\(\int (\frac {-9+b^2}{4 c}+b x+c x^2)^5 \, dx\) [76]

Optimal result

Integrand size = 23, antiderivative size = 109 \[ \int \left (\frac {-9+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {81 (3-b-2 c x)^6}{128 c^6}-\frac {405 (3-b-2 c x)^7}{896 c^6}+\frac {135 (3-b-2 c x)^8}{1024 c^6}-\frac {5 (3-b-2 c x)^9}{256 c^6}+\frac {3 (3-b-2 c x)^{10}}{2048 c^6}-\frac {(3-b-2 c x)^{11}}{22528 c^6} \]

[Out]

81/128*(-2*c*x-b+3)^6/c^6-405/896*(-2*c*x-b+3)^7/c^6+135/1024*(-2*c*x-b+3)^8/c^6-5/256*(-2*c*x-b+3)^9/c^6+3/20
48*(-2*c*x-b+3)^10/c^6-1/22528*(-2*c*x-b+3)^11/c^6

Rubi [A] (verified)

Time = 0.10 (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} \[ \int \left (\frac {-9+b^2}{4 c}+b x+c x^2\right )^5 \, dx=-\frac {(-b-2 c x+3)^{11}}{22528 c^6}+\frac {3 (-b-2 c x+3)^{10}}{2048 c^6}-\frac {5 (-b-2 c x+3)^9}{256 c^6}+\frac {135 (-b-2 c x+3)^8}{1024 c^6}-\frac {405 (-b-2 c x+3)^7}{896 c^6}+\frac {81 (-b-2 c x+3)^6}{128 c^6} \]

[In]

Int[((-9 + b^2)/(4*c) + b*x + c*x^2)^5,x]

[Out]

(81*(3 - b - 2*c*x)^6)/(128*c^6) - (405*(3 - b - 2*c*x)^7)/(896*c^6) + (135*(3 - b - 2*c*x)^8)/(1024*c^6) - (5
*(3 - b - 2*c*x)^9)/(256*c^6) + (3*(3 - b - 2*c*x)^10)/(2048*c^6) - (3 - 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} (-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*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.82 \[ \int \left (\frac {-9+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {\left (-9+b^2\right )^5 x}{1024 c^5}+\frac {5 b \left (-9+b^2\right )^4 x^2}{512 c^4}+\frac {15 \left (-9+b^2\right )^3 \left (-1+b^2\right ) x^3}{256 c^3}+\frac {15 b \left (-9+b^2\right )^2 \left (-3+b^2\right ) x^4}{64 c^2}+\frac {3 \left (-9+b^2\right ) \left (27-42 b^2+7 b^4\right ) x^5}{32 c}+\frac {3}{16} b \left (135-70 b^2+7 b^4\right ) x^6+\frac {15}{56} \left (27-42 b^2+7 b^4\right ) c x^7+\frac {15}{8} b \left (-3+b^2\right ) c^2 x^8+\frac {5}{4} \left (-1+b^2\right ) c^3 x^9+\frac {1}{2} b c^4 x^{10}+\frac {c^5 x^{11}}{11} \]

[In]

Integrate[((-9 + b^2)/(4*c) + b*x + c*x^2)^5,x]

[Out]

((-9 + b^2)^5*x)/(1024*c^5) + (5*b*(-9 + b^2)^4*x^2)/(512*c^4) + (15*(-9 + b^2)^3*(-1 + b^2)*x^3)/(256*c^3) +
(15*b*(-9 + b^2)^2*(-3 + b^2)*x^4)/(64*c^2) + (3*(-9 + b^2)*(27 - 42*b^2 + 7*b^4)*x^5)/(32*c) + (3*b*(135 - 70
*b^2 + 7*b^4)*x^6)/16 + (15*(27 - 42*b^2 + 7*b^4)*c*x^7)/56 + (15*b*(-3 + b^2)*c^2*x^8)/8 + (5*(-1 + b^2)*c^3*
x^9)/4 + (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 = 2.26 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.50

