Integrand size = 23, antiderivative size = 109 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {(1-b-2 c x)^6}{384 c^6}-\frac {5 (1-b-2 c x)^7}{896 c^6}+\frac {5 (1-b-2 c x)^8}{1024 c^6}-\frac {5 (1-b-2 c x)^9}{2304 c^6}+\frac {(1-b-2 c x)^{10}}{2048 c^6}-\frac {(1-b-2 c x)^{11}}{22528 c^6} \] Output:
1/384*(-2*c*x-b+1)^6/c^6-5/896*(-2*c*x-b+1)^7/c^6+5/1024*(-2*c*x-b+1)^8/c^ 6-5/2304*(-2*c*x-b+1)^9/c^6+1/2048*(-2*c*x-b+1)^10/c^6-1/22528*(-2*c*x-b+1 )^11/c^6
Time = 0.03 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.89 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {\left (-1+b^2\right )^5 x}{1024 c^5}+\frac {5 b \left (-1+b^2\right )^4 x^2}{512 c^4}+\frac {5 \left (-1+b^2\right )^3 \left (-1+9 b^2\right ) x^3}{768 c^3}+\frac {5 b \left (-1+b^2\right )^2 \left (-1+3 b^2\right ) x^4}{64 c^2}+\frac {\left (-1+b^2\right ) \left (1-14 b^2+21 b^4\right ) x^5}{32 c}+\frac {1}{48} b \left (15-70 b^2+63 b^4\right ) x^6+\frac {5}{56} \left (1-14 b^2+21 b^4\right ) c x^7+\frac {5}{8} b \left (-1+3 b^2\right ) c^2 x^8+\frac {5}{36} \left (-1+9 b^2\right ) c^3 x^9+\frac {1}{2} b c^4 x^{10}+\frac {c^5 x^{11}}{11} \] Input:
Integrate[((-1 + b^2)/(4*c) + b*x + c*x^2)^5,x]
Output:
((-1 + b^2)^5*x)/(1024*c^5) + (5*b*(-1 + b^2)^4*x^2)/(512*c^4) + (5*(-1 + b^2)^3*(-1 + 9*b^2)*x^3)/(768*c^3) + (5*b*(-1 + b^2)^2*(-1 + 3*b^2)*x^4)/( 64*c^2) + ((-1 + b^2)*(1 - 14*b^2 + 21*b^4)*x^5)/(32*c) + (b*(15 - 70*b^2 + 63*b^4)*x^6)/48 + (5*(1 - 14*b^2 + 21*b^4)*c*x^7)/56 + (5*b*(-1 + 3*b^2) *c^2*x^8)/8 + (5*(-1 + 9*b^2)*c^3*x^9)/36 + (b*c^4*x^10)/2 + (c^5*x^11)/11
Time = 0.58 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1084, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (\frac {b^2-1}{4 c}+b x+c x^2\right )^5 \, dx\) |
| \(\Big \downarrow \) 1084 |
| \(\displaystyle \frac {\int \left (\left (\frac {b-1}{2}+c x\right )^{10}+\left (\frac {b-1}{2}+c x\right )^5-\frac {5}{512} (-b-2 c x+1)^9+\frac {5}{128} (-b-2 c x+1)^8-\frac {5}{64} (-b-2 c x+1)^7+\frac {5}{64} (-b-2 c x+1)^6\right )dx}{c^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {(-b-2 c x+1)^{11}}{22528 c}+\frac {(-b-2 c x+1)^{10}}{2048 c}-\frac {5 (-b-2 c x+1)^9}{2304 c}+\frac {5 (-b-2 c x+1)^8}{1024 c}-\frac {5 (-b-2 c x+1)^7}{896 c}+\frac {(-b-2 c x+1)^6}{384 c}}{c^5}\) |
Input:
Int[((-1 + b^2)/(4*c) + b*x + c*x^2)^5,x]
Output:
((1 - b - 2*c*x)^6/(384*c) - (5*(1 - b - 2*c*x)^7)/(896*c) + (5*(1 - b - 2 *c*x)^8)/(1024*c) - (5*(1 - b - 2*c*x)^9)/(2304*c) + (1 - b - 2*c*x)^10/(2 048*c) - (1 - b - 2*c*x)^11/(22528*c))/c^5
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[(b/2 - q/2 + c*x)^p*(b/2 + q /2 + c*x)^p, x], x], x] /; !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c}, x] && IntegerQ[p] && NiceSqrtQ[b^2 - 4*a*c]
Leaf count of result is larger than twice the leaf count of optimal. \(272\) vs. \(2(97)=194\).
