9} + 103488 \, b^{5} c^{5} x^{6} - 443520 \, b c^{7} x^{8} + 51744 \, b^{6} c^{4} x^{5} - 887040 \, b^{2} c^{6} x^{7} + 18480 \, b^{7} c^{3} x^{4} - 1034880 \, b^{3} c^{5} x^{6} + 4620 \, b^{8} c^{2} x^{3} - 776160 \, b^{4} c^{4} x^{5} + 570240 \, c^{6} x^{7} + 770 \, b^{9} c x^{2} - 388080 \, b^{5} c^{3} x^{4} + 1995840 \, b c^{5} x^{6} + 77 \, b^{10} x - 129360 \, b^{6} c^{2} x^{3} + 2993760 \, b^{2} c^{4} x^{5} - 27720 \, b^{7} c x^{2} + 2494800 \, b^{3} c^{3} x^{4} - 3465 \, b^{8} x + 1247400 \, b^{4} c^{2} x^{3} - 1796256 \, c^{4} x^{5} + 374220 \, b^{5} c x^{2} - 4490640 \, b c^{3} x^{4} + 62370 \, b^{6} x - 4490640 \, b^{2} c^{2} x^{3} - 2245320 \, b^{3} c x^{2} - 561330 \, b^{4} x + 3367980 \, c^{2} x^{3} + 5051970 \, b c x^{2} + 2525985 \, b^{2} x - 4546773 \, x}{78848 \, c^{5}} \]
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}-9\right ) c^{3}}{4}+\left (4 b^{2} c^{2}+2 \left (\frac {3 b^{2}}{2}-\frac {9}{2}\right ) c^{2}\right ) c \right ) x^{9}}{9}+\frac {\left (\left (b^{2}-9\right ) b \,c^{2}+\left (4 b^{2} c^{2}+2 \left (\frac {3 b^{2}}{2}-\frac {9}{2}\right ) c^{2}\right ) b +\left (\left (b^{2}-9\right ) b c +4 \left (\frac {3 b^{2}}{2}-\frac {9}{2}\right ) b c \right ) c \right ) x^{8}}{8}+\frac {\left (\left (\left (b^{2}-9\right ) b c +4 \left (\frac {3 b^{2}}{2}-\frac {9}{2}\right ) b c \right ) b +\left (2 \left (b^{2}-9\right ) b^{2}+\frac {\left (b^{2}-9\right )^{2}}{8}+\left (\frac {3 b^{2}}{2}-\frac {9}{2}\right )^{2}\right ) c +\frac {\left (b^{2}-9\right ) \left (4 b^{2} c^{2}+2 \left (\frac {3 b^{2}}{2}-\frac {9}{2}\right ) c^{2}\right )}{4 c}\right ) x^{7}}{7}+\frac {\left (\left (2 \left (b^{2}-9\right ) b^{2}+\frac {\left (b^{2}-9\right )^{2}}{8}+\left (\frac {3 b^{2}}{2}-\frac {9}{2}\right )^{2}\right ) b +\left (\frac {\left (b^{2}-9\right )^{2} b}{4 c}+\frac {\left (b^{2}-9\right ) \left (\frac {3 b^{2}}{2}-\frac {9}{2}\right ) b}{c}\right ) c +\frac {\left (b^{2}-9\right ) \left (\left (b^{2}-9\right ) b c +4 \left (\frac {3 b^{2}}{2}-\frac {9}{2}\right ) b c \right )}{4 c}\right ) x^{6}}{6}+\frac {\left (\left (\frac {\left (b^{2}-9\right )^{2} b}{4 c}+\frac {\left (b^{2}-9\right ) \left (\frac {3 b^{2}}{2}-\frac {9}{2}\right ) b}{c}\right ) b +\left (\frac {\left (b^{2}-9\right )^{2} b^{2}}{4 c^{2}}+\frac {\left (b^{2}-9\right )^{2} \left (\frac {3 b^{2}}{2}-\frac {9}{2}\right )}{8 c^{2}}\right ) c +\frac {\left (b^{2}-9\right ) \left (2 \left (b^{2}-9\right ) b^{2}+\frac {\left (b^{2}-9\right )^{2}}{8}+\left (\frac {3 b^{2}}{2}-\frac {9}{2}\right )^{2}\right )}{4 c}\right ) x^{5}}{5}+\frac {\left (\left (\frac {\left (b^{2}-9\right )^{2} b^{2}}{4 c^{2}}+\frac {\left (b^{2}-9\right )^{2} \left (\frac {3 b^{2}}{2}-\frac {9}{2}\right )}{8 c^{2}}\right ) b +\frac {\left (b^{2}-9\right )^{3} b}{16 c^{2}}+\frac {\left (b^{2}-9\right ) \left (\frac {\left (b^{2}-9\right )^{2} b}{4 c}+\frac {\left (b^{2}-9\right ) \left (\frac {3 b^{2}}{2}-\frac {9}{2}\right ) b}{c}\right )}{4 