\(\int (\frac {-16+b^2}{4 c}+b x+c x^2)^5 \, dx\) [91]

Optimal result

Integrand size = 23, antiderivative size = 109 \[ \int \left (\frac {-16+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {8 (4-b-2 c x)^6}{3 c^6}-\frac {10 (4-b-2 c x)^7}{7 c^6}+\frac {5 (4-b-2 c x)^8}{16 c^6}-\frac {5 (4-b-2 c x)^9}{144 c^6}+\frac {(4-b-2 c x)^{10}}{512 c^6}-\frac {(4-b-2 c x)^{11}}{22528 c^6} \] Output:

8/3*(-2*c*x-b+4)^6/c^6-10/7*(-2*c*x-b+4)^7/c^6+5/16*(-2*c*x-b+4)^8/c^6-5/1 
44*(-2*c*x-b+4)^9/c^6+1/512*(-2*c*x-b+4)^10/c^6-1/22528*(-2*c*x-b+4)^11/c^ 
6
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.89 \[ \int \left (\frac {-16+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {\left (-16+b^2\right )^5 x}{1024 c^5}+\frac {5 b \left (-16+b^2\right )^4 x^2}{512 c^4}+\frac {5 \left (-16+b^2\right )^3 \left (-16+9 b^2\right ) x^3}{768 c^3}+\frac {5 b \left (-16+b^2\right )^2 \left (-16+3 b^2\right ) x^4}{64 c^2}+\frac {\left (-16+b^2\right ) \left (256-224 b^2+21 b^4\right ) x^5}{32 c}+\frac {1}{48} b \left (3840-1120 b^2+63 b^4\right ) x^6+\frac {5}{56} \left (256-224 b^2+21 b^4\right ) c x^7+\frac {5}{8} b \left (-16+3 b^2\right ) c^2 x^8+\frac {5}{36} \left (-16+9 b^2\right ) c^3 x^9+\frac {1}{2} b c^4 x^{10}+\frac {c^5 x^{11}}{11} \] Input:

Integrate[((-16 + b^2)/(4*c) + b*x + c*x^2)^5,x]
 

Output:

((-16 + b^2)^5*x)/(1024*c^5) + (5*b*(-16 + b^2)^4*x^2)/(512*c^4) + (5*(-16 
 + b^2)^3*(-16 + 9*b^2)*x^3)/(768*c^3) + (5*b*(-16 + b^2)^2*(-16 + 3*b^2)* 
x^4)/(64*c^2) + ((-16 + b^2)*(256 - 224*b^2 + 21*b^4)*x^5)/(32*c) + (b*(38 
40 - 1120*b^2 + 63*b^4)*x^6)/48 + (5*(256 - 224*b^2 + 21*b^4)*c*x^7)/56 + 
(5*b*(-16 + 3*b^2)*c^2*x^8)/8 + (5*(-16 + 9*b^2)*c^3*x^9)/36 + (b*c^4*x^10 
)/2 + (c^5*x^11)/11
 

Rubi [A] (verified)

Time = 0.57 (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.

\(\displaystyle \int \left (\frac {b^2-16}{4 c}+b x+c x^2\right )^5 \, dx\)

\(\Big \downarrow \) 1084

\(\displaystyle \frac {\int \left (\left (\frac {b-4}{2}+c x\right )^{10}-\frac {5}{128} (-b-2 c x+4)^9+\frac {5}{8} (-b-2 c x+4)^8-5 (-b-2 c x+4)^7+20 (-b-2 c x+4)^6-32 (-b-2 c x+4)^5\right )dx}{c^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {(-b-2 c x+4)^{11}}{22528 c}+\frac {(-b-2 c x+4)^{10}}{512 c}-\frac {5 (-b-2 c x+4)^9}{144 c}+\frac {5 (-b-2 c x+4)^8}{16 c}-\frac {10 (-b-2 c x+4)^7}{7 c}+\frac {8 (-b-2 c x+4)^6}{3 c}}{c^5}\)

