\(\int (\frac {-4+b^2}{4 c}+b x+c x^2)^5 \, dx\) [75]

Optimal result

Integrand size = 23, antiderivative size = 109 \[ \int \left (\frac {-4+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {(2-b-2 c x)^6}{12 c^6}-\frac {5 (2-b-2 c x)^7}{56 c^6}+\frac {5 (2-b-2 c x)^8}{128 c^6}-\frac {5 (2-b-2 c x)^9}{576 c^6}+\frac {(2-b-2 c x)^{10}}{1024 c^6}-\frac {(2-b-2 c x)^{11}}{22528 c^6} \]

[Out]

1/12*(-2*c*x-b+2)^6/c^6-5/56*(-2*c*x-b+2)^7/c^6+5/128*(-2*c*x-b+2)^8/c^6-5/576*(-2*c*x-b+2)^9/c^6+1/1024*(-2*c
*x-b+2)^10/c^6-1/22528*(-2*c*x-b+2)^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} \[ \int \left (\frac {-4+b^2}{4 c}+b x+c x^2\right )^5 \, dx=-\frac {(-b-2 c x+2)^{11}}{22528 c^6}+\frac {(-b-2 c x+2)^{10}}{1024 c^6}-\frac {5 (-b-2 c x+2)^9}{576 c^6}+\frac {5 (-b-2 c x+2)^8}{128 c^6}-\frac {5 (-b-2 c x+2)^7}{56 c^6}+\frac {(-b-2 c x+2)^6}{12 c^6} \]

[In]

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

[Out]

(2 - b - 2*c*x)^6/(12*c^6) - (5*(2 - b - 2*c*x)^7)/(56*c^6) + (5*(2 - b - 2*c*x)^8)/(128*c^6) - (5*(2 - b - 2*
c*x)^9)/(576*c^6) + (2 - b - 2*c*x)^10/(1024*c^6) - (2 - 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} (-2+b)+c x\right )^5 \left (\frac {2+b}{2}+c x\right )^5 \, dx}{c^5} \\ & = \frac {\int \left (32 \left (\frac {1}{2} (-2+b)+c x\right )^5+80 \left (\frac {1}{2} (-2+b)+c x\right )^6+80 \left (\frac {1}{2} (-2+b)+c x\right )^7+40 \left (\frac {1}{2} (-2+b)+c x\right )^8+10 \left (\frac {1}{2} (-2+b)+c x\right )^9+\left (\frac {1}{2} (-2+b)+c x\right )^{10}\right ) \, dx}{c^5} \\ & = \frac {(2-b-2 c x)^6}{12 c^6}-\frac {5 (2-b-2 c x)^7}{56 c^6}+\frac {5 (2-b-2 c x)^8}{128 c^6}-\frac {5 (2-b-2 c x)^9}{576 c^6}+\frac {(2-b-2 c x)^{10}}{1024 c^6}-\frac {(2-b-2 c x)^{11}}{22528 c^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.89 \[ \int \left (\frac {-4+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {\left (-4+b^2\right )^5 x}{1024 c^5}+\frac {5 b \left (-4+b^2\right )^4 x^2}{512 c^4}+\frac {5 \left (-4+b^2\right )^3 \left (-4+9 b^2\right ) x^3}{768 c^3}+\frac {5 b \left (-4+b^2\right )^2 \left (-4+3 b^2\right ) x^4}{64 c^2}+\frac {\left (-4+b^2\right ) \left (16-56 b^2+21 b^4\right ) x^5}{32 c}+\frac {1}{48} b \left (240-280 b^2+63 b^4\right ) x^6+\frac {5}{56} \left (16-56 b^2+21 b^4\right ) c x^7+\frac {5}{8} b \left (-4+3 b^2\right ) c^2 x^8+\frac {5}{36} \left (-4+9 b^2\right ) c^3 x^9+\frac {1}{2} b c^4 x^{10}+\frac {c^5 x^{11}}{11} \]

[In]

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

[Out]

