\(\int (\frac {-1+b^2}{4 c}+b x+c x^2)^5 \, dx\) [74]

Optimal result

Integrand size = 23, antiderivative size = 109 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {(1-b-2 c x)^6}{384 c^6}-\frac {5 (1-b-2 c x)^7}{896 c^6}+\frac {5 (1-b-2 c x)^8}{1024 c^6}-\frac {5 (1-b-2 c x)^9}{2304 c^6}+\frac {(1-b-2 c x)^{10}}{2048 c^6}-\frac {(1-b-2 c x)^{11}}{22528 c^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {624, 45} \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=-\frac {(-b-2 c x+1)^{11}}{22528 c^6}+\frac {(-b-2 c x+1)^{10}}{2048 c^6}-\frac {5 (-b-2 c x+1)^9}{2304 c^6}+\frac {5 (-b-2 c x+1)^8}{1024 c^6}-\frac {5 (-b-2 c x+1)^7}{896 c^6}+\frac {(-b-2 c x+1)^6}{384 c^6} \]

[In]

Int[((-1 + b^2)/(4*c) + b*x + c*x^2)^5,x]

[Out]

(1 - b - 2*c*x)^6/(384*c^6) - (5*(1 - b - 2*c*x)^7)/(896*c^6) + (5*(1 - b - 2*c*x)^8)/(1024*c^6) - (5*(1 - b -
2*c*x)^9)/(2304*c^6) + (1 - b - 2*c*x)^10/(2048*c^6) - (1 - b - 2*c*x)^11/(22528*c^6)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 624

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[1/c^p, Int[Simp[
b/2 - q/2 + c*x, x]^p*Simp[b/2 + q/2 + c*x, x]^p, x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGt
Q[p, 0] && PerfectSquareQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.89 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {\left (-1+b^2\right )^5 x}{1024 c^5}+\frac {5 b \left (-1+b^2\right )^4 x^2}{512 c^4}+\frac {5 \left (-1+b^2\right )^3 \left (-1+9 b^2\right ) x^3}{768 c^3}+\frac {5 b \left (-1+b^2\right )^2 \left (-1+3 b^2\right ) x^4}{64 c^2}+\frac {\left (-1+b^2\right ) \left (1-14 b^2+21 b^4\right ) x^5}{32 c}+\frac {1}{48} b \left (15-70 b^2+63 b^4\right ) x^6+\frac {5}{56} \left (1-14 b^2+21 b^4\right ) c x^7+\frac {5}{8} b \left (-1+3 b^2\right ) c^2 x^8+\frac {5}{36} \left (-1+9 b^2\right ) c^3 x^9+\frac {1}{2} b c^4 x^{10}+\frac {c^5 x^{11}}{11} \]

[In]

Integrate[((-1 + b^2)/(4*c) + b*x + c*x^2)^5,x]

[Out]

((-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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(272\) vs. \(2(97)=194\).