Input:

Int[((-16 + b^2)/(4*c) + b*x + c*x^2)^5,x]
 

Output:

((8*(4 - b - 2*c*x)^6)/(3*c) - (10*(4 - b - 2*c*x)^7)/(7*c) + (5*(4 - b - 
2*c*x)^8)/(16*c) - (5*(4 - b - 2*c*x)^9)/(144*c) + (4 - b - 2*c*x)^10/(512 
*c) - (4 - b - 2*c*x)^11/(22528*c))/c^5
 

Defintions of rubi rules used

rule 1084

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (warning: unable to verify)

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

Time = 0.58 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.50

method	result	size
norman	\(\frac {\left (\frac {5}{4} b^{2} c^{7}-\frac {20}{9} c^{7}\right ) x^{9}+\left (\frac {15}{8} b^{3} c^{6}-10 b \,c^{6}\right ) x^{8}+\left (\frac {15}{8} b^{4} c^{5}-20 c^{5} b^{2}+\frac {160}{7} c^{5}\right ) x^{7}+\left (\frac {21}{16} b^{5} c^{4}-\frac {70}{3} b^{3} c^{4}+80 b \,c^{4}\right ) x^{6}+\left (\frac {21}{32} b^{6} c^{3}-\frac {35}{2} b^{4} c^{3}+120 c^{3} b^{2}-128 c^{3}\right ) x^{5}+\left (\frac {15}{64} b^{7} c^{2}-\frac {35}{4} b^{5} c^{2}+100 b^{3} c^{2}-320 b \,c^{2}\right ) x^{4}+\left (\frac {5}{512} b^{9}-\frac {5}{8} b^{7}+15 b^{5}-160 b^{3}+640 b \right ) x^{2}+\left (\frac {15}{256} b^{8} c -\frac {35}{12} b^{6} c +50 b^{4} c -320 b^{2} c +\frac {1280}{3} c \right ) x^{3}+\frac {c^{9} x^{11}}{11}+\frac {b \,c^{8} x^{10}}{2}+\frac {\left (b^{10}-80 b^{8}+2560 b^{6}-40960 b^{4}+327680 b^{2}-1048576\right ) x}{1024 c}}{c^{4}}\)	\(273\)
gosper	\(\frac {x \left (64512 c^{10} x^{10}+354816 c^{9} b \,x^{9}+887040 x^{8} b^{2} c^{8}+1330560 b^{3} c^{7} x^{7}+1330560 x^{6} b^{4} c^{6}-1576960 x^{8} c^{8}+931392 x^{5} b^{5} c^{5}-7096320 b \,c^{7} x^{7}+465696 b^{6} c^{4} x^{4}-14192640 x^{6} c^{6} b^{2}+166320 b^{7} c^{3} x^{3}-16558080 x^{5} b^{3} c^{5}+41580 x^{2} b^{8} c^{2}-12418560 b^{4} c^{4} x^{4}[phone] x^{6} c^{6}+6930 b^{9} c x -6209280 b^{5} c^{3} x^{3}[phone] x^{5} b \,c^{5}+693 b^{10}-2069760 x^{2} b^{6} c^{2}[phone] x^{4} b^{2} c^{4}-443520 b^{7} c x [phone] b^{3} c^{3} x^{3}-55440 b^{8}[phone] c^{2} x^{2} b^{4}-90832896 c^{4} x^{4}[phone] x c \,b^{5}-227082240 b \,c^{3} x^{3}+1774080 b^{6}-227082240 b^{2} c^{2} x^{2}-113541120 b^{3} c x -28385280 b^{4}[phone] c^{2} x^{2}[phone] c b x [phone] b^{2}-726663168\right )}{709632 c^{5}}\)	\(319\)