Time = 0.59 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.50
| method | result | size |
| norman | \(\frac {\left (\frac {5}{4} b^{2} c^{7}-\frac {5}{36} c^{7}\right ) x^{9}+\left (\frac {15}{8} b^{3} c^{6}-\frac {5}{8} b \,c^{6}\right ) x^{8}+\left (\frac {15}{8} b^{4} c^{5}-\frac {5}{4} c^{5} b^{2}+\frac {5}{56} c^{5}\right ) x^{7}+\left (\frac {21}{16} b^{5} c^{4}-\frac {35}{24} b^{3} c^{4}+\frac {5}{16} b \,c^{4}\right ) x^{6}+\left (\frac {21}{32} b^{6} c^{3}-\frac {35}{32} b^{4} c^{3}+\frac {15}{32} c^{3} b^{2}-\frac {1}{32} c^{3}\right ) x^{5}+\left (\frac {15}{64} b^{7} c^{2}-\frac {35}{64} b^{5} c^{2}+\frac {25}{64} b^{3} c^{2}-\frac {5}{64} b \,c^{2}\right ) x^{4}+\left (\frac {5}{512} b^{9}-\frac {5}{128} b^{7}+\frac {15}{256} b^{5}-\frac {5}{128} b^{3}+\frac {5}{512} b \right ) x^{2}+\left (\frac {15}{256} b^{8} c -\frac {35}{192} b^{6} c +\frac {25}{128} b^{4} c -\frac {5}{64} b^{2} c +\frac {5}{768} c \right ) x^{3}+\frac {c^{9} x^{11}}{11}+\frac {b \,c^{8} x^{10}}{2}+\frac {\left (b^{10}-5 b^{8}+10 b^{6}-10 b^{4}+5 b^{2}-1\right ) x}{1024 c}}{c^{4}}\) | \(273\) |
| gosper | \(\frac {x \left (64512 c^{10} x^{10}+354816 c^{9} b \,x^{9}+887040 x^{8} b^{2} c^{8}+1330560 b^{3} c^{7} x^{7}+1330560 x^{6} b^{4} c^{6}-98560 x^{8} c^{8}+931392 x^{5} b^{5} c^{5}-443520 b \,c^{7} x^{7}+465696 b^{6} c^{4} x^{4}-887040 x^{6} c^{6} b^{2}+166320 b^{7} c^{3} x^{3}-1034880 x^{5} b^{3} c^{5}+41580 x^{2} b^{8} c^{2}-776160 b^{4} c^{4} x^{4}+63360 x^{6} c^{6}+6930 b^{9} c x -388080 b^{5} c^{3} x^{3}+221760 x^{5} b \,c^{5}+693 b^{10}-129360 x^{2} b^{6} c^{2}+332640 x^{4} b^{2} c^{4}-27720 b^{7} c x +277200 b^{3} c^{3} x^{3}-3465 b^{8}+138600 c^{2} x^{2} b^{4}-22176 c^{4} x^{4}+41580 x c \,b^{5}-55440 b \,c^{3} x^{3}+6930 b^{6}-55440 b^{2} c^{2} x^{2}-27720 b^{3} c x -6930 b^{4}+4620 c^{2} x^{2}+6930 c b x +3465 b^{2}-693\right )}{709632 c^{5}}\) | \(319\) |
| parallelrisch | \(\frac {64512 c^{10} x^{11}+354816 c^{9} b \,x^{10}+887040 x^{9} b^{2} c^{8}+1330560 b^{3} c^{7} x^{8}+1330560 x^{7} b^{4} c^{6}-98560 x^{9} c^{8}+931392 x^{6} b^{5} c^{5}-443520 b \,c^{7} x^{8}+465696 b^{6} c^{4} x^{5}-887040 x^{7} c^{6} b^{2}+166320 b^{7} c^{3} x^{4}-1034880 x^{6} b^{3} c^{5}+41580 x^{3} b^{8} c^{2}-776160 c^{4} b^{4} x^{5}+63360 x^{7} c^{6}+6930 