Time = 2.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.50

method	result	size
norman	\(\frac {\left (\frac {5}{4} b^{2} c^{7}-\frac {5}{36} c^{7}\right ) x^{9}+\left (\frac {15}{8} b^{3} c^{6}-\frac {5}{8} b \,c^{6}\right ) x^{8}+\left (\frac {15}{8} b^{4} c^{5}-\frac {5}{4} b^{2} c^{5}+\frac {5}{56} c^{5}\right ) x^{7}+\left (\frac {21}{16} b^{5} c^{4}-\frac {35}{24} c^{4} b^{3}+\frac {5}{16} b \,c^{4}\right ) x^{6}+\left (\frac {15}{64} b^{7} c^{2}-\frac {35}{64} b^{5} c^{2}+\frac {25}{64} c^{2} b^{3}-\frac {5}{64} b \,c^{2}\right ) x^{4}+\left (\frac {21}{32} c^{3} b^{6}-\frac {35}{32} b^{4} c^{3}+\frac {15}{32} b^{2} c^{3}-\frac {1}{32} c^{3}\right ) x^{5}+\left (\frac {5}{512} b^{9}-\frac {5}{128} b^{7}+\frac {15}{256} b^{5}-\frac {5}{128} b^{3}+\frac {5}{512} b \right ) x^{2}+\left (\frac {15}{256} b^{8} c -\frac {35}{192} b^{6} c +\frac {25}{128} b^{4} c -\frac {5}{64} b^{2} c +\frac {5}{768} c \right ) x^{3}+\frac {c^{9} x^{11}}{11}+\frac {b \,c^{8} x^{10}}{2}+\frac {\left (b^{10}-5 b^{8}+10 b^{6}-10 b^{4}+5 b^{2}-1\right ) x}{1024 c}}{c^{4}}\)	\(273\)
gosper	\(\frac {x \left (64512 c^{10} x^{10}+354816 c^{9} b \,x^{9}+887040 x^{8} b^{2} c^{8}+1330560 b^{3} c^{7} x^{7}+1330560 x^{6} b^{4} c^{6}-98560 x^{8} c^{8}+931392 x^{5} b^{5} c^{5}-443520 b \,c^{7} x^{7}+465696 b^{6} c^{4} x^{4}-887040 x^{6} b^{2} c^{6}+166320 b^{7} c^{3} x^{3}-1034880 x^{5} b^{3} c^{5}+41580 x^{2} b^{8} c^{2}-776160 b^{4} c^{4} x^{4}+63360 x^{6} c^{6}+6930 b^{9} c x -388080 x^{3} b^{5} c^{3}+221760 x^{5} b \,c^{5}+693 b^{10}-129360 x^{2} c^{2} b^{6}+332640 b^{2} c^{4} x^{4}-27720 b^{7} c x +277200 x^{3} b^{3} c^{3}-3465 b^{8}+138600 x^{2} b^{4} c^{2}-22176 c^{4} x^{4}+41580 b^{5} c x -55440 b \,c^{3} x^{3}+6930 b^{6}-55440 b^{2} c^{2} x^{2}-27720 b^{3} c x -6930 b^{4}+4620 c^{2} x^{2}+6930 b c x +3465 b^{2}-693\right )}{709632 c^{5}}\)	\(319\)
parallelrisch	\(\frac {64512 c^{10} x^{11}+354816 c^{9} b \,x^{10}+887040 x^{9} b^{2} c^{8}+1330560 b^{3} c^{7} x^{8}+1330560 x^{7} b^{4} c^{6}-98560 x^{9} c^{8}+931392 x^{6} b^{5} c^{5}-443520 b \,c^{7} x^{8}+465696 b^{6} c^{4} x^{5}-887040 x^{7} b^{2} c^{6}+166320 b^{7} c^{3} x^{4}-1034880 x^{6} b^{3} c^{5}+41580 x^{3} b^{8} c^{2}-776160 x^{5} b^{4} c^{4}+63360 x^{7} c^{6}+6930 b^{9} c \,x^{2}-388080 b^{5} c^{3} x^{4}+221760 x^{6} b \,c^{5}+693 b^{10} x -129360 x^{3} c^{2} b^{6}+332640 b^{2} c^{4} x^{5}-27720 b^{7} c \,x^{2}+277200 c^{3} b^{3} x^{4}-3465 b^{8} x +138600 b^{4} c^{2} x^{3}-22176 c^{4} x^{5}+41580 b^{5} c \,x^{2}-55440 b \,c^{3} x^{4}+6930 b^{6} x -55440 b^{2} c^{2} x^{3}-27720 b^{3} c \,x^{2}-6930 b^{4} x +4620 c^{2} x^{3}+6930 c b \,x^{2}+3465 b^{2} x -693 x}{709632 c^{5}}\)	\(335\)
risch	\(\frac {5 b \,x^{6}}{16}-\frac {35 b^{3} x^{6}}{24}-\frac {x}{1024 c^{5}}+\frac {21 b^{5} x^{6}}{16}-\frac {x^{5}}{32 c}+\frac {15 b^{4} c \,x^{7}}{8}+\frac {21 b^{6} x^{5}}{32 c}+\frac {15 b^{2} x^{5}}{32 c}+\frac {5 c^{3} x^{9} b^{2}}{4}+\frac {15 b^{7} x^{4}}{64 c^{2}}-\frac {35 b^{5} x^{4}}{64 c^{2}}+\frac {5 b^{9} x^{2}}{512 c^{4}}-\frac {5 b^{7} x^{2}}{128 c^{4}}+\frac {15 b^{5} x^{2}}{256 c^{4}}-\frac {5 b^{3} x^{2}}{128 c^{4}}+\frac {15 x^{3} b^{8}}{256 c^{3}}-\frac {35 x^{3} b^{6}}{192 c^{3}}+\frac {b \,c^{4} x^{10}}{2}+\frac {15 b^{3} c^{2} x^{8}}{8}-\frac {5 b \,x^{4}}{64 c^{2}}-\frac {5 c^{2} b \,x^{8}}{8}+\frac {5 b^{6} x}{512 c^{5}}-\frac {5 b^{4} x}{512 c^{5}}+\frac {c^{5} x^{11}}{11}-\frac {35 x^{5} b^{4}}{32 c}+\frac {25 b^{3} x^{4}}{64 c^{2}}+\frac {5 b \,x^{2}}{512 c^{4}}-\frac {5 b^{8} x}{1024 c^{5}}+\frac {5 c \,x^{7}}{56}+\frac {5 x^{3}}{768 c^{3}}-\frac {5 c^{3} x^{9}}{36}-\frac {5 b^{2} x^{3}}{64 c^{3}}+\frac {25 b^{4} x^{3}}{128 c^{3}}+\frac {b^{10} x}{1024 c^{5}}+\frac {5 b^{2} x}{1024 c^{5}}-\frac {5 b^{2} c \,x^{7}}{4}\)	\(343\)
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 b^{2} c^{3}+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}}\)	\(648\)