parallelrisch	\(\frac {64512 c^{10} x^{11}+354816 c^{9} b \,x^{10}+887040 x^{9} b^{2} c^{8}+1330560 b^{3} c^{7} x^{8}+1330560 x^{7} b^{4} c^{6}-1576960 x^{9} c^{8}+931392 x^{6} b^{5} c^{5}-7096320 b \,c^{7} x^{8}+465696 b^{6} c^{4} x^{5}-14192640 x^{7} c^{6} b^{2}+166320 b^{7} c^{3} x^{4}-16558080 x^{6} b^{3} c^{5}+41580 x^{3} b^{8} c^{2}-12418560 c^{4} b^{4} x^{5}[phone] x^{7} c^{6}+6930 b^{9} c \,x^{2}-6209280 b^{5} c^{3} x^{4}[phone] x^{6} b \,c^{5}+693 b^{10} x -2069760 x^{3} b^{6} c^{2}[phone] b^{2} c^{4} x^{5}-443520 b^{7} c \,x^{2}[phone] b^{3} c^{3} x^{4}-55440 b^{8} x [phone] b^{4} c^{2} x^{3}-90832896 c^{4} x^{5}[phone] b^{5} c \,x^{2}-227082240 b \,c^{3} x^{4}+1774080 b^{6} x -227082240 b^{2} c^{2} x^{3}-113541120 b^{3} c \,x^{2}-28385280 b^{4} x [phone] x^{3} c^{2}[phone] b c \,x^{2}[phone] b^{2} x -726663168 x}{709632 c^{5}}\)	\(335\)
risch	\(80 b \,x^{6}-\frac {70 b^{3} x^{6}}{3}-20 b^{2} c \,x^{7}+\frac {15 x^{3} b^{8}}{256 c^{3}}-\frac {35 x^{3} b^{6}}{12 c^{3}}+\frac {15 c^{2} b^{3} x^{8}}{8}-\frac {160 b^{3} x^{2}}{c^{4}}+\frac {5 b^{9} x^{2}}{512 c^{4}}-10 c^{2} b \,x^{8}+\frac {21 b^{6} x^{5}}{32 c}+\frac {120 b^{2} x^{5}}{c}-\frac {1024 x}{c^{5}}+\frac {1280 x^{3}}{3 c^{3}}-\frac {128 x^{5}}{c}-\frac {20 c^{3} x^{9}}{9}+\frac {160 c \,x^{7}}{7}+\frac {50 b^{4} x^{3}}{c^{3}}-\frac {5 b^{7} x^{2}}{8 c^{4}}+\frac {15 b^{5} x^{2}}{c^{4}}+\frac {c^{5} x^{11}}{11}+\frac {b^{10} x}{1024 c^{5}}-\frac {5 b^{8} x}{64 c^{5}}+\frac {5 b^{6} x}{2 c^{5}}-\frac {40 b^{4} x}{c^{5}}+\frac {15 b^{7} x^{4}}{64 c^{2}}-\frac {35 b^{5} x^{4}}{4 c^{2}}-\frac {320 b \,x^{4}}{c^{2}}+\frac {5 c^{3} x^{9} b^{2}}{4}+\frac {b \,c^{4} x^{10}}{2}-\frac {320 b^{2} x^{3}}{c^{3}}+\frac {320 b^{2} x}{c^{5}}+\frac {640 b \,x^{2}}{c^{4}}+\frac {21 x^{6} b^{5}}{16}-\frac {35 b^{4} x^{5}}{2 c}+\frac {100 b^{3} x^{4}}{c^{2}}+\frac {15 b^{4} c \,x^{7}}{8}\)	\(343\)
orering	\(\frac {x \left (64512 c^{10} x^{10}+354816 c^{9} b \,x^{9}+887040 x^{8} b^{2} c^{8}+1330560 b^{3} c^{7} x^{7}+1330560 x^{6} b^{4} c^{6}-1576960 x^{8} c^{8}+931392 x^{5} b^{5} c^{5}-7096320 b \,c^{7} x^{7}+465696 b^{6} c^{4} x^{4}-14192640 x^{6} c^{6} b^{2}+166320 b^{7} c^{3} x^{3}-16558080 x^{5} b^{3} c^{5}+41580 x^{2} b^{8} c^{2}-12418560 b^{4} c^{4} x^{4}[phone] x^{6} c^{6}+6930 b^{9} c x -6209280 b^{5} c^{3} x^{3}[phone] x^{5} b \,c^{5}+693 b^{10}-2069760 x^{2} b^{6} c^{2}[phone] x^{4} b^{2} c^{4}-443520 b^{7} c x [phone] b^{3} c^{3} x^{3}-55440 b^{8}[phone] c^{2} x^{2} b^{4}-90832896 c^{4} x^{4}[phone] x c \,b^{5}-227082240 b \,c^{3} x^{3}+1774080 b^{6}-227082240 b^{2} c^{2} x^{2}-113541120 b^{3} c x -28385280 b^{4}[phone] c^{2} x^{2}[phone] c b x [phone] b^{2}-726663168\right ) \left (\frac {b^{2}-16}{4 c}+b x +c \,x^{2}\right )^{5}}{693 \left (2 c x +b +4\right )^{5} \left (2 c x +b -4\right )^{5}}\)	\(355\)