b^{9} c \,x^{2}-388080 b^{5} c^{3} x^{4}+221760 x^{6} b \,c^{5}+693 b^{10} x -129360 x^{3} b^{6} c^{2}+332640 b^{2} c^{4} x^{5}-27720 b^{7} c \,x^{2}+277200 b^{3} c^{3} x^{4}-3465 b^{8} x +138600 b^{4} c^{2} x^{3}-22176 c^{4} x^{5}+41580 b^{5} c \,x^{2}-55440 b \,c^{3} x^{4}+6930 b^{6} x -55440 b^{2} c^{2} x^{3}-27720 b^{3} c \,x^{2}-6930 b^{4} x +4620 x^{3} c^{2}+6930 b c \,x^{2}+3465 b^{2} x -693 x}{709632 c^{5}}\) | \(335\) |
| risch | \(\frac {5 b \,x^{6}}{16}-\frac {35 b^{3} x^{6}}{24}-\frac {5 b^{2} c \,x^{7}}{4}+\frac {15 x^{3} b^{8}}{256 c^{3}}-\frac {35 x^{3} b^{6}}{192 c^{3}}+\frac {15 c^{2} b^{3} x^{8}}{8}-\frac {5 b^{3} x^{2}}{128 c^{4}}+\frac {5 b^{9} x^{2}}{512 c^{4}}-\frac {5 c^{2} b \,x^{8}}{8}+\frac {21 b^{6} x^{5}}{32 c}+\frac {15 b^{2} x^{5}}{32 c}-\frac {x}{1024 c^{5}}+\frac {5 x^{3}}{768 c^{3}}-\frac {x^{5}}{32 c}-\frac {5 c^{3} x^{9}}{36}+\frac {5 c \,x^{7}}{56}+\frac {25 b^{4} x^{3}}{128 c^{3}}-\frac {5 b^{7} x^{2}}{128 c^{4}}+\frac {15 b^{5} x^{2}}{256 c^{4}}+\frac {c^{5} x^{11}}{11}+\frac {b^{10} x}{1024 c^{5}}-\frac {5 b^{8} x}{1024 c^{5}}+\frac {5 b^{6} x}{512 c^{5}}-\frac {5 b^{4} x}{512 c^{5}}+\frac {15 b^{7} x^{4}}{64 c^{2}}-\frac {35 b^{5} x^{4}}{64 c^{2}}-\frac {5 b \,x^{4}}{64 c^{2}}+\frac {5 c^{3} x^{9} b^{2}}{4}+\frac {b \,c^{4} x^{10}}{2}-\frac {5 b^{2} x^{3}}{64 c^{3}}+\frac {5 b^{2} x}{1024 c^{5}}+\frac {5 b \,x^{2}}{512 c^{4}}+\frac {21 x^{6} b^{5}}{16}-\frac {35 b^{4} x^{5}}{32 c}+\frac {25 b^{3} x^{4}}{64 c^{2}}+\frac {15 b^{4} c \,x^{7}}{8}\) | \(343\) |
| orering | \(\frac {x \left (64512 c^{10} x^{10}+354816 c^{9} b \,x^{9}+887040 x^{8} b^{2} c^{8}+1330560 b^{3} c^{7} x^{7}+1330560 x^{6} b^{4} c^{6}-98560 x^{8} c^{8}+931392 x^{5} b^{5} c^{5}-443520 b \,c^{7} x^{7}+465696 b^{6} c^{4} x^{4}-887040 x^{6} c^{6} b^{2}+166320 b^{7} c^{3} x^{3}-1034880 x^{5} b^{3} c^{5}+41580 x^{2} b^{8} c^{2}-776160 b^{4} c^{4} x^{4}+63360 x^{6} c^{6}+6930 b^{9} c x -388080 b^{5} c^{3} x^{3}+221760 x^{5} b \,c^{5}+693 b^{10}-129360 x^{2} b^{6} c^{2}+332640 x^{4} b^{2} c^{4}-27720 b^{7} c x +277200 b^{3} c^{3} x^{3}-3465 b^{8}+138600 c^{2} x^{2} b^{4}-22176 c^{4} x^{4}+41580 x c \,b^{5}-55440 b \,c^{3} x^{3}+6930 b^{6}-55440 b^{2} c^{2} x^{2}-27720 b^{3} c x -6930 b^{4}+4620 c^{2} x^{2}+6930 c b x +3465 b^{2}-693\right ) \left (\frac {b^{2}-1}{4 c}+b x +c \,x^{2}\right )^{5}}{693 \left (2 c x +b +1\right )^{5} \left (2 c x +b -1\right )^{5}}\) | \(355\) |