method	result	size
norman	\(\frac {\left (\frac {5}{4} b^{2} c^{7}-\frac {5}{4} c^{7}\right ) x^{9}+\left (\frac {15}{8} b^{3} c^{6}-\frac {45}{8} b \,c^{6}\right ) x^{8}+\left (\frac {15}{8} b^{4} c^{5}-\frac {45}{4} b^{2} c^{5}+\frac {405}{56} c^{5}\right ) x^{7}+\left (\frac {21}{16} b^{5} c^{4}-\frac {105}{8} c^{4} b^{3}+\frac {405}{16} b \,c^{4}\right ) x^{6}+\left (\frac {15}{64} b^{7} c^{2}-\frac {315}{64} b^{5} c^{2}+\frac {2025}{64} c^{2} b^{3}-\frac {3645}{64} b \,c^{2}\right ) x^{4}+\left (\frac {21}{32} c^{3} b^{6}-\frac {315}{32} b^{4} c^{3}+\frac {1215}{32} b^{2} c^{3}-\frac {729}{32} c^{3}\right ) x^{5}+\left (\frac {5}{512} b^{9}-\frac {45}{128} b^{7}+\frac {1215}{256} b^{5}-\frac {3645}{128} b^{3}+\frac {32805}{512} b \right ) x^{2}+\left (\frac {15}{256} b^{8} c -\frac {105}{64} b^{6} c +\frac {2025}{128} b^{4} c -\frac {3645}{64} b^{2} c +\frac {10935}{256} c \right ) x^{3}+\frac {c^{9} x^{11}}{11}+\frac {b \,c^{8} x^{10}}{2}+\frac {\left (b^{10}-45 b^{8}+810 b^{6}-7290 b^{4}+32805 b^{2}-59049\right ) x}{1024 c}}{c^{4}}\)	\(273\)
gosper	\(\frac {x \left (7168 c^{10} x^{10}+39424 c^{9} b \,x^{9}+98560 x^{8} b^{2} c^{8}+147840 b^{3} c^{7} x^{7}+147840 x^{6} b^{4} c^{6}-98560 x^{8} c^{8}+103488 x^{5} b^{5} c^{5}-443520 b \,c^{7} x^{7}+51744 b^{6} c^{4} x^{4}-887040 x^{6} b^{2} c^{6}+18480 b^{7} c^{3} x^{3}-1034880 x^{5} b^{3} c^{5}+4620 x^{2} b^{8} c^{2}-776160 b^{4} c^{4} x^{4}+570240 x^{6} c^{6}+770 b^{9} c x -388080 x^{3} b^{5} c^{3}+1995840 x^{5} b \,c^{5}+77 b^{10}-129360 x^{2} c^{2} b^{6}+2993760 b^{2} c^{4} x^{4}-27720 b^{7} c x +2494800 x^{3} b^{3} c^{3}-3465 b^{8}+1247400 x^{2} b^{4} c^{2}-1796256 c^{4} x^{4}+374220 b^{5} c x -4490640 b \,c^{3} x^{3}+62370 b^{6}-4490640 b^{2} c^{2} x^{2}-2245320 b^{3} c x -561330 b^{4}+3367980 c^{2} x^{2}+5051970 b c x +2525985 b^{2}-4546773\right )}{78848 c^{5}}\)	\(319\)
parallelrisch	\(\frac {7168 c^{10} x^{11}+39424 c^{9} b \,x^{10}+98560 x^{9} b^{2} c^{8}+147840 b^{3} c^{7} x^{8}+147840 x^{7} b^{4} c^{6}-98560 x^{9} c^{8}+103488 x^{6} b^{5} c^{5}-443520 b \,c^{7} x^{8}+51744 b^{6} c^{4} x^{5}-887040 x^{7} b^{2} c^{6}+18480 b^{7} c^{3} x^{4}-1034880 x^{6} b^{3} c^{5}+4620 x^{3} b^{8} c^{2}-776160 x^{5} b^{4} c^{4}+570240 x^{7} c^{6}+770 b^{9} c \,x^{2}-388080 b^{5} c^{3} x^{4}+1995840 x^{6} b \,c^{5}+77 b^{10} x -129360 x^{3} c^{2} b^{6}+2993760 b^{2} c^{4} x^{5}-27720 b^{7} c \,x^{2}+2494800 c^{3} b^{3} x^{4}-3465 b^{8} x +1247400 b^{4} c^{2} x^{3}-1796256 c^{4} x^{5}+374220 b^{5} c \,x^{2}-4490640 b \,c^{3} x^{4}+62370 b^{6} x -4490640 b^{2} c^{2} x^{3}-2245320 b^{3} c \,x^{2}-561330 b^{4} x +3367980 c^{2} x^{3}+5051970 c b \,x^{2}+2525985 b^{2} x -4546773 x}{78848 c^{5}}\)	\(335\)