default	\(\frac {c^{5} x^{11}}{11}+\frac {b \,c^{4} x^{10}}{2}+\frac {\left (256 \left (b^{2}-16\right ) c^{3}+4096 c^{3} b^{2}+4 c \left (32 \left (24 b^{2}-128\right ) c^{2}+1024 b^{2} c^{2}\right )\right ) x^{9}}{9216}+\frac {\left (1024 \left (b^{2}-16\right ) c^{2} b +4 b \left (32 \left (24 b^{2}-128\right ) c^{2}+1024 b^{2} c^{2}\right )+4 c \left (256 \left (b^{2}-16\right ) c b +64 \left (24 b^{2}-128\right ) b c \right )\right ) x^{8}}{8192}+\frac {\left (\frac {\left (b^{2}-16\right ) \left (32 \left (24 b^{2}-128\right ) c^{2}+1024 b^{2} c^{2}\right )}{c}+4 b \left (256 \left (b^{2}-16\right ) c b +64 \left (24 b^{2}-128\right ) b c \right )+4 c \left (32 \left (b^{2}-16\right )^{2}+512 \left (b^{2}-16\right ) b^{2}+\left (24 b^{2}-128\right )^{2}\right )\right ) x^{7}}{7168}+\frac {\left (\frac {\left (b^{2}-16\right ) \left (256 \left (b^{2}-16\right ) c b +64 \left (24 b^{2}-128\right ) b c \right )}{c}+4 b \left (32 \left (b^{2}-16\right )^{2}+512 \left (b^{2}-16\right ) b^{2}+\left (24 b^{2}-128\right )^{2}\right )+4 c \left (\frac {64 \left (b^{2}-16\right )^{2} b}{c}+\frac {16 \left (b^{2}-16\right ) b \left (24 b^{2}-128\right )}{c}\right )\right ) x^{6}}{6144}+\frac {\left (\frac {\left (b^{2}-16\right ) \left (32 \left (b^{2}-16\right )^{2}+512 \left (b^{2}-16\right ) b^{2}+\left (24 b^{2}-128\right )^{2}\right )}{c}+4 b \left (\frac {64 \left (b^{2}-16\right )^{2} b}{c}+\frac {16 \left (b^{2}-16\right ) b \left (24 b^{2}-128\right )}{c}\right )+4 c \left (\frac {2 \left (b^{2}-16\right )^{2} \left (24 b^{2}-128\right )}{c^{2}}+\frac {64 \left (b^{2}-16\right )^{2} b^{2}}{c^{2}}\right )\right ) x^{5}}{5120}+\frac {\left (\frac {\left (b^{2}-16\right ) \left (\frac {64 \left (b^{2}-16\right )^{2} b}{c}+\frac {16 \left (b^{2}-16\right ) b \left (24 b^{2}-128\right )}{c}\right )}{c}+4 b \left (\frac {2 \left (b^{2}-16\right )^{2} \left (24 b^{2}-128\right )}{c^{2}}+\frac {64 \left (b^{2}-16\right )^{2} b^{2}}{c^{2}}\right )+\frac {64 \left (b^{2}-16\right )^{3} b}{c^{2}}\right ) x^{4}}{4096}+\frac {\left (\frac {\left (b^{2}-16\right ) \left (\frac {2 \left (b^{2}-16\right )^{2} \left (24 b^{2}-128\right )}{c^{2}}+\frac {64 \left (b^{2}-16\right )^{2} b^{2}}{c^{2}}\right )}{c}+\frac {64 b^{2} \left (b^{2}-16\right )^{3}}{c^{3}}+\frac {4 \left (b^{2}-16\right )^{4}}{c^{3}}\right ) x^{3}}{3072}+\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}}\)	\(648\)