((-4 + b^2)^5*x)/(1024*c^5) + (5*b*(-4 + b^2)^4*x^2)/(512*c^4) + (5*(-4 + b^2)^3*(-4 + 9*b^2)*x^3)/(768*c^3) +
(5*b*(-4 + b^2)^2*(-4 + 3*b^2)*x^4)/(64*c^2) + ((-4 + b^2)*(16 - 56*b^2 + 21*b^4)*x^5)/(32*c) + (b*(240 - 280
*b^2 + 63*b^4)*x^6)/48 + (5*(16 - 56*b^2 + 21*b^4)*c*x^7)/56 + (5*b*(-4 + 3*b^2)*c^2*x^8)/8 + (5*(-4 + 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 = 2.23 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.50

method	result	size
norman	\(\frac {\left (\frac {5}{4} b^{2} c^{7}-\frac {5}{9} c^{7}\right ) x^{9}+\left (\frac {15}{8} b^{3} c^{6}-\frac {5}{2} b \,c^{6}\right ) x^{8}+\left (\frac {15}{8} b^{4} c^{5}-5 b^{2} c^{5}+\frac {10}{7} c^{5}\right ) x^{7}+\left (\frac {21}{16} b^{5} c^{4}-\frac {35}{6} c^{4} b^{3}+5 b \,c^{4}\right ) x^{6}+\left (\frac {15}{64} b^{7} c^{2}-\frac {35}{16} b^{5} c^{2}+\frac {25}{4} c^{2} b^{3}-5 b \,c^{2}\right ) x^{4}+\left (\frac {21}{32} c^{3} b^{6}-\frac {35}{8} b^{4} c^{3}+\frac {15}{2} b^{2} c^{3}-2 c^{3}\right ) x^{5}+\left (\frac {5}{512} b^{9}-\frac {5}{32} b^{7}+\frac {15}{16} b^{5}-\frac {5}{2} b^{3}+\frac {5}{2} b \right ) x^{2}+\left (\frac {15}{256} b^{8} c -\frac {35}{48} b^{6} c +\frac {25}{8} b^{4} c -5 b^{2} c +\frac {5}{3} c \right ) x^{3}+\frac {c^{9} x^{11}}{11}+\frac {b \,c^{8} x^{10}}{2}+\frac {\left (b^{10}-20 b^{8}+160 b^{6}-640 b^{4}+1280 b^{2}-1024\right ) x}{1024 c}}{c^{4}}\)	\(273\)
gosper	\(\frac {x \left (64512 c^{10} x^{10}+354816 c^{9} b \,x^{9}+887040 x^{8} b^{2} c^{8}+1330560 b^{3} c^{7} x^{7}+1330560 x^{6} b^{4} c^{6}-394240 x^{8} c^{8}+931392 x^{5} b^{5} c^{5}-1774080 b \,c^{7} x^{7}+465696 b^{6} c^{4} x^{4}-3548160 x^{6} b^{2} c^{6}+166320 b^{7} c^{3} x^{3}-4139520 x^{5} b^{3} c^{5}+41580 x^{2} b^{8} c^{2}-3104640 b^{4} c^{4} x^{4}+1013760 x^{6} c^{6}+6930 b^{9} c x -1552320 x^{3} b^{5} c^{3}+3548160 x^{5} b \,c^{5}+693 b^{10}-517440 x^{2} c^{2} b^{6}+5322240 b^{2} c^{4} x^{4}-110880 b^{7} c x +4435200 x^{3} b^{3} c^{3}-13860 b^{8}+2217600 x^{2} b^{4} c^{2}-1419264 c^{4} x^{4}+665280 b^{5} c x -3548160 b \,c^{3} x^{3}+110880 b^{6}-3548160 b^{2} c^{2} x^{2}-1774080 b^{3} c x -443520 b^{4}+1182720 c^{2} x^{2}+1774080 b c x +887040 b^{2}-709632\right )}{709632 c^{5}}\)	\(319\)