[In]

int((1/4*(b^2-1)/c+b*x+c*x^2)^5,x,method=_RETURNVERBOSE)

[Out]

((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*b^2*c^5+5/56*c^5)*x^7+(21/16*b^5*c^
4-35/24*c^4*b^3+5/16*b*c^4)*x^6+(15/64*b^7*c^2-35/64*b^5*c^2+25/64*c^2*b^3-5/64*b*c^2)*x^4+(21/32*c^3*b^6-35/3
2*b^4*c^3+15/32*b^2*c^3-1/32*c^3)*x^5+(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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (85) = 170\).

Time = 0.26 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.14 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 98560 \, {\left (9 \, b^{2} - 1\right )} c^{8} x^{9} + 443520 \, {\left (3 \, b^{3} - b\right )} c^{7} x^{8} + 63360 \, {\left (21 \, b^{4} - 14 \, b^{2} + 1\right )} c^{6} x^{7} + 14784 \, {\left (63 \, b^{5} - 70 \, b^{3} + 15 \, b\right )} c^{5} x^{6} + 22176 \, {\left (21 \, b^{6} - 35 \, b^{4} + 15 \, b^{2} - 1\right )} c^{4} x^{5} + 55440 \, {\left (3 \, b^{7} - 7 \, b^{5} + 5 \, b^{3} - b\right )} c^{3} x^{4} + 4620 \, {\left (9 \, b^{8} - 28 \, b^{6} + 30 \, b^{4} - 12 \, b^{2} + 1\right )} c^{2} x^{3} + 6930 \, {\left (b^{9} - 4 \, b^{7} + 6 \, b^{5} - 4 \, b^{3} + b\right )} c x^{2} + 693 \, {\left (b^{10} - 5 \, b^{8} + 10 \, b^{6} - 10 \, b^{4} + 5 \, b^{2} - 1\right )} x}{709632 \, c^{5}} \]

[In]

integrate((1/4*(b^2-1)/c+b*x+c*x^2)^5,x, algorithm="fricas")

[Out]

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 + 6
930*(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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (95) = 190\).

Time = 0.09 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.32 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {b c^{4} x^{10}}{2} + \frac {c^{5} x^{11}}{11} + x^{9} \cdot \left (\frac {5 b^{2} c^{3}}{4} - \frac {5 c^{3}}{36}\right ) + x^{8} \cdot \left (\frac {15 b^{3} c^{2}}{8} - \frac {5 b c^{2}}{8}\right ) + x^{7} \cdot \left (\frac {15 b^{4} c}{8} - \frac {5 b^{2} c}{4} + \frac {5 c}{56}\right ) + x^{6} \cdot \left (\frac {21 b^{5}}{16} - \frac {35 b^{3}}{24} + \frac {5 b}{16}\right ) + \frac {x^{5} \cdot \left (21 b^{6} - 35 b^{4} + 15 b^{2} - 1\right )}{32 c} + \frac {x^{4} \cdot \left (15 b^{7} - 35 b^{5} + 25 b^{3} - 5 b\right )}{64 c^{2}} + \frac {x^{3} \cdot \left (45 b^{8} - 140 b^{6} + 150 b^{4} - 60 b^{2} + 5\right )}{768 c^{3}} + \frac {x^{2} \cdot \left (5 b^{9} - 20 b^{7} + 30 b^{5} - 20 b^{3} + 5 b\right )}{512 c^{4}} + \frac {x \left (b^{10} - 5 b^{8} + 10 b^{6} - 10 b^{4} + 5 b^{2} - 1\right )}{1024 c^{5}} \]

[In]

integrate((1/4*(b**2-1)/c+b*x+c*x**2)**5,x)

[Out]

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 - 6
0*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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (85) = 170\).