c}\right ) x^{4}}{4}+\frac {\left (\frac {\left (b^{2}-9\right )^{3} b^{2}}{16 c^{3}}+\frac {\left (b^{2}-9\right ) \left (\frac {\left (b^{2}-9\right )^{2} b^{2}}{4 c^{2}}+\frac {\left (b^{2}-9\right )^{2} \left (\frac {3 b^{2}}{2}-\frac {9}{2}\right )}{8 c^{2}}\right )}{4 c}+\frac {\left (b^{2}-9\right )^{4}}{256 c^{3}}\right ) x^{3}}{3}+\frac {5 \left (b^{2}-9\right )^{4} b \,x^{2}}{512 c^{4}}+\frac {\left (b^{2}-9\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 \[ \frac {1}{11} \, c^{5} x^{11} + \frac {1}{2} \, b c^{4} x^{10} + \frac {10}{9} \, b^{2} c^{3} x^{9} + \frac {5}{4} \, b^{3} c^{2} x^{8} + \frac {5}{7} \, b^{4} c x^{7} + \frac {1}{6} \, b^{5} x^{6} + \frac {5 \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} {\left (b^{2} - 9\right )}^{4}}{1536 \, c^{4}} + \frac {{\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} {\left (b^{2} - 9\right )}^{3}}{192 \, c^{3}} + \frac {{\left (20 \, c^{3} x^{7} + 70 \, b c^{2} x^{6} + 84 \, b^{2} c x^{5} + 35 \, b^{3} x^{4}\right )} {\left (b^{2} - 9\right )}^{2}}{224 \, c^{2}} + \frac {{\left (70 \, c^{4} x^{9} + 315 \, b c^{3} x^{8} + 540 \, b^{2} c^{2} x^{7} + 420 \, b^{3} c x^{6} + 126 \, b^{4} x^{5}\right )} {\left (b^{2} - 9\right )}}{504 \, c} + \frac {{\left (b^{2} - 9\right )}^{5} x}{1024 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 176, normalized size = 1.61 \[ \frac {c^5\,x^{11}}{11}+\frac {5\,c^3\,x^9\,\left (b^2-1\right )}{4}+\frac {x\,{\left (b^2-9\right )}^5}{1024\,c^5}+\frac {3\,b\,x^6\,\left (7\,b^4-70\,b^2+135\right )}{16}+\frac {15\,c\,x^7\,\left (7\,b^4-42\,b^2+27\right )}{56}+\frac {b\,c^4\,x^{10}}{2}+\frac {3\,x^5\,\left (7\,b^6-105\,b^4+405\,b^2-243\right )}{32\,c}+\frac {15\,b\,c^2\,x^8\,\left (b^2-3\right )}{8}+\frac {15\,x^3\,\left (b^2-1\right )\,{\left (b^2-9\right )}^3}{256\,c^3}+\frac {5\,b\,x^2\,{\left (b^2-9\right )}^4}{512\,c^4}+\frac {15\,b\,x^4\,\left (b^2-3\right )\,{\left (b^2-9\right )}^2}{64\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.20, size = 253, normalized size = 2.32 \[ \frac {b c^{4} x^{10}}{2} + \frac {c^{5} x^{11}}{11} + x^{9} \left (\frac {5 b^{2} c^{3}}{4} - \frac {5 c^{3}}{4}\right ) + x^{8} \left (\frac {15 b^{3} c^{2}}{8} - \frac {45 b c^{2}}{8}\right ) + x^{7} \left (\frac {15 b^{4} c}{8} - \frac {45 b^{2} c}{4} + \frac {405 c}{56}\right ) + x^{6} \left (\frac {21 b^{5}}{16} - \frac {105 b^{3}}{8} + \frac {405 b}{16}\right ) + \frac {x^{5} \left (21 b^{6} - 315 b^{4} + 1215 b^{2} - 729\right )}{32 c} + \frac {x^{4} \left (15 b^{7} - 315 b^{5} + 2025 b^{3} - 3645 b\right )}{64 c^{2}} + \frac {x^{3} \left (15 b^{8} - 420 b^{6} + 4050 b^{4} - 14580 b^{2} + 10935\right )}{256 c^{3}} + \frac {x^{2} \left (5 b^{9} - 180 b^{7} + 2430 b^{5} - 14580 b^{3} + 32805 b\right )}{512 c^{4}} + \frac {x \left (b^{10} - 45 b^{8} + 810 b^{6} - 7290 b^{4} + 32805 b^{2} - 59049\right )}{1024 c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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