parallelrisch	\(\frac {64512 c^{10} x^{11}+354816 c^{9} b \,x^{10}+887040 x^{9} b^{2} c^{8}+1330560 b^{3} c^{7} x^{8}+1330560 x^{7} b^{4} c^{6}-394240 x^{9} c^{8}+931392 x^{6} b^{5} c^{5}-1774080 b \,c^{7} x^{8}+465696 b^{6} c^{4} x^{5}-3548160 x^{7} b^{2} c^{6}+166320 b^{7} c^{3} x^{4}-4139520 x^{6} b^{3} c^{5}+41580 x^{3} b^{8} c^{2}-3104640 x^{5} b^{4} c^{4}+1013760 x^{7} c^{6}+6930 b^{9} c \,x^{2}-1552320 b^{5} c^{3} x^{4}+3548160 x^{6} b \,c^{5}+693 b^{10} x -517440 x^{3} c^{2} b^{6}+5322240 b^{2} c^{4} x^{5}-110880 b^{7} c \,x^{2}+4435200 c^{3} b^{3} x^{4}-13860 b^{8} x +2217600 b^{4} c^{2} x^{3}-1419264 c^{4} x^{5}+665280 b^{5} c \,x^{2}-3548160 b \,c^{3} x^{4}+110880 b^{6} x -3548160 b^{2} c^{2} x^{3}-1774080 b^{3} c \,x^{2}-443520 b^{4} x +1182720 c^{2} x^{3}+1774080 c b \,x^{2}+887040 b^{2} x -709632 x}{709632 c^{5}}\)	\(335\)
risch	\(5 b \,x^{6}-\frac {35 b^{3} x^{6}}{6}-\frac {x}{c^{5}}+\frac {21 b^{5} x^{6}}{16}-\frac {2 x^{5}}{c}+\frac {15 b^{4} c \,x^{7}}{8}+\frac {21 b^{6} x^{5}}{32 c}+\frac {15 b^{2} x^{5}}{2 c}+\frac {5 c^{3} x^{9} b^{2}}{4}+\frac {15 b^{7} x^{4}}{64 c^{2}}-\frac {35 b^{5} x^{4}}{16 c^{2}}+\frac {5 b^{9} x^{2}}{512 c^{4}}-\frac {5 b^{7} x^{2}}{32 c^{4}}+\frac {15 b^{5} x^{2}}{16 c^{4}}-\frac {5 b^{3} x^{2}}{2 c^{4}}+\frac {15 x^{3} b^{8}}{256 c^{3}}-\frac {35 x^{3} b^{6}}{48 c^{3}}+\frac {b \,c^{4} x^{10}}{2}+\frac {15 b^{3} c^{2} x^{8}}{8}-\frac {5 b \,x^{4}}{c^{2}}-\frac {5 c^{2} b \,x^{8}}{2}+\frac {5 b^{6} x}{32 c^{5}}-\frac {5 b^{4} x}{8 c^{5}}+\frac {c^{5} x^{11}}{11}-\frac {35 x^{5} b^{4}}{8 c}+\frac {25 b^{3} x^{4}}{4 c^{2}}+\frac {5 b \,x^{2}}{2 c^{4}}-\frac {5 b^{8} x}{256 c^{5}}+\frac {10 c \,x^{7}}{7}+\frac {5 x^{3}}{3 c^{3}}-\frac {5 c^{3} x^{9}}{9}-\frac {5 b^{2} x^{3}}{c^{3}}+\frac {25 b^{4} x^{3}}{8 c^{3}}+\frac {b^{10} x}{1024 c^{5}}+\frac {5 b^{2} x}{4 c^{5}}-5 b^{2} c \,x^{7}\)	\(343\)