Time = 0.19 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.15 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {1}{11} \, c^{5} x^{11} + \frac {1}{2} \, b c^{4} x^{10} + \frac {10}{9} \, b^{2} c^{3} x^{9} + \frac {5}{4} \, b^{3} c^{2} x^{8} + \frac {5}{7} \, b^{4} c x^{7} + \frac {1}{6} \, b^{5} x^{6} + \frac {5 \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} {\left (b^{2} - 1\right )}^{4}}{1536 \, c^{4}} + \frac {{\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} {\left (b^{2} - 1\right )}^{3}}{192 \, c^{3}} + \frac {{\left (20 \, c^{3} x^{7} + 70 \, b c^{2} x^{6} + 84 \, b^{2} c x^{5} + 35 \, b^{3} x^{4}\right )} {\left (b^{2} - 1\right )}^{2}}{224 \, c^{2}} + \frac {{\left (70 \, c^{4} x^{9} + 315 \, b c^{3} x^{8} + 540 \, b^{2} c^{2} x^{7} + 420 \, b^{3} c x^{6} + 126 \, b^{4} x^{5}\right )} {\left (b^{2} - 1\right )}}{504 \, c} + \frac {{\left (b^{2} - 1\right )}^{5} x}{1024 \, c^{5}} \]

[In]

integrate((1/4*(b^2-1)/c+b*x+c*x^2)^5,x, algorithm="maxima")

[Out]

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 + 1/192*(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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (85) = 170\).

Time = 0.27 (sec) , antiderivative size = 334, normalized size of antiderivative = 3.06 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 887040 \, b^{2} c^{8} x^{9} + 1330560 \, b^{3} c^{7} x^{8} + 1330560 \, b^{4} c^{6} x^{7} - 98560 \, c^{8} x^{9} + 931392 \, b^{5} c^{5} x^{6} - 443520 \, b c^{7} x^{8} + 465696 \, b^{6} c^{4} x^{5} - 887040 \, b^{2} c^{6} x^{7} + 166320 \, b^{7} c^{3} x^{4} - 1034880 \, b^{3} c^{5} x^{6} + 41580 \, b^{8} c^{2} x^{3} - 776160 \, b^{4} c^{4} x^{5} + 63360 \, c^{6} x^{7} + 6930 \, b^{9} c x^{2} - 388080 \, b^{5} c^{3} x^{4} + 221760 \, b c^{5} x^{6} + 693 \, b^{10} x - 129360 \, b^{6} c^{2} x^{3} + 332640 \, b^{2} c^{4} x^{5} - 27720 \, b^{7} c x^{2} + 277200 \, b^{3} c^{3} x^{4} - 3465 \, b^{8} x + 138600 \, b^{4} c^{2} x^{3} - 22176 \, c^{4} x^{5} + 41580 \, b^{5} c x^{2} - 55440 \, b c^{3} x^{4} + 6930 \, b^{6} x - 55440 \, b^{2} c^{2} x^{3} - 27720 \, b^{3} c x^{2} - 6930 \, b^{4} x + 4620 \, c^{2} x^{3} + 6930 \, b c x^{2} + 3465 \, b^{2} x - 693 \, x}{709632 \, c^{5}} \]

[In]

integrate((1/4*(b^2-1)/c+b*x+c*x^2)^5,x, algorithm="giac")

[Out]

1/709632*(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 - 38
8080*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 + 69
30*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

Mupad [B] (verification not implemented)

Time = 9.16 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.69 \[ \int \left (\frac {-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {c^5\,x^{11}}{11}+\frac {x\,{\left (b^2-1\right )}^5}{1024\,c^5}+\frac {b\,x^6\,\left (63\,b^4-70\,b^2+15\right )}{48}+\frac {5\,c\,x^7\,\left (21\,b^4-14\,b^2+1\right )}{56}+\frac {b\,c^4\,x^{10}}{2}+\frac {5\,c^3\,x^9\,\left (9\,b^2-1\right )}{36}+\frac {x^5\,\left (21\,b^6-35\,b^4+15\,b^2-1\right )}{32\,c}+\frac {5\,b\,c^2\,x^8\,\left (3\,b^2-1\right )}{8}+\frac {5\,b\,x^2\,{\left (b^2-1\right )}^4}{512\,c^4}+\frac {5\,x^3\,{\left (b^2-1\right )}^3\,\left (9\,b^2-1\right )}{768\,c^3}+\frac {5\,b\,x^4\,{\left (b^2-1\right )}^2\,\left (3\,b^2-1\right )}{64\,c^2} \]

[In]

int((b*x + c*x^2 + (b^2/4 - 1/4)/c)^5,x)

[Out]

(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)