risch	\(\frac {405 b \,x^{6}}{16}-\frac {105 b^{3} x^{6}}{8}-\frac {59049 x}{1024 c^{5}}+\frac {21 b^{5} x^{6}}{16}-\frac {729 x^{5}}{32 c}+\frac {15 b^{4} c \,x^{7}}{8}+\frac {21 b^{6} x^{5}}{32 c}+\frac {1215 b^{2} x^{5}}{32 c}+\frac {5 c^{3} x^{9} b^{2}}{4}+\frac {15 b^{7} x^{4}}{64 c^{2}}-\frac {315 b^{5} x^{4}}{64 c^{2}}+\frac {5 b^{9} x^{2}}{512 c^{4}}-\frac {45 b^{7} x^{2}}{128 c^{4}}+\frac {1215 b^{5} x^{2}}{256 c^{4}}-\frac {3645 b^{3} x^{2}}{128 c^{4}}+\frac {15 x^{3} b^{8}}{256 c^{3}}-\frac {105 x^{3} b^{6}}{64 c^{3}}+\frac {b \,c^{4} x^{10}}{2}+\frac {15 b^{3} c^{2} x^{8}}{8}-\frac {3645 b \,x^{4}}{64 c^{2}}-\frac {45 c^{2} b \,x^{8}}{8}+\frac {405 b^{6} x}{512 c^{5}}-\frac {3645 b^{4} x}{512 c^{5}}+\frac {c^{5} x^{11}}{11}-\frac {315 x^{5} b^{4}}{32 c}+\frac {2025 b^{3} x^{4}}{64 c^{2}}+\frac {32805 b \,x^{2}}{512 c^{4}}-\frac {45 b^{8} x}{1024 c^{5}}+\frac {405 c \,x^{7}}{56}+\frac {10935 x^{3}}{256 c^{3}}-\frac {5 c^{3} x^{9}}{4}-\frac {3645 b^{2} x^{3}}{64 c^{3}}+\frac {2025 b^{4} x^{3}}{128 c^{3}}+\frac {b^{10} x}{1024 c^{5}}+\frac {32805 b^{2} x}{1024 c^{5}}-\frac {45 b^{2} c \,x^{7}}{4}\)	\(343\)
default	\(\frac {c^{5} x^{11}}{11}+\frac {b \,c^{4} x^{10}}{2}+\frac {\left (256 \left (b^{2}-9\right ) c^{3}+4096 b^{2} c^{3}+4 c \left (32 \left (24 b^{2}-72\right ) c^{2}+1024 b^{2} c^{2}\right )\right ) x^{9}}{9216}+\frac {\left (1024 \left (b^{2}-9\right ) c^{2} b +4 b \left (32 \left (24 b^{2}-72\right ) c^{2}+1024 b^{2} c^{2}\right )+4 c \left (256 \left (b^{2}-9\right ) c b +64 \left (24 b^{2}-72\right ) b c \right )\right ) x^{8}}{8192}+\frac {\left (\frac {\left (b^{2}-9\right ) \left (32 \left (24 b^{2}-72\right ) c^{2}+1024 b^{2} c^{2}\right )}{c}+4 b \left (256 \left (b^{2}-9\right ) c b +64 \left (24 b^{2}-72\right ) b c \right )+4 c \left (32 \left (b^{2}-9\right )^{2}+512 \left (b^{2}-9\right ) b^{2}+\left (24 b^{2}-72\right )^{2}\right )\right ) x^{7}}{7168}+\frac {\left (\frac {\left (b^{2}-9\right ) \left (256 \left (b^{2}-9\right ) c b +64 \left (24 b^{2}-72\right ) b c \right )}{c}+4 b \left (32 \left (b^{2}-9\right )^{2}+512 \left (b^{2}-9\right ) b^{2}+\left (24 b^{2}-72\right )^{2}\right )+4 c \left (\frac {64 \left (b^{2}-9\right )^{2} b}{c}+\frac {16 \left (b^{2}-9\right ) b \left (24 b^{2}-72\right )}{c}\right )\right ) x^{6}}{6144}+\frac {\left (\frac {\left (b^{2}-9\right ) \left (32 \left (b^{2}-9\right )^{2}+512 \left (b^{2}-9\right ) b^{2}+\left (24 b^{2}-72\right )^{2}\right )}{c}+4 b \left (\frac {64 \left (b^{2}-9\right )^{2} b}{c}+\frac {16 \left (b^{2}-9\right ) b \left (24 b^{2}-72\right )}{c}\right )+4 c \left (\frac {2 \left (b^{2}-9\right )^{2} \left (24 b^{2}-72\right )}{c^{2}}+\frac {64 \left (b^{2}-9\right )^{2} b^{2}}{c^{2}}\right )\right ) x^{5}}{5120}+\frac {\left (\frac {\left (b^{2}-9\right ) \left (\frac {64 \left (b^{2}-9\right )^{2} b}{c}+\frac {16 \left (b^{2}-9\right ) b \left (24 b^{2}-72\right )}{c}\right )}{c}+4 b \left (\frac {2 \left (b^{2}-9\right )^{2} \left (24 b^{2}-72\right )}{c^{2}}+\frac {64 \left (b^{2}-9\right )^{2} b^{2}}{c^{2}}\right )+\frac {64 \left (b^{2}-9\right )^{3} b}{c^{2}}\right ) x^{4}}{4096}+\frac {\left (\frac {\left (b^{2}-9\right ) \left (\frac {2 \left (b^{2}-9\right )^{2} \left (24 b^{2}-72\right )}{c^{2}}+\frac {64 \left (b^{2}-9\right )^{2} b^{2}}{c^{2}}\right )}{c}+\frac {64 b^{2} \left (b^{2}-9\right )^{3}}{c^{3}}+\frac {4 \left (b^{2}-9\right )^{4}}{c^{3}}\right ) x^{3}}{3072}+\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}}\)	\(648\)