| default | \(\frac {c^{5} x^{11}}{11}+\frac {b \,c^{4} x^{10}}{2}+\frac {\left (256 \left (b^{2}-1\right ) c^{3}+4096 c^{3} b^{2}+4 c \left (32 \left (24 b^{2}-8\right ) c^{2}+1024 b^{2} c^{2}\right )\right ) x^{9}}{9216}+\frac {\left (1024 \left (b^{2}-1\right ) c^{2} b +4 b \left (32 \left (24 b^{2}-8\right ) c^{2}+1024 b^{2} c^{2}\right )+4 c \left (256 \left (b^{2}-1\right ) c b +64 \left (24 b^{2}-8\right ) b c \right )\right ) x^{8}}{8192}+\frac {\left (\frac {\left (b^{2}-1\right ) \left (32 \left (24 b^{2}-8\right ) c^{2}+1024 b^{2} c^{2}\right )}{c}+4 b \left (256 \left (b^{2}-1\right ) c b +64 \left (24 b^{2}-8\right ) b c \right )+4 c \left (32 \left (b^{2}-1\right )^{2}+512 \left (b^{2}-1\right ) b^{2}+\left (24 b^{2}-8\right )^{2}\right )\right ) x^{7}}{7168}+\frac {\left (\frac {\left (b^{2}-1\right ) \left (256 \left (b^{2}-1\right ) c b +64 \left (24 b^{2}-8\right ) b c \right )}{c}+4 b \left (32 \left (b^{2}-1\right )^{2}+512 \left (b^{2}-1\right ) b^{2}+\left (24 b^{2}-8\right )^{2}\right )+4 c \left (\frac {64 \left (b^{2}-1\right )^{2} b}{c}+\frac {16 \left (b^{2}-1\right ) b \left (24 b^{2}-8\right )}{c}\right )\right ) x^{6}}{6144}+\frac {\left (\frac {\left (b^{2}-1\right ) \left (32 \left (b^{2}-1\right )^{2}+512 \left (b^{2}-1\right ) b^{2}+\left (24 b^{2}-8\right )^{2}\right )}{c}+4 b \left (\frac {64 \left (b^{2}-1\right )^{2} b}{c}+\frac {16 \left (b^{2}-1\right ) b \left (24 b^{2}-8\right )}{c}\right )+4 c \left (\frac {2 \left (b^{2}-1\right )^{2} \left (24 b^{2}-8\right )}{c^{2}}+\frac {64 \left (b^{2}-1\right )^{2} b^{2}}{c^{2}}\right )\right ) x^{5}}{5120}+\frac {\left (\frac {\left (b^{2}-1\right ) \left (\frac {64 \left (b^{2}-1\right )^{2} b}{c}+\frac {16 \left (b^{2}-1\right ) b \left (24 b^{2}-8\right )}{c}\right )}{c}+4 b \left (\frac {2 \left (b^{2}-1\right )^{2} \left (24 b^{2}-8\right )}{c^{2}}+\frac {64 \left (b^{2}-1\right )^{2} b^{2}}{c^{2}}\right )+\frac {64 \left (b^{2}-1\right )^{3} b}{c^{2}}\right ) x^{4}}{4096}+\frac {\left (\frac {\left (b^{2}-1\right ) \left (\frac {2 \left (b^{2}-1\right )^{2} \left (24 b^{2}-8\right )}{c^{2}}+\frac {64 \left (b^{2}-1\right )^{2} b^{2}}{c^{2}}\right )}{c}+\frac {64 b^{2} \left (b^{2}-1\right )^{3}}{c^{3}}+\frac {4 \left (b^{2}-1\right )^{4}}{c^{3}}\right ) x^{3}}{3072}+\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}}\) | \(648\) |