default	\(\frac {c^{5} x^{11}}{11}+\frac {b \,c^{4} x^{10}}{2}+\frac {\left (256 \left (b^{2}-4\right ) c^{3}+4096 b^{2} c^{3}+4 c \left (32 \left (24 b^{2}-32\right ) c^{2}+1024 b^{2} c^{2}\right )\right ) x^{9}}{9216}+\frac {\left (1024 \left (b^{2}-4\right ) c^{2} b +4 b \left (32 \left (24 b^{2}-32\right ) c^{2}+1024 b^{2} c^{2}\right )+4 c \left (256 \left (b^{2}-4\right ) c b +64 \left (24 b^{2}-32\right ) b c \right )\right ) x^{8}}{8192}+\frac {\left (\frac {\left (b^{2}-4\right ) \left (32 \left (24 b^{2}-32\right ) c^{2}+1024 b^{2} c^{2}\right )}{c}+4 b \left (256 \left (b^{2}-4\right ) c b +64 \left (24 b^{2}-32\right ) b c \right )+4 c \left (32 \left (b^{2}-4\right )^{2}+512 \left (b^{2}-4\right ) b^{2}+\left (24 b^{2}-32\right )^{2}\right )\right ) x^{7}}{7168}+\frac {\left (\frac {\left (b^{2}-4\right ) \left (256 \left (b^{2}-4\right ) c b +64 \left (24 b^{2}-32\right ) b c \right )}{c}+4 b \left (32 \left (b^{2}-4\right )^{2}+512 \left (b^{2}-4\right ) b^{2}+\left (24 b^{2}-32\right )^{2}\right )+4 c \left (\frac {64 \left (b^{2}-4\right )^{2} b}{c}+\frac {16 \left (b^{2}-4\right ) b \left (24 b^{2}-32\right )}{c}\right )\right ) x^{6}}{6144}+\frac {\left (\frac {\left (b^{2}-4\right ) \left (32 \left (b^{2}-4\right )^{2}+512 \left (b^{2}-4\right ) b^{2}+\left (24 b^{2}-32\right )^{2}\right )}{c}+4 b \left (\frac {64 \left (b^{2}-4\right )^{2} b}{c}+\frac {16 \left (b^{2}-4\right ) b \left (24 b^{2}-32\right )}{c}\right )+4 c \left (\frac {2 \left (b^{2}-4\right )^{2} \left (24 b^{2}-32\right )}{c^{2}}+\frac {64 \left (b^{2}-4\right )^{2} b^{2}}{c^{2}}\right )\right ) x^{5}}{5120}+\frac {\left (\frac {\left (b^{2}-4\right ) \left (\frac {64 \left (b^{2}-4\right )^{2} b}{c}+\frac {16 \left (b^{2}-4\right ) b \left (24 b^{2}-32\right )}{c}\right )}{c}+4 b \left (\frac {2 \left (b^{2}-4\right )^{2} \left (24 b^{2}-32\right )}{c^{2}}+\frac {64 \left (b^{2}-4\right )^{2} b^{2}}{c^{2}}\right )+\frac {64 \left (b^{2}-4\right )^{3} b}{c^{2}}\right ) x^{4}}{4096}+\frac {\left (\frac {\left (b^{2}-4\right ) \left (\frac {2 \left (b^{2}-4\right )^{2} \left (24 b^{2}-32\right )}{c^{2}}+\frac {64 \left (b^{2}-4\right )^{2} b^{2}}{c^{2}}\right )}{c}+\frac {64 b^{2} \left (b^{2}-4\right )^{3}}{c^{3}}+\frac {4 \left (b^{2}-4\right )^{4}}{c^{3}}\right ) x^{3}}{3072}+\frac {5 \left (b^{2}-4\right )^{4} b \,x^{2}}{512 c^{4}}+\frac {\left (b^{2}-4\right )^{5} x}{1024 c^{5}}\)	\(648\)