[In]

int((1/4*(b^2-9)/c+b*x+c*x^2)^5,x,method=_RETURNVERBOSE)

[Out]

((5/4*b^2*c^7-5/4*c^7)*x^9+(15/8*b^3*c^6-45/8*b*c^6)*x^8+(15/8*b^4*c^5-45/4*b^2*c^5+405/56*c^5)*x^7+(21/16*b^5
*c^4-105/8*c^4*b^3+405/16*b*c^4)*x^6+(15/64*b^7*c^2-315/64*b^5*c^2+2025/64*c^2*b^3-3645/64*b*c^2)*x^4+(21/32*c
^3*b^6-315/32*b^4*c^3+1215/32*b^2*c^3-729/32*c^3)*x^5+(5/512*b^9-45/128*b^7+1215/256*b^5-3645/128*b^3+32805/51
2*b)*x^2+(15/256*b^8*c-105/64*b^6*c+2025/128*b^4*c-3645/64*b^2*c+10935/256*c)*x^3+1/11*c^9*x^11+1/2*b*c^8*x^10
+1/1024*(b^10-45*b^8+810*b^6-7290*b^4+32805*b^2-59049)/c*x)/c^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (85) = 170\).

Time = 0.43 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.08 \[ \int \left (\frac {-9+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {7168 \, c^{10} x^{11} + 39424 \, b c^{9} x^{10} + 98560 \, {\left (b^{2} - 1\right )} c^{8} x^{9} + 147840 \, {\left (b^{3} - 3 \, b\right )} c^{7} x^{8} + 21120 \, {\left (7 \, b^{4} - 42 \, b^{2} + 27\right )} c^{6} x^{7} + 14784 \, {\left (7 \, b^{5} - 70 \, b^{3} + 135 \, b\right )} c^{5} x^{6} + 7392 \, {\left (7 \, b^{6} - 105 \, b^{4} + 405 \, b^{2} - 243\right )} c^{4} x^{5} + 18480 \, {\left (b^{7} - 21 \, b^{5} + 135 \, b^{3} - 243 \, b\right )} c^{3} x^{4} + 4620 \, {\left (b^{8} - 28 \, b^{6} + 270 \, b^{4} - 972 \, b^{2} + 729\right )} c^{2} x^{3} + 770 \, {\left (b^{9} - 36 \, b^{7} + 486 \, b^{5} - 2916 \, b^{3} + 6561 \, b\right )} c x^{2} + 77 \, {\left (b^{10} - 45 \, b^{8} + 810 \, b^{6} - 7290 \, b^{4} + 32805 \, b^{2} - 59049\right )} x}{78848 \, c^{5}} \]

[In]

integrate((1/4*(b^2-9)/c+b*x+c*x^2)^5,x, algorithm="fricas")

[Out]

1/78848*(7168*c^10*x^11 + 39424*b*c^9*x^10 + 98560*(b^2 - 1)*c^8*x^9 + 147840*(b^3 - 3*b)*c^7*x^8 + 21120*(7*b
^4 - 42*b^2 + 27)*c^6*x^7 + 14784*(7*b^5 - 70*b^3 + 135*b)*c^5*x^6 + 7392*(7*b^6 - 105*b^4 + 405*b^2 - 243)*c^
4*x^5 + 18480*(b^7 - 21*b^5 + 135*b^3 - 243*b)*c^3*x^4 + 4620*(b^8 - 28*b^6 + 270*b^4 - 972*b^2 + 729)*c^2*x^3
+ 770*(b^9 - 36*b^7 + 486*b^5 - 2916*b^3 + 6561*b)*c*x^2 + 77*(b^10 - 45*b^8 + 810*b^6 - 7290*b^4 + 32805*b^2
- 59049)*x)/c^5

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (99) = 198\).