Input:
int((1/4*(b^2-1)/c+b*x+c*x^2)^5,x,method=_RETURNVERBOSE)
Output:
((5/4*b^2*c^7-5/36*c^7)*x^9+(15/8*b^3*c^6-5/8*b*c^6)*x^8+(15/8*b^4*c^5-5/4 *c^5*b^2+5/56*c^5)*x^7+(21/16*b^5*c^4-35/24*b^3*c^4+5/16*b*c^4)*x^6+(21/32 *b^6*c^3-35/32*b^4*c^3+15/32*c^3*b^2-1/32*c^3)*x^5+(15/64*b^7*c^2-35/64*b^ 5*c^2+25/64*b^3*c^2-5/64*b*c^2)*x^4+(5/512*b^9-5/128*b^7+15/256*b^5-5/128* b^3+5/512*b)*x^2+(15/256*b^8*c-35/192*b^6*c+25/128*b^4*c-5/64*b^2*c+5/768* c)*x^3+1/11*c^9*x^11+1/2*b*c^8*x^10+1/1024*(b^10-5*b^8+10*b^6-10*b^4+5*b^2 -1)/c*x)/c^4
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (85) = 170\).
Time = 0.08 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.14 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 98560 \, {\left (9 \, b^{2} - 1\right )} c^{8} x^{9} + 443520 \, {\left (3 \, b^{3} - b\right )} c^{7} x^{8} + 63360 \, {\left (21 \, b^{4} - 14 \, b^{2} + 1\right )} c^{6} x^{7} + 14784 \, {\left (63 \, b^{5} - 70 \, b^{3} + 15 \, b\right )} c^{5} x^{6} + 22176 \, {\left (21 \, b^{6} - 35 \, b^{4} + 15 \, b^{2} - 1\right )} c^{4} x^{5} + 55440 \, {\left (3 \, b^{7} - 7 \, b^{5} + 5 \, b^{3} - b\right )} c^{3} x^{4} + 4620 \, {\left (9 \, b^{8} - 28 \, b^{6} + 30 \, b^{4} - 12 \, b^{2} + 1\right )} c^{2} x^{3} + 6930 \, {\left (b^{9} - 4 \, b^{7} + 6 \, b^{5} - 4 \, b^{3} + b\right )} c x^{2} + 693 \, {\left (b^{10} - 5 \, b^{8} + 10 \, b^{6} - 10 \, b^{4} + 5 \, b^{2} - 1\right )} x}{709632 \, c^{5}} \] Input:
integrate((1/4*(b^2-1)/c+b*x+c*x^2)^5,x, algorithm="fricas")
Output:
1/709632*(64512*c^10*x^11 + 354816*b*c^9*x^10 + 98560*(9*b^2 - 1)*c^8*x^9 + 443520*(3*b^3 - b)*c^7*x^8 + 63360*(21*b^4 - 14*b^2 + 1)*c^6*x^7 + 14784 *(63*b^5 - 70*b^3 + 15*b)*c^5*x^6 + 22176*(21*b^6 - 35*b^4 + 15*b^2 - 1)*c ^4*x^5 + 55440*(3*b^7 - 7*b^5 + 5*b^3 - b)*c^3*x^4 + 4620*(9*b^8 - 28*b^6 + 30*b^4 - 12*b^2 + 1)*c^2*x^3 + 6930*(b^9 - 4*b^7 + 6*b^5 - 4*b^3 + b)*c* x^2 + 693*(b^10 - 5*b^8 + 10*b^6 - 10*b^4 + 5*b^2 - 1)*x)/c^5
Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (95) = 190\).