[In]

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

[Out]

((5/4*b^2*c^7-5/9*c^7)*x^9+(15/8*b^3*c^6-5/2*b*c^6)*x^8+(15/8*b^4*c^5-5*b^2*c^5+10/7*c^5)*x^7+(21/16*b^5*c^4-3
5/6*c^4*b^3+5*b*c^4)*x^6+(15/64*b^7*c^2-35/16*b^5*c^2+25/4*c^2*b^3-5*b*c^2)*x^4+(21/32*c^3*b^6-35/8*b^4*c^3+15
/2*b^2*c^3-2*c^3)*x^5+(5/512*b^9-5/32*b^7+15/16*b^5-5/2*b^3+5/2*b)*x^2+(15/256*b^8*c-35/48*b^6*c+25/8*b^4*c-5*
b^2*c+5/3*c)*x^3+1/11*c^9*x^11+1/2*b*c^8*x^10+1/1024*(b^10-20*b^8+160*b^6-640*b^4+1280*b^2-1024)/c*x)/c^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (85) = 170\).

Time = 0.45 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.16 \[ \int \left (\frac {-4+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 98560 \, {\left (9 \, b^{2} - 4\right )} c^{8} x^{9} + 443520 \, {\left (3 \, b^{3} - 4 \, b\right )} c^{7} x^{8} + 63360 \, {\left (21 \, b^{4} - 56 \, b^{2} + 16\right )} c^{6} x^{7} + 14784 \, {\left (63 \, b^{5} - 280 \, b^{3} + 240 \, b\right )} c^{5} x^{6} + 22176 \, {\left (21 \, b^{6} - 140 \, b^{4} + 240 \, b^{2} - 64\right )} c^{4} x^{5} + 55440 \, {\left (3 \, b^{7} - 28 \, b^{5} + 80 \, b^{3} - 64 \, b\right )} c^{3} x^{4} + 4620 \, {\left (9 \, b^{8} - 112 \, b^{6} + 480 \, b^{4} - 768 \, b^{2} + 256\right )} c^{2} x^{3} + 6930 \, {\left (b^{9} - 16 \, b^{7} + 96 \, b^{5} - 256 \, b^{3} + 256 \, b\right )} c x^{2} + 693 \, {\left (b^{10} - 20 \, b^{8} + 160 \, b^{6} - 640 \, b^{4} + 1280 \, b^{2} - 1024\right )} x}{709632 \, c^{5}} \]

[In]

integrate((1/4*(b^2-4)/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 - 4)*c^8*x^9 + 443520*(3*b^3 - 4*b)*c^7*x^8 + 633
60*(21*b^4 - 56*b^2 + 16)*c^6*x^7 + 14784*(63*b^5 - 280*b^3 + 240*b)*c^5*x^6 + 22176*(21*b^6 - 140*b^4 + 240*b
^2 - 64)*c^4*x^5 + 55440*(3*b^7 - 28*b^5 + 80*b^3 - 64*b)*c^3*x^4 + 4620*(9*b^8 - 112*b^6 + 480*b^4 - 768*b^2
+ 256)*c^2*x^3 + 6930*(b^9 - 16*b^7 + 96*b^5 - 256*b^3 + 256*b)*c*x^2 + 693*(b^10 - 20*b^8 + 160*b^6 - 640*b^4
+ 1280*b^2 - 1024)*x)/c^5

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (95) = 190\).

Time = 0.09 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.29 \[ \int \left (\frac {-4+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {b c^{4} x^{10}}{2} + \frac {c^{5} x^{11}}{11} + x^{9} \cdot \left (\frac {5 b^{2} c^{3}}{4} - \frac {5 c^{3}}{9}\right ) + x^{8} \cdot \left (\frac {15 b^{3} c^{2}}{8} - \frac {5 b c^{2}}{2}\right ) + x^{7} \cdot \left (\frac {15 b^{4} c}{8} - 5 b^{2} c + \frac {10 c}{7}\right ) + x^{6} \cdot \left (\frac {21 b^{5}}{16} - \frac {35 b^{3}}{6} + 5 b\right ) + \frac {x^{5} \cdot \left (21 b^{6} - 140 b^{4} + 240 b^{2} - 64\right )}{32 c} + \frac {x^{4} \cdot \left (15 b^{7} - 140 b^{5} + 400 b^{3} - 320 b\right )}{64 c^{2}} + \frac {x^{3} \cdot \left (45 b^{8} - 560 b^{6} + 2400 b^{4} - 3840 b^{2} + 1280\right )}{768 c^{3}} + \frac {x^{2} \cdot \left (5 b^{9} - 80 b^{7} + 480 b^{5} - 1280 b^{3} + 1280 b\right )}{512 c^{4}} + \frac {x \left (b^{10} - 20 b^{8} + 160 b^{6} - 640 b^{4} + 1280 b^{2} - 1024\right )}{1024 c^{5}} \]