Time = 0.08 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.32 \[ \int \left (\frac {-9+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {b c^{4} x^{10}}{2} + \frac {c^{5} x^{11}}{11} + x^{9} \cdot \left (\frac {5 b^{2} c^{3}}{4} - \frac {5 c^{3}}{4}\right ) + x^{8} \cdot \left (\frac {15 b^{3} c^{2}}{8} - \frac {45 b c^{2}}{8}\right ) + x^{7} \cdot \left (\frac {15 b^{4} c}{8} - \frac {45 b^{2} c}{4} + \frac {405 c}{56}\right ) + x^{6} \cdot \left (\frac {21 b^{5}}{16} - \frac {105 b^{3}}{8} + \frac {405 b}{16}\right ) + \frac {x^{5} \cdot \left (21 b^{6} - 315 b^{4} + 1215 b^{2} - 729\right )}{32 c} + \frac {x^{4} \cdot \left (15 b^{7} - 315 b^{5} + 2025 b^{3} - 3645 b\right )}{64 c^{2}} + \frac {x^{3} \cdot \left (15 b^{8} - 420 b^{6} + 4050 b^{4} - 14580 b^{2} + 10935\right )}{256 c^{3}} + \frac {x^{2} \cdot \left (5 b^{9} - 180 b^{7} + 2430 b^{5} - 14580 b^{3} + 32805 b\right )}{512 c^{4}} + \frac {x \left (b^{10} - 45 b^{8} + 810 b^{6} - 7290 b^{4} + 32805 b^{2} - 59049\right )}{1024 c^{5}} \]

[In]

integrate((1/4*(b**2-9)/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**3/4) + x**8*(15*b**3*c**2/8 - 45*b*c**2/8) + x**7*
(15*b**4*c/8 - 45*b**2*c/4 + 405*c/56) + x**6*(21*b**5/16 - 105*b**3/8 + 405*b/16) + x**5*(21*b**6 - 315*b**4
+ 1215*b**2 - 729)/(32*c) + x**4*(15*b**7 - 315*b**5 + 2025*b**3 - 3645*b)/(64*c**2) + x**3*(15*b**8 - 420*b**
6 + 4050*b**4 - 14580*b**2 + 10935)/(256*c**3) + x**2*(5*b**9 - 180*b**7 + 2430*b**5 - 14580*b**3 + 32805*b)/(
512*c**4) + x*(b**10 - 45*b**8 + 810*b**6 - 7290*b**4 + 32805*b**2 - 59049)/(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 = 0.20 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.15 \[ \int \left (\frac {-9+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {1}{11} \, c^{5} x^{11} + \frac {1}{2} \, b c^{4} x^{10} + \frac {10}{9} \, b^{2} c^{3} x^{9} + \frac {5}{4} \, b^{3} c^{2} x^{8} + \frac {5}{7} \, b^{4} c x^{7} + \frac {1}{6} \, b^{5} x^{6} + \frac {5 \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} {\left (b^{2} - 9\right )}^{4}}{1536 \, c^{4}} + \frac {{\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} {\left (b^{2} - 9\right )}^{3}}{192 \, c^{3}} + \frac {{\left (20 \, c^{3} x^{7} + 70 \, b c^{2} x^{6} + 84 \, b^{2} c x^{5} + 35 \, b^{3} x^{4}\right )} {\left (b^{2} - 9\right )}^{2}}{224 \, c^{2}} + \frac {{\left (70 \, c^{4} x^{9} + 315 \, b c^{3} x^{8} + 540 \, b^{2} c^{2} x^{7} + 420 \, b^{3} c x^{6} + 126 \, b^{4} x^{5}\right )} {\left (b^{2} - 9\right )}}{504 \, c} + \frac {{\left (b^{2} - 9\right )}^{5} x}{1024 \, c^{5}} \]