Time = 0.11 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.32 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {b c^{4} x^{10}}{2} + \frac {c^{5} x^{11}}{11} + x^{9} \cdot \left (\frac {5 b^{2} c^{3}}{4} - \frac {5 c^{3}}{36}\right ) + x^{8} \cdot \left (\frac {15 b^{3} c^{2}}{8} - \frac {5 b c^{2}}{8}\right ) + x^{7} \cdot \left (\frac {15 b^{4} c}{8} - \frac {5 b^{2} c}{4} + \frac {5 c}{56}\right ) + x^{6} \cdot \left (\frac {21 b^{5}}{16} - \frac {35 b^{3}}{24} + \frac {5 b}{16}\right ) + \frac {x^{5} \cdot \left (21 b^{6} - 35 b^{4} + 15 b^{2} - 1\right )}{32 c} + \frac {x^{4} \cdot \left (15 b^{7} - 35 b^{5} + 25 b^{3} - 5 b\right )}{64 c^{2}} + \frac {x^{3} \cdot \left (45 b^{8} - 140 b^{6} + 150 b^{4} - 60 b^{2} + 5\right )}{768 c^{3}} + \frac {x^{2} \cdot \left (5 b^{9} - 20 b^{7} + 30 b^{5} - 20 b^{3} + 5 b\right )}{512 c^{4}} + \frac {x \left (b^{10} - 5 b^{8} + 10 b^{6} - 10 b^{4} + 5 b^{2} - 1\right )}{1024 c^{5}} \] Input:
integrate((1/4*(b**2-1)/c+b*x+c*x**2)**5,x)
Output:
b*c**4*x**10/2 + c**5*x**11/11 + x**9*(5*b**2*c**3/4 - 5*c**3/36) + x**8*( 15*b**3*c**2/8 - 5*b*c**2/8) + x**7*(15*b**4*c/8 - 5*b**2*c/4 + 5*c/56) + x**6*(21*b**5/16 - 35*b**3/24 + 5*b/16) + x**5*(21*b**6 - 35*b**4 + 15*b** 2 - 1)/(32*c) + x**4*(15*b**7 - 35*b**5 + 25*b**3 - 5*b)/(64*c**2) + x**3* (45*b**8 - 140*b**6 + 150*b**4 - 60*b**2 + 5)/(768*c**3) + x**2*(5*b**9 - 20*b**7 + 30*b**5 - 20*b**3 + 5*b)/(512*c**4) + x*(b**10 - 5*b**8 + 10*b** 6 - 10*b**4 + 5*b**2 - 1)/(1024*c**5)
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (85) = 170\).
Time = 0.03 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.15 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {1}{11} \, c^{5} x^{11} + \frac {1}{2} \, b c^{4} x^{10} + \frac {10}{9} \, b^{2} c^{3} x^{9} + \frac {5}{4} \, b^{3} c^{2} x^{8} + \frac {5}{7} \, b^{4} c x^{7} + \frac {1}{6} \, b^{5} x^{6} + \frac {5 \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} {\left (b^{2} - 1\right )}^{4}}{1536 \, c^{4}} + \frac {{\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} {\left (b^{2} - 1\right )}^{3}}{192 \, c^{3}} + \frac {{\left (20 \, c^{3} x^{7} + 70 \, b c^{2} x^{6} + 84 \, b^{2} c x^{5} + 35 \, b^{3} x^{4}\right )} {\left (b^{2} - 1\right )}^{2}}{224 \, c^{2}} + \frac {{\left (70 \, c^{4} x^{9} + 315 \, b c^{3} x^{8} + 540 \, b^{2} c^{2} x^{7} + 420 \, b^{3} c x^{6} + 126 \, b^{4} x^{5}\right )} {\left (b^{2} - 1\right )}}{504 \, c} + \frac {{\left (b^{2} - 1\right )}^{5} x}{1024 \, c^{5}} \] Input:
integrate((1/4*(b^2-1)/c+b*x+c*x^2)^5,x, algorithm="maxima")
Output:
1/11*c^5*x^11 + 1/2*b*c^4*x^10 + 10/9*b^2*c^3*x^9 + 5/4*b^3*c^2*x^8 + 5/7* b^4*c*x^7 + 1/6*b^5*x^6 + 5/1536*(2*c*x^3 + 3*b*x^2)*(b^2 - 1)^4/c^4 + 1/1 92*(6*c^2*x^5 + 15*b*c*x^4 + 10*b^2*x^3)*(b^2 - 1)^3/c^3 + 1/224*(20*c^3*x ^7 + 70*b*c^2*x^6 + 84*b^2*c*x^5 + 35*b^3*x^4)*(b^2 - 1)^2/c^2 + 1/504*(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)* (b^2 - 1)/c + 1/1024*(b^2 - 1)^5*x/c^5
Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (85) = 170\).