[In]

integrate((1/4*(b**2-4)/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**3/9) + x**8*(15*b**3*c**2/8 - 5*b*c**2/2) + x**7*(
15*b**4*c/8 - 5*b**2*c + 10*c/7) + x**6*(21*b**5/16 - 35*b**3/6 + 5*b) + x**5*(21*b**6 - 140*b**4 + 240*b**2 -
64)/(32*c) + x**4*(15*b**7 - 140*b**5 + 400*b**3 - 320*b)/(64*c**2) + x**3*(45*b**8 - 560*b**6 + 2400*b**4 -
3840*b**2 + 1280)/(768*c**3) + x**2*(5*b**9 - 80*b**7 + 480*b**5 - 1280*b**3 + 1280*b)/(512*c**4) + x*(b**10 -
20*b**8 + 160*b**6 - 640*b**4 + 1280*b**2 - 1024)/(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 \[ \int \left (\frac {-4+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {1}{11} \, c^{5} x^{11} + \frac {1}{2} \, b c^{4} x^{10} + \frac {10}{9} \, b^{2} c^{3} x^{9} + \frac {5}{4} \, b^{3} c^{2} x^{8} + \frac {5}{7} \, b^{4} c x^{7} + \frac {1}{6} \, b^{5} x^{6} + \frac {5 \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} {\left (b^{2} - 4\right )}^{4}}{1536 \, c^{4}} + \frac {{\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} {\left (b^{2} - 4\right )}^{3}}{192 \, c^{3}} + \frac {{\left (20 \, c^{3} x^{7} + 70 \, b c^{2} x^{6} + 84 \, b^{2} c x^{5} + 35 \, b^{3} x^{4}\right )} {\left (b^{2} - 4\right )}^{2}}{224 \, c^{2}} + \frac {{\left (70 \, c^{4} x^{9} + 315 \, b c^{3} x^{8} + 540 \, b^{2} c^{2} x^{7} + 420 \, b^{3} c x^{6} + 126 \, b^{4} x^{5}\right )} {\left (b^{2} - 4\right )}}{504 \, c} + \frac {{\left (b^{2} - 4\right )}^{5} x}{1024 \, c^{5}} \]

[In]

integrate((1/4*(b^2-4)/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 - 4)^4/c^4 + 1/192*(6*c^2*x^5 + 15*b*c*x^4 + 10*b^2*x^3)*(b^2 - 4)^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 - 4)^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 - 4)/c + 1/1024*(b^2 - 4)^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 \[ \int \left (\frac {-4+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 887040 \, b^{2} c^{8} x^{9} + 1330560 \, b^{3} c^{7} x^{8} + 1330560 \, b^{4} c^{6} x^{7} - 394240 \, c^{8} x^{9} + 931392 \, b^{5} c^{5} x^{6} - 1774080 \, b c^{7} x^{8} + 465696 \, b^{6} c^{4} x^{5} - 3548160 \, b^{2} c^{6} x^{7} + 166320 \, b^{7} c^{3} x^{4} - 4139520 \, b^{3} c^{5} x^{6} + 41580 \, b^{8} c^{2} x^{3} - 3104640 \, b^{4} c^{4} x^{5} + 1013760 \, c^{6} x^{7} + 6930 \, b^{9} c x^{2} - 1552320 \, b^{5} c^{3} x^{4} + 3548160 \, b c^{5} x^{6} + 693 \, b^{10} x - 517440 \, b^{6} c^{2} x^{3} + 5322240 \, b^{2} c^{4} x^{5} - 110880 \, b^{7} c x^{2} + 4435200 \, b^{3} c^{3} x^{4} - 13860 \, b^{8} x + 2217600 \, b^{4} c^{2} x^{3} - 1419264 \, c^{4} x^{5} + 665280 \, b^{5} c x^{2} - 3548160 \, b c^{3} x^{4} + 110880 \, b^{6} x - 3548160 \, b^{2} c^{2} x^{3} - 1774080 \, b^{3} c x^{2} - 443520 \, b^{4} x + 1182720 \, c^{2} x^{3} + 1774080 \, b c x^{2} + 887040 \, b^{2} x - 709632 \, x}{709632 \, c^{5}} \]