[In]

integrate((1/4*(b^2-9)/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 - 9)^4/c^4 + 1/192*(6*c^2*x^5 + 15*b*c*x^4 + 10*b^2*x^3)*(b^2 - 9)^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 - 9)^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 - 9)/c + 1/1024*(b^2 - 9)^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 = 0.26 (sec) , antiderivative size = 334, normalized size of antiderivative = 3.06 \[ \int \left (\frac {-9+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {7168 \, c^{10} x^{11} + 39424 \, b c^{9} x^{10} + 98560 \, b^{2} c^{8} x^{9} + 147840 \, b^{3} c^{7} x^{8} + 147840 \, b^{4} c^{6} x^{7} - 98560 \, c^{8} x^{9} + 103488 \, b^{5} c^{5} x^{6} - 443520 \, b c^{7} x^{8} + 51744 \, b^{6} c^{4} x^{5} - 887040 \, b^{2} c^{6} x^{7} + 18480 \, b^{7} c^{3} x^{4} - 1034880 \, b^{3} c^{5} x^{6} + 4620 \, b^{8} c^{2} x^{3} - 776160 \, b^{4} c^{4} x^{5} + 570240 \, c^{6} x^{7} + 770 \, b^{9} c x^{2} - 388080 \, b^{5} c^{3} x^{4} + 1995840 \, b c^{5} x^{6} + 77 \, b^{10} x - 129360 \, b^{6} c^{2} x^{3} + 2993760 \, b^{2} c^{4} x^{5} - 27720 \, b^{7} c x^{2} + 2494800 \, b^{3} c^{3} x^{4} - 3465 \, b^{8} x + 1247400 \, b^{4} c^{2} x^{3} - 1796256 \, c^{4} x^{5} + 374220 \, b^{5} c x^{2} - 4490640 \, b c^{3} x^{4} + 62370 \, b^{6} x - 4490640 \, b^{2} c^{2} x^{3} - 2245320 \, b^{3} c x^{2} - 561330 \, b^{4} x + 3367980 \, c^{2} x^{3} + 5051970 \, b c x^{2} + 2525985 \, b^{2} x - 4546773 \, x}{78848 \, c^{5}} \]

[In]

integrate((1/4*(b^2-9)/c+b*x+c*x^2)^5,x, algorithm="giac")

[Out]

1/78848*(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 - 985
60*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 + 623
70*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 + 252598
5*b^2*x - 4546773*x)/c^5

Mupad [B] (verification not implemented)

Time = 9.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.61 \[ \int \left (\frac {-9+b^2}{4 c}+b x+c x^2\right )^5 \, dx=\frac {c^5\,x^{11}}{11}+\frac {5\,c^3\,x^9\,\left (b^2-1\right )}{4}+\frac {x\,{\left (b^2-9\right )}^5}{1024\,c^5}+\frac {3\,b\,x^6\,\left (7\,b^4-70\,b^2+135\right )}{16}+\frac {15\,c\,x^7\,\left (7\,b^4-42\,b^2+27\right )}{56}+\frac {b\,c^4\,x^{10}}{2}+\frac {3\,x^5\,\left (7\,b^6-105\,b^4+405\,b^2-243\right )}{32\,c}+\frac {15\,b\,c^2\,x^8\,\left (b^2-3\right )}{8}+\frac {15\,x^3\,\left (b^2-1\right )\,{\left (b^2-9\right )}^3}{256\,c^3}+\frac {5\,b\,x^2\,{\left (b^2-9\right )}^4}{512\,c^4}+\frac {15\,b\,x^4\,\left (b^2-3\right )\,{\left (b^2-9\right )}^2}{64\,c^2} \]

[In]

int((b*x + c*x^2 + (b^2/4 - 9/4)/c)^5,x)

[Out]

(c^5*x^11)/11 + (5*c^3*x^9*(b^2 - 1))/4 + (x*(b^2 - 9)^5)/(1024*c^5) + (3*b*x^6*(7*b^4 - 70*b^2 + 135))/16 + (
15*c*x^7*(7*b^4 - 42*b^2 + 27))/56 + (b*c^4*x^10)/2 + (3*x^5*(405*b^2 - 105*b^4 + 7*b^6 - 243))/(32*c) + (15*b
*c^2*x^8*(b^2 - 3))/8 + (15*x^3*(b^2 - 1)*(b^2 - 9)^3)/(256*c^3) + (5*b*x^2*(b^2 - 9)^4)/(512*c^4) + (15*b*x^4
*(b^2 - 3)*(b^2 - 9)^2)/(64*c^2)