Time = 0.15 (sec) , antiderivative size = 334, normalized size of antiderivative = 3.06 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 887040 \, b^{2} c^{8} x^{9} + 1330560 \, b^{3} c^{7} x^{8} + 1330560 \, b^{4} c^{6} x^{7} - 98560 \, c^{8} x^{9} + 931392 \, b^{5} c^{5} x^{6} - 443520 \, b c^{7} x^{8} + 465696 \, b^{6} c^{4} x^{5} - 887040 \, b^{2} c^{6} x^{7} + 166320 \, b^{7} c^{3} x^{4} - 1034880 \, b^{3} c^{5} x^{6} + 41580 \, b^{8} c^{2} x^{3} - 776160 \, b^{4} c^{4} x^{5} + 63360 \, c^{6} x^{7} + 6930 \, b^{9} c x^{2} - 388080 \, b^{5} c^{3} x^{4} + 221760 \, b c^{5} x^{6} + 693 \, b^{10} x - 129360 \, b^{6} c^{2} x^{3} + 332640 \, b^{2} c^{4} x^{5} - 27720 \, b^{7} c x^{2} + 277200 \, b^{3} c^{3} x^{4} - 3465 \, b^{8} x + 138600 \, b^{4} c^{2} x^{3} - 22176 \, c^{4} x^{5} + 41580 \, b^{5} c x^{2} - 55440 \, b c^{3} x^{4} + 6930 \, b^{6} x - 55440 \, b^{2} c^{2} x^{3} - 27720 \, b^{3} c x^{2} - 6930 \, b^{4} x + 4620 \, c^{2} x^{3} + 6930 \, b c x^{2} + 3465 \, b^{2} x - 693 \, x}{709632 \, c^{5}} \] Input:
integrate((1/4*(b^2-1)/c+b*x+c*x^2)^5,x, algorithm="giac")
Output:
1/709632*(64512*c^10*x^11 + 354816*b*c^9*x^10 + 887040*b^2*c^8*x^9 + 13305 60*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 + 6 3360*c^6*x^7 + 6930*b^9*c*x^2 - 388080*b^5*c^3*x^4 + 221760*b*c^5*x^6 + 69 3*b^10*x - 129360*b^6*c^2*x^3 + 332640*b^2*c^4*x^5 - 27720*b^7*c*x^2 + 277 200*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)/c^5
Time = 9.84 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.69 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {c^5\,x^{11}}{11}+\frac {x\,{\left (b^2-1\right )}^5}{1024\,c^5}+\frac {b\,x^6\,\left (63\,b^4-70\,b^2+15\right )}{48}+\frac {5\,c\,x^7\,\left (21\,b^4-14\,b^2+1\right )}{56}+\frac {b\,c^4\,x^{10}}{2}+\frac {5\,c^3\,x^9\,\left (9\,b^2-1\right )}{36}+\frac {x^5\,\left (21\,b^6-35\,b^4+15\,b^2-1\right )}{32\,c}+\frac {5\,b\,c^2\,x^8\,\left (3\,b^2-1\right )}{8}+\frac {5\,b\,x^2\,{\left (b^2-1\right )}^4}{512\,c^4}+\frac {5\,x^3\,{\left (b^2-1\right )}^3\,\left (9\,b^2-1\right )}{768\,c^3}+\frac {5\,b\,x^4\,{\left (b^2-1\right )}^2\,\left (3\,b^2-1\right )}{64\,c^2} \] Input:
int((b*x + c*x^2 + (b^2/4 - 1/4)/c)^5,x)
Output:
(c^5*x^11)/11 + (x*(b^2 - 1)^5)/(1024*c^5) + (b*x^6*(63*b^4 - 70*b^2 + 15) )/48 + (5*c*x^7*(21*b^4 - 14*b^2 + 1))/56 + (b*c^4*x^10)/2 + (5*c^3*x^9*(9 *b^2 - 1))/36 + (x^5*(15*b^2 - 35*b^4 + 21*b^6 - 1))/(32*c) + (5*b*c^2*x^8 *(3*b^2 - 1))/8 + (5*b*x^2*(b^2 - 1)^4)/(512*c^4) + (5*x^3*(b^2 - 1)^3*(9* b^2 - 1))/(768*c^3) + (5*b*x^4*(b^2 - 1)^2*(3*b^2 - 1))/(64*c^2)
Time = 0.22 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.92 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {x \left (64512 c^{10} x^{10}+354816 b \,c^{9} x^{9}+887040 b^{2} c^{8} x^{8}+1330560 b^{3} c^{7} x^{7}+1330560 b^{4} c^{6} x^{6}-98560 c^{8} x^{8}+931392 b^{5} c^{5} x^{5}-443520 b \,c^{7} x^{7}+465696 b^{6} c^{4} x^{4}-887040 b^{2} c^{6} x^{6}+166320 b^{7} c^{3} x^{3}-1034880 b^{3} c^{5} x^{5}+41580 b^{8} c^{2} x^{2}-776160 b^{4} c^{4} x^{4}+63360 c^{6} x^{6}+6930 b^{9} c x -388080 b^{5} c^{3} x^{3}+221760 b \,c^{5} x^{5}+693 b^{10}-129360 b^{6} c^{2} x^{2}+332640 b^{2} c^{4} x^{4}-27720 b^{7} c x +277200 b^{3} c^{3} x^{3}-3465 b^{8}+138600 b^{4} c^{2} x^{2}-22176 c^{4} x^{4}+41580 b^{5} c x -55440 b \,c^{3} x^{3}+6930 b^{6}-55440 b^{2} c^{2} x^{2}-27720 b^{3} c x -6930 b^{4}+4620 c^{2} x^{2}+6930 b c x +3465 b^{2}-693\right )}{709632 c^{5}} \] Input:
int((1/4*(b^2-1)/c+b*x+c*x^2)^5,x)
Output:
(x*(693*b**10 + 6930*b**9*c*x + 41580*b**8*c**2*x**2 - 3465*b**8 + 166320* b**7*c**3*x**3 - 27720*b**7*c*x + 465696*b**6*c**4*x**4 - 129360*b**6*c**2 *x**2 + 6930*b**6 + 931392*b**5*c**5*x**5 - 388080*b**5*c**3*x**3 + 41580* b**5*c*x + 1330560*b**4*c**6*x**6 - 776160*b**4*c**4*x**4 + 138600*b**4*c* *2*x**2 - 6930*b**4 + 1330560*b**3*c**7*x**7 - 1034880*b**3*c**5*x**5 + 27 7200*b**3*c**3*x**3 - 27720*b**3*c*x + 887040*b**2*c**8*x**8 - 887040*b**2 *c**6*x**6 + 332640*b**2*c**4*x**4 - 55440*b**2*c**2*x**2 + 3465*b**2 + 35 4816*b*c**9*x**9 - 443520*b*c**7*x**7 + 221760*b*c**5*x**5 - 55440*b*c**3* x**3 + 6930*b*c*x + 64512*c**10*x**10 - 98560*c**8*x**8 + 63360*c**6*x**6 - 22176*c**4*x**4 + 4620*c**2*x**2 - 693))/(709632*c**5)