Input:

int((1/4*(b^2-16)/c+b*x+c*x^2)^5,x,method=_RETURNVERBOSE)
 

Output:

((5/4*b^2*c^7-20/9*c^7)*x^9+(15/8*b^3*c^6-10*b*c^6)*x^8+(15/8*b^4*c^5-20*c 
^5*b^2+160/7*c^5)*x^7+(21/16*b^5*c^4-70/3*b^3*c^4+80*b*c^4)*x^6+(21/32*b^6 
*c^3-35/2*b^4*c^3+120*c^3*b^2-128*c^3)*x^5+(15/64*b^7*c^2-35/4*b^5*c^2+100 
*b^3*c^2-320*b*c^2)*x^4+(5/512*b^9-5/8*b^7+15*b^5-160*b^3+640*b)*x^2+(15/2 
56*b^8*c-35/12*b^6*c+50*b^4*c-320*b^2*c+1280/3*c)*x^3+1/11*c^9*x^11+1/2*b* 
c^8*x^10+1/1024*(b^10-80*b^8+2560*b^6-40960*b^4+327680*b^2-1048576)/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 = 0.08 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.16 \[ \int \left (\frac {-16+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 98560 \, {\left (9 \, b^{2} - 16\right )} c^{8} x^{9} + 443520 \, {\left (3 \, b^{3} - 16 \, b\right )} c^{7} x^{8} + 63360 \, {\left (21 \, b^{4} - 224 \, b^{2} + 256\right )} c^{6} x^{7} + 14784 \, {\left (63 \, b^{5} - 1120 \, b^{3} + 3840 \, b\right )} c^{5} x^{6} + 22176 \, {\left (21 \, b^{6} - 560 \, b^{4} + 3840 \, b^{2} - 4096\right )} c^{4} x^{5} + 55440 \, {\left (3 \, b^{7} - 112 \, b^{5} + 1280 \, b^{3} - 4096 \, b\right )} c^{3} x^{4} + 4620 \, {\left (9 \, b^{8} - 448 \, b^{6} + 7680 \, b^{4} - 49152 \, b^{2} + 65536\right )} c^{2} x^{3} + 6930 \, {\left (b^{9} - 64 \, b^{7} + 1536 \, b^{5} - 16384 \, b^{3} + 65536 \, b\right )} c x^{2} + 693 \, {\left (b^{10} - 80 \, b^{8} + 2560 \, b^{6} - 40960 \, b^{4} + 327680 \, b^{2} - 1048576\right )} x}{709632 \, c^{5}} \] Input:

integrate((1/4*(b^2-16)/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 - 16)*c^8*x^9 
 + 443520*(3*b^3 - 16*b)*c^7*x^8 + 63360*(21*b^4 - 224*b^2 + 256)*c^6*x^7 
+ 14784*(63*b^5 - 1120*b^3 + 3840*b)*c^5*x^6 + 22176*(21*b^6 - 560*b^4 + 3 
840*b^2 - 4096)*c^4*x^5 + 55440*(3*b^7 - 112*b^5 + 1280*b^3 - 4096*b)*c^3* 
x^4 + 4620*(9*b^8 - 448*b^6 + 7680*b^4 - 49152*b^2 + 65536)*c^2*x^3 + 6930 
*(b^9 - 64*b^7 + 1536*b^5 - 16384*b^3 + 65536*b)*c*x^2 + 693*(b^10 - 80*b^ 
8 + 2560*b^6 - 40960*b^4 + 327680*b^2 - 1048576)*x)/c^5
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (97) = 194\).

Time = 0.07 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.28 \[ \int \left (\frac {-16+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {b c^{4} x^{10}}{2} + \frac {c^{5} x^{11}}{11} + x^{9} \cdot \left (\frac {5 b^{2} c^{3}}{4} - \frac {20 c^{3}}{9}\right ) + x^{8} \cdot \left (\frac {15 b^{3} c^{2}}{8} - 10 b c^{2}\right ) + x^{7} \cdot \left (\frac {15 b^{4} c}{8} - 20 b^{2} c + \frac {160 c}{7}\right ) + x^{6} \cdot \left (\frac {21 b^{5}}{16} - \frac {70 b^{3}}{3} + 80 b\right ) + \frac {x^{5} \cdot \left (21 b^{6} - 560 b^{4} + 3840 b^{2} - 4096\right )}{32 c} + \frac {x^{4} \cdot \left (15 b^{7} - 560 b^{5} + 6400 b^{3} - 20480 b\right )}{64 c^{2}} + \frac {x^{3} \cdot \left (45 b^{8} - 2240 b^{6} + 38400 b^{4} - 245760 b^{2} + 327680\right )}{768 c^{3}} + \frac {x^{2} \cdot \left (5 b^{9} - 320 b^{7} + 7680 b^{5} - 81920 b^{3} + 327680 b\right )}{512 c^{4}} + \frac {x \left (b^{10} - 80 b^{8} + 2560 b^{6} - 40960 b^{4} + 327680 b^{2} - 1048576\right )}{1024 c^{5}} \] Input:

integrate((1/4*(b**2-16)/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 - 20*c**3/9) + x**8*( 
15*b**3*c**2/8 - 10*b*c**2) + x**7*(15*b**4*c/8 - 20*b**2*c + 160*c/7) + x 
**6*(21*b**5/16 - 70*b**3/3 + 80*b) + x**5*(21*b**6 - 560*b**4 + 3840*b**2 
 - 4096)/(32*c) + x**4*(15*b**7 - 560*b**5 + 6400*b**3 - 20480*b)/(64*c**2 
) + x**3*(45*b**8 - 2240*b**6 + 38400*b**4 - 245760*b**2 + 327680)/(768*c* 
*3) + x**2*(5*b**9 - 320*b**7 + 7680*b**5 - 81920*b**3 + 327680*b)/(512*c* 
*4) + x*(b**10 - 80*b**8 + 2560*b**6 - 40960*b**4 + 327680*b**2 - 1048576) 
/(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 = 0.03 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.15 \[ \int \left (\frac {-16+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {1}{11} \, c^{5} x^{11} + \frac {1}{2} \, b c^{4} x^{10} + \frac {10}{9} \, b^{2} c^{3} x^{9} + \frac {5}{4} \, b^{3} c^{2} x^{8} + \frac {5}{7} \, b^{4} c x^{7} + \frac {1}{6} \, b^{5} x^{6} + \frac {5 \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} {\left (b^{2} - 16\right )}^{4}}{1536 \, c^{4}} + \frac {{\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} {\left (b^{2} - 16\right )}^{3}}{192 \, c^{3}} + \frac {{\left (20 \, c^{3} x^{7} + 70 \, b c^{2} x^{6} + 84 \, b^{2} c x^{5} + 35 \, b^{3} x^{4}\right )} {\left (b^{2} - 16\right )}^{2}}{224 \, c^{2}} + \frac {{\left (70 \, c^{4} x^{9} + 315 \, b c^{3} x^{8} + 540 \, b^{2} c^{2} x^{7} + 420 \, b^{3} c x^{6} + 126 \, b^{4} x^{5}\right )} {\left (b^{2} - 16\right )}}{504 \, c} + \frac {{\left (b^{2} - 16\right )}^{5} x}{1024 \, c^{5}} \] Input:

integrate((1/4*(b^2-16)/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 - 16)^4/c^4 + 1/ 
192*(6*c^2*x^5 + 15*b*c*x^4 + 10*b^2*x^3)*(b^2 - 16)^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 - 16)^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 - 16)/c + 1/1024*(b^2 - 16)^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 = 0.16 (sec) , antiderivative size = 334, normalized size of antiderivative = 3.06 \[ \int \left (\frac {-16+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 887040 \, b^{2} c^{8} x^{9} + 1330560 \, b^{3} c^{7} x^{8} + 1330560 \, b^{4} c^{6} x^{7} - 1576960 \, c^{8} x^{9} + 931392 \, b^{5} c^{5} x^{6} - 7096320 \, b c^{7} x^{8} + 465696 \, b^{6} c^{4} x^{5} - 14192640 \, b^{2} c^{6} x^{7} + 166320 \, b^{7} c^{3} x^{4} - 16558080 \, b^{3} c^{5} x^{6} + 41580 \, b^{8} c^{2} x^{3} - 12418560 \, b^{4} c^{4} x^{5} + 16220160 \, c^{6} x^{7} + 6930 \, b^{9} c x^{2} - 6209280 \, b^{5} c^{3} x^{4} + 56770560 \, b c^{5} x^{6} + 693 \, b^{10} x - 2069760 \, b^{6} c^{2} x^{3} + 85155840 \, b^{2} c^{4} x^{5} - 443520 \, b^{7} c x^{2} + 70963200 \, b^{3} c^{3} x^{4} - 55440 \, b^{8} x + 35481600 \, b^{4} c^{2} x^{3} - 90832896 \, c^{4} x^{5} + 10644480 \, b^{5} c x^{2} - 227082240 \, b c^{3} x^{4} + 1774080 \, b^{6} x - 227082240 \, b^{2} c^{2} x^{3} - 113541120 \, b^{3} c x^{2} - 28385280 \, b^{4} x + 302776320 \, c^{2} x^{3} + 454164480 \, b c x^{2} + 227082240 \, b^{2} x - 726663168 \, x}{709632 \, c^{5}} \] Input:

integrate((1/4*(b^2-16)/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 - 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 - 4435 
20*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 + 302776 
320*c^2*x^3 + 454164480*b*c*x^2 + 227082240*b^2*x - 726663168*x)/c^5
 

Mupad [B] (verification not implemented)

Time = 9.72 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.69 \[ \int \left (\frac {-16+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {c^5\,x^{11}}{11}+\frac {x\,{\left (b^2-16\right )}^5}{1024\,c^5}+\frac {b\,x^6\,\left (63\,b^4-1120\,b^2+3840\right )}{48}+\frac {5\,c\,x^7\,\left (21\,b^4-224\,b^2+256\right )}{56}+\frac {b\,c^4\,x^{10}}{2}+\frac {5\,c^3\,x^9\,\left (9\,b^2-16\right )}{36}+\frac {x^5\,\left (21\,b^6-560\,b^4+3840\,b^2-4096\right )}{32\,c}+\frac {5\,b\,c^2\,x^8\,\left (3\,b^2-16\right )}{8}+\frac {5\,b\,x^2\,{\left (b^2-16\right )}^4}{512\,c^4}+\frac {5\,x^3\,{\left (b^2-16\right )}^3\,\left (9\,b^2-16\right )}{768\,c^3}+\frac {5\,b\,x^4\,{\left (b^2-16\right )}^2\,\left (3\,b^2-16\right )}{64\,c^2} \] Input:

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

Output:

(c^5*x^11)/11 + (x*(b^2 - 16)^5)/(1024*c^5) + (b*x^6*(63*b^4 - 1120*b^2 + 
3840))/48 + (5*c*x^7*(21*b^4 - 224*b^2 + 256))/56 + (b*c^4*x^10)/2 + (5*c^ 
3*x^9*(9*b^2 - 16))/36 + (x^5*(3840*b^2 - 560*b^4 + 21*b^6 - 4096))/(32*c) 
 + (5*b*c^2*x^8*(3*b^2 - 16))/8 + (5*b*x^2*(b^2 - 16)^4)/(512*c^4) + (5*x^ 
3*(b^2 - 16)^3*(9*b^2 - 16))/(768*c^3) + (5*b*x^4*(b^2 - 16)^2*(3*b^2 - 16 
))/(64*c^2)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.92 \[ \int \left (\frac {-16+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {x \left (64512 c^{10} x^{10}+354816 b \,c^{9} x^{9}+887040 b^{2} c^{8} x^{8}+1330560 b^{3} c^{7} x^{7}+1330560 b^{4} c^{6} x^{6}-1576960 c^{8} x^{8}+931392 b^{5} c^{5} x^{5}-7096320 b \,c^{7} x^{7}+465696 b^{6} c^{4} x^{4}-14192640 b^{2} c^{6} x^{6}+166320 b^{7} c^{3} x^{3}-16558080 b^{3} c^{5} x^{5}+41580 b^{8} c^{2} x^{2}-12418560 b^{4} c^{4} x^{4}[phone] c^{6} x^{6}+6930 b^{9} c x -6209280 b^{5} c^{3} x^{3}[phone] b \,c^{5} x^{5}+693 b^{10}-2069760 b^{6} c^{2} x^{2}[phone] b^{2} c^{4} x^{4}-443520 b^{7} c x [phone] b^{3} c^{3} x^{3}-55440 b^{8}[phone] b^{4} c^{2} x^{2}-90832896 c^{4} x^{4}[phone] b^{5} c x -227082240 b \,c^{3} x^{3}+1774080 b^{6}-227082240 b^{2} c^{2} x^{2}-113541120 b^{3} c x -28385280 b^{4}[phone] c^{2} x^{2}[phone] b c x [phone] b^{2}-726663168\right )}{709632 c^{5}} \] Input:

int((1/4*(b^2-16)/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 - 55440*b**8 + 166320 
*b**7*c**3*x**3 - 443520*b**7*c*x + 465696*b**6*c**4*x**4 - 2069760*b**6*c 
**2*x**2 + 1774080*b**6 + 931392*b**5*c**5*x**5 - 6209280*b**5*c**3*x**3 + 
 10644480*b**5*c*x + 1330560*b**4*c**6*x**6 - 12418560*b**4*c**4*x**4 + 35 
481600*b**4*c**2*x**2 - 28385280*b**4 + 1330560*b**3*c**7*x**7 - 16558080* 
b**3*c**5*x**5 + 70963200*b**3*c**3*x**3 - 113541120*b**3*c*x + 887040*b** 
2*c**8*x**8 - 14192640*b**2*c**6*x**6 + 85155840*b**2*c**4*x**4 - 22708224 
0*b**2*c**2*x**2 + 227082240*b**2 + 354816*b*c**9*x**9 - 7096320*b*c**7*x* 
*7 + 56770560*b*c**5*x**5 - 227082240*b*c**3*x**3 + 454164480*b*c*x + 6451 
2*c**10*x**10 - 1576960*c**8*x**8 + 16220160*c**6*x**6 - 90832896*c**4*x** 
4 + 302776320*c**2*x**2 - 726663168))/(709632*c**5)