[In]

integrate((1/4*(b^2-4)/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
- 394240*c^8*x^9 + 931392*b^5*c^5*x^6 - 1774080*b*c^7*x^8 + 465696*b^6*c^4*x^5 - 3548160*b^2*c^6*x^7 + 166320
*b^7*c^3*x^4 - 4139520*b^3*c^5*x^6 + 41580*b^8*c^2*x^3 - 3104640*b^4*c^4*x^5 + 1013760*c^6*x^7 + 6930*b^9*c*x^
2 - 1552320*b^5*c^3*x^4 + 3548160*b*c^5*x^6 + 693*b^10*x - 517440*b^6*c^2*x^3 + 5322240*b^2*c^4*x^5 - 110880*b
^7*c*x^2 + 4435200*b^3*c^3*x^4 - 13860*b^8*x + 2217600*b^4*c^2*x^3 - 1419264*c^4*x^5 + 665280*b^5*c*x^2 - 3548
160*b*c^3*x^4 + 110880*b^6*x - 3548160*b^2*c^2*x^3 - 1774080*b^3*c*x^2 - 443520*b^4*x + 1182720*c^2*x^3 + 1774
080*b*c*x^2 + 887040*b^2*x - 709632*x)/c^5

Mupad [B] (verification not implemented)

Time = 9.17 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.69 \[ \int \left (\frac {-4+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {c^5\,x^{11}}{11}+\frac {x\,{\left (b^2-4\right )}^5}{1024\,c^5}+\frac {b\,x^6\,\left (63\,b^4-280\,b^2+240\right )}{48}+\frac {5\,c\,x^7\,\left (21\,b^4-56\,b^2+16\right )}{56}+\frac {b\,c^4\,x^{10}}{2}+\frac {5\,c^3\,x^9\,\left (9\,b^2-4\right )}{36}+\frac {x^5\,\left (21\,b^6-140\,b^4+240\,b^2-64\right )}{32\,c}+\frac {5\,b\,c^2\,x^8\,\left (3\,b^2-4\right )}{8}+\frac {5\,b\,x^2\,{\left (b^2-4\right )}^4}{512\,c^4}+\frac {5\,x^3\,{\left (b^2-4\right )}^3\,\left (9\,b^2-4\right )}{768\,c^3}+\frac {5\,b\,x^4\,{\left (b^2-4\right )}^2\,\left (3\,b^2-4\right )}{64\,c^2} \]

[In]

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

[Out]

(c^5*x^11)/11 + (x*(b^2 - 4)^5)/(1024*c^5) + (b*x^6*(63*b^4 - 280*b^2 + 240))/48 + (5*c*x^7*(21*b^4 - 56*b^2 +
16))/56 + (b*c^4*x^10)/2 + (5*c^3*x^9*(9*b^2 - 4))/36 + (x^5*(240*b^2 - 140*b^4 + 21*b^6 - 64))/(32*c) + (5*b
*c^2*x^8*(3*b^2 - 4))/8 + (5*b*x^2*(b^2 - 4)^4)/(512*c^4) + (5*x^3*(b^2 - 4)^3*(9*b^2 - 4))/(768*c^3) + (5*b*x
^4*(b^2 - 4)^2*(3*b^2 - 4))/(